27
SEVENTH EDITI ON Introduction to So li d State Physics CHARLES KIT TEL
SEVENTH EDITION
Introduction to Solid State Physics
CHARLES KIT TEL
14 Diamagnetism and Paramagnetism
LANGEVIN DIAMAGNETISM EQUATION 417
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS 419
PARAMAGNETISM 420
QUANTUM THEORY OF PARAMAGNETISM 420 Rare earth ions 423 Hund rules 424 Iron group ions 425 Crystal field splitting 426 Quenching of the orbital angular momentum 426 Spectroscopie splitting factor 429 Van Vleck temperature-independent paramagnetism 430
COOLING BY ISENTROPIC DEMAGNETIZATION 431 Nuclear demagnetization 432
PARAMAGNETIC SUSCEPTffiILITY OF CONDUCTION ELECTRONS 433
SUMMARY 436
PROBLEMS 436
1 Diamagnetic susceptibility of atomic hydrogen 436 2 Hund rules 437 3 Triplet excited states 437 4 Heat capacity from internaI degrees of freedom 438 5 Pauli spin susceptibility 438 6 Conduction electron ferromagnetism 438 7 Two-Ievel system 440 8 Paramagnetism of S =1 system 440
REFERENCES 440
NOTATION In the problems treated in this chapter the magnetic field B is always closely equal to the applied field Ba so that we write B for Ba in most instances
--------t + Or---------~T_--------------------------------
Pauli paramagnetism (metals) Temperature
Diamagnetism
Figure 1 Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substances
416
CHAPT ER 14 DIAMAGNET ISM AND PARAMAGNETISM
Magnetism is inseparable from quantum mechanics for a strictly classical
system in thermal equilibrium can display no magnetic moment even in a magnetic field The magnetic moment of a free atom has three principal sources the spin with which electrons are endowed their orbital angular moshymentum about the nucleus and the change in the orbital moment induced by an applied magne tic field
The first two effects give paramagnetic contributions to the magnetization and the third gives a diamagne tic contribution In the ground Is state of the
hydrogen atpm the orbital moment is zero and the magnetic moment is that of the electron spin along with a small induced diamagnetic moment In the 1S2
state ofhelium the sp in and orbital moments are both zero and there is only an induced moment Atoms with filled electron shells have zero spin and zero orbital moment these moments are associated with unfilled shells
The magnetization M is defined as the magnetic moment per unit volume
The magnetic susceptibility pe r unit volume is defined as
M (SI) X = -LoM (1) (CeS) x = 13 B
where B is the macroscopic magne tic field intensity In both systems of units X
is dimensionless We shall sometimes for convenience refer to MIB as the susshy
ceptibility without specifying the system of units Quite freque ntly a susceptibility is defi ned referred to unit mass Or to a
mole of the substance The molar susceptibility is written as XM the magnetic moment per gram is sometimes written as CT Subs tances with a negative magshynetic susceptibility are called diamagnetic Substances with a positive susceptishybility are called paramagnetic as in Fig 1
O rdered arrays of magnetic moments are discussed in Chapter 15 the arrays may be ferromagnetic ferrimagnetic antiferromagnetic helical or more complex in form Nuclear magnetic moments give r ise to nuclear
3paramagnetism Magnetic moments of nuclei are of the order of 10- times smaller than the magnetic moment of the electron
LANGEVIN DIAMAGNETISM EQUATION
Diamagnetism is associated with the tendency of electrical charges parshytially to shield the in terior of a body from an applied magnetic field In electroshymagnetism we are familiar with Lenzs law when the fl ux th rough an electrical circuit is changed an induced current is set up in such a direction as to oppose the flux change
417
418
In a superconductor or in an electron orbit within an atom the induced current persists as long as the field is present The magnetic fie ld of the induced current is opposite to the applied field and the magnetic moment associated with the current is a diamagnetic moment Even in a normal metal there is a diamagnetic contribution from the conduction electrons and this diamagshynetism is not destroyed by collisions of the electrons
The usual treatment of the diamagnetism of atoms and ions employs the Larmor theorem in a magnetic field the motion of the electrons around a central nucleus is to the first order in B the same as a possible motion in the absence of B except for the superposition of a precession of the electrons with angular frequency
(ces) w = eB2mc (SI) w = eB2m (2)
If the field is applied slowly the motion in the rotating reference system will be the same as the original motion in the rest system before the application of the field
If the average electron current around the nucleus is zero initially the application of the magnetic field will cause a finite current around the nushycleus The current is equivalent to a magnetic moment opposite to the applied field It is assumed that the Larmor frequency (2) is much lower than the freshyquency of the original motion in the central field This condition is not satisfied in free carrier cyclotron resonance and the cyclotron frequency is twice the frequency (2)
The Larmor precession of Z electrons is equivalent to an electric current
1 eB)(SI) 1 = (charge)(revolutions per unit time) = (- Ze) (- -- (3)271 2m
The magnetic moment IL of a current loop is given by the product (current) X (area of the loop) The are a of the loop of radius p is 7TP2 We have
ZtfB (S I) JI = - 4m (pl) (4)
Here (p2) = (x2) + (y2) is the mean square of the perpendicular distance of the electron from the field axis thro tigh the nucleus The mean square distance of the electrons from the nucleus is (r2) = (x2) + (y2) + (Z2) For a spherically symmetrical distribution of charge we have (x2) = (y2) = (Z2) so that (r 2) =
i(p2)
From (4) the diamagnetic susceptibility per unit volume is if N is the number of atoms per unit volume
2 = NIL = _ NZe (r2)(ces) X (5)B 6mc2
419 14 Diamagnetism and Paramagnetism
2 x = ILQNIl- = ILQNZe (r2 )(SI)
B 6m
This is the classical Langevin result The problem of calculating the diamagnetic susceptibili ty of an isolated
atom is reduced to the calculation of (r 2) for the electron distribution within the atom The distribution can be calculated by quantum mechanics
Experimental values for neutral atoms are most easily obtained for the inert gases Typical experimental values of the molar susceptibilities are the following
He Ne Ar Kr Xe
XM in CGS in 10-6 cm3lrnole -19 -72 -194 -280 -430
In dielectric solids the diamagnetic contribution of the ion cores is deshyscribed roughly by the Langevin result The contribution of conduction elecshytrons is more complicated as is evident from the de Haas-van Alphen effect discussed in Chapter 9
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS
From (G 18) the effect of a magnetic field is to add to the hamiltonian the terms
ieh e2
A2J-C = -(V A + Amiddot V) + -- (6)2mc 2mc2
for an atomic electron these tenns may usually be treated as a small perturbashytion If the magnetic field is uniform and in the z direction we may write
A x = -~yB Ay = h B Az = 0 (7)
and (6) becomes
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2
The first term on the right is proportional to the orbital angular mUlnenshytum component Lz if r is measured from the nucleus In mononuclear systems this term gives rise only to paramagnetism The second term gives for a spherishycally symmetric system a contribution
2 2 E = e B 2
-12 (r ) (9) n1C2
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
14 Diamagnetism and Paramagnetism
LANGEVIN DIAMAGNETISM EQUATION 417
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS 419
PARAMAGNETISM 420
QUANTUM THEORY OF PARAMAGNETISM 420 Rare earth ions 423 Hund rules 424 Iron group ions 425 Crystal field splitting 426 Quenching of the orbital angular momentum 426 Spectroscopie splitting factor 429 Van Vleck temperature-independent paramagnetism 430
COOLING BY ISENTROPIC DEMAGNETIZATION 431 Nuclear demagnetization 432
PARAMAGNETIC SUSCEPTffiILITY OF CONDUCTION ELECTRONS 433
SUMMARY 436
PROBLEMS 436
1 Diamagnetic susceptibility of atomic hydrogen 436 2 Hund rules 437 3 Triplet excited states 437 4 Heat capacity from internaI degrees of freedom 438 5 Pauli spin susceptibility 438 6 Conduction electron ferromagnetism 438 7 Two-Ievel system 440 8 Paramagnetism of S =1 system 440
REFERENCES 440
NOTATION In the problems treated in this chapter the magnetic field B is always closely equal to the applied field Ba so that we write B for Ba in most instances
--------t + Or---------~T_--------------------------------
Pauli paramagnetism (metals) Temperature
Diamagnetism
Figure 1 Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substances
416
CHAPT ER 14 DIAMAGNET ISM AND PARAMAGNETISM
Magnetism is inseparable from quantum mechanics for a strictly classical
system in thermal equilibrium can display no magnetic moment even in a magnetic field The magnetic moment of a free atom has three principal sources the spin with which electrons are endowed their orbital angular moshymentum about the nucleus and the change in the orbital moment induced by an applied magne tic field
The first two effects give paramagnetic contributions to the magnetization and the third gives a diamagne tic contribution In the ground Is state of the
hydrogen atpm the orbital moment is zero and the magnetic moment is that of the electron spin along with a small induced diamagnetic moment In the 1S2
state ofhelium the sp in and orbital moments are both zero and there is only an induced moment Atoms with filled electron shells have zero spin and zero orbital moment these moments are associated with unfilled shells
The magnetization M is defined as the magnetic moment per unit volume
The magnetic susceptibility pe r unit volume is defined as
M (SI) X = -LoM (1) (CeS) x = 13 B
where B is the macroscopic magne tic field intensity In both systems of units X
is dimensionless We shall sometimes for convenience refer to MIB as the susshy
ceptibility without specifying the system of units Quite freque ntly a susceptibility is defi ned referred to unit mass Or to a
mole of the substance The molar susceptibility is written as XM the magnetic moment per gram is sometimes written as CT Subs tances with a negative magshynetic susceptibility are called diamagnetic Substances with a positive susceptishybility are called paramagnetic as in Fig 1
O rdered arrays of magnetic moments are discussed in Chapter 15 the arrays may be ferromagnetic ferrimagnetic antiferromagnetic helical or more complex in form Nuclear magnetic moments give r ise to nuclear
3paramagnetism Magnetic moments of nuclei are of the order of 10- times smaller than the magnetic moment of the electron
LANGEVIN DIAMAGNETISM EQUATION
Diamagnetism is associated with the tendency of electrical charges parshytially to shield the in terior of a body from an applied magnetic field In electroshymagnetism we are familiar with Lenzs law when the fl ux th rough an electrical circuit is changed an induced current is set up in such a direction as to oppose the flux change
417
418
In a superconductor or in an electron orbit within an atom the induced current persists as long as the field is present The magnetic fie ld of the induced current is opposite to the applied field and the magnetic moment associated with the current is a diamagnetic moment Even in a normal metal there is a diamagnetic contribution from the conduction electrons and this diamagshynetism is not destroyed by collisions of the electrons
The usual treatment of the diamagnetism of atoms and ions employs the Larmor theorem in a magnetic field the motion of the electrons around a central nucleus is to the first order in B the same as a possible motion in the absence of B except for the superposition of a precession of the electrons with angular frequency
(ces) w = eB2mc (SI) w = eB2m (2)
If the field is applied slowly the motion in the rotating reference system will be the same as the original motion in the rest system before the application of the field
If the average electron current around the nucleus is zero initially the application of the magnetic field will cause a finite current around the nushycleus The current is equivalent to a magnetic moment opposite to the applied field It is assumed that the Larmor frequency (2) is much lower than the freshyquency of the original motion in the central field This condition is not satisfied in free carrier cyclotron resonance and the cyclotron frequency is twice the frequency (2)
The Larmor precession of Z electrons is equivalent to an electric current
1 eB)(SI) 1 = (charge)(revolutions per unit time) = (- Ze) (- -- (3)271 2m
The magnetic moment IL of a current loop is given by the product (current) X (area of the loop) The are a of the loop of radius p is 7TP2 We have
ZtfB (S I) JI = - 4m (pl) (4)
Here (p2) = (x2) + (y2) is the mean square of the perpendicular distance of the electron from the field axis thro tigh the nucleus The mean square distance of the electrons from the nucleus is (r2) = (x2) + (y2) + (Z2) For a spherically symmetrical distribution of charge we have (x2) = (y2) = (Z2) so that (r 2) =
i(p2)
From (4) the diamagnetic susceptibility per unit volume is if N is the number of atoms per unit volume
2 = NIL = _ NZe (r2)(ces) X (5)B 6mc2
419 14 Diamagnetism and Paramagnetism
2 x = ILQNIl- = ILQNZe (r2 )(SI)
B 6m
This is the classical Langevin result The problem of calculating the diamagnetic susceptibili ty of an isolated
atom is reduced to the calculation of (r 2) for the electron distribution within the atom The distribution can be calculated by quantum mechanics
Experimental values for neutral atoms are most easily obtained for the inert gases Typical experimental values of the molar susceptibilities are the following
He Ne Ar Kr Xe
XM in CGS in 10-6 cm3lrnole -19 -72 -194 -280 -430
In dielectric solids the diamagnetic contribution of the ion cores is deshyscribed roughly by the Langevin result The contribution of conduction elecshytrons is more complicated as is evident from the de Haas-van Alphen effect discussed in Chapter 9
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS
From (G 18) the effect of a magnetic field is to add to the hamiltonian the terms
ieh e2
A2J-C = -(V A + Amiddot V) + -- (6)2mc 2mc2
for an atomic electron these tenns may usually be treated as a small perturbashytion If the magnetic field is uniform and in the z direction we may write
A x = -~yB Ay = h B Az = 0 (7)
and (6) becomes
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2
The first term on the right is proportional to the orbital angular mUlnenshytum component Lz if r is measured from the nucleus In mononuclear systems this term gives rise only to paramagnetism The second term gives for a spherishycally symmetric system a contribution
2 2 E = e B 2
-12 (r ) (9) n1C2
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
--------t + Or---------~T_--------------------------------
Pauli paramagnetism (metals) Temperature
Diamagnetism
Figure 1 Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substances
416
CHAPT ER 14 DIAMAGNET ISM AND PARAMAGNETISM
Magnetism is inseparable from quantum mechanics for a strictly classical
system in thermal equilibrium can display no magnetic moment even in a magnetic field The magnetic moment of a free atom has three principal sources the spin with which electrons are endowed their orbital angular moshymentum about the nucleus and the change in the orbital moment induced by an applied magne tic field
The first two effects give paramagnetic contributions to the magnetization and the third gives a diamagne tic contribution In the ground Is state of the
hydrogen atpm the orbital moment is zero and the magnetic moment is that of the electron spin along with a small induced diamagnetic moment In the 1S2
state ofhelium the sp in and orbital moments are both zero and there is only an induced moment Atoms with filled electron shells have zero spin and zero orbital moment these moments are associated with unfilled shells
The magnetization M is defined as the magnetic moment per unit volume
The magnetic susceptibility pe r unit volume is defined as
M (SI) X = -LoM (1) (CeS) x = 13 B
where B is the macroscopic magne tic field intensity In both systems of units X
is dimensionless We shall sometimes for convenience refer to MIB as the susshy
ceptibility without specifying the system of units Quite freque ntly a susceptibility is defi ned referred to unit mass Or to a
mole of the substance The molar susceptibility is written as XM the magnetic moment per gram is sometimes written as CT Subs tances with a negative magshynetic susceptibility are called diamagnetic Substances with a positive susceptishybility are called paramagnetic as in Fig 1
O rdered arrays of magnetic moments are discussed in Chapter 15 the arrays may be ferromagnetic ferrimagnetic antiferromagnetic helical or more complex in form Nuclear magnetic moments give r ise to nuclear
3paramagnetism Magnetic moments of nuclei are of the order of 10- times smaller than the magnetic moment of the electron
LANGEVIN DIAMAGNETISM EQUATION
Diamagnetism is associated with the tendency of electrical charges parshytially to shield the in terior of a body from an applied magnetic field In electroshymagnetism we are familiar with Lenzs law when the fl ux th rough an electrical circuit is changed an induced current is set up in such a direction as to oppose the flux change
417
418
In a superconductor or in an electron orbit within an atom the induced current persists as long as the field is present The magnetic fie ld of the induced current is opposite to the applied field and the magnetic moment associated with the current is a diamagnetic moment Even in a normal metal there is a diamagnetic contribution from the conduction electrons and this diamagshynetism is not destroyed by collisions of the electrons
The usual treatment of the diamagnetism of atoms and ions employs the Larmor theorem in a magnetic field the motion of the electrons around a central nucleus is to the first order in B the same as a possible motion in the absence of B except for the superposition of a precession of the electrons with angular frequency
(ces) w = eB2mc (SI) w = eB2m (2)
If the field is applied slowly the motion in the rotating reference system will be the same as the original motion in the rest system before the application of the field
If the average electron current around the nucleus is zero initially the application of the magnetic field will cause a finite current around the nushycleus The current is equivalent to a magnetic moment opposite to the applied field It is assumed that the Larmor frequency (2) is much lower than the freshyquency of the original motion in the central field This condition is not satisfied in free carrier cyclotron resonance and the cyclotron frequency is twice the frequency (2)
The Larmor precession of Z electrons is equivalent to an electric current
1 eB)(SI) 1 = (charge)(revolutions per unit time) = (- Ze) (- -- (3)271 2m
The magnetic moment IL of a current loop is given by the product (current) X (area of the loop) The are a of the loop of radius p is 7TP2 We have
ZtfB (S I) JI = - 4m (pl) (4)
Here (p2) = (x2) + (y2) is the mean square of the perpendicular distance of the electron from the field axis thro tigh the nucleus The mean square distance of the electrons from the nucleus is (r2) = (x2) + (y2) + (Z2) For a spherically symmetrical distribution of charge we have (x2) = (y2) = (Z2) so that (r 2) =
i(p2)
From (4) the diamagnetic susceptibility per unit volume is if N is the number of atoms per unit volume
2 = NIL = _ NZe (r2)(ces) X (5)B 6mc2
419 14 Diamagnetism and Paramagnetism
2 x = ILQNIl- = ILQNZe (r2 )(SI)
B 6m
This is the classical Langevin result The problem of calculating the diamagnetic susceptibili ty of an isolated
atom is reduced to the calculation of (r 2) for the electron distribution within the atom The distribution can be calculated by quantum mechanics
Experimental values for neutral atoms are most easily obtained for the inert gases Typical experimental values of the molar susceptibilities are the following
He Ne Ar Kr Xe
XM in CGS in 10-6 cm3lrnole -19 -72 -194 -280 -430
In dielectric solids the diamagnetic contribution of the ion cores is deshyscribed roughly by the Langevin result The contribution of conduction elecshytrons is more complicated as is evident from the de Haas-van Alphen effect discussed in Chapter 9
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS
From (G 18) the effect of a magnetic field is to add to the hamiltonian the terms
ieh e2
A2J-C = -(V A + Amiddot V) + -- (6)2mc 2mc2
for an atomic electron these tenns may usually be treated as a small perturbashytion If the magnetic field is uniform and in the z direction we may write
A x = -~yB Ay = h B Az = 0 (7)
and (6) becomes
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2
The first term on the right is proportional to the orbital angular mUlnenshytum component Lz if r is measured from the nucleus In mononuclear systems this term gives rise only to paramagnetism The second term gives for a spherishycally symmetric system a contribution
2 2 E = e B 2
-12 (r ) (9) n1C2
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
CHAPT ER 14 DIAMAGNET ISM AND PARAMAGNETISM
Magnetism is inseparable from quantum mechanics for a strictly classical
system in thermal equilibrium can display no magnetic moment even in a magnetic field The magnetic moment of a free atom has three principal sources the spin with which electrons are endowed their orbital angular moshymentum about the nucleus and the change in the orbital moment induced by an applied magne tic field
The first two effects give paramagnetic contributions to the magnetization and the third gives a diamagne tic contribution In the ground Is state of the
hydrogen atpm the orbital moment is zero and the magnetic moment is that of the electron spin along with a small induced diamagnetic moment In the 1S2
state ofhelium the sp in and orbital moments are both zero and there is only an induced moment Atoms with filled electron shells have zero spin and zero orbital moment these moments are associated with unfilled shells
The magnetization M is defined as the magnetic moment per unit volume
The magnetic susceptibility pe r unit volume is defined as
M (SI) X = -LoM (1) (CeS) x = 13 B
where B is the macroscopic magne tic field intensity In both systems of units X
is dimensionless We shall sometimes for convenience refer to MIB as the susshy
ceptibility without specifying the system of units Quite freque ntly a susceptibility is defi ned referred to unit mass Or to a
mole of the substance The molar susceptibility is written as XM the magnetic moment per gram is sometimes written as CT Subs tances with a negative magshynetic susceptibility are called diamagnetic Substances with a positive susceptishybility are called paramagnetic as in Fig 1
O rdered arrays of magnetic moments are discussed in Chapter 15 the arrays may be ferromagnetic ferrimagnetic antiferromagnetic helical or more complex in form Nuclear magnetic moments give r ise to nuclear
3paramagnetism Magnetic moments of nuclei are of the order of 10- times smaller than the magnetic moment of the electron
LANGEVIN DIAMAGNETISM EQUATION
Diamagnetism is associated with the tendency of electrical charges parshytially to shield the in terior of a body from an applied magnetic field In electroshymagnetism we are familiar with Lenzs law when the fl ux th rough an electrical circuit is changed an induced current is set up in such a direction as to oppose the flux change
417
418
In a superconductor or in an electron orbit within an atom the induced current persists as long as the field is present The magnetic fie ld of the induced current is opposite to the applied field and the magnetic moment associated with the current is a diamagnetic moment Even in a normal metal there is a diamagnetic contribution from the conduction electrons and this diamagshynetism is not destroyed by collisions of the electrons
The usual treatment of the diamagnetism of atoms and ions employs the Larmor theorem in a magnetic field the motion of the electrons around a central nucleus is to the first order in B the same as a possible motion in the absence of B except for the superposition of a precession of the electrons with angular frequency
(ces) w = eB2mc (SI) w = eB2m (2)
If the field is applied slowly the motion in the rotating reference system will be the same as the original motion in the rest system before the application of the field
If the average electron current around the nucleus is zero initially the application of the magnetic field will cause a finite current around the nushycleus The current is equivalent to a magnetic moment opposite to the applied field It is assumed that the Larmor frequency (2) is much lower than the freshyquency of the original motion in the central field This condition is not satisfied in free carrier cyclotron resonance and the cyclotron frequency is twice the frequency (2)
The Larmor precession of Z electrons is equivalent to an electric current
1 eB)(SI) 1 = (charge)(revolutions per unit time) = (- Ze) (- -- (3)271 2m
The magnetic moment IL of a current loop is given by the product (current) X (area of the loop) The are a of the loop of radius p is 7TP2 We have
ZtfB (S I) JI = - 4m (pl) (4)
Here (p2) = (x2) + (y2) is the mean square of the perpendicular distance of the electron from the field axis thro tigh the nucleus The mean square distance of the electrons from the nucleus is (r2) = (x2) + (y2) + (Z2) For a spherically symmetrical distribution of charge we have (x2) = (y2) = (Z2) so that (r 2) =
i(p2)
From (4) the diamagnetic susceptibility per unit volume is if N is the number of atoms per unit volume
2 = NIL = _ NZe (r2)(ces) X (5)B 6mc2
419 14 Diamagnetism and Paramagnetism
2 x = ILQNIl- = ILQNZe (r2 )(SI)
B 6m
This is the classical Langevin result The problem of calculating the diamagnetic susceptibili ty of an isolated
atom is reduced to the calculation of (r 2) for the electron distribution within the atom The distribution can be calculated by quantum mechanics
Experimental values for neutral atoms are most easily obtained for the inert gases Typical experimental values of the molar susceptibilities are the following
He Ne Ar Kr Xe
XM in CGS in 10-6 cm3lrnole -19 -72 -194 -280 -430
In dielectric solids the diamagnetic contribution of the ion cores is deshyscribed roughly by the Langevin result The contribution of conduction elecshytrons is more complicated as is evident from the de Haas-van Alphen effect discussed in Chapter 9
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS
From (G 18) the effect of a magnetic field is to add to the hamiltonian the terms
ieh e2
A2J-C = -(V A + Amiddot V) + -- (6)2mc 2mc2
for an atomic electron these tenns may usually be treated as a small perturbashytion If the magnetic field is uniform and in the z direction we may write
A x = -~yB Ay = h B Az = 0 (7)
and (6) becomes
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2
The first term on the right is proportional to the orbital angular mUlnenshytum component Lz if r is measured from the nucleus In mononuclear systems this term gives rise only to paramagnetism The second term gives for a spherishycally symmetric system a contribution
2 2 E = e B 2
-12 (r ) (9) n1C2
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
418
In a superconductor or in an electron orbit within an atom the induced current persists as long as the field is present The magnetic fie ld of the induced current is opposite to the applied field and the magnetic moment associated with the current is a diamagnetic moment Even in a normal metal there is a diamagnetic contribution from the conduction electrons and this diamagshynetism is not destroyed by collisions of the electrons
The usual treatment of the diamagnetism of atoms and ions employs the Larmor theorem in a magnetic field the motion of the electrons around a central nucleus is to the first order in B the same as a possible motion in the absence of B except for the superposition of a precession of the electrons with angular frequency
(ces) w = eB2mc (SI) w = eB2m (2)
If the field is applied slowly the motion in the rotating reference system will be the same as the original motion in the rest system before the application of the field
If the average electron current around the nucleus is zero initially the application of the magnetic field will cause a finite current around the nushycleus The current is equivalent to a magnetic moment opposite to the applied field It is assumed that the Larmor frequency (2) is much lower than the freshyquency of the original motion in the central field This condition is not satisfied in free carrier cyclotron resonance and the cyclotron frequency is twice the frequency (2)
The Larmor precession of Z electrons is equivalent to an electric current
1 eB)(SI) 1 = (charge)(revolutions per unit time) = (- Ze) (- -- (3)271 2m
The magnetic moment IL of a current loop is given by the product (current) X (area of the loop) The are a of the loop of radius p is 7TP2 We have
ZtfB (S I) JI = - 4m (pl) (4)
Here (p2) = (x2) + (y2) is the mean square of the perpendicular distance of the electron from the field axis thro tigh the nucleus The mean square distance of the electrons from the nucleus is (r2) = (x2) + (y2) + (Z2) For a spherically symmetrical distribution of charge we have (x2) = (y2) = (Z2) so that (r 2) =
i(p2)
From (4) the diamagnetic susceptibility per unit volume is if N is the number of atoms per unit volume
2 = NIL = _ NZe (r2)(ces) X (5)B 6mc2
419 14 Diamagnetism and Paramagnetism
2 x = ILQNIl- = ILQNZe (r2 )(SI)
B 6m
This is the classical Langevin result The problem of calculating the diamagnetic susceptibili ty of an isolated
atom is reduced to the calculation of (r 2) for the electron distribution within the atom The distribution can be calculated by quantum mechanics
Experimental values for neutral atoms are most easily obtained for the inert gases Typical experimental values of the molar susceptibilities are the following
He Ne Ar Kr Xe
XM in CGS in 10-6 cm3lrnole -19 -72 -194 -280 -430
In dielectric solids the diamagnetic contribution of the ion cores is deshyscribed roughly by the Langevin result The contribution of conduction elecshytrons is more complicated as is evident from the de Haas-van Alphen effect discussed in Chapter 9
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS
From (G 18) the effect of a magnetic field is to add to the hamiltonian the terms
ieh e2
A2J-C = -(V A + Amiddot V) + -- (6)2mc 2mc2
for an atomic electron these tenns may usually be treated as a small perturbashytion If the magnetic field is uniform and in the z direction we may write
A x = -~yB Ay = h B Az = 0 (7)
and (6) becomes
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2
The first term on the right is proportional to the orbital angular mUlnenshytum component Lz if r is measured from the nucleus In mononuclear systems this term gives rise only to paramagnetism The second term gives for a spherishycally symmetric system a contribution
2 2 E = e B 2
-12 (r ) (9) n1C2
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
419 14 Diamagnetism and Paramagnetism
2 x = ILQNIl- = ILQNZe (r2 )(SI)
B 6m
This is the classical Langevin result The problem of calculating the diamagnetic susceptibili ty of an isolated
atom is reduced to the calculation of (r 2) for the electron distribution within the atom The distribution can be calculated by quantum mechanics
Experimental values for neutral atoms are most easily obtained for the inert gases Typical experimental values of the molar susceptibilities are the following
He Ne Ar Kr Xe
XM in CGS in 10-6 cm3lrnole -19 -72 -194 -280 -430
In dielectric solids the diamagnetic contribution of the ion cores is deshyscribed roughly by the Langevin result The contribution of conduction elecshytrons is more complicated as is evident from the de Haas-van Alphen effect discussed in Chapter 9
QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS
From (G 18) the effect of a magnetic field is to add to the hamiltonian the terms
ieh e2
A2J-C = -(V A + Amiddot V) + -- (6)2mc 2mc2
for an atomic electron these tenns may usually be treated as a small perturbashytion If the magnetic field is uniform and in the z direction we may write
A x = -~yB Ay = h B Az = 0 (7)
and (6) becomes
2iehB(d d) e2BJ-C = -- x- - y- + --(x2 + y2) (8)
2mc dy dx 8mc2
The first term on the right is proportional to the orbital angular mUlnenshytum component Lz if r is measured from the nucleus In mononuclear systems this term gives rise only to paramagnetism The second term gives for a spherishycally symmetric system a contribution
2 2 E = e B 2
-12 (r ) (9) n1C2
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
The moment is netic
in with
ta is in
lar oxygen and organic ltgt0
4 Metalslt
The 1l15~U moment of an atom or ion in free space is given
where the total angular momentum IiL and liS angular momentalt
The constant 1icircs the ratio of the moment to the angular momenshytum l is called the
a g defined by
For an g = 2 as For a free atom the g factor is the Landeacute equation
g = l + ~-------~------
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
421 14 Diamagnetism and Paramagnetism
4s 100 1
ms IJz 1075 0
8 050If 02
( 2ILB - IL s
B025-- 1 e c -2 J o1 I ii 1
o 05 10 15 20 ILBlkBT
Figure 2 Energy level splitting for one electron in a magnetic field B directed along the positive z Figure 3 Fractional populations of a two-level axis For an electron the magnetic moment JL is system in thermal equilibrium at temperature T opposite in sign to the spin S so that JL = in a magnetic field B The magnetic moment is -gJLBS In th e low energy state the magnetic proportional ta the difference between the two moment is paraIJel ta the magnetic field curves
The Bohr magneton J-tB is defined as eh2mc in ces and eh2m in SI It is closely equal to the spin magnetic moment of a free electron
The energy levels of the system in a magnetic field are
U = - P B = mjgJ-tBB (14)
where mj is the azimuthal quantum number and has the values J J - l - J For a single spin with no orbi tal moment we have mj = plusmn i and g = 2
whence U = plusmn J-tBB This splitting is shown in Fig 2 If a system has only two levels the equilibrium populations are with
T == kBT NI exp(J-tBIT)
(15)N exp(jLBiT) + exp( - jLBIT)
Nz exp( - J-tB IT) (16)
N exp(J-tBIT) + exp( - jLBiT)
here N j Nz are the populations of the lower and upper levels and N = N j + N2 is the total number of atoms The fractional populations are plotshyted in Fig 3
The projection of the magnetic moment of the upper state along the field direction is - J-t and of the lower state is J-t The resultant magnetization for N atoms per unit volume is with x == J-tBkBT
eX - e-X M = (NI - N2)J-t = NJ-t middot x + _ = NJ-t tanh x (17)
e e
For x ~ l tanh x = x and we have
M =NJ-t(J-tBkBT) (18)
In a magnetic field an atom with angular momentum quantum number J has 2J + 1 equally spaced energy levels The magnetization (Fig 4) is given by
M = NgJJ-tB Bj(x) (x == gJJ-tBBkBT ) (19)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
422
700 1TIIInITTDoFPI5F~FiTiumli
BIT in kG deg- L
Figure 4 Plot of magnetic moment versus BIT for spherical samples of (1) potassium chromium alum (II) ferric ammonium alum and (III) gadolinium sulfate octahydrate Over 995 magnetic saturation is achieved at 13 K and about 50000 gauss (ST) After W E Henry
where the Brillouin function BI is defined by
2J + 1 ((2J + l)x) 1 ( x )B(x) = ctnh - - ctnh - (20) 2J 2J 2J 2J
Equation (17) is a special case of (20) for J = t For x lts l we have
1 x x3
ctnh x = - + - - - + (21) x 3 45
and the susceptibility is
M NJ(J + 1)g2JL~ C -= (22)B 3kBT T
Here p is the effective number of Bohr magnetons defined as
p == gU(J + 1)F 2 (23)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
14 Dianwgnetism and Paranwgnetism
40~--------~----~~-~------~~
s
i
Temperature Je
Figure 5 Plot of lX vs T for a gadolinium salt Gd(CzH5 S04h straight line the Curie law (Aftel L C Jackson and Onnes)
Rare Earth Ions
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
Even in the no other
atom state is characshy
maximum S allowed exclusion
maximum value of the momentum consistent with of S
is to IL - SI when the shell is more than half fulL
ruIe L 0 so
different
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
425 14 Diamagnetism and Paramagnetism
Table l Effective magneton numbers p for trivalent lanthanide group ions
(Near room tempe rature)
---shy p(calc) = p(exp) Ion Configuration Basic level gU(] + 1)]JJ2 approximate
__=l
c eacute+ 4P5s2p6 2F s I2 2 54 24
Pr3 + 4j25s2p6 3H 4 3 58 3 5 Nd3+ 4P5s2
p6 41912 362 35 Pm3+ 4f 45s2p6 514 2 68 Sm3 + 4fs5s2p 6 6H sf2 084 15 Eu3+ 4f65s2p6 7F o 0 34 Gd3+ 4F5s2
p6 8S712 794 80 Tb3+ 4jB5s2p6 7F
6 972 95 D y 3+ 4f95s2p6 6H 1SI2 1063 106 Ho3+ 4po5s2p6 sIs 1060 104 Er3+ 4f1l5s2p6 41 1S12 959 95 Tm3+ 4P25s2p6 3H
6 7 57 73 Yb3+ 4P35s2
p6 2F7i2 454 45
The second Hund rule is best approached by model calculations Pauling and Wilson l for example give a calculation of the spectral terms that arise fro m the configuration p2 The third Hund rule is a consequence of the sign of the spin-orbit interaction For a single electron the energy is lowest when the spin is antiparallel to the orbital angular momentum But the Iow energy pairs mL
ms are progressively used up as we add electrons to the shell by the exclusion principle when the shell is more th an half full the state of lowest energy necesshysarily has the spin parallel ta the orbit
Consider two examples of the Hund fuIes The ion c eacute+ has a single f electron an f electron has l = 3 and s = i Because the f shell is less than half full the ] value by the preceding rule is IL - SI = L - = l The ion Pr3+ has two f electrons one of the mIes tells us that the spins add to give S = 1 Both f electrons cannot have ml = 3 without violating the Pauli exclusion principle so that the maximum L consistent with the Pauli principle is not 6 but 5 The] value is IL - si = 5 - 1 = 4
Iron Group Ions
Table 2 shows that ~he experimental magneton numbers for salts of the iron transition group of the peltiodic table are in poor agreement with (18) The values often agree quite weil with magneton numbers p = 2[S(S + 1)]112 calcu-
IL Pauling and E B Wilson Introduction to quantum mechanics McGraw-Hill 1935 pp 239-246
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
426
Table 2 E ffective magneton numbers for iron group ions
Config- Basic p(calc) = p(calc) = Ion uration level gU(] + 1)]112 2[$($ + 1)]112 p(exp)a
Ti3+ y4+ 3d l 2D 3I2 155 1 73 18 y 3+ 3d2 3F2 163 283 28 Cr3+ y2+ 3d3 4F 32 0 77 387 38 Mn3+ Cr+ 3d4 5DO 0 490 49 F e3+ Mn 2+ 3d5 6551 2 592 592 59 Fe2+ Co2+
3d6
3d7
5D4
4F 92
670 663
490 387
54 48
Ni2+ 3d8 3F 4 559 283 32 Cu2 + 3d9 2D52 355 173 19
Representative values
lated as if the orbital moment were not there at ail We say that the orbital moments are quenched
Crystal Field Splitting
The difference in behavior of the rare earth and the iron group salts is that the 4f shell responsible for paramagnetism in the rare earth ions lies deep inside the ions within the 5s and 5p sheIls whereas in the iron group ions the 3d shell responsible for paramagnetism is the outermost shell The 3d shell experiences the intense inhomogeneous electric field produced by neighboring ions This inhomogeneous electric field is called the crystal field The interacshytion of the paramagnetic ions with the crystal field has two major effects the coupling of L and S vectors is largely broken up so that the states are nO longer specified by their J values further the 2L + l sublevels belonging to a given L which are degenerate in the free ion may nOw be split by the crystal field as in Fig 6 This split ting diminishes the contribution of the orbital motion to the magnetic moment
Quenching of the Orbital Angular Momentum
In an electric field directed toward a fixed nucleus the plane of a classical orbit is fixed in space so that aIl the orbital angular momentum components Lxgt Ly Lz are constant In quantum theory one angular momentum component usually taken as Lz and the square of the total orbital angular momentum L2 are constant in a central field In a noncentral field the plane of the orbit will move about the angular momentum components are no longer constant and may average to zero In a crystal Lz will no longer be a constant of the motion although to a good approximation L2 may continue to be constant When Lz averages to zero the orbital angular momentum is said to be quenched The
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
427 14 Diamagnetism and Paramagnetism
===== PPy
y - ---pzy
reg reg (a) (b) (c) (d)
Figure 6 Consider an atom with orbital angular momentum L = l placed in the uniaxial crystalline electric field of the two positive ions along the z axis In the free atom the states mL = plusmn l 0 have identical energies-they are degenerate In the crystal the atom has a lower energy when the electron cloud is close to positive ions as in (a) th an when it is oriented midway between them as in (b) and (c) The wavefunctions that give rise to these charge densities are of the form zf(r) xf(r) and yf(r) and are called the Pz Px Py orbitaIs respectively In an axially symmetric field as shown the Px and Py orbitaIs are degenerate The energy levels referred to the free atom (dotted ine) are shown in (d) If the electric field does not have axial symmetry ail three states will have different energies
magne tic moment of astate is given by the average value of the magnetic moment operator I-tB(L + 2S) In a magnetic field along the z direction the orbital contribution to the magnetic moment is proportion al to the quantum expectation value of L z the orbital magnetic moment is quenched if the meshychanical moment Lz is quenched
When the spin-orbit interaction energy is introduced the spin may drag sorne orbital moment along with it If the sign of the interaction favors paraUel orientation of the spin and orbital magnetic moments the total magnetic moshyment will be larger than for the spin alone and the g value will be larger than 2 The experimental results are in agreement with the known variation of sign of the spin-orbit interaction g gt 2 when the 3d shell is more than half full g = 2 when the shell is half full and g lt 2 when the shell is less than half full
We consider a single electron wi th orbital quantum number L = 1 moving about a nucleus the whole being placed in an inhomogeneous crystalline elecshytric field We omit electron spin
In a crystal of orthorhombic symmetry the charges on neighboring ions will produce an electrostatic potential cp about the nucleus of thJ form
ecp = AX2 + By2 - (A + B )Z2 (24)
where A and B are constants This expression is the lowest degree polynomial in x y z which is a solution of the Laplace equation V2cp = 0 and compatible with the symmetry of the crystal
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
428
Uy = yf(r) Uz = zf(r)
are normalized
= 2Ui
= 0
Consider
dx dy dz (28)
the integral the diagonal matrix
elements
+ dx dy dz (29)
where dx dz
The their angular lobes
o This effect is momentum
age is zero in magnetic moment also
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
ParamilgnetIcircttm
(30)
- Agraveagravel
the hetween
g
g
1966 extensive See L Orgel Introduction to transition references are given by D Sturge Phys
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
430
Van Vleck Temperature-Independent Paramagnetism
We conside r an atomic or molecular system which has no magnetic moshyment in the ground state by which we mean that the diagonal matrix element of the magnetic moment operator JLz is zero
Suppose that there is a nondiagonal matrix element (slJLzIO) of the magnetic moment operator connecting the ground state degwith the excited state s of energy Acirc = Es - Eo above the ground state Then by standard perturbation theory the wavefunction of the ground state in a weak field (JLzB ~ Acirc) becomes
(32)
and the wavefunction of the excited state becomes
(33)
The perturbed ground state now has a moment
(34)
and the upper state has a moment
(35)
There are two interesting cases to consider Case (a) Acirc ~ kBT The surplus population in the ground state over the
excited state is approximately equal to NAcirc2kBT so that the resultant magnetishyzation is
M = 2BI(slJLzIO)1 2 NAcirc (36)
Acirc 2kBT
which gives for the susceptibility
(37)
Here N is the number of molecules per unit volume This contribution is of the usuaI Curie form although the mechanism of magnetization here is by polarizashytion of the states of the system whereas with free spins the mechanism of magnetization is the redistribution of ions among the spin states We note that the splitting Acirc does not enter in (37)
Case (h) Acirc kBT Here the population is nearly aIl in the ground state so that
M = 2NBI(slJLzIOgt1 2
(38)Acirc
The susceptibility is
(39)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
431 Diamagnetism P aramagnetism
type of contribution known as Van Vleck
COOLING DY
The first metbcd
the
partly lined is also lowered if
1)
in
3The method was suggested by P Debye Ann Giauque Am Chem Soc 49 1864 (1927) For many purposes SUI)plantt~d by the
dilution which operates solution in He play the raIe of atoms in a gas and
12
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
432
Spin
Total
Spin
Lattice Time- Time-
Before 1 New equilibrium Be ore cw equilibrium
Time at which Time at which magnetic fie ld magnetic field
is removed is lemoved
Figure 7 During adiabatic demagnetization the total entropy of the specimen is constant For effective cooling the initial entropy of the lattice should be small in comparison with the entropy of the spin sys tem
The steps carried out in the cooling process are shown in Fig 8 The field is applied at temperature Tl with the specimen in good thermal contact with the surroundings giving the isothermal path ab The specimen is then insushylated (la- = 0) and the fi eld removed the specimen follows the constant enshytropy path he ending up at temperature T2 The thermal contact at Tl is proshyvided by helium gas and the thermal contact is broken by removing the gas with a pump
Nuclear Demagnetization
The population of a magne tic sublevel is a function only of fLB lkBT hence of BIT The spin-system entropy is a function only of the population distribushytion hence the spin entropy is a function only of BIT IfBtgt is the effective field that corresponds to the local interactions the final temperature T2 reached in an adiabatic demagnetization experiment is
11 T2 = Tl (BtgtIB) (41)
whe re B is the initial field and Tl the initial temperature Because nuclear magne tic moments are weak nuclear magnetic interacshy
tions are much weaker than similar electronic interactions We expect to reach a temperature 100 times lower with a nuclear paramagnet than with an electron paramagnet The initial temperature Tl of the nuclear stage in a nuclear spinshycooling experiment must be lower than in an electron spin-cooling experiment If we start at B = 50 kG and Tl = 001 K then fLBlkBTl = 05 and the enshy
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
433 14 Diamagrwtism and Paramagfletism
B =0 BA = 100 gauss
07r---------------------------------------------------------~ 06
~ ~ ~ 05
sect S ~ 4 ~
g ~ 0 3
~ S Qi ~
01
o6 L ~ 10 15 do ~5 j J T mK
middotigure 8 Entropy for a pin 1 sys tem as a funetion of te mperature assuming an internaI random magne tic field Be of 100 gauss The specimen is magnetized iso thermally along ab and is th en insulated thermally The external magnetie field is turned off along be In order to keep the figure on a reasonable seale the initial temperature Tl is lower th an wouId be used in practice and so is the external magnetic fi eld
tropy decrease on magnetization is over 10 percent of the maximum spin enshytropy This is sufficient to overwhelm the lattice and from (41) we estimate a final temperature T2 = 10-7 K The first4 nuclear cooling experiment was carshyried out on Cu nudei in the metal starting from a fi rst stage at about 002 K as attained by electronic cooling The lowest temperature reached was 12 x 10- 6 K
The results in Fig 9 fit a line of the fonn of(41) Tz = T1(31 B) with B in gauss so that B11 = 31 gauss This is the effective interaction field of the magshynetic moments of the Cu nuclei The motivation for using nud ei in a metal is that conduction electrons help ensure rapid thermal contact of lattice and nushydei at the tempe rature of the first stage The present record5 for a spin tempershyature is 280 pK in rhodium
PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRONS
We are going to try to show how on the basis of these stati stics the fact that many
metals are diamagnetic or only weakly paramagnetic can be brought into agreeshy
ment with tb e existence of a magnetic mome nt of tbe e lectrons
W Pauli 1927
Classical fr ee electron theory gives an unsatisfactory account of the parashymagnetic susceptibility of the conduction electrons An electron has associated with it a magnetic moment of one Bohr magneton -La One might expect that
4N Kurti F N H Robinson F E Simon and D A Spohr Nature 178 450 (1956) for reviews see N middot Kurti Cryogenies 1 2 (1960) Adv in Cryogenie Engineering 8 1 (1963)
sp J Hakonen et al Phys Rev Lett 70 2818 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
434
Initial magnetic field in kG
lonr---T5--------~lrO--------~20~---3TO~ 9
8
7
1 6
~ 5 10e 4 u
Euml S 3
lL-__L-~~~~~~--------~--~ 03 06 2
Initial BIT in 106 GK
Figure 9 Nuclear demagnetizations of copper nuclei in the metal starting from 0012 K and various fields (After M V Hobden and N KurtL)
the conduction electrons would make a Curie-type paramagnetic contribution (22) to the magnetization of the metal M = N-L~BlkB T Instead it is observed that the magnetization of most normal nonferromagnetic metals is independent of temperature
Pauli showed that the application of the Fermi-Dirac distribution (Chapshyter 6) w6uld correct the theory as required We firs t give a qualitative explanashytion of the situation The result (18) tells us that the probabili ty an atom will be lined up parallel to the field B exceeds the probability of the antiparallel orienshytation by roughly -LBlkB T For N atoms per unit volume this gives a net magshynetization = N-L2BlkBT the standard result
Most conduction electrons in a metal however have no possibility of turning over when a field is applied because most orbitais in the Fermi sea with parallel spin are already occupied Only the electrons within a range kBT
of the top of the Fermi distribution have a chance to turn over in the field thus only the fraction TIT F of the total number of electrons contribute to the suscepshytibility Hence
N-L2B T N-L2 M =---=--B
kBT TF kBTF
which is independent of temperature and of the observed order of magnitude We now calculate the expression for the paramagnetic susceptibility of a
free electron gas at T ~ TF We follow the method of calculation suggested by Fig 10 An alternate derivation is the subject of Problem 5
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
--
435 14 Diamagnetism and Paramagnetism
Total energy kinetic + magne tic of electrons
l 1 ~ Parallel ta field
Dffi~~~ ~ Density of 1 orbitaislt o~~
1
(a) (b)
Figure 10 Pauli paramagnetism at absolu te zero the orbitais in the shaded regions in (a) are occupied The numbers of electrons in the up and down band will adjust ta make the energies equal at the Fermi level The chemical potential (Fermi level) of the moment up electrons is equal to that of the moment down electrons In (b) we show the excess of moment up electrons in the magnetic field
The concentration of electrons with magnetic moments parallel to the magnetic field is
l JF l l EF lN+ = - dE D (E + fJ-B ) == - dE D(E) + - fJ-B D(EF)
2 - l-B 2 0 2
written for absolute zero Here ~D(E + fJ-B ) is the densitv of orbitaIs of one 2 bull
spin orientation with allowance fo r the downward shift of energy by - fJ-B The approximation is written for kBT lt EF bull
The concentration of electrons with magnetic moments antiparallei to the magnetic field is
l JEF l llFN_ = - dE D(E - fJ-B) == - dE D (E) - - fJ-B D(EF) 21-B 20 2
The magnetization is given by M = fJ-(N + - N _) so that
3N fJ-2 M = fJ-2 D (EF)B = - k B (42)
2 BTF
with D(EF) = 3N2EF = 3N2kBTF from Chapter 6 The result (42) gives the Pauli spin magnetization of the conduction electrons for kBT lt EF bull
In deriving the paramagnetic susceptibility we have supposed that the spatial motion of the electrons is not affected by the magnetic field But the wavefunctions are modified by the magnetic fie ld Landau has shown that for
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
436
B
(43)
the
by
The UUU1HlltUy high for transition Ipl~rn heat
of atomic Z is X atomic (Langevin)
the maximum S consistent with this S The
and IL - S if the shell is Jess
is
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
437 14 Diamagnetism and Paramagnetism
8 0 r iT T TtS 1 1 1 IIT shy
70
60
~ 50 ~
-r--r-2_ w
E ~ --~_ I~
1
8 Cr __ 40 Vg ~ 0 ~~ -~w l
Vgt ~30
f-- - r--_ shy
20 I r- -- ~Nb
_J-_+-_r-zr- v - - -~-_ Rhl11
10 Na ~ K-- -+--1f--+_-J-Hr r--- - - Ta 1Rbf---T--t-- l
J J J00 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
T in K
Figure 11 Temperature dependence of the magnetic susceptibility of metals (Courtesy of C J Kriessman )
2 Huml mles Apply the Hund rules to find the ground state (the basic level in the notation ofTable 1) of (a) Eu ++ in the configuration 4[1 5S2p6 (b) Yb3+ (c) Tb3+ The results fo r (b) and (c) are in Table 1 but you should give the separate steps in applying the rules
3 Triplet excited states Some organic molecules have a triplet (S = 1) excited state at an energy kBil above a singlet (S = 0) ground state (a) Find an expression for the magnetic moment (J-L ) in a fie ld B (b) Show that the susceptibility for T p il is approximately independent of il (c) With the help of a diagram of energy levels versus field and a rough sketch of entropy versus field explain how this system might be cooled by adiabatic magnetization (not demagnetization)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
438
4 Consider two-Ievel system with and Iower states the splitting may arise from
Show that the hoat capacity per system is
c
capacity interaction between nuclear and electronic
electron spin order) 1lL111 are often detected experishy
in the heat capacity in the region T P Agrave
interaetions (see with fields al50
spin of a conduction eleetron gas at absoshyanother method
be the eoneentrations eleetrons Show that in a magnetie field B the total energy of the spin-up band in a free eleetron gas is
+()
where in zero magnetic field Find a similar + E - with respect to and solve
for the value of in the approximation ~ 1 Go to show that the
in agreement with
6 approximate the eHeet of intershyaetions among the eonduction electrons if assume that eleetrons with parallel
with each other vith energy is positive while electrons with not interact with each other Show with the of Problem 5
(1 + ()
find a similar expression for the total energy and for in the limit ~ 1 Show that the magnetization is
so the interaction enhances the susceptibility (c) Show that with B = 0 the total energy is unstable at 0 when V gt this is satisfied a neUc state ( 0) will have a lower energy th an paramagnetic state Because of the assumption t ~ l this is a sufficient condition for but it may not be a neccssary condition It is known
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
439
1 eNT = 43 x
0002 0004 0006 0008 001
14 Dinmafnetism and Paramagnetism
05 r-j--------r-----------i
Figure 12 Heat capacity of a two-level system as a function of Tt where t is the level splitting The Schottky anomaly is a very useful tool for determining energy level splittings of ions in rareshyearth and transition-group metals compounds and alloys
0008
0006 0
1
(3 E
0004E S
h u 0002
Figure 13 The normal-state heat capacity of gallium at T lt 021 K The nuclear quadrupole (G T 2) and conduction electron (G 0 T) contributions dominate the heat capacity at very low ct
temperatures (After K Phillips)
degl~ 03
S egraveJ p 8 02
01
00
Level21 j Level l
4 5 6 x = Tlt
TO in KJ
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
u= c=
7 Two-level system The result of Problem 4 is often seen in another form If the two energy levels are at agrave and -il that the energy and heat capacity are
of agrave are proportional to the temshy
to the heat capacity of dilute 1519 It is al50 used in the
8 Itystem Find the magnetization 1 moment
as a function field and temperature for a system of spins with S n (b) Show that in the li mit li-B lt kT result is shy
A Abragam and B Bleaney Electron resonance tom Dover 1986 B G Casimir Magnetism and very tempe ratu res DoveT 1961 A c1assic
Darby and K R Taylor Physics of rare earth Halsted 1972 A J Freeman The actinides electronic structure and related properties Academie 1974 R D Hudson Princip les and Elsevier 1972
North-Holland 1970 Knoepfel Pused Lounasmaa and methods below 1 K Academie Press 1974
Introduction ta transition metal 2nd ed Wiley 1966 Van Vleck The theory Oxford 1932 derivashy
tions of basic theorems G K White 3rd Oxford 1987 R White Quantum theory A J Freeman and G H Lander actinides North-
Holland 1984-1993 Sturge Jahn-Teller effect in solids Solid state 91 (1967)
OBrien and C C Chancey The effect An introduction and current reshyview Amer J Physics 61 (1993)
