Magnetic materials

Magnetic property• The response of the materials to external magnetic field

• All the materials are magnetic, only the degree of response varies, which is measured in terms of their magnetization (strong or weak)

• The parameters used to study the magnetic behaviors of the materials are as follows:

1.Magnetic dipoles & magnetic moment:

Magnetic dipoles are analogous to electric dipoles ; consists of a north pole and a south pole of strength ‘m’ each separated by a small distance ‘l’

Magneic moment = mxl

For a circular current loop equivalent to a magnetic

dipole, magnetic moment μM = NIxA ( amp. m2)

& torque on the dipole τ = μM XB

Where ‘I’ is the current in the loop

and ‘A’ is the area of the loop,N is no. of turns

N S

l

μM

I

Right hand

Screw rule

• 2. Magnetisation = dipole moment / volume

M = μ / V ( amp. / met.)

• 3.Magnetic susceptibility = magnetization / mag. field strength

χ = M / H (no unit)

• 4.Magnetic permeability = magnetic induction / mag. field strength

μ = B / H (Wb / amp. met. = H/met)

• μ =μ0 μr

• 5.Relative permeability μr = μ / μ0,

• μ0 = absolute permeability = 4πX 10 -7 H/m

• 6.Relation between H,B & M is

B = μ0 (M+H) = μ0 (χH + H) = μ0 (1 + χ) H

• B = μ H = μ0 μr H so

• 7. μr =(1 + χ)

Diamagnetism Paramagnetism Ferromagnetism

1.Normally referred as non-magnetic as the response is very weak2.In ext. magnetic field magnetic moment induced in a direction opposite to applied field– repelled by the fieldH=0, M=0 H=H→

M= - M←3. Permeability μ<1 4. susceptibility χ <05. Susceptibility does not depend on temperature5. Ex; Cu, Ag, Hg, Au, Zn,SC

1.Normally referred as non-magnetic as the response is very weak2. They posses permanent magnetic moments, which are randomly oriented in the absence of ext. magnetic field., hence net magnetization is zero. When a field is applied moments get aligned in the field direction, giving positive magnetization

H=0, M=0 H=H→ M= M3. Permeability μ>1 4. susceptibility χ is positive, small and temp. dependantχ = C/ T →Curie law5.Ex; Al, Cr, Na, Ti, Zr

1.Referred as magnetic as response is strong(exchange coupling)2. Posses permanent dipoles3. Show spontaneous magnetization—even in the absence of ext. field, magnetization shown is high, when field is applied, M increases4.Permeability μ>1 5.susceptibility χ is positive,

large and temp. dependant

χ = C/ T-θ →Curie – Weiss law

Ex; Fe, Co, Ni

ferromagnetic domains

show spontaneous

Classification of magnetic materials

Origin of magnetic moment

• The three sources of magnetic moment in an atom are

• 1. orbital motion of the electron 2.spin motion of the electron

• 3. nuclear spin

• If the vector sum of all the contribution is zero then net magnetization is zero

• 1. orbital motion of the electron

• Motion of the electron (charged particle)around the nucleus in a circular orbit (orbital motion) is equivalent to a circular current and behaves as a magnetic dipole.

• Associated magnetic moment is μM = IxA ( amp. m2)

• I= - q/T , where T is the time period for one rotation = 2 π r / v

• ‘v’ is the velocity of the electron in the orbit

• μM = (- q v / 2 π r ) (π r2) = - qvr/2 = - q/ 2m ( mvr) = - (q/ 2m ) L

• Orbital magnetic moment μorb = - (q/ 2m ) L

2.spin motion of the electron

• Similarly, the spin motion of the electron around their own axis give rise to spin magnetic moment

•

Total magnetic moment due to electron motion inside the atom is

• μM = - (q/ 2m ) (L+S) = - (q/ 2m ) J, J is the total angular momentum ( from |L+S| to |L- S|

• Quantum no. associated with ‘L’ is √* l(l+1) ħ+ , ‘l’ is the orbital quantum no.

• l=0---s shell, l=1---p shell, l=2-----d shell, l=3------f shell

• Quantum no. associated with ‘S’ ± ħ/2

• Total magnetic moment = μM = - g(q/ 2m ) J

• If magnetic field is applied along z-direction , the component of the total magnetic moment in that direction is, μM = -g(qħ/2m)mj

μspin = - (q/ 2m ) S

• Where mj is the magnetic quantum no. Varying from (J to -J)

• & g is called Lande’s g-factor

g =

• Calculation rules

1.if electrons are in the s-orbit, orbital magnetic moment is zero (l=0)

2. for completely filled shell, orbital magnetic moment is zero (l=0)

As ml = l to –l (s shell,l=0, p shell, l=1, d shell, l=2 and f shell, l=3..)

3. partially filled p, d and f shells contribute to orbital magnetic moment

4.If all electrons are paired, spin magnetic momentum is zero

3. Nuclear spin

Due to the spin of nucleus, a magnetic moment is associated which is very small as compared to the electronic contribution as heavy mass is involved (10-3 times) is masked by electronic mag. Mom.

)1(2

)1()1()1(1

JJ

LLSSJJ

Bohr magnetron

• If there is only a single electron it will have only spin magnetic moment μs = -2 (qħ / 2m) ±1/2 , as g=2 and mj= ±1/2

• This is the fundamental magnetic moment called Bohr magnetron ‘μB‘

• All the magnetic moments are expressed in terms of ‘μB‘

• ‘μB‘ = = 9.27x 10-24 Am2

• Ex;-1. hydrogen atom

One electron in s shell. So l=0 and s= +1/2 or -1/2 implies that orbital magnetic mom. Is zero and spin mag. Mom. Is same as the Bohr magnetron (‘μB‘ is the magnetic moment possesed by hydrogen electron)

2. Helium atom ?...calculate

m

q

2

• For other atoms apply the rules to calculate

• Hund’s rule: 1. spins of electrons remain parallel to each other to the max. Extent

• 2. max . Value of L is consistent with the spin S

• 3. if shell is less than half filled J= |L- S| ,

if more than half filled J= |L+ S| &

if exactly half filled then L=0 and J=S

Langevin’s theory of diamagnetism

• motion of the electron in the orbit around the nucleus is equivalent to a current in the closed circuit.

• When magnetic field is applied electric field is induced in the circuit.

• The induced emf is opposite to the applied field and electrons are accelerated in the opposite direction.

• Acceleration a =- eEi/m, Ei = Vi/d, where ‘V’i is induced emf and ‘d’ is the path covered by the electron in the orbit, d=2πρ

• ‘ρ’ is the radius of projection of the orbit in the X-Y plane.

• Vi = induced emf = - (rate of change of magnetic flux.)

• If MF is applied along Z-direction, flux= B(πρ2)

• Putting all these, we get acceleration a= -

• Or,

2()2(

Bdt

d

m

e

dt

dB

m

e

dt

dv

)2(

2

B

dBm

edv

02

Or, v2-v1= ∆v= (eρ/ 2m) B indicates that the velocity of the electron in the

orbit changes, so angular momentum also changes

the magnetic moment.

μM = IxA = -eA/T

= -eA ∆v / 2π ρ

= - eρ/2 ( eρ/2m)B

= - (e2 ρ2/4m ) B

As the plane of the orbit varies continuously due to applied MF we take average

value of ρav2 = xav

2+yav2

if ‘r’ is the radius of the atom, then rav2= xav

2+yav2+zav

2

For spherical symmetric atom xav2 =yav

2 = zav2 = rav

2 / 3

So, ρav2 = (2/3) rav

2

• Hence, μM = - (e2 rav2 /6m ) B

• If there are ‘Z’ no. of electrons, then, μM = - Z(e2 rav2 /6m ) B

• If N is no. of atoms per vol.

• Then magnetization M= - NZ(e2 rav2 /4m ) B

• Negative sign indicates that Magnetization is opposite to applied field

• So the diamagnetic susceptibility

χ = μ0 M/B

or,

It indicates that Susceptibility depends on average mean square radius and independent of temperature

χ = -NZ(μ0 e2 rav

2 /6m)

Theories of Paramagnetism

Classical theory (Langevin's):

Let ‘n’ = no. of dipoles in a system = n0 exp ( - E/ kβT) (Boltzmann’s distribution formula)

‘B’ = magnetic field applied

‘τ’ = torque experienced by the dipoles in the magnetic field

= μM X B = μM B sinθ

‘E’ = energy of the dipoles

= ∫ τ dθ

= ∫ μM B sinθ dθ = - μM B cosθ = - μM .B

Now we have ‘n’ = n0 exp (μM B cosθ / kβT )

Total dipole moment for all these dipoles =

< μM > = ; μMcosθ is the component of ‘μM‘

along B

n

M dn0

dn

dnM cos

B

μM

θ

but dn=

So, < μM > =

If we put cosθ = x, dx= - sinθdθ & μMB / kβ T = a

then the above expression becomes

< μM > =

= μM [ coth a- 1/a ]

dTk

B

Tk

Bn MM sincos

exp0

0

0

sin)/cosexp(

sin)/cosexp(cos

dTkB

dTkB

M

MM

1

1

1

1

)exp(

)exp(

dxax

dxaxxM

• < μM > = μM L (a) ; L(a) is called Langevin’s function

• Now magnetisation

• Case-I :- ‘a’ is large—when ‘T’ is small or ‘B’ is large

L(a) ≈ 1 hence M = N μM = Ms = saturation magnetisation

• Case-II :- ‘a’ is small—when ‘T’ is large or ‘B’ is small

coth = 1/a + a/3 – a2/45+ -------- ≈ 1/a + a/3

L(a) = a/3

magnetisation M = Nμm μmB/ 3kβ T = N μm2 B/ 3kβ T

Susceptibility

or, χ = C/T ------------Curie law

M= N μM L(a)

χ = M/ H = N μm2 μ0 / 3kβ T

Quantum Theory :

• The energy of the system in the magnetic field applied in Z-direction is E = - μ z.B

• μ z= -gmjμB

• so, E = 2mjμB B

• For a system of one electron (l=0) , mj =+ ½ or – ½

• E = - μB B when mj =- ½ & E = μB B when mj = + ½

• One corresponding to parallel and other to antiparallel

spin magnetic moment orientation w.r.t.

Magnetic field

Difference in energy between the two

∆E= 2 μB B

• Let out of ‘N’ total atoms per volume

In the system N1 are parallel and N2

are antiparallel

mj =- ½

mj =+ ½

B=0

B ≠0

E1

E

E2

Z- dir

• Distribution of atoms at thermal equilibrium in the two corresponding states are given by Maxwell –Boltzmann distribution function

• N1/ N = e –E1

/ kβ

T / N & N2/ N = e –E2

/ k β

T /N

• Net magnetization M = (N1-N2) μz

• Or M =

• M = N μB tanh ( μBB/ kβ T)

• In case μBB < < kβ T

• M= N μB ( μBB/ kβ T) = N μ2BB/ kβ T

• χ = M / H or

• χ = C/T------ Curie law

Tk

B

Tk

B

Tk

B

Tk

B

BBB

BB

ee

eeN

χ = N μ0 μ2B/ kβ T

Ferromagnetic theory ( Weiss )• As per Weiss in ferromagnetic materials spontaneous

magnetization is observed, which is due to a strong internal field arising from an exchange interaction between the magnetic moments in the neighborhood domains

• exchange interaction between two atoms ‘I’ and ‘j’

• = U= -2 J SiSj

‘J’ is called the exchange integral

• Internal field ‘H’ is proportional to the magnetisation

• H int α M

• Or H int = λM

• Htot= H appl + Hint

• H tot= H appl + λM

• As, χ = M / H

= M / H appl + λM = C / T by Curie law

• C/ T = M/ H appl. +λ M

• C H appl. = MT - C λ M or, M = C Happl. / T – C λ or, M = C H / T-Tc

• Susceptibility -- ---- Curie-Weiss law

• Ferromagnetic domains:

B

χ = M / H = C / T-Tc

Domain wall ≈ 10-2 μm

Ferromagnetic hysteresis

Ms

Ms(0)

TTc

Ms= saturation magnetisation

Mr = remanent magnetisation

Hc= coercive field

Soft and hard FM materialsSoft ferromagnetic Hard ferromagnetic

1. Can be easily magnetized or demagnetized

2. Thin and long hysteresis loop3. High permeability and low coercive field4. Large susceptibility & low remanent mag.5. As area of the loop is small, magnetic

energy loss per volume is less during magnetisation and demagnetisation

6. Application: electromagnet, in motors, generators, dynamos and switching circuits

7. Ex: Fe-Si alloy , Fe-Co-Mn alloy and Fe-Ni alloy

1. Can not be magnetised or demagnetised easily

2. Wide hysteresis loop3. Low permeability and high coercive field4. small susceptibility & high remanent mag.5. Large area of the loop indicates, magnetic

energy loss per volume is high during magnetisation and demagnetisation

6. For permanent magnet in speakers, clocks

7. Rare earth alloys with Mn, Fe, Co, Ni

SOFT & HARD FERROMAGNETIC HYSTERESIS LOOP

Ferrimagnetic & Antiferromagnetic materials

• Ferrimagnetic material are special class of ferromagnetic material called ‘ ferrites’ with high permeability, saturation magnetisation and show hysteresis (square loop)

• Suitable for high frequency application and magnetic devices

• Some spin magnetic moments are in opposite direction in the magnetised state, but are of different magnitudes, giving rise to net finite magnetic moment.

• Molecular formula: Me2+ Fe23+O3 ;

Me is a divalent atom like, Fe, Mn, Zn,Cd,Cu,Ni,Co,Mg

• Crystal structure: Inverse spinel

• In the cubic cell ‘8’ atoms per unit cell.

• In the unit cell, 32 O-2 ions , 16 Fe 3+ ions, and 8 Me2+ ions

• 8- Fe 3+ ions, 8- Me2+ ions are surrounded by 6 oxygen ion—octahedral site and spins are parallel

• Rest 8- Fe 3+ ions are surrounded by 4 oxygen ions—tetrahedralsite and spins antiparallel

• Hence net spin moment of Fe 3+ ions cancel ( 8 up spin and 8 down spin)

• Only , 8 Me2+ ions contribute to magnetic moment.

• As spin magnetic moment is μ = g μBS

• since g=2, magnetic moment of one divalent atom μdi = 2μBS

in a unit cell 8 such atoms are present

( for Fe2+ , s= 2, for Co =3/2,Ni=1, Cu=1/2, Mn=5/2 )

• total moment in unit cell= 8 x μdi

• Magnetisation M = total moment per volume = 8 x μdi / a3

Where a is lattice parameter

• Applications: resistivity of ferrites are very high so applied for high frequency application (eddy current energy loss less)

• Mixed ferrites are produced by combining two suitable divalent atoms

• Ferrites have square hysteresis loop. So used for digital storage device ( two values of magnetisation +Ms & - Ms; so 1 or 0 )

• Soft ferrites are used for high freq. Transformer core, computer memory, hard disc, floppy disk audio video cassette, recorder head

• Hard ferrites are used for permanent magnets in generator, motor, loud speaker, telephone

• Non-volatile memory called magnetic bubbles

• Antiferromagnetism: when exchange interaction between

adjacent or neighboring domains give rise to ordered antiparallel spin arrangement, below a temp. called Neel temp. ex.-MnO,MnS,FeCl2, CoO. Net moment or magnetisation is Zero

χ= C / (T +θ )

TTN

χ

Fe3+ Fe3+

Me2+

S= 5/2 S= 5/2S= 2

octahedral tetrahedral

Mn2+= 3d5 μ = g μBS = 2 x 5/2 x μB = 5μB

Fe2+ = 3d6 , μ = 4 μB

Co2+= 3d7, μ = 3 μB

Ni 2+= 3d8, μ = 2μB

Cu2+= 3d9 μ = 1μB

Magnetite has the empirical formula Fe3O4 , or Fe2+(Fe3+O2)2 , “ferrous ferrite” .Its formula as a spinel would be Fe3+

tetFe2+octFe3+

octO4 , where "tet" and "oct"stand for tetrahedral and octahedral coordinations by the oxide anions . In theabove model , the blue spheres represent the tetrahedral iron(III) cations , andthe red spheres are the octahedrally coordinated iron(II) and (III) cations . Theoxide anions are shown as the green spheres . Because of the fortuitousinverse nature of the magnetite structure , ferrous and ferric cations are bothin the similar octahedral coordination by oxides . In "normal" spinels , such asthe mineral spinel itself (magnesium aluminate) , the A cation is tetrahedraland the M cations are both octahedral

Ferrites = Me2+O Fe3+2O3

Me= Fe, Mn, Co, Ni, Cu, Mg, Zn, Cd