PY3P05

o  Fine structure

o  Spin-orbit interaction.

o  Relativistic kinetic energy correction

o  Hyperfine structure

o  The Lamb shift.

o  Nuclear moments.

PY3P05

o  Fine structure of H-atom is due to spin-orbit interaction:

o  If L is parallel to S => J is a maximum => high energy configuration.

o  Angular momenta are described in terms of quantum numbers, s, l and j:

!

"Eso =#Z!

2m2cr3ˆ S $ ˆ L

+Ze -e !

ˆ L

!

ˆ S

is a max

!

ˆ J

+Ze -e !

ˆ L

!

ˆ S

is a min

!

ˆ J

!

ˆ J = ˆ L + ˆ S ˆ J 2 = ( ˆ L + ˆ S )( ˆ L + ˆ S ) = ˆ L ̂ L + ˆ S ̂ S + 2 ˆ S " ˆ L

ˆ S " ˆ L = 12

( ˆ J " ˆ J # ˆ L " ˆ L # ˆ S " ˆ S )

=> ˆ S " ˆ L = !2

2[ j( j +1) # l(l +1) # s(s +1)]

!

"#Eso =$Z!3

4m2c1r3[ j( j +1) % l(l +1) % s(s+1)]

PY3P05

o  For practical purposes, convenient to express spin-orbit coupling as

where is the spin-orbit coupling constant. Therefore, for the 2p electron:

E 2p1

j = 3/2

j = 1/2

+1/2a Angular momenta aligned

-a Angular momenta opposite

!

"Eso =a23232

+1#

$ %

&

' ( )1(1+1) ) 1

212

+1#

$ %

&

' (

*

+ ,

-

. / = +

12a

"Eso =a21212

+1#

$ %

&

' ( )1(1+1) ) 1

212

+1#

$ %

&

' (

*

+ ,

-

. / = )a

!

"Eso =a2j j +1( ) # l(l +1) # s s+1( )[ ]

!

a = Ze2µ0!2 /8"m2r3

PY3P05

o  The spin-orbit coupling constant is directly measurable from the doublet structure of spectra.

o  If we use the radius rn of the nth Bohr radius as a rough approximation for r (from Lectures 1-2):

o  Spin-orbit coupling increases sharply with Z. Difficult for observed for H-atom, as Z = 1: 0.14 Å (H!), 0.08 Å (H"), 0.07 Å (H#).

o  Evaluating the quantum mechanical form,

o  Therefore, using this and s = 1/2:

!

r = 4"#0n2!2

mZe2

=> a ~ Z4

n6

!

a ~ Z 4

n3 l(l +1)(2l +1)[ ]

!

"Eso =Z 2# 4

2n3mc 2 [ j( j +1) $ l(l +1) $ 3/4]

l(l +1)(2l +1)

PY3P05

o  Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime.

o  Each level is labeled by L, S and J: 2S+1LJ

o  L = 0 => S o  L = 1 => P o  L = 2 =>D o  L = 3 =>F

o  If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, 2P3/2 and 2P1/2

o  2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin. o  If S = 0 => multiplicity is 1: singlet term. o  If S = 1/2 => multiplicity is 2: doublet term. o  If S = 1 => multiplicity is 3: triplet term.

o  Most useful when dealing with multi-electron atoms.

PY3P05

o  The energy level diagram can also be drawn as a term diagram.

o  Each term is evaluated using: 2S+1LJ

o  For H, the levels of the 2P term arising from spin-orbit coupling are given below:

E 2p1 (2P)

2P3/2

2P1/2

+1/2a Angular momenta aligned

-a Angular momenta opposite

PY3P05

o  Spectral lines of H composed of closely spaced doublets. Splitting is due to interactions between electron spin S and the orbital angular momentum L => spin-orbit coupling.

o  H! line is single line according to the Bohr or Schrödinger theory. Occurs at 656.47 nm for H and 656.29 nm for D (isotope shift, !"~0.2 nm).

o  Spin-orbit coupling produces fine-structure splitting of ~0.016 nm. Corresponds to an internal magnetic field on the electron of about 0.4 Tesla.

H!

PY3P05

o  According to special relativity, the kinetic energy of an electron of mass m and velocity v is:

o  The first term is the standard non-relativistic expression for kinetic energy. The second term is the lowest-order relativistic correction to this energy.

o  Therefore, the correction to the Hamiltonian is

o  Using the fact that we can write

!

T =p2

2m"

p4

8m3c 2+ ...

!

"Hrel = #1

8m3c 2p4

!

p2

2m= E "V

!

"Hrel = #1

2mc 2E 2 # 2EV +V 2( )

PY3P05

o  With V = -Z2e / r , applying first-order perturbation theory to previous Hamiltonian reduces the problem of finding the expectation values of r -1 and r -2.

o  Produces an energy shift comparable to spin-orbit effect.

o  Note that !Erel ~ "4 => (1/137)4 ~ 10-8

o  A complete relativistic treatment of the electron involves the solving the Dirac equation.

!

"Erel = #Z 2$ 4

n3mc 2 1

2l +1#38n

%

& '

(

) *

PY3P05

o  For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much smaller than the gross structure.

Enlj = En + $EFS

o  Gross structure determined by En from Schrödinger equation. The fine structure is determined by

o  As En = -Z2E0/n2, where E0 = 1/2!2mc2, we can write

o  Gives the energy of the gross and fine structure of the hydrogen atom.

!

"EFS = "Eso + "Erel = #Z 4$ 4

2n3mc 2 1

2l +1#38n

%

& '

(

) *

!

EH"atom = "Z 2E0

n21+

Z 2# 2

n1

j +1/2"34n

$

% &

'

( )

$

% &

'

( )

PY3P05

o  Energy correction only depends on j, which is of the order of !2 ~ 10-4 times smaller that the principle energy splitting.

o  All levels are shifted down from the Bohr energies.

o  For every n>1 and l, there are two states corresponding to j = l ± 1/2.

o  States with same n and j but different l, have the same energies (does not hold when Lamb shift is included). i.e., are degenerate.

o  Using incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics!

PY3P05

o  Spectral lines give information on nucleus. Main effects are isotope shift and hyperfine structure.

o  According to Schrödinger and Dirac theory, states with same n and j but different l are degenerate. However, Lamb and Retherford showed in 1947 that 22S1/2 (n = 2, l = 0, j = 1/2) and 22P1/2 (n = 2, l = 1, j = 1/2) of are not degenerate. Energy difference is ~4.4 x 10-6 eV.

o  Experiment proved that even states with the same total angular momentum J are energetically different.

PY3P05

1.  Excite H-atoms to 22S1/2 metastable state by e- bombardment. Forbidden to spontaneuosly decay to 12S1/2 optically.

2.  Cause transitions to 22P1/2 state using tunable microwaves. Transitions only occur when microwaves tuned to transition frequency. These atoms then decay emitting Ly! line.

3.  Measure number of atoms in 22S1/2 state from H-atom collisions with tungsten (W) target. When excitation to 22P1/2, current drops.

4.  Excited H atoms (22S1/2 metastable state) cause secondary electron emission and current from the target. Dexcited H atoms (12S1/2 ground state) do not.

PY3P05

o  According to Dirac and Schrödinger theory, states with the same n and j quantum numbers but different l quantum numbers ought to be degenerate. Lamb and Retherford showed that 2 S1/2 (n=2, l=0, j=1/2) and 2P1/2 (n=2, l=1, j=1/2) states of hydrogen atom were not degenerate, but that the S state had slightly higher energy by E/h = 1057.864 MHz.

o  Effect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized.

o  For further info, see http://www.pha.jhu.edu/~rt19/hydro/node8.html

PY3P05

Energy scale Energy (eV) Effects

Gross structure 1-10 electron-nuclear attraction Electron kinetic energy Electron-electron repulsion

Fine structure 0.001 - 0.01 Spin-orbit interaction Relativistic corrections

Hyperfine structure 10-6 - 10-5 Nuclear interactions

PY3P05

o  Hyperfine structure results from magnetic interaction between the electron’s total angular momentum (J) and the nuclear spin (I).

o  Angular momentum of electron creates a magnetic field at the nucleus which is proportional to J.

o  Interaction energy is therefore

o  Magnitude is very small as nuclear dipole is ~2000 smaller than electron (µ~1/m).

o  Hyperfine splitting is about three orders of magnitude smaller than splitting due to fine structure.

!

"Ehyperfine = # ˆ µ nucleus $ ˆ B electron % ˆ I $ ˆ J

PY3P05

o  Like electron, the proton has a spin angular momentum and an associated intrinsic dipole moment

o  The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and hence the effect is small.

o  Resulting energy correction can be shown to be:

o  Total angular momentum including nuclear spin, orbital angular momentum and electron spin is

where

o  The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has spin 1/2,.

o  Hence every energy level associated with a particular set of quantum numbers n, l, and j will be split into two levels of slightly different energy, depending on the relative orientation of the proton magnetic dipole with the electron state.

!

ˆ µ p = gpeM

ˆ I

!

ˆ F = ˆ I + ˆ J

!

F = f ( f +1)!Fz = mf !

!

"E p =gpe

2

mMc 2r3ˆ I # ˆ J

PY3P05

o  The energy splitting of the hyperfine interaction is given by

where a is the hyperfine structure constant.

o  E.g., consider the ground state of H-atom. Nucleus consists of a single proton, so I = 1/2. The hydrogen ground state is the 1s 2S1/2 term, which has J = 1/2. Spin of the electron can be parallel (F = 1) or antiparallel (F = 0). Transitions between these levels occur at 21 cm (1420 MHz).

o  For ground state of the hydrogen atom (n=1), the energy separation between the states of F = 1 and F = 0 is 5.9 x 10-6 eV. Prove this!

F = 1

F = 0

21 cm radio map of the Milky Way

!

"EHFS =a2

f f +1( ) # i(i +1) # j j +1( )[ ]

!

PY3P05

o  Selection rules determine the allowed transitions between terms.

$n = any integer $l = ±1 $j = 0, ±1 $f = 0, ±1

PY3P05

o  Gross structure: o  Covers largest interactions within the atom:

o  Kinetic energy of electrons in their orbits. o  Attractive electrostatic potential between positive nucleus and negative electrons o  Repulsive electrostatic interaction between electrons in a multi-electron atom.

o  Size of these interactions gives energies in the 1-10 eV range and upwards. o  Determine whether a photon is IR, visible, UV or X-ray.

o  Fine structure: o  Spectral lines often come as multiplets. E.g., H! line.

=> smaller interactions within atom, called spin-orbit interaction. o  Electrons in orbit about nucleus give rise to magnetic moment

of magnitude µB, which electron spin interacts with. Produces small shift in energy.

o  Hyperfine structure: o  Fine-structure lines are split into more multiplets. o  Caused by interactions between electron spin and nucleus spin. o  Nucleus produces a magnetic moment of magnitude

~µB/2000 due to nuclear spin. o  E.g., 21-cm line in radio astronomy.