Download o  Fine structure Spin-orbit interaction. Relativistic kinetic energy correction

o  Fine structure
o  Spin-orbit interaction.
o  Relativistic kinetic energy correction
o  Hyperfine structure
o  The Lamb shift.
o  Nuclear moments.
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o  Fine structure of H-atom is due to spin-orbit interaction:
Jˆ is a max
Z! ˆ ˆ
"E so = #
S$L
2m 2cr 3
Lˆ
o  If L is parallel to S => J is a maximum => high energy
configuration.
!
+Ze
!
o  Angular momenta are described in terms of quantum
numbers, s, l and j:
Jˆ = Lˆ + Sˆ
Jˆ 2 = ( Lˆ + Sˆ )( Lˆ + Sˆ ) = Lˆ Lˆ + SˆSˆ + 2 Sˆ " Lˆ
1
Sˆ " Lˆ = ( Jˆ " Jˆ # Lˆ " Lˆ # Sˆ " Sˆ )
2
!2
=> Sˆ " Lˆ = [ j( j + 1) # l(l + 1) # s(s + 1)]
2
"#E so = $
!
-e
!
!
!
Sˆ
Jˆ is a min
Lˆ
!
+Ze
-e
!
Sˆ
Z! 3 1
[ j( j + 1) % l(l + 1) % s(s + 1)]
4m 2c r 3
!
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o 
For practical purposes, convenient to express spin-orbit coupling as
a
[ j ( j + 1) # l(l + 1) # s(s + 1)]
2
where a = Ze 2µ0 ! 2 /8"m 2 r 3 is the spin-orbit coupling constant. Therefore, for the 2p
electron:
! "E = a * 3 # 3 + 1& )1(1+ 1) ) 1 # 1 + 1&- = + 1 a
(
%
(/
, %
so
2 +2 $ 2 '
2 $ 2 '.
2
!
"E so =
a *1 # 1 &
1 # 1 &"E so = , % + 1( )1(1+ 1) ) % + 1(/ = )a
2 +2 $ 2 '
2 $ 2 '.
!E
j = 3/2
+1/2a
Angular momenta aligned
j = 1/2
-a
Angular momenta opposite
2p1
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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):
n 2!2
r = 4 "#0
2
mZe
=> a ~
Z4
n6
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:
!
"E so =
a~
Z4
n 3 [ l(l + 1)(2l + 1)]
Z 2# 4
[ j( j + 1) $ l(l + 1) $ 3/4]
mc 2
3
2n
l(l + 1)(2l + 1)
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!
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:
o 
o 
o 
o 
2S+1L
J
L = 0 => S
L = 1 => P
L = 2 =>D
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.
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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:
2P
E
3/2
+1/2a
Angular momenta aligned
2p1 (2P)
2P
1/2
-a
Angular momenta opposite
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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 
H!
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.
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o  According to special relativity, the kinetic energy of an electron of mass m and
velocity v is:
2
4
T=
p
p
"
+ ...
2m 8m 3c 2
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
"H rel = #
o  Using the fact that
1
p4
8m 3c 2
p2
= E " V we can write
2m
!
1
2
2
"H rel = #
2 ( E # 2EV + V )
2mc
!
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!
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.
"E rel = #
% 1
Z 2$ 4
3(
mc 2 '
# *
3
&
n
2l + 1 8n )
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.
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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
4 4
"E FS = "E so + "E rel = #
% 1
Z $
3(
mc 2 '
# *
3
& 2l + 1 8n )
2n
o  As En = -Z2E0/n2, where E0 = 1/2!2mc2, we can write
!
E H "atom = "
Z 2 E 0 $ Z 2# 2 $ 1
3 ''
1+
" ))
&
2 &
n %
n % j + 1/2 4n ((
! of the gross and fine structure of the hydrogen atom.
o  Gives the energy
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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!
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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.
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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.
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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
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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
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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 "E hyperfine = #µˆ nucleus $ Bˆ electron % Iˆ $ Jˆ
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.
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o 
o 
Like electron, the proton has a spin angular momentum and an associated intrinsic dipole
moment
e
µˆ p = g p Iˆ
M
The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and
hence the effect is small.
gpe 2 ˆ ˆ
I#J
mMc 2 r 3
o 
! be shown to be:
Resulting energy correction can
o 
Total angular momentum including nuclear spin, orbital angular momentum and electron spin
is
Fˆ = Iˆ + Jˆ
!
F
=
f
(
f
+
1)!
where
"E p =
Fz = m f !
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.
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o 
The energy splitting of the hyperfine interaction is
given by
a
"E HFS = [ f ( f + 1) # i(i + 1) # j ( j + 1)]
2
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
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o  Selection rules determine the allowed transitions between terms.
$n = any integer
$l = ±1
$j = 0, ±1
$f = 0, ±1
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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.
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