Transcript spin-orbit coupling

Ch4 Fine structure of atoms
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Magnetic moments
Spin of the electron
Stern-Gerlach experiment
Spectrum of the alkali atoms
Spin-orbit coupling (interaction)
The Zeeman effect
Some words
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Fine structure
Electron spin
Orbital motion
Magnetic moment
Alkali atoms
Valence
intrinsic
Magneton
Atomic state
• Spin-orbit coupling
(interaction)
• Frame of reference
• Larmor precession
• Lande g factor
• Alkali metals
• diffuse
• Stern-Gerlach
experiment
• Zeeman’s effect
Magnetic moment of the orbital
motion
• An electron moving in an orbit is equivalent
to a circular current, which
possesses
a

e
e

l  
(Bohr magneton )
magnetic moment.
2m
2m
• The magnetic moment associated with the
orbital angular momentum is quantised.
• An applied magnetic field B acts on the
orbital magnetic moment by trying to align
the vectors μl and B. The electrons precess
about the direction of the field, called
Larmor precession. There is also a potential
energy E=- μl ·B.
B
Electron spin
• Electron spin was introduced by Uhlenbeck
and Goudsmit in 1925. They proposed that
the electron possesses an intrinsic angular
momentum independent of any orbital
angular momentum it might have. Four
quantum numbers: n, l, ml, ms.
• The splitting of many spectral lines in a
magnetic field can only be explained if the
electron
has
a
spin
s.
angulare momentum

| s | s ( s  1)    g
s
s
S
2m
The total angular momentum
& magnetic moment
j
  
j  l  s ,   l   s
l
s
 j  l cos( j , l )   s cos( j , s )
  g j J B
3 1 sˆ 2  lˆ 2
gj   (
)
2
ˆj
2 2
μl
μS
μ
μj
Fine structure
• All energy levels except the s states of oneelectron atoms are spit into two substates.
This produces a doublet or multiplet
structure of the spectral lines, namely, fine
structure.
• It can not be explained by Coulomb
interaction between the nucleus and the
electrons. It results from a magnetic
interaction between the orbital magnetic
moment and spin magnetic moment of the
electron, called spin-orbit coupling.
Spin-orbit coupling (I)
• An electron revolving about a nucleus finds
itself in a magnetic field produced by the
nucleus which is circling about it in its own
frame of reference.
• This magnetic field then acts upon the
electron’s own spin magnetic moment to
produce substates in terms of energy. S=1/2
could make some single state (n,l,m) split
Ze 2
into two substates. V   B  1
sˆ  lˆ
ls
S
Nuc
40 2me 2c 2 r 3
(Z ) 4 me c 2 [ j ( j  1)  l (l  1)  34 ]

4n 3
l (l  12 )(l  1)
Spin-orbit coupling (II)
• The coupling of two magnetic moments in an atom
leads to an addition of the two angular momenta to
yield a total angular momentum. j=l+s
• The total angular momentum has the magnitude of
with j=|l±1/2| for a single electron system. The
quantum number j is the quantum number of the
total angular momentum
.
j ( j  1)
• For a p electron, l=1, s=1/2, there are two states
with j=1/2, 3/2, respectively.
The symbolism for atomic states
• The atomic state is denoted by a symbol, and
the orbital angular momentum is indicated by
upper case letters S,P,D,F.
• The primary quantum number n is written as
an integer in front of the letter, and the total
angular momentum quantum number j as a
subscript.
• The multiplicity 2s+1 (the number of j values)
is indicated by a superscript to the left of the
letter.
n2 s 1l j 22P1/ 2 22P3/ 2 (n  1,l  1, j  1/ 2, 3/ 2)
Lande g factor
• It is a measure of the ratio of the
magnetic moment (in Bohr magnetons)
to the angular momentum (in units of h
bar).
 j   j ( j  1) g j  B
 jz  m j g j  B g j 
j
B
j ( j  1)
• For the orbit angular momentum, j=l,
gl=1. For spin angular momentum, j=s,
gs=2.
3 1 sˆ  lˆ
g   (
)
ˆ
2 2 momentum,
j
• For the total angular
g
factor is given by:
2
j
2
2
The alkali atoms
• The alkali atoms have a
weakly bound outer
electron, the so-called
valence electron, and all
other (Z-1) electrons are
in closed shells.
• The yellow D line in the
spectrum of the sodium
(Na) atom, i.e. the
transition 3P3S,is a
doublet: D1=589nm,
D2=589.6nm.
Alkali metals
• Selection rules:
• The first primary series: nP2S, P:
l  1, , j  1,0
double substates , 
doublet,
two lines are
getting closer with n increased.
• The sharp secondary series: nS2P,
doublet, two lines are separated uniformly
with n increased.
• The diffuse secondary series nD2P, are
triplet, D & P are both split into substates.
2D 2P , 2D  2P , 2D  2P
5/2
3/2
3/2
3/2
3/2
1/2;
2D 2P (?)
5/2
1/2
Spectrum of alkali metals
First primary series
Sharp secondary series
Diffuse secondary series
Stern-Gerlach experiment (I)
• Space quantization was first explicitly
demonstrated by Stern & Gerlach in 1921.
• They directed a beam of neutral silver atoms
from an oven through a set of collimating
slits into an inhomogeneous magnetic field B
which exerts a force on the magnetic
moments.
• When the magnetic moment is parallel to B,
it moves in the direction of increasing field
strength, while an anti-parallel magnetic
moment moves towards lower field strength.
Stern-Gerlach experiment (II)
dB
dB
Fz   z
 m j g j  B
dz
dz
• They found that the initial beam split
into two distinct parts which correspond
to the two opposite spin orientation in
the magnetic field.
• For hydrogen atoms in ground state,
n=1, l=0, j=s, mj=-1/2, +1/2, gj=2, mjgj
=±1,
Stern-Gerlach apparatus
The Zeeman effect
• The total angular momentum J is the vector
sum of orbital angular momentum and spin
angular momentum. J=L+S
J
j ( j  1) J z  m j 
• When there is no external magnetic field,the
total angular momentum is conserved. When
an external B is applied, J precesses about
the direction of B, causing additional
potential energy mgμBB (2j+1 levels).
• Selection rules: m  0,1