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Chapter 6
Fine structure
R
r
-e
+Ze
-(Z-1)e
Term scheme (Grotrian
diagram) of the Sodium atom.
Double D lines
(yellow)
Fine structure in the optical spectra of atoms
Many of the lines in the spectra of alkali atoms are double, and are called
doublets. They occur because all the energy terms En,l of atoms with single
valence electrons, except for the s terms, are split into two terms. This splitting
cannot be understood in terms of the theory discussed so far. It is
fundamentally different from the lifting of orbital degeneracy discussed in
chapter 5. If the orbital degeneracy has already been lifted, there must be a
new effect involved, one which has not yet been taken into account. So far, we
have not yet discussed the magnetic properties of atoms. To explain the
doublet structure, the magnetic properties of atoms should not be neglected:
Main idea:
 A magnetic moment l is associated with the orbital angular moment;
 The electron also has a spin S, which is also associated with a magnetic
moment s;
 The two magnetic moments l and s interact. They can be parallel or
antiparallel to each other. The two configurations have slightly different
binding energies, which leads to the fine structure of the spectrum.
A magnetic dipole
We know from electrodynamics that a circular electric current
generates a magnetic dipole field. The magnetic dipole
moment µl of a conducting loop is defined as:
µl
I
A = area

l  I  A

A magnetic dipole
If we bring this magnetic dipole into a homogeneous magnetic field
B, a torque  is applied to the dipole:

  B


µl
I
A = area
The magnetic potential energy of the dipole:
Vmag
 
   B    B cos 
The magnetic moment can be defined either in terms of the torque
in the field or the potential energy.
Magnetic moment of the orbital motion
An electron moving in an orbit is equivalent to a circular
electric current. The orbiting electron will do have a magnetic
dipole moment. In atomic and nuclear physics, the magnetic
moment is often defined as the torque in a uniform field of
strength H (not of strength B).

    B,


  0 I  A


Where B = µ0 H, the induction constant µ0 = 410-7 Vs/Am
Calculation of the orbital moment
r
2r
v
T
e

e 
l  
l
2m
The circulating electron has
an angular momentum l and
a magnetic dipole moment
µl . For a negative charge,
the vectors l and µl point in
opposite directions.
For an electron, the charge q = -e, the velocity v moving in a
circular orbit. If the time for a single revolution is T= 2/, a
current:
q
e
I
T

2
Calculation of the orbital moment
The magnetic moment of the circular current:
  IA   er
1
2
2
If we introduce the orbital angular momentum |l| = mvr = mr2,
then the orbital angular momentum:
e 
 
l
2m0

Where m0 is the rest mass of electron.
If the charge q is positive, the vectors µl and l point in the same
direction; if it is negative, as with the electron, they point in
opposite directions.
In atoms with a certain quantum state, the magnitude of the
orbital angular momentum is determined by the orbital
quantum number l:

h
l  l (l  1)
 l (l  1)
2
The magnitude of the magnetic moment of an orbit with the
angular momentum quantum number l:
e 
e
 
l 
 l (l  1)   B l (l  1)
2m0
2m0

Where µB is called Bohr Magneton, the unit of magnetic moment
in atoms:
e
B 
  9.274078 10  24 Am2
2m0
The orbital magnetic moment is related to the orbital angular
momentum. It is also expressed:
gl  B 

l


Where gl is called g factor or Lande factor. It is dimensionless
and here has the numerical value gl = 1. The g factor was
introduced by Landé. In the presence of spin-orbit coupling, the
coupling between the spin magnetic moment and orbital
magnetic moment, the g factor was used to characterise the ratio
of the magnetic moment and the total angular momentum.
The g factor is defined as the ratio of the magnetic moment to the
corresponding angular momentum in units of µB and ħ,
respectively:

l

gl    
l B
Gyroscope and its precession
If the gyroscope is tipped, the gimbals will try to reorient to keep
the spin axis of the rotor in the same direction. If released in this
orientation, the gyroscope will precess in the direction shown
because of the torque exerted by gravity on the gyroscope.
Precession and orientation in a magnetic field
An applied field with the magnetic flux density B0 acts on the orbital magnetic
moment µl by trying to align the vectors and B0 parallel to one another, since
the potential energy is a minimum in this orientation. The electrons, which are
moving in their orbits, behave mechanically like gyroscopes and carry out the
usual precession about the direction of the field. The precession frequency of
the electron orbit, the Larmor frequency, is:
L
B0

l B sin  g l  B
L 


B  B
l sin 
l sin 

t
|l|sin
l

l
l+l
Where  is called the gyromagnetic ratio.
The Larmor frequency is independent of
the angle .
Directional quantisation
Directional quantisation: The orientation of the angular momentum in space is not
random. The solution of the Schrödinger equation implies that when one axis is
established, one of the component of the angular momentum is quantised. The
axis can be determined by a magnetic field. Therefore only discrete values of ,
the angle between B and l or µl, are allowed.
The components of the angular momentum in z direction:
lz = mlħ,
ml = 0, ±1,…, ±l
Where ml is the magnetic quantum number, and it can have 2l+1 different values.
l is the orbital quantum number. The largest possible component of l in the z
direction is lħ.
The magnetic moment associated with the orbital angular momentum is
correspondingly quantised. For its component in the z direction is:
l , z
e

l z  ml  B
2m0
The experimental demonstration of the existence of a directional quantisation
was provided by the Stern and Gerlach experiment.
For example：
l 2
B(z)
L  l (l  1)   2(2  1)   6 
LZ  ml 
ml  0 ,  1,  2 ,    ,  l
LZ  0,  ,  2
l , z  ml  B  (0,1,2)  B
L 6
Lz  2
m=2

m=1
0
m=0

m = -1
 2
m = -2
Spin and magnetic moment of the electron
Electron spin was introduced by Uhlenbeck and Goudsmit in 1925
to explain spectroscopic observations. In fact, the splitting of many
spectral lines in a magnetic field can only be explained if the
electron has a spin angular momentum s.
Goudsmit on the discovery of electron spin.mht
What is the electron spin.mht
The spin is the intrinsic nature of electron, which has its own spin
angular momentum s and associated magnetic moment µs.
The spin angular momentum of the electron:

s  s(s  1)  
s=1/2 is the spin quantum number. The associated spin
magnetic moment:

e 
s   gs
s
2m0
gs is the so-called g factor of the electron, and its value has been empirically
determined to be gs = 2.0023.
Dirac showed in 1928 that the spin of the electron is a necessary
consequence of a relativistic quantum theory (the Schrödinger theory is nonrelativistic). The g factor gs = 2 could also be thus derived. The slight
difference between the predicted value of 2 and the empirical value can only
be understood if the interaction of the electron with its own radiation field is
taken into account through quantum electrodynamics.
Schematically shown the spin and magnetic moment
of the electron
Spin

s  s(s  1)  
Charge
-e
Mass
m0
Magnetic
Moment

e 
s   gs
s
2m0
Directional quantisation of the spin in the magnetic field
Similar to the orbital angular momentum and magnetic moment,
the orientation of the spin angular momentum is also directionally
quantised in the presence of an external magnetic field B. The spin
can only have two orientations in a defined z axis: parallel and
antiparallel (or spin up or spin down). Its components in this
defined z direction are
sz = ms ħ
with ms = ±1/2;
ms is the magnetic quantum number of the spin.
The orientation of the associated magnetic moment is also
directional quantised. The z component is
µs,z = -gsmsµB.
The electron spin has two possible orientation in a
magnetic field in the z direction. These are
characterised by the quantum number ms = ±1/2
B, z
s z   12 
ms = +1/2

s  3/ 4 
s z   12 
ms = -1/2
Similar to the orbital motion, the gyromagnetic ratio
of the electron was defined as the ratio between the
magnetic moment and the angular moment of the
electron:


e
 s    gs
s
2m0
The relation between gyromagnetic ratio and g factor:
e
 s  gs
  gs B
2m0
It will see that the easiest and most definitive way to calculate
the magnetic properties of atoms is often to make use of
measurements of the ratio  or g.
Detection of directional quantisation by
Stern and Gerlach
In 1921, the deflection of atomic beams in inhomogeneous
magnetic fields made it possible:
 The experimental demonstration of directional quantisation and
 The direct measurement of the magnetic moments of atoms.
Stern and Gerlach experiment
Left panel: the atomic beam passes through an inhomogeneous magnetic field.
One observes the splitting of the beam into two components. Right panel:
observed intensity distribution of an atomic beam with and without an applied
magnetic field. In the first experiments of Stern and Gerlach, it was a beam of
silver atoms which was generated in an atomic beam furnace and collimated by a
series of slits. Later, hydrogen atoms form a gas discharge were also used.
Inhomogeneous B
The collimated beam passes
through a highly inhomogeneous
magnetic field, with the direction
of the beam perpendicular to the
direction of the field and of the
gradient. The directions of the
field and gradient are the same.
Without the field, the vectors of the magnetic moments and angular momenta of
the atoms are randomly oriented in space.
In a homogeneous field, these vectors precess around the field direction z. The
inhomogeneous field exerts an additional force on the magnetic moments. The
direction and magnitude of this force depends on the relative orientation between
the magnetic field and the magnetic dipole. A magnetic dipole which is oriented
parallel to the magnetic field moves in the direction of increasing field strength,
while an antiparallel dipole moves towards lower field strength. A dipole which
is perpendicular to the field does not move.
The deflection force can be derived from the potential energy
in the magnetic field:
Fz 
Vmag
z
dB
dB
 z

cos  ,
dz
dz
Vmag
 
   B
Where  is the angle between the magnetic moment and the direction of the
field gradient.
In classical mechanics, any orientation  is allowed. Atoms with magnetic
moments perpendicular to the filed gradient are not deflected. Those in which
the vectors are parallel are deflected the most, and all possible intermediate
values can occur. In the classical picture one thus expects a continuum of
possible deflections. With H and Ag atoms, however, two rather sharp peaks
were observed on the detector.
Conclusions from Stern and Gerlach experiments
 There is a directional quantisation. There are only discrete possibilities for
the orientation relative to a filed B0, in this case two, parallel and antiparallel.
 In general this method provides observed values for atomic magnetic
moments if the magnitude of the field gradient is known.
 The s electron has an orbital angular momentum l = 0 and one observes only
spin magnetism.
 For all atoms which have an s electron in the outermost position, the angular
momenta and magnetic moments of all inner electrons cancel each other, which
gives the same value for the deflection force. One measures only the effect of
the outermost s electron.
 Like gyroscopes, atoms maintain the magnitude and direction of their angular
momenta in the course of their motion in space.
Fine structure and spin-orbit coupling
Fine structure: all energy terms except of the s state are split
into two substates, which produces a doublet or multiplet
structure of the spectral lines.
The fine structure cannot be explained with the Coulomb
interaction between the nucleus and the electrons. It results
from a magnetic interaction between the orbital magnetic
moment and the intrinsic moments of the electron, the spinorbital coupling. Depending on whether the two moments are
parallel or antiparallel, the energy term is shifted somewhat.
The coupling of the magnetic moments leads to an addition
of the two angular momenta to yield a total angular
momentum.
Conclusions for the spin-orbit coupling
 l and s add to give a total angular momentum j with the vector
model j = l + s, and j will precess in the magnetic field.
z, B
jz
j
l
s
 j has the magnitude
j ( j  1)   with j = |l ± s| and s = ±1/2.
Here j is a new quantity: the quantum number of the total
angular momentum.
 Similar to l, there is also a directional quantisation for j.
The z component must obey the condition:
jz = mjħ,
mj = j, j-1,…, -j
(2j+1 possibility)
For example, a state with j = 3/2 is fourfold degenerate.
 A magnetic moment µj is associated with j;
 For optical transitions, a selection rule Δj = 0 or ±1 is
valid; however, a transition from j = 0 to j = 0 is always
forbidden. This selection rule may be considered to be an
empirical result, derived from the observed spectra.
Calculation of spin-orbit splitting in the
Bohr model
To calculate the energy difference between the parallel and
antiparallel orientations of the orbital angular momentum
and the spin, the simple Bohr model will be used.
The motion of the electron around the nucleus generates a
magnetic field Bl at the site of the electron. This field
interacts with the magnetic moment of the electron. To
determine the magnitude of this magnetic field, we borrow
from relativity theory and assume that the electron is
stationary and that the nucleus moves instead.
l
v
+
(µs)z
-
r
(µs)z
-r
+ v
-
l
The magnetic field of the moving charge +Ze is found from
the Biot-Savart law to be


Ze 0 
Ze 0 
Bl  
[v  (r )] 
l
3
3
4r
4r m0
Where the angular momentum
 

 
l  r  m0v  m0v  r
The magnetic field which is generated by the relative motion
of the nucleus and the electron is thus proportional and
parallel to the orbital angular momentum of the electron. We
still require the back transformation to the center-of-mass
system of the atom, in which the nucleus is essentially at rest
and the electron orbits around it. A factor ½ occurs in this
back transformation, the so-called Thomas factor, which can
only be justified by a complete relativistic calculation. The
particle in its orbit is accelerated, and from the viewpoint of
the proton, the rest system of the electron rotates one
additional time about its axis during each revolution around
the orbit. The back transformation is therefore complicated
and will not be calculated in detail here.

Ze 0 
Bl 
l
3
8r m0
The magnetic moment of the electron process about
the magnetic field Bl produced by the orbital motion.
Bl, z
µs
l
+
r
s
The interaction energy between the
spin and the orbital field, with gs = 2 :

Vl , s    s  Bl
e  
2
( s  Bl )
2m0
Ze 2  0  

(s  l )
2 3
8m0 r

To get the order of magnitude, we set Z = 1, r = 0.1 nm, and obtain
Vl,s  10-4 eV. The field produced by the orbital motion at the
position of the electron is about 1 tesla = 104 Gauss.
The energy of spin-orbit interaction can be written in
the form:
 
a   a 
Vl , s  2 l  s  2 l s cos(l , s )


Ze 2 0 2
Where a is the spin-orbit coupling constant a 
8m0 r 3
The scalar product l · s may be expressed in terms of the vectors l
and s by using the law of cosines:

l
j = s+l
s
2
2
a 2
Vl , s  2 ( j  l  s )
2
a
 [ j ( j  1)  l (l  1)  s ( s  1)]
2
This form is directly measurable by
determination of the doublet structure in
the optical spectra.
If we use the radius rn of the nth Bohr radius as a
rough approximation for r:
40  2 n 2
rn 
Ze 2 m0
Z4
a 6
n
We must remember, however, that there are no fixed orbits in
the quantum theoretical description of the atom. Therefore it is
necessary to replace r –3 by the corresponding quantum
theoretical average
2
1 / r   ( / r )dV
3
3
Where  is the wavefunction and dV the volume. Then we obtain:
4
Z
a 3
n l (l  12 )(l  1)
The energy terms with considering
of spin-orbit coupling?
In general, the simbolism is n 2S+1LJ. The upper case
letters S, L and J apply to several-electron atoms,
while the corresponding lower case letters apply to
single-electron atoms. Where S(or s) is spin quantum
number, L(or l) is the orbital angular momentum
quantum number, and J(or j) is the total angular
momentum quantum number.
2 2S1/2 for a state in which the valence electron has the
quantum numbers n = 2, l = 0, j = ½.
2 2P1/2 For states in which the valence electron has
2 2P3/2 the quantum number n = 2, l = 1, j = ½ or 3/2,
respectively.
The spin-orbit interaction splits each level into two.
For example the sodium D lines, the 3P state splits
into 3P1/2 and 3P3/2 states.
For 3P1/2 state, s = ½, l = 1, j = ½, Vl,s = -a;
For 3P3/2 state, s = ½, l = 1, j = 3/2, Vl,s = a/2.
3P
a/2
-a
3P1/2
3P3/2
For D state splits into D3/2 and D5/2 states
For 3D3/2 state, s = ½, l = 2, j = 3/2;
For 3D5/2 state, s = ½, l = 2, j = 5/2.
Summarize for the fine structure of oneelectron states:
 Interaction of the electron with the orbital momentum or
the orbital moment splits each level into two.
 For s terms there is no splitting, because there is no
magnetic field with which the spin can align itself.
 Levels with higher values of the quantum number j have
higher energies.
 The fine structure splitting Vl,s is proportional to the
fourth power of the nuclear charge Z.
 The slitting is greatest for the smallest principal quantum
number n.
Level scheme of the Alkali atoms
Term scheme for alkali atoms, i.e.
one-electron states, including the
spin-orbit splitting. The terms are
displaced with respect to those of
the H atom. The fine structure
splitting decreases with increasing
values of n and l. A few allowed
transitions are indicated.
The selection rules:
l = ±1, j = ±1, 0
Optical transitions are thus allowed only if the angular
momentum changes. The total angular momentum angular j,
however, can remain the same. This would happen if the orbital
angular momentum and the spin changed in opposite directions.
The first principal series of the alkali atoms arises from
transitions between the lowest n2S1/2 term (i.e. n = 2, 3, 4, 5, 6
for Li, Na, K, Rb, Cs) and the P terms 2P1/2 and 2P3/2. Since the
s terms are single-valued, one sees pairs of lines. The same
holds for the sharp secondary series, which consists of
transitions between the two lowest P terms and all higher 2S1/2
lines. The lines of the diffuse secondary series, however, are
triple, because both the P and the D terms are double.
2D
5/2
2D
3/2
2P
3/2
2P
1/2
Allowed transitions
forbidden transitions
Fine structure in the hydrogen atom
Since the wavefunctions of the H atom are known explicitly, its
fine structure can be exactly calculated. For the H atom, both the
relativity correction and the spin orbit interaction are small
compared to the energy terms En,l, but the two are of comparable
magnitude:
En,l,j = En,l + Erel + El,s
The two correction terms, the one for the relativistic mass change
Erel and the other for the spin-orbit coupling El,s, together give the
fine structure correction EFS = Erel + El,s. The complete calculation
was carried by Dirac.
EFS
En 2  1
3  2


   Z
n  j  1 / 2 4n 
Where the Sommerfeld
fine structure constant:

e2
40 c
(or
0c 2
e )
4
The energy shift of H atom with respect to the calculated
energy terms En,l is of the order of 2, i.e. (1/137)2, and is thus
difficult to measure. According to Dirac’s calculation, the fine
structure energy of the H atom depends only on j, not on l. this
means that terms with differing l (for the same n) and the same j
have the same energies. They are energetically degenerate.
The fine structure energies of heavier
atoms are larger and are thus easier to
observe. Their calculations, however, are
far more difficult, because the exact
calculation of the wavefunctions of
atoms with more than one electron is far
more complex.
The Lamb shift
In the years 1947-1952, Lamb and Retherford showed
that even the relativistic Dirac theory did not completely
describe the H atom. They used the methods of highfrequency and microwave spectroscopy to observe very
small energy shifts and splitting in the spectrum of
atomic hydrogen. In other words, they used the
absorption by H atoms of electromagnetic radiation from
high-frequency transmitters or Klystron tubes they could,
in this way, observe energy difference between terms
with the same j, namely differences of 0.03cm-1 — this
corresponds to a difference of 900 MHz — between the
terms 22S1/2 and 22P1/2.
The Lamb shift: fine structure of the n = 2 level in the H atom
according to Bohr, Dirac and quantum electrodynamics
taking into account the Lamb shift. The j degeneracy is lifted.
Levels with the same quantum numbers n and j, but different l, are not
exactly the same. Rather, all S1/2 terms are higher than the corresponding P1/2
terms by an amount equal to about 10% of the energy difference (P3/2 -P1/2),
and the P3/2 terms are higher than the D3/2 terms by about 2‰ of (D5/2 – D3/2)
homework
Pp202, 12.1, 12.2, 12.3, 12.5, 12.7, 12.9