Phy368 amp spin orbit coupling

1

PHY368| ATOMIC AND MOLECULAR PHYSICS
COMSATS INSTITUTE OF INFORMATION AND TECHNOLOGY

TABLE OF CONTENT
•

SPIN-ORBIT COUPLING

•

SCHEMES FOR SPIN ORBIT COUPLING

•

L-S COUPLING SCHEME
➢ TYPES OF INTERACTION IN L-S COUPLING
•

Orbit-Orbit Interaction

•

Spin-Spin Interaction

•

Spin-Orbit Interaction

➢ VECTOR MODEL OF L-S COUPLING SCHEME
➢ TERM SYMBOL FOR DIFFERENT SPIN ORBIT STATES
•

FINE STRUCTURE OF L-S COUPLING

•

SELECTION RULES

•

TRANSITION BETWEEN L-S COUPLING SCHEME

•

REFERENCES

2

SPIN ORBIT COUPLING
The spin angular momentum of electron can interact with its orbital angular momentum.
This lead to splitting of different energy levels which can lead to different transition
energies. This effect is known as spin orbit coupling. In simple words we can explain it as
how a particles spin and orbital angular momentum interacts together.

The total angular momentum is obtained due to the interaction between spin and orbital
angular momentum. The value of this total momentum is maximum when both the spin and
orbital momentum is parallel. We can get the total angular momentum due to the following
coupling equation

Here, J represents the total angular quantum number, and L gives the orbital quantum
number

SCHEMES OF SPIN-ORBIT COUPLING
•

L-S coupling scheme

•

J-J coupling scheme

Here in this assignment, we will discuss only the L-S coupling scheme.

L-S COUPLING SCHEME
In the Russell Saunders scheme (named after Henry Norris Russell, 1877-1957 a Princeton
Astronomer, and Frederick Albert Saunders, 1875-1963 a Harvard Physicist and published
in Astrophysics Journal, 61, 38, 1925)

DEFINITION
In multi-electron atoms, the spin angular momentum of individual electrons adds or couples
to give the resultant spin angular momentum (s) likewise orbital angular momentum of
individual electrons couples to give the resultant angular momentum (L). Then L and S
couple to give total angular momentum J.
J=L+S

3

TYPES OF INTERACTION IN L-S COUPLING
Orbit-Orbit Interaction
Consider orbital motion of two electrons, ℓ1 and ℓ2 are their quantum numbers, ℓ1*h/2π and
ℓ2*h/2π are their angular momenta. ℓ1* and ℓ2* are quantized with respect to each other in
such a way that they form resultant L*
where L*=[L(L+1)]1/2 and
L=0, 1, 2, 3, 4, 5, 6, 7,…..for S, P, D, F, G, H, I, J,…
Consider an electron in the p orbit and other in the d orbit. Here two vectors ℓ 1*=√2 and
ℓ 2*=√6 may orient themselves in any one of three positions L*= √2, √6 and √12
corresponding to L=1, 2, 3 or P, D, F terms. All integral values of L from ℓ2- ℓ1 to ℓ2+ ℓ1
are allowed.

Spin-Spin Interaction
With two electrons each having a spin angular momentum of s*h/2π
where s*= [s(s+1)]1/2 and s=1/2.
There are two ways in which a spin resultant S*h/2π may be formed.
Consider s1 and s2 are spin quantum number of two electrons. This implies resultant
quantum numbers are S=0 and S=1. S=0 gives rise to singlet and S=1 to triplet.

Spin-Orbit interaction:
Coupling occurs between the resultant spin and orbital momenta of an electron which gives
rise to J the total angular momentum quantum number .The Russell Saunders term symbol
that results from these considerations is given by:
(2s+1)

Lj

4

VECTOR MODEL OF L-S COUPLING

Fig 1(a) Shows coupling of orbital angular moments.

Fig 1(b) Shows coupling of

spin momenta
Fig 1© Show L and S COUPLING

TERM SYMBOL FOR DIFFERENT SPIN ORBIT STATES:
Table 1 recapitulates the situation for an excited helium atom in the 1s 2p configuration.
To derive
the term symbols for the different spin-orbit states, we first need to calculate the total orbital
angular momentum and total spin from those of the individual electrons, respectively. Then

5

L and S are combined to find the total angular momentum of this group of electrons, J. The
possible combinations
for L, S and J are given as following:
L=| ℓ1− ℓ2|…… ℓ1+ ℓ2 (in integer steps)
S=|s1−s2|……s1+s2 (in integer steps)
J=|L−S|…..L+S (in integer steps)

TABLE 1.1

In this case, there is a singlet state (1P1) and a triplet of states (3P0, 3P1, 3P2). The spin
multiplicity indicates how many states there are in each group.
In principle, any other electron configuration can be treated in exactly the same way,
although the
number of different terms can become quite large. Table 2 shows a simple example of a
configuration with two valence electrons, 2p 3p. total orb The orbital angular momentum
and the total spin are calculated from those of the individual electrons. In this case, three
possible values for L rather than just one is obtained. Spin-wise, the situation is the same
for any two-electron configuration since the spin of a single electron is always s=1/2. We
then have to look at all possible permutations of the total orbital angular momenta and the
total spin, as shown in the table 1.2. For each permutation, we calculate the total angular

6

momentum, J, and work out the term symbol as before. are three separate singlets, two
proper triplets consisting of three states each, and one additional triplet state, 3S1.

TABLE 1.2

FINE STRICTURE OF L-S COUPLING SCHEME:
For light atoms, the spins and orbital angular momenta of individual electrons are found to
interact with each other strongly enough that you can combine them to form a resultant spin
S and resultant orbital angular momentum L (this is called Russell-Saunders or LS
coupling). The S and L are combined to produce a total angular momentum quantum
number J, and it is found that higher J values lie lower in energy
When an external interaction such as a magnetic field is applied, then further splitting of
the energy levels occurs, and that splitting is characterized in terms of the magnetic
quantum number associated with the z-component of angular momentum. This splitting is

7

called the Zeeman effect. Effects on energy levels from applied electric fields are
called Stark effects.
By using information from Table 1.1 and Table 1.2 we can draw the Fine structure For LS COUPLING in the 4p4d orbital as follow.

Fig 2.1

SELECTION RULES FOR L-S coupling:
We have found that the selection rules for a single electron moving from one atomic orbital
to another are
Δl=±1
Δml=0,±1
For an atom as a whole in the limit of L-S coupling, the Russell-Saunders
selection rules are

8

ΔS=0
ΔL=0,±1
ΔJ=0,±1
However, the J=0 to J=0 transition is forbidden
ΔmJ=0,±1
However, the mJ=0 to mj=0 is forbidden if ΔJ=0

TRANSITIONS BETWEEN L-S COUPLING SCHEME:
BY using Information from Table 1.1 and Table 1.2
The transitions between L-S COUPLING SCHEME FOR L-S COUPLING SCHEME can
be drawn for magnesium according to the selection rule as follow

9

REFERENCES:
YouTube VIDEO LINK 1:
https://youtu.be/Jyvr5TIgvMkYouTube
VIDEO LINK 2:
https://youtu.be/kcvtKnDb3iYBOOK 1:

BOOKS:
Basics of Atomic and Molecular Spectroscopy by J.Michael Hollas
Atoms, Molecules and Photons by W.Demtröder

WEBSITE LINKS

10

https://theory.physics.manchester.ac.uk/
https://chem.libretexts.org/

11

…