Ls coupling

Russal Saunders coupling, L-s coupling, J-J coupling, Spin orbital coupling

1. 1




INTRODUCTION:
Usually we understand from the word coupling is the act of combining two things, here we are
going to discuss the coupling in quantum physics. There are two types of coupling. LS coupling
and JJ coupling. Now let’s see that what we understand from the coupling in the quantum physics.
Russell-Saunders of L-S coupling notation for atomic states is a system of notating
observed states for atoms, based upon a particular conception of the angular momentum
possessed by each state, and the sources of such angular momenta in the atom. In general, when
one attempts to measure the angular momentum associated with an atom (or molecule for that
matter), only the total angular momentum will be measurable, along with its projection along
the observation axis (z-axis). Thus, if more than one source of angular momentum such as spin
or orbital motion contribute angular momentum, we will only observe the total angular
momentum, with the contribution from this source included.
The total angular momentum of an atom (a vector, J v) can be shown to be
consistent with a vector addition of all the various sources of angular momentum in the atom.
Thus, while we can only measure the total angular momentum for the atom, we deduce the
source of the angular momentum in its various amounts and orientations, by postulating that
contributions come from various sources that “couple” or combine together to give the resulting
total.
For atomic term symbols, we normally restrict our attention to the total angular
momentum arising from electrons, in the form of spin angular momentum (SAM) and orbital
angular momentum (OAM). Total angular momentum due to these two sources would be
described by the quantum numbers S and L, along with Ms and Ml if these two forms of angular
momentum did not interact in the atom. In these cases we would say that S and L are “good”
quantum numbers, i.e., the observed behavior of the system exactly matches what we would
predict using S and L When both are present, the magnetic moments associated with each form
of angular momentum interact to “couple” in various ways, much as magnets can be aligned
classically in a variety of configurations of varying energy. The difference here is that the
orientations of the coupled magnetic moments due to spin angular momentum and orbital
angular momentum can only take certain orientations with respect to each other, such that each
preserves its allowed orientations with respect to the z- or magnetization axis, just as it would in
an uncoupled system.
LS coupling

2. 2

When atoms are “light”, usually meaning up to the end of the first-row transition
metals, the total angular momentum of atoms acts as if S and L were preserved, and only weakly
perturbed by interaction between one another. Then L-S coupling is used.
TOTAL ANGULAR MOMENT UM IN MANY ELECTRON ATOMS:
The addition of orbital angular momenta for a many electron system is much more
involved than that of one electron system. When more than one electron contributes orbital and
spin angular momenta to the total angular momentum J of an atom J is still the vector sum of
these individual momenta. Generally two types of coupling known as Russel-Saunders os L-S
coupling and J-J coupling occurs. We now describe L-S coupling.
L AND S OR RUSSELL–SAUNDERS COUPLING:
This is also called as normal coupling as this occurs most frequently. When more
than one electron contributes orbital and spin angular momenta to the total angular momentum
J of an atom, J is still the vector sum of these individual momenta. In these schemes, the usual
pattern for all but the heaviest atoms is that the orbital angular momenta L of the various
electrons are coupled together into a single resultant L. the spin angular momenta Si are also
coupled together into another single resultant S. The momenta L and S then interact via the spin-
orbit effect to form the total angular momentum J. This scheme is called LS coupling. In the L-S
coupling scheme, the orbital angular momenta L1, L2, L3,……… of the electrons combine vectorially
to form a total orbital angular momentum L and similar is the case for S. Thus,
??????= ??????
1+??????
2+??????
3+⋯………
??????= ??????
1+??????
2+??????
3+⋯………
The momenta L and S combine to form the total angular momentum J as
??????=??????+??????
It is the convention to use capital letters L, S and J for the angular momentum quantum numbers
of a many electron system. These are simply integers and half integers. To avoid confusion, ℏ is
added while writing the magnitude of angular momentum vectors. These vectors follow the
addition rules associated with quantized quantities. This procedure would be valid only if
coupling between the individual Li and Si vectors is weak. Then the individual Li vectors would
couple and give the L vector while the Si vectors couple and give the S vector. It is found that this
type of coupling is obeyed by a large number of elements, particularly lighter ones.
When two orbital angular momenta having quantum numbers l1 and l2 combine, the allowed
values of L are

3. 3

??????=(??????
1+??????
2),(??????
1+??????
2−1),…..,|??????
1−??????
2|
Similarly the allowed values are calculated for S and J. For a given value of L, the allowed values
of J are
??????=(??????+??????),(??????+??????−1),….,|??????−??????|
For L>S, there are 2S+1 values of J. However, for L<S, the number of possible values of J are 2L+1.
The value of 2S+1 is called the multiplicity of the state. For a two electron system (s1=s2=1/2), the
total spin angular momentum quantum number S=0 (spin antiparallel) or 1 (spin parallel). When
S=0, we have 2S+1=1 and such states are referred to as singlet states. On the other hand, when
S=1 and 2S+1=3, these are referred to as triplet state. Often the singlet and triplet levels are
grouped separately.
To describe states conveniently, one requires a notation. The symbol n Lj used for a single
electron is changed as follows.
State symbol or Term symbol = n
2S+1
Lj
Where,
The superscript 2S+1 represents the multiplicity of the state,
Subscript J is the total angular momentum quantum number, and
L stand for S,P,D,……. Representing the orbital angular momentum quantum number which has
already been discussed. This way of representing states is known as the term symbol of the state.
Thus, if S=1/2, L=1, J=3/2, 1/2. The corresponding states will be
2
P3/2 and
2
P1/2, read as doublet P
three halves and doublet P half. Multiplicities associated with different number of electrons in
the outer shell are given by the rule which states that the terms of atoms or ions with even
number of valence electrons have odd multiplicities and vice versa.
J-J COUPLING:
In case of heavier elements, particularly in excited states, the coupling between the various s-
vector and l vectors of individual electrons become weak and a tendency to form a J-vector by
the l and s vectors for each separate electron is markedly developed. In the extreme case, l and
s for each electron combine to form a separate J and all the J-vectors are then vectorially added
to form J. Thus for each active electron J1=l1 + s1; J2=l2 +s2; J3=l3 + s3,……etc. and J=J1 + J2 + J3 + …
= ∑Ji. This coupling scheme is known as J-J coupling which is distinctly different from L-S
coupling scheme.

4. 4

Pure J-J coupling is seldom found and in most of the known cases L-S coupling is effective.
However, there are many heavy elements that have spectra that require for their interpretation
a coupling intermediate between the L-S and J-J.
EXAMPLE:
Consider two electrons, one in the 4P and the other in the 4d subshell. Obtain the L, S and J
values and the spectroscopic symbols for this two electron system.
SOLUTION:
For the two electrons, l1 = 1, l2 = 2, s1 = ½, s2 = 1/2
The possible values of L; l1 +l2, l1 +l2 -1,….|l1 – l2| = 3, 2, 1
Similarly, S = 1, 0
The possible values of J :
L = 3, S = 1, J = 4, 3, 2 L = 2, S = 0, J = 2
L = 3, S = 0, J = 3 L = 1, S = 1, J = 2, 1, 0
L = 2, S = 1, J = 3, 2, 1 L = 1, S = 0, J = 1
Spectroscopic symbols for two electrons-one in 4p and the outer in 4d subshells.
S L J Spectroscopic
symbol

0 (Singlet) 1 1 4
1
P1
2 2 4
1
D2
3 3 4
1
F3
1 4
3
P2
1 (triplet) 1 1 4
3
P1 4
3
P2,1,0
0 4
3
P0
3 4
3
D3
1 (triplet) 2 2 4
3
D2 4
3
D3,2,1
1 4
3
D1
4 4
3
F4
1 (triplet) 3 3 4
3
F3 4
3
F4,3,2
2 4
3
F2

5. 5

EXAMPLE:
A state is denoted as
4
D5/2. (i) what are the values of L, S and J? (ii) What is the minimum
number of electrons which could give rise to this?
SOLUTION:
(I).For the state
4
D5/2, 2S+1 = 4
S = 3/2, L = 2, J = 5/2
(II). Minimum number of electrons which could give S = 3/2 is 3, since each electron has s = ½.
EXAMPLE:
Find the values of L and S of the ground state of nitrogen.
SOLUTION:
The electronic configuration of nitrogen is 1s
2
2s
2
2p
3
. Therefore, the maximum possible value
of Ms = 3/2.
To maximize ML, we assign the maximum value of ml = 1 to the first electron. The maximum
value of ml left for the second electron is zero. Then the maximum value of ml left for the third
electron is -1. Hence,
ML,max = 1+ 0 + (-1) = 0 or L = 0
For the ground state of nitrogen, L = 0, S = 3/2.
APPENDIX:
COUPLING:
Coupling is the act of combining two things.
FOUR QUANTUM NUMBERS:
 The principal quantum number (n) describes the size of the orbital.
 The angular quantum number (l) describes the shape of the orbital.
 The magnetic quantum number (m) describe the orientation in space of a particular
orbital.
 The Spin quantum number (m) describe the spin.

6. 6

SINGLET STATE:
In quantum mechanics, a singlet is a quantum state of a system with a spin of 0, such that there
is only one allowed value of the spin component, 0. for a singlet, L=2(0)+1=1 only one state
exists (singlet).
DOUBLET STATE:
If we have a single electron, S=1/2 then, for a doublet, L=2(1/2)+1=2 two states exist (doublet)
TRIPLET STATE:
A pair of spin -1/2 particles can be combined to form one of three states of total spin 1 called
the triplet. For a triplet, L=2(1)+1=3 three states exist (triplet).
REFERENCES:
 Modern Physics by G. Aruldhas (university of kerala ) & P. Rajagopal (Mahatma Gandhi
University).
 Principles of Modern Physics by Ajay K Saxena.
 Modern Physics by S. L. Kakani & Shubhra Kakani.
 Physics of atoms and molecules by B. H. Bransden and C. J. Joachain.
 Wikipedia

__X___________________________________________________________________________
M. Usman Mustafa
IU13S6BA035
BS 5
th
Session (2013-17)