LS coupling

LS coupling or Russell–Saunders coupling

1. By: Muhammad Naeem

2. Coupling
•Couplingistheactofcombiningtwothings.
•Couplingsaremechanicalelementsthat‘couples’twodrive
elementswhichenablesmotiontobetransferredfromone
elementtoanother.
•Inordertotransmittorquebetweentwoshaftsthateither
tendtolieinthesamelineorslightlymisaligned,a
couplingisused.

3. Coupling in Physics
➢Inphysics,twosystemsarecouplediftheyareinteracting
witheachother.
•Therearetwotypesofcoupling
i.L-Scoupling
ii.J-Jcoupling

4. L-S coupling or Russell-Saunders coupling
Inatomicspectroscopy,Russell–Saunderscoupling,
alsoknownasLScoupling,specifiesacouplingschemeof
electronicspin-andorbital-angularmomenta.Thecoupling
schemeisnamedafterH.N.RussellandF.A.Saunders(1925).

5. Total angular Momentum of an atom
•Totalangularmomentumassociatedwiththeatomis
measuredalongitsprojectionalongobservationaxis(z-axis).
•Ifmorethanonesourceofangular
momentum(spinororbitalmotion
contributestoA.M),wewillonly
observethetotalangularmomentum
fromthissourceincluded.

6. Total angular Momentum of an atom
•Contributions come from various sources that “couple” or
combine together to give the resulting total.
•Here we’ll discuss the angular momentum for many electron
system.

7. Quantum Numbers
➢Theprincipalquantumnumber(n)describesthesizeoftheorbital.
➢Theangularquantumnumber(l)describestheshapeoftheorbital.
➢Themagneticquantumnumber(m
l)describetheorientationin
spaceofaparticularorbital.
➢TheSpinquantumnumber(m
s)describethespin.

8. Total Angular Momentum
Inthisscheme,theusualpatternforallatomsisthattheorbitalangular
momentaLofthevariouselectronsarecoupledtogetherintoasingle
resultantL(combinevectorially).
??????=??????
1+??????
2+??????
3+…………
Thespinangularmomentas
iarealsocoupledtogetherintoanothersingle
resultantS.
S=??????
1+??????
2+??????
3+……….

9. Total Angular Momentum
So,
L=σ�
?????? And S=σ??????
??????
➢ThemomentaLandStheninteractviathespin-orbiteffecttoformthe
totalangularmomentumJ.ThisschemeiscalledLScoupling.
•ThemomentaLandScombinetoformthetotalangularmomentumJas
??????=??????+??????

10. TheangularmomentummagnitudesL,S,JandtheirzcomponentsL
z,S
z
andJ
zareallquantizedwiththeirrespectivequantumnumbersL,S,J,M
L,
M
s,andM
J.Hence
�=�(�+1)ℏ
�
??????=�
??????ℏ
??????=??????(??????+1)ℏ
??????
??????=�
??????ℏ
??????=??????(??????+1)ℏ
??????
??????=�
??????ℏ

11. ➢BothLandM
Larealwaysintegersor0,whiletheotherquantum
numbersarehalfintegralifanoddnumberofelectronsisinvolvedand
integralor0ifevennumberofelectronsisinvolved.
➢Whentwoorbitalangularmomentahavingquantumnumbersl
1andl
2
combine,theallowedvaluesofLare
??????=(??????
1+??????
2),(??????
1+??????
2−1),…..,|??????
1−??????
2|
➢SimilarlytheallowedvaluesarecalculatedforSandJ.

12. ➢ForagivenvalueofL,theallowedvaluesofJare
??????=??????+??????,??????+??????−1,….,|??????−??????|
➢ForL>S,thereare2S+1valuesofJ.
➢ForL<S,thenumberofpossiblevaluesofJare2L+1.
➢Thevalueof2S+1iscalledthemultiplicityofthestate.
➢WhenS=0,wehave2S+1=1,suchstatesarereferredtoassingletstates
➢WhenS=1/2and2S+1=2,thesearereferredtoasdoubletstate.
➢WhenS=1and2S+1=3,thesearereferredtoastripletstate.

13. Term symbols
➢To describe states conveniently, one requires a
notation. The symbol nL
jused for a single electron is
changed as follows.
•State symbol or Term symbol = n
2S+1
L
j
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, …….

14. ❑HowtoreadL-Scouplingnotation?
•IfS=1/2,L=1,J=3/2,1/2.Thecorrespondingstateswillbe
2
P
3/2and
2
P
1/2,readasdoubletPthreehalvesanddoubletP
halfrespectively.
•Multiplicitiesassociatedwithdifferentnumberofelectrons
intheoutershellaregivenbytherulewhichstatesthatthe
termsofatomsorionswithevennumberofvalence
electronshaveoddmultiplicitiesandviceversa.

15. L-S coupling’ Selection Rules
ΔS = 0
ΔL = 0,±1
ΔJ = 0,±1
However, the J=o to J=0 transition is forbidden
Δm
J= 0,±1
However, the m
J= 0 to m
J= 0 is forbidden if ΔJ = 0

16. Reference
•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& ShubhraKakani.
•Physics of atoms and molecules by B. H. Bransdenand C. J.
Joachain.
•Wikipedia