# Russel-Saunder Coupling| L S Coupling | Spin Orbit Coupling

Russell–Saunders coupling is also known as LS coupling or Spin Orbit Coupling, specifies a coupling scheme of electronic spin- and orbital-angular momenta. Russell-Saunders coupling is useful mainly for the lighter atoms (atomic number less than 57) because in these atoms spin-orbit coupling is weak. For heavier atoms j-j coupling gives a better approximation for atomic wave function.

In this coupling scheme, it is presumed that the orbital angular momenta (l) of the individual electrons added to form a resultant orbital angular momentum 'L'. Similarly, the individual spin angular momenta (s) couple to produce a resultant spin angular momentum 'S'. Then 'L' and 'S' combine to form the total angular momentum 'J'.

This scheme of coupling may be visualized in terms of a vector model of angular momentum.

J = L + S

When two angular momenta are parallel, the value of 'J' is high

J = l + 1/2

When two angular momenta are in opposite directions, the value of 'J' is low

J = l − 1/2

J can also be expressed in terms of total angular momentum quantum number(j)

J = √j(j+1).h/2π

where 'J' = total angular momentum quantum number (non-negative integer or half integer)

The accepted values of 'J' can be obtained by using Clebsch- Gordan series.

J = |L + S| to |L − S|

If there is a single electron outside a closed shell then J = j and 'j' is either l + 1/2 or l − 1/2

The ground state term for free ion with P

L = orbital quantum number
= 1(+1) + 1(0) + 1(-1) = 0

so, term is S

S = total spin quantum number

= ½ + ½ + ½ = 3/2

so, spin multiplicity (2S + 1) = 4

J = total angular momentum quantum number

= |L + S| ... |L - S|

= (0 + 3/2) ... (0 – 3/2)

= 3/2 ... 3/2

= 3/2

so, The ground state term for a free ion with P

L = l

Where l

If two orbital angular momenta are parallel, then 'L' (l

Value of 'L' is assigned by term is given below in term symbol.

Example for d

The value of 'L' = 2 + 1 = 3

so the term symbol is F for d

Total spin angular momentum (S) = Total available spin = s

Where s

Hence total spin angular momentum is sum of spin of each unpaired electron.

Paired electrons spins are cancel due to the opposite spin.

Using value of 'S', spin multiplicity can be dermine as-

2S + 1 = 2(2/2) + 1 = 3 for two unpaired electrons with parallel spin.

As s = n/2

so, 2s + 1 = = n + 1

In this coupling scheme, it is presumed that the orbital angular momenta (l) of the individual electrons added to form a resultant orbital angular momentum 'L'. Similarly, the individual spin angular momenta (s) couple to produce a resultant spin angular momentum 'S'. Then 'L' and 'S' combine to form the total angular momentum 'J'.

**Due to L-S Coupling or Russel-Saunder Coupling, spectroscopic states are produced**. The degeneracy of 'J' is 2J + 1.This scheme of coupling may be visualized in terms of a vector model of angular momentum.

J = L + S

*For less than half filled configuration, minimum value of 'J' is ground state and for more than half filled configuration, maximum value of 'J' is ground state*.When two angular momenta are parallel, the value of 'J' is high

J = l + 1/2

When two angular momenta are in opposite directions, the value of 'J' is low

J = l − 1/2

J can also be expressed in terms of total angular momentum quantum number(j)

J = √j(j+1).h/2π

where 'J' = total angular momentum quantum number (non-negative integer or half integer)

The accepted values of 'J' can be obtained by using Clebsch- Gordan series.

J = |L + S| to |L − S|

If there is a single electron outside a closed shell then J = j and 'j' is either l + 1/2 or l − 1/2

The ground state term for free ion with P

^{3}configuration-+1 | 0 | −1 |
---|---|---|

↑ | ↑ | ↑ |

so, term is S

S = total spin quantum number

= ½ + ½ + ½ = 3/2

so, spin multiplicity (2S + 1) = 4

J = total angular momentum quantum number

= |L + S| ... |L - S|

= (0 + 3/2) ... (0 – 3/2)

= 3/2 ... 3/2

= 3/2

so, The ground state term for a free ion with P

^{3}configuration is^{2S + 1}S_{J}=^{4}S_{3/2}### l-l Coupling

If there are more than one electrons in subshell then the value of 'L' is obtained by coupling of individual orbital angular quantum numbers. In other words we can say that interaction of orbital angular momentum(l) of individual electrons gives total orbital angular momentum(L).L = l

_{1}+ l_{2}+ l_{3}+ ...Where l

_{1}, l_{2}, l_{3},... are the orbital angular momentum of individual electrons 1,2,3,... respectively.If two orbital angular momenta are parallel, then 'L' (l

_{1}+ l_{2}) is maximum and if two orbital angular momenta are opposed, then 'L'(l_{1}− l_{2}) is minimum.Value of 'L' is assigned by term is given below in term symbol.

Value of L | Term |
---|---|

0 | S |

1 | P |

2 | D |

3 | F |

4 | G |

5 | H |

6 | I |

7 | K |

Example for d

^{2}system-+2 | +1 | 0 | −1 | −2 |
---|---|---|---|---|

↑ | ↑ |

so the term symbol is F for d

^{2}system.### s-s Coupling

The value of the total spin angular momentum (S) is obtained from the unpaired electrons in the outer orbit of the atom.Total spin angular momentum (S) = Total available spin = s

_{1}+ s_{2}+ s_{3}+...Where s

_{1}, s_{2}, s_{3}are the spin quantum number of electrons.Hence total spin angular momentum is sum of spin of each unpaired electron.

Paired electrons spins are cancel due to the opposite spin.

Using value of 'S', spin multiplicity can be dermine as-

2S + 1 = 2(2/2) + 1 = 3 for two unpaired electrons with parallel spin.

As s = n/2

so, 2s + 1 = = n + 1

## j-j Coupling

For heavier elements with larger nuclear charge, the spin-orbit interactions become as strong as the interactions between individual spins or orbital angular momenta. In those cases the spin and orbital angular momenta of individual electrons tend to couple to form individual electron angular momenta.In j-j coupling, the orbital angular momentum 'l', and spin angula momentum 's' of each electron are first coupled to form a total angular momentum 'j' for that electron. These single-electron total angular momenta are then combined into a total angular momentum 'J', for the group of electrons. This is in contrast to LS coupling, where the total orbital angular momentum L, and total spin S, of the system are calculated first and then combined to the total angular momentum 'J', of the whole system.

j

_{i}= l

_{i}+ s

_{i}

J = ∑ j

_{i}

Russel-Saunders coupling is more important for transition metals than j-j coupling.

In general, the results from both coupling schemes (L-S and J-J) are different.