**6. Number of microstates**

6 Advanced Aspects of Spectroscopy

Hence, L = 3 -1 = 2 i.e., D; S = 2; 2S+1 = 5; and J = L- S = 0; Term symbol = 5D0

Hence, L = +2+1+0-1 = 2 i.e., D ; S = 1 /2 ; 2S+1 = 2 ; and J = L+ S = 3/2 ; Term symbol = 2D5/2

Spin multiplicity indicates the number of orientations in the external field. If the spin multiplicity is three, there will be three orientations in the magnetic field.- parallel, perpendicular and opposed. There are similar orientations in the angular momentum in an

The spectroscopic term symbols for dn configurations are given in the Table-1. The terms are read as follows: The left-hand superscript of the term symbol is read as singlet, doublet, triplet, quartet, quintet, sextet, septet, octet, etc., for spin multiplicity values of 1, 2, 3, 4, 5, 6, 7, 8, etc., respectively.1S0 (singlet S nought); 2S1/2 (doublet S one–half); 3P2 (triplet P two ); 5I8 (quintet I eight). It is seen from the Table-1 that dn and d10-n have same term symbols, if we

dn Term dn Term

It is also found that empty sub -shell configurations such as p0, d0, f0, etc., and full filled subshell configurations such as p6, d10, f14, etc., have always the term symbol 1S0 since the resultant spin and angular momenta are equal to zero. All the inert gases have term symbols for their ground state 1S0 .Similarly all alkali metals reduce to one electron problems since closed shell core contributes nothing to L , S and J; their ground state term symbol is given by 2S1/2. Hence d electrons are only of importance in deciding term symbols of transition

We have seen that the degeneracy with regard to spin is its multiplicity which is given by (2S+1). The total spin multiplicity is denoted by Ms running from S to -S. Similarly orbital

d10 d9 d8 d6

1S0 2D5/2 3F4 5D4

ignore J values. Here n stands for the number of electrons in dn configuration.

1S0 2D3/2 3F2 5D0 6S5/2

d0 d1 d2 d4 d5

For d4 configuration:

For d9 configuration:

external field.

**Table 1.** Term symbols

**5. Total degeneracy** 

metals.

The electrons may be filled in orbitals by different arrangements since the orbitals have different ml values and electrons may also occupy singly or get paired. Each different type of electronic arrangement gives rise to a microstate. Thus each electronic configuration will have a fixed number of microstates. The numbers of microstates for p2 configuration are given in Table-2 (for both excited and ground states).


**Table 2.** Number of microstates for p2 configuration

Each vertical column is one micro state. Thus for p 2 configuration, there are 15 microstates. In the above diagram, the arrangement of singlet states of paired configurations given in A (see below) is not different from that given in B and hence only one arrangement for each ml value.

The number of microstates possible for any electronic configuration may be calculated from the formula,

Number of microstates = n! / r! (n - r)!

Where n is the twice the number of orbitals, r is the number of electrons and ! is the factorial.

For p2 configuration, n= 3x2 =6; r = 2; n – r = 4

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720; 2! = 2 x 1 =2; 4! = 4 x 3 x 2 x 1 = 24

Substituting in the formula, the number of microstates is 15.

Similarly for a d2 configuration, the number of microstates is given by 10! / 2! (10 – 2)!

$$\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \left(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\right)} = 45$$

Thus a d2 configuration will have 45 microstates. Microstates of different dn configuration are given in Table-3.


**Table 3.** Microstates of different dn configuration
