0198506961.pdf

Exercises for Chapter 5 95

theLS-coupling scheme describes this atom. Show

the electric dipole transitions that are allowed be-

tween the levels.

(5.5)TheLS-coupling scheme

3s3p, Mg 3s3p, Fe14+

2.1850 23.386

2.1870 23.966

2.1911 25.378

3.5051 35.193

The table gives the energy levels, in units of

106 m−^1 measured from the ground state, of the

3s3p configuration in neutral magnesium (Z =

12) and the magnesium-like ion Fe14+. Suggest,

with reasons, further quantum numbers to iden-

tify these levels. Calculate the ratio of the spin–

orbit interaction energies in the 3s3p configura-

tion of Mg and Fe14+, and explain your result.

Discuss the occurrence in the solar spectrum of

a strong line at 41.726 nm that originates from

Fe14+. Would you expect a corresponding tran-

sition in neutral Mg?

(5.6)LS-coupling for configurations with equivalent
electrons

(a) List the values of the magnetic quantum num-

bersml 1 ,ms 1 ,ml 2 andms 2 for the two elec-

trons in annp^2 configuration to show that fif-

teen degenerate states exist within the central-

field approximation. Write down the values of

ML=ml 1 +ml 2 andMS=ms 1 +ms 2 associ-

ated with each of these states to show that the

only possible terms in theLS-coupling scheme

are^1 S,^3 Pand^1 D.

(b) The 1s^2 2s^2 2p^2 configuration of doubly-ionized

oxygen has levels at relative positions 0, 113,

307, 20 271 and 43 184 (in units of cm−^1 )

above the ground state, and its spectrum con-

tains weak emission lines at 19 964 cm−^1 and

20 158 cm−^1. Identify the quantum numbers

for each of the levels and discuss the extent to

which theLS-coupling scheme describes this

multiplet.

(c) For six d-electrons, in the same sub-shell,

write a list of the values of themsandml

for the individual electrons corresponding to

MS=2andML=2.Briefly discuss why this

is the maximum value ofMS,andwhyML
 2

for this particular value ofMS.(Hence from

Hund’s rules the^5 Dtermhasthelowesten-

ergy.) Specify the lowest-energy term for each

of the five configurationsnd,nd^2 ,nd^3 ,nd^4

andnd^5.

(5.7)Transition fromLS-tojj-coupling

3p4s, Si

J Energy (10^6 m−^1 )

0 3.968

1 3.976

2 3.996

1 4.099

3p7s, Si

J Energy (10^6 m−^1 )

0 6.154

1 6.160

2 6.182

1 6.188

The table givesJ-values and energies (in units of

106 m−^1 measured from the ground state) of the

levels in the 3p4s and 3p7s configurations of sili-

con. Suggest further quantum numbers to identify

the levels.

Why do the two configurations have nearly the

same value ofEJ=2−EJ=0but quite different en-

ergy separations between the twoJ=1states?

(5.8)Angular-momentum coupling schemes

4p5s, germanium

J Energy (10^6 m−^1 )

03.75

13.77

23.91

14.00

5p6s, tin

J Energy (10^6 m−^1 )

03.47

13.49

23.86

13.93

The table gives theJ-values and energies (in units

of 10^6 m−^1 measured from the ground state) of the

levels in the configurations 4p5s in Ge and 5p6s in

Sn. Data for the 3p4s configuration in Si are given

in the previous exercise. How well does theLS-

coupling scheme describe the energy levels of the

np(n+ 1)s configurations withn=3,4and5?

Give a physical reason for the observed trends in

the energy levels.

One of theJ= 1 levels in Ge has a Land ́eg-factor

ofgJ=1.06. Which level would you expect this

to be and why?

(5.9)Selection rules in theLS-coupling scheme

State the selection rules that determine the config-

urations, terms and levels that can be connected

by an electric dipole transition in theLS-coupling

approximation. Explain which rules are rigorous,

and which depend on the validity of the coupling