0198506961.pdf

64 The alkalis

Table 4.2The effective principal quantum numbers and quantum defects for the

ground configuration of the alkalis. Note that the quantum defects do depend slightly

onn(see Exercise 4.3), so the value given in this table for the 3s-electron in sodium

differs slightly from the value given in the text (δs=1.35) that applies forn>5.

Element Configuration n∗ δs

Li 2s 1.59 0.41

Na 3s 1.63 1.37

K 4s 1.77 2.23

Rb 5s 1.81 3.19

Cs 6s 1.87 4.13

(from eqn 4.1) are remarkably similar for all the ground configurations

of the alkalis, as shown in Table 4.2.

In potassium the lowering of the energy for the s-electrons leads to the

4s sub-shell filling before 3d. By caesium (spelt cesium in the US) the

6s configuration has lower energy than 4f (δf0 for Cs). The exercises

give other examples, and quantum defects are tabulated in Kuhn (1969)

and Woodgate (1980), amongst others.

4.3 The central-field approximation

The previous section showed that the modification of Bohr’s formula by

the quantum defects gives reasonably accurate values for the energies

of the levels in alkalis. We described an alkali metal atom as a single

electron orbiting around a core with a net charge of +1e, i.e. the nucleus

surrounded byN−1 electrons. This is a top-down approach where we

consider just the energy required to remove the valence electron from

the rest of the atom; this binding energy is equivalent to the ionization

energy of the atom. In this section we start from the bottom up and

consider the energy of all the electrons. The Hamiltonian forNelectrons

in the Coulomb potential of a charge +Zeis

H=

∑N

i=1







−

^2

2 m

∇^2 i−

Ze^2 / 4 π 0

ri

+

∑N

j>i

e^2 / 4 π 0

rij







. (4.2)

The first two terms are the kinetic energy and potential energy for each

electron in the Coulomb field of a nucleus of chargeZ. The term with

rij=|ri−rj|in the denominator is the electrostatic repulsion between

the two electrons atriandrj. The sum is taken over all electrons with

(^8) For example, lithium has three inter- j>ito avoid double counting. (^8) This electrostatic repulsion is too large
actions between the three electrons, in-
versely proportional tor 12 ,r 13 andr 23 ;
summing over alljfor each value ofi
would give six terms.
to be treated as a perturbation; indeed, at large distances the repulsion
cancels out most of the attraction to the nucleus. To proceed further
we make the physically reasonable assumption that a large part of the
repulsion between the electrons can be treated as a central potential