0198506961.pdf

66 The alkalis

1

11

Fig. 4.3The change-over from the short- to the long-range is not calculated but is

drawn to be a reasonable guess, using the following criteria. The typical radius of

the 1s wavefunction around the nucleus of charge +Ze=+11eis abouta 0 /11, and

soZeffwill start to drop at this distance. We know thatZeff∼1 at the distance

at which the 3d wavefunction has appreciable probability since that eigenstate has

nearly the same energy as in hydrogen. The form of the functionZeff(r)canbe

found quantitatively by the Thomas–Fermi method described in Woodgate (1980).

At large distances the otherN−1 electrons screen most of the nuclear

charge so that the field is equivalent to that of charge +1e:

E(r)→

e

4 π 0 r^2

̂r. (4.9)

These two limits can be incorporated in a central field of the form

ECF(r)→

Zeffe

4 π 0 r^2

̂r. (4.10)

TheeffectiveatomicnumberZeff(r) has limiting values ofZeff(0) =Z

(^9) This is not necessarily the best way to andZeff(r)→1asr →∞,assketchedinFig.4.3. (^9) The potential
parametrise the problem for numerical
calculations but it is useful for under-
standing the underlying physical prin-
ciples.
energy of an electron in the central field is obtained by integrating from
infinity:
VCF(r)=e
∫r
∞
|ECF(r′)|dr′. (4.11)
The form of this potential is shown in Fig. 4.4.
So far, in our discussion of the sodium atom in terms of the wave-
function of the valence electron in a central field we have neglected
Fig. 4.4The form of the potential en-
ergy of an electron in the central-field
approximation (e^2 M=e^2 / 4 π 0 ). This
approximate sketch for a sodium atom
shows that the potential energy crosses
over fromVCF(r)=−e^2 M/rat long
range to− 11 e^2 M/r+Voffset; the con-
stantVoffsetcomes from the integration
in eqn 4.11 (ifZeff(r)=11forallrthen
Voffset= 0 but this is not the case). For
electrons withl>0 the effective poten-
tial should also include the term that
arises from the angular momentum, as
shown in Fig. 4.5.