Biomedical Engineering Reference
In-Depth Information
The wave functions for the atoms Be, B, C, N, O, F and Ne are written as simple analytic
expressions with several parameters. The best values of these parameters are then determ-
ined by the variation method. In the final wave functions the effective quantum number
is very nearly two, the radial node is so small as to have little effect upon the charge
distribution, the coefficient in the exponential is related to an empirical 'mean effective
charge'.
14.11.2
Slater's Rules
Finally we return to the work of Slater (1930), and once again you might like to read his
abstract.
In analogy with the method of Zener for the atoms Li to F, simple rules are set up giving
approximate analytical atomic wavefunctions for all the atoms, in any stage of ionization.
These are applied to X-ray levels, sizes of atoms and ions, diamagnetic susceptibility etc. In
connection with ferromagnetism, it is shown that if this really depends on the existence of
incomplete shells within the atoms, rather far apart in the crystal, then the metals most likely
to show it would be Fe, Co, Ni and alloys of Mn and Cu (Heuser alloys).
Slater extended Zener's shielding constants for Li to F to the other atoms by adjusting the
values until he got agreement with experimental results of stripped atom and X-ray levels,
atom sizes and the other quantities mentioned. He noticed that Zener's wavefunctions had
radial nodes but argued that they were unimportant since they come much closer to the
nucleus than for hydrogen. Consequently he decided to ignore them altogether and wrote
a radial part as
r
n
∗
−
1
exp
Z
r
−
s
−
(14.30)
n
∗
where
n
∗
is an effective quantum number and
s
the shielding constant. Parameters
n
∗
and
s
are found by simple rules (that have become known as Slater's rules) as shown in Table 14.7
and discussed below.
Table 14.7
Slater
n
∗
values
n
n
∗
1
1
2
2
3
3
4
3.7
5
4.0
6
4.2
s
, the electrons are divided into the following groups, each having
a different shielding constant: 1s; 2s, 2p; 3s, 3p; 3d; 4s, 4p; 4d; 4f; 5s, 5p; etc. That is,
the s and p of a given
n
are grouped together but the d and f are separated. The shells are
considered to be arranged from inside out in the order named.
The shielding constant
s
is formed, for any group of electrons, from the following
contributions.
For determining
Z
−
