QUANTUM THEORY 
67 
! 
iken as 
b (46-4) 
1:7-1). 
J of its 
L, - *2> 
system 
re often 
Lumber, 
larately 
is being 
fference 
idering. 
electron 
excited 
though 
changes 
tie same 
removal 
f atoms 
r (1 - x) 
dectron, 
em into 
nmarked 
of these 
becomes 
p which 
ntinuity 
we must 
r rite ctq/t 
V irtually 
slectrons 
r excited 
ibution.) 
.(47-2). 
... is no 
tities are 
partly guess-work. But a certain amount of experimental knowledge is 
available for most elements. In particular the values of y for the principal 
energy-levels in the complete atom are known from measurements of the 
frequency of the radiation emitted during transitions in accordance with 
(36-1). The extent to which these are modified in the incomplete ions must 
be estimated by us as best we can. 
An apparent difficulty arises because the series on the right of (46-4) 
is divergent, the exponentials tending to unity for large values of n so 
that the series behaves like (n + 1 ). But the later terms of the series 
are fictitious. The semiaxis of the orbit increases proportionately to n 2 
so that in any practical problem the orbit for large values of n will extend 
into regions no longer under the predominant attraction of the nucleus. 
The series in (47-1) is therefore not really infinite but stops at an outermost 
orbit beyond which the electron would be regarded as belonging to another 
atom. The arbitrary convention employed in fixing this limit applies also 
to the left side of the equation, since we cannot say whether an atom is 
ionised or not unless we have a definite rule for assigning each distant 
electron of negative energy to its proper atom. The simple calculation 
breaks down in three ways: ( 1 ) the field of the nucleus is shielded by 
the electrons (free or bound) surrounding it, ( 2 ) the periodicity becomes 
imperfect so that the quantisation fades away, (3) it reckons any region 
of space many times over as part of the field of every nucleus in the 
assemblage*. 
Examples of the use of (47-1) for calculating the degree of ionisation 
at given temperature and density are given in Chapter x. 
Theory of Weights of States. 
48. Let q x , q 2 ..., p x , p 2 ... be Hamiltonian coordinates specifying the 
state of a system at time s and satisfying (42-3). For convenience we con 
sider six variables as in § 42 but the investigation is the same for any 
number of degrees of freedom. 
Consider the states comprised in a range q x to q x + 8q x , . .., p 3 to 
p 3 + 8p 3 . Such a range will be called a cell and the volume of the cell is 
defined to be 
V = 8q x 8q 2 8q 3 8p x 8p 2 8p 3 , 
or more generally for a cell of any shape 
V = [JJfJJdq x dq 2 dq 3 dp x dp 2 dp 3 (48-1). 
* It will be understood that the failure of (46-4) does not stand in contradiction 
to what we have already said as to the universal validity of (46-2). Equation (47-1) 
is universally valid provided that the ionisation energies fa, fa ••• are calculated 
with reference to the actual circumstances of the atoms ; they may be different from 
the values for isolated atoms. 
5-2