66 
QUANTUM THEORY 
the distance 8 will be the same for both classes. Since 8 may be taken as 
large as we please the calculation for undisturbed systems is valid. 
For more complex systems (46-3) is modified, and we must write (46-4) 
in the more general form 
(1 - x)/x = (To (¿ 2 / 2 t 7 mRTf {q 1 e~ Xl ! RT + q 2 e~**! RT + ...} ...(47-1). 
Here x refers to the removal of a particular electron, the energy of its 
removal from the normal and successive excited orbits being — xi > — Xz > 
etc. Strictly speaking q x , q 2 ... are all unity since in the complex system 
no two orbits will have precisely the same energy; but in practice we often 
group together the orbits with the same principal quantum number, 
ignoring the slight differences of y. Also we ought strictly to treat separately 
the systems in which electrons other than the one whose removal is being 
considered are excited, because their excitation will make some difference 
to the energies xi> Xz ••• 5 hut in practice this is scarcely worth considering. 
The excitation of the other electrons occurs whether the particular electron 
is present or not, that is to say, both N (1 — x) and Nx include excited 
systems; the approximation does not omit the excited systems, though 
it treats them not quite rigorously by amalgamating their energy-changes 
with those of the normal systems. 
When a number r of electrons in symmetrical orbits require the same 
energy — x for their removal it would be inconvenient to treat the removal 
as t distinct ionisations. For example, let N be the number of atoms 
stripped of their M and higher electrons. Dividing these into N (1 — x) 
atoms retaining a marked L electron and Nx ionised as to this electron, 
x is given by (47-1); but we are more concerned to divide them into 
N (1 — y) retaining all the L electrons and Ny ionised as to an unmarked 
L electron. 
Considering the N (1 — y) un-ionised atoms the proportion of these 
with a highly excited L electron in given space-velocity range becomes 
multiplied by r, since any one of the r electrons of the group which 
happens to be in this range will count. To secure the former continuity 
with the systems consisting of ionised atoms and free electrons, we must 
also have r times as many of these systems. Evidently if we write ct 0 /t 
instead of o - 0 in the former equations the balance will be secured. (Virtually 
we assign to each of the L electrons a partial pressure of the free electrons 
equal to 1 /r of the whole electron pressure, so that its highly excited 
states grade continuously into its share of the free electron distribution.) 
Accordingly making this substitution in (47-1) 
—M = To (MZirmRT)* (q.e-xdRT + ...) (47-2). 
V r 
In the complex systems the calculation of Xi , X 2 <Zi > dz • • • is no 
longer straightforward, and theoretical estimates of these quantities are 
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