QUANTUM THEORY
63
(44-5).
•riding to
its. Thus
. But by
assume that there is circular symmetry about the radius*. Then for
complete spherical symmetry the number of velocities in any zone of
directions 6 to 6 + dd (measured from the radial direction) is proportional
to the area of the zone 2 tt sin Odd, i.e. to d (cos 6) and therefore to du,
since u—V cos 9.
Now integrate with respect to u. As n' goes from 0 to n, e' goes from
(44-6).
0 to 1 ; w goes from 0 as far as it can, viz. to V —the premature stoppage
being due to the fact that orbits of too low eccentricity lie wholly outside
r. Thus ± u goes from V to 0 , and allowing for the double sign the integral
of du is 2 V . Hence the integrated result is
his range
Bn 5 . 2 Vr 2 dr 07
cyfF e ~ xn,m d Xn- rr = 4 t TprHr (45-3),
-iV 77 -yra-
.(44-7).
where p is the densityf of the electrons at r with energy in a range dy n .
By (42-61) and (42-62)
rfi/a'- (2K/mpi)\
and using the values of p, and K (42-1) and (42-62) the result (45-3) reduces
to
p = B^e-^¥d Xn .
Then
If — ifj ( r ) is the potential energy of an electron at r due to the field of
the nucleus
X« = - « H r ) + |wF 2 ,
so that dx n = mVdV.
?y Xn
Hence P = B % e ~ XnlRT 47tV2(IV ( 45 ' 4 )-
.(45-1).
Since the distribution of velocities has been shown to have spherical
symmetry we can divide the shell 4^rV 2 dV of “velocity-space” uniformly
Lve n'.
• r + dr is
is 7Td^l ¡jfi.
into its rectangular elements dudvdw just as a spherical shell 47 t r 2 dr is
split into its elements dxdydz; so that the density of electrons with
velocities between u, v, w and u + du, v + dv, w + dw is
B Yi e~x n l RT dudvdw,
h A
i of their
ell at any
and the number in a range of space and velocity dxdydz dudvdw is
B ™ e~* n l RT dxdydz dudvdw (45-5).
..(45-2).
* This could be proved by considering the distribution of cos i, but it appears
to be sufficiently obvious.
f The use of the torm density is convenient, but it must be understood that the
nal to du.
symmetry.
We may
shell r to r + dr is for each electron referred to a different nucleus. These shells are
here thought of as superposed. Or, if preferred, instead of considering a large
number of different atoms we can deal with the time-average for one atom.
/