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. 
/