QUANTUM THEORY 
55 
/ 
Hence by (40-61) 
q 2 c 2 b 2 
.(40-63), 
q x 8ttv 12 2 8p 
where 1 /b 21 is interpreted simply as the average duration of state 2 before 
a spontaneous relapse occurs, and 8p is the width of the spectral line 
emitted by such relapses*. 
The number of atoms in a gram is 1 / Ah , where A is the atomic weight 
and h the mass of a hydrogen atom. Hence the mass absorption coefficient 
k, or absorption coefficient per gm. per sq. cm. is 
k 
a 
ATh 
<h 
.(40-64). 
q x 8 ttv 12 2 AnS p 
The coefficients in (40-63) and (40-64) refer to the absorption of mono 
chromatic radiation of frequency p 12 by material composed wholly of 
atoms in state 1 . Also if the atom has more than one electron which by 
excitation can absorb frequency p 12 , the coefficients become multiplied by 
the corresponding factor. 
Planck’s Law (37-9) can now be given fully as 
( v> ¿T-^hv/RT T (40-7). 
By Stefan’s Law 
aT 4 = I" I (p, T) dp 
877 -h /ETA f 00 x 3 dx 
(x — JipjRT). 
.(40-8). 
h J J 0 e x - 1 
The integral is equal to 7 t 4 /15, so that 
87 r 5 R i 
a 15 c z h 3 
Some of the chief properties of Planck’s Law may be stated here for 
reference. The mean frequency of the radiation is given by 
3 x^dx f 00 x z dx 
0 e x ^ 1 ~ Jo 1 
= 24 (l - 5 + 2 - 5 + 3 - 5 + ...) -r 6 (l - 4 + 2 - 4 + 3 - 4 + ...) 
= 3-8322, 
so that hv = 3-83 RT (40-91). 
The number of quanta per cubic centimetre is 
'RT z x 2 dx 
he ) 1 0 e® — 1 * 
x = 
877 
* More precisely a is the average absorption coefficient over any width 8p 
sufficient to cover the absorption line, so that a8p gives the whole absorption of the 
line. If we were to take 8p to cover only part of the absorption line the coefficient 
b 21 would refer to a fraction only of the emissions and would not then be equal to 
the reciprocal of the duration of the excited state.