Some Orbits of an Electron. 



207 



several rounds may be made before it approaches that of the 

 nucleus, unless p or the velocity are very small. 



When the force between the particles is repulsive the 

 integration of the equations is very similar, but now in all 

 cases there are of course two asymptotes, and the only effect 

 is that the particle is not so much deflected as it would be if 

 its mass were constant. These orbits do not seem to be of 

 any great interest. 



4. We must now consider the numerical values for the 

 various orbits. In the first place a = e^/mc 2 . Let E=n0 

 so that n is the number of electrons in the atom. Then 

 a — n x 2'S x 10 ~ 13 . Take n= 100, which is probably roughly 

 the value it would have for a heavy element, say lead. We 

 shall take a=2'S X 10 _u throughout. A simple proportional 

 reduction will give it in' other cases. To show the general 

 behaviour of the orbits I take four values of the velocity, 

 0*01 c, 0'lc, 0*9 c, and 0'99c and find in each case the value 

 of p which gives the critical orbit. This determines what 

 particles cannot escape. Speaking loosely, it may be sup- 



posed that the spiral part of the orbit begins when 0= ~ ; 



to find the shape of the spiral I therefore find r for 



IT 57T 97T 



= 



I also find c(t 5lt — t\ and c(h n — totr 



It 



as 



W^off t\ 

 *4* \~2 2) \ 2 2 



these are roughly the first two wave-lengths of the emitted 

 radiation, provided that this radiation does not itself entirely 

 modify the orbits. It will be seen that when the velocity is 

 small the critical value of p increases in inverse proportion 

 to it and is of a size quite comparable to that of an atom. 

 All the quantities are, of course, measured in centimetres. 



Table I. The Critical Orbits. 



V/c 



•01 



•1 





•9 



•99 





*)<*" 



2-8X10- 9 



2-8x10" 



•10 



1-35 xlO" 11 



4-0x10" 



-12 



P • 







2-2X10- 11 

 9-lxlO" 13 



2-8X10" 13 



2-0x10" 

 8-9x10" 



2-8x10" 



-li 



-13 

 -13 



4-6xl0~ 12 

 3-2xl0" 13 



1-1 XlO" 13 



1-4x10- 

 1-0x10' 



3-5x10" 



-12 



2 



"13 



'ott ••••; 



-14 



7 9ir 



2 





£( t Ort- 

 iz 



2 



tn) 



2 



torr) 



2 



3-8X10" 11 

 3-2xl0~ 12 



3-4x10- 

 3-2x10" 



-11 

 -12 



8-8 xlO" 12 

 1-2X10" 12 



2-7X10" 

 3-8XUT 



-12 

 -13 



