DISTRIBUTION OF RADIOACTIVITY IN DEPTH 137 



Early in 1906 the distribution of temperature in a globe heated ex- 

 clusively by radium was discussed almost simultaneously by Mr J. 

 Koenigsberger ^^ and Mr E. J. Strutt.'^* 



Mr Koenigsberger, starting from Fourier's equation for the conduction 

 of heat in three dimensions, discusses several distributions of radium: 

 (1) a uniform distribution throughout the globe, (2) a uniformly active 

 shell, ( 3 ) a sphere in which the activity fades out gradually in depth, and 

 (4) an active shell buried beneath an inactive layer of rock J* 



'aphys. Zeitsch., vol. 7, 1906, p. 297. 



" Proceedings of the Royal Society of London, series A, vol. 77, 1906, p. 472. 

 '* The partial differential equation representing the conduction of heat in a sphere is 

 by Riemann's Partielle DlfCerentialgleichungen, section 61 : 



d (ur) k d" (ur) 



dt c dr^ ' 



where u is the excess of temperature, as compared with the surface, of a point at a 

 distance r from the center ; k is the conductivity ; c the thermal capacity, and t the 

 time. If q is the quantity of heat liberated per second per cubic centimeter by radio- 

 activity, the rate at which the temperature would rise is q/c. If the earth is in ther- 

 mal equilibrium, the heat lost by conduction per unit of time must be equal and oppo- 

 site to q/c. Hence 



dn q 



"dt~ ~~ V ' 

 and since r does not vary with t, 



du _d (ur) _ qr 

 dt dt c ' 



Consequently 



d^ (ur) _ qr 

 dr^ F 



represents the distribution of temperature in any radioactive sphere in which the 

 radium is symmetrically distributed with reference to the center. 



When q and Tc are constant for such a portion of the sphere as is radioactive, the 

 integral of this expression contains two arbitrary constants. If the problem to be dis- 

 cussed is that of a superficial shell of radioactive matter, let R be the outer radius of 

 the earth and Ro the inner radius of the shell. Then du/dr must vanish, for r = Rq 

 and u must be zero at the surface or when r = R. The complete solution is then 



ku _ R"- — r2 R-'o / I l\ du__ q_ / . R\ \ 



q ~ B 3 \V~ R I ' dr ^~Vcy' W I 



for a sphere which is radioactive throughout -Ro = o in these formulas. If, on the 

 other hand, the shell is thin, let R — r— x, where x is the distance from the surface, 

 and let R — Ro= s the thickness of the shell. Then x/R will be a small quantity and 

 the formulas reduce to the form employed by Mr Strutt and which will be used here in 

 the text. As Mr Koenigsberger shows, q may, if desired, be regarded as a function 

 of r; for example, q' may be a constant, such that 



5' r" = g ; 



so that the radioactivity vanishes at the center of the earth and is greatest at the sur- 

 face. The differential equation may still be Integrated with ease. Again, k may be 

 supposed to vary with the temperature. I may observe, however, that it can not really 

 vary by such a law a.9 k = k(, (I — au), for then when u = 1/ a, k = o, and this 

 would mean that a body heated to a temperature l/a would never cool at all ! 



In a sphere of initially uniform temperature cooling according to Fourier's law, the 

 temperature gradient is a maximum at the surface and decreases continuously with in- 

 creasing depth. By an inadvertence, Mr Koenigsberger states that In this case the 

 gradient increases with depth. 



