708 BELL SYSTEM TECHNICAL JOURNAL 



electrons if there were one in each compartment. If we set the vol- 

 ume of the sphere equal to N¥IVG a.nd solve for p, we get the radius 

 pm of the sphere which would just contain all the electrons if there 

 were G of them in each compartment. But this is the maximum 

 value of the momentum of the electrons, it is the momentum of the 

 fastest of the electrons. The corresponding speed Vm of the fastest of 

 the electrons is pm/m, therefore is given by the expression : 



Vm = — [-r ] (71a) 



m \47r/ 



and the corresponding kinetic energy ^niv,,? is the same as Wi. 



It is expedient to set down for future reference the mean values of 

 speed V and of several integer powers of v, for a gas distributed according 

 to the Fermi law at the absolute zero. The general formula for 

 the mean value of any power v^ of v is this: 



— \ r 1 r""* 



j,s = - j v'f(v)dv = - {4:TrGlm%^) \ v'+Hv 



«J ^ Jo 



and in particular: 



Fi = 3/2i'„.; V = 3z'„/4; i^ = SvJjS; ? = ^vj. (72a) 



The corresponding values for the Maxwell distribution are these; 



Fi = 2(m/27r/^r)i/2; v^ = (2kT/m); (72b) 



V = 2(2kT/Trmy''~; v^ = ^ir{2kTlTrmyi\ 



Plotted as functions of e, the distribution-function / in the co- 

 ordinates and momenta starts out as an horizontal straight line at a 

 distance Gjh^ from the axis of abscissae, while the distribution-function 

 F in the energy starts out as a concave-upward parabolic arc; these 

 continue as far as the abscissa e = Wi, and from then on the curves 

 coincide with the axis of abscissae. 



The foregoing statements are valid for absolute zero; what happens 

 as the temperature rises? Sommerfeld has proved that the sharp 

 angles in the distribution-curve are very gradually and slowly rounded 

 off, the curve always traversing the midpoint of the vertical arc BC 

 (Fig. 1). The far end of the curve sinks down to the axis of abscissae 

 in the fashion of the Maxwell law. Even at room-temperature and 

 even far above, however, the distribution departs so little from the 

 absolute-zero form that many phenomena may be interpreted in a 

 qualitative way, simply by imagining the absolute-zero distribution — 

 the completely degenerate distribution, it is called — to persist all 



