696 BELL SYSTEM TECHNICAL JOURNAL 



From this equation v^ax can be found if NIV and the velocities Ct 

 and Cl are known. Knowing j/max and the distribution in frequency, 

 Debye summed the specific heats of all the oscillators and obtained 

 the specific heat of the solid. According to this theory the specific 

 heat vanishes at T = and is proportional to T^ near T = 0. At high 

 temperatures it approaches the classical value of 3)R. A measure of 

 the temperature at which the classical value is closely approached is 

 given by the maximum frequency of atomic vibration j'^axi when ^T is 

 greater than Ai'max, all the modes of vibration including the highest 

 make substantial contributions to the specific heat. The temperature 

 at which this occurs is known as the Debye temperature and denoted 

 by the symbol Qd] obviously Od = hv^^jk. The specific heat given 

 by Debye's equation is a function of TJOd only and can thus be rep- 

 resented by the expression CiT/dD); so that by this theory all crystals 

 should have the same curve for specific heat versus temperature except 

 for changes in the temperature scale corresponding to the different 

 values of their Debye temperatures. 



TABLE III 

 Debye Temperatures in Degrees Kelvin Used in Figure 21 



In Fig. 21 is shown a compilation of specific heat data." For each 

 substance a value of do (given in Table III) has been chosen so as to 

 obtain the best agreement with experiment and the values of the specific 

 heat have then been plotted as a function of T/do- The Debye theory 

 relates to specific heat at constant volume and in it no allowance is 

 made for the energy due to thermal expansion. The experimental 

 points are derived from measurements of specific heat at constant 

 pressure which have been transformed by using a thermodynamical 

 relationship so as to give specific heat at constant volume. 



For these curves 6d was chosen so as to obtain the best fit. It is, 

 however, possible to calculate do from theory by using the elastic 

 constants of the material to evaluate Ct and Cl and then substituting 

 in Eq. (8). For sodium the elastic constants have been calculated 

 entirely from theory by the methods described in the section on 



"Taken from E. Schroedinger, Handbuch der Physik, Vol. X, p. 307 (1926). 



