THE QUANTUM PHYSICS OF SOLIDS 



675 



this number dN] it will depend upon E and be proportional to dE and 

 we may write 



dN = N{E)dE, (3) 



where the function N{E) represents the "number of quantum states 

 per unit energy" at E. This equation, like many in statistical mechan- 

 ics requires special interpretation, because if dE is small enough — less 

 than the spacing between levels in the band — it may include no levels. 

 If, however, we always use small but not infinitesimal values for dE, 

 so many levels will be included in it that equation (3) is quite satis- 

 factory. In Fig. 13(2 we represent qualitatively the distribution in 



N(E) 



^ N (E) ^ 



NUMBER OF QUANTUM STATES PER UNIT ENERGY 



Fig. 13 — Distribution of energy states in energy. 



(a) For two separate energy bands. 



(b) For overlapping energy bands. 



(c) For free electrons. 



N (E) 



energy for two energy bands. We plot N(E) horizontally so as to 

 retain the vertical scale for E. The area under the curve for the 2p 

 levels is three times that for the 2^. This is because the number of 

 states in the 2p and 2^ bands are respectively six times and two times 

 the number of atoms in the crystal. The Is band lies too low to be 

 shown on this figure; its levels will be concentrated over a.very narrow 

 range in energy in keeping with the small splitting suggested in Fig. 12. 

 It is possible for the energy band arising from one atomic energy 

 level to overlap the energy bands arising from other atomic levels. 

 We shall be concerned below with several cases where this occurs for 

 various crystals. When it does occur the states in the bands become 

 mixed up and it is no longer possible .to decide which atomic level was 



