674 BELL SYSTEM TECHNICAL JOURNAL 



dated with a wave form, shown dashed. This wave form is in every 

 case of such a wave-length that it has an integral number of half wave- 

 lengths along the edge of the crystal.^" The number of half wave- 

 lengths is a suitable quantum number for the wave functions in the 

 crystal and a more general consideration of it in the next paper will 

 lead us into the theory of the "Brillouin zone" and the zone structure 

 of energy bands. The second subject concerns the transmission prop- 

 erties of the crystal. The set of coupled circuits of Fig. 11 constitutes 

 a length of transmission line. A line of this type is a simple filter net- 

 work and as such it has bands of frequency in which power will be 

 transmitted and bands in which it will not. The allowed frequencies 

 lie in the transmitting bands. The system of coupled mechanical 

 vibrators likewise constitutes a mechanical filter. Just as the mechan- 

 ical and electrical systems can transmit power in their allowed bands, 

 a crystal can transmit an electron whose energy is in an allowed band. 

 The electrons in an allowed band, however, can produce a net current 

 only if the band is partially filled. Electrons in wholly filled energy 

 bands, although individually representing tiny currents to and fro in 

 the crystal, can produce — we shall find — no net current as their indi- 

 vidual currents cancel out in pairs. On this basis the theorist ex- 

 plains the difference between metals and insulators as follows: in a 

 metal some of the energy bands are partly filled, but in an insulator 

 each energy band is either completely filled or completely empty. 



Distributions of Quantum States in Energy Bands 



When there are a very large number of atoms in the crystal, it is 

 impractical to represent the energy levels by distinct lines as was done 

 for the case of six atoms in Fig. 12 and another scheme must be used. 

 For a crystal of macroscopic dimensions the number of levels in the 

 band is of the order of 10^*, that is a million million million million. 

 When so many levels are placed so close together, a continuous band of 

 allowed energies is suggested. Actually, of course, only a discrete set 

 of allowed energies is possible, the total number in the band being 

 that required by the conservation of states. We shall now consider 

 the distribution in energy of these quantum states; that is, how many 

 lie in a given range of energy between E and E + dE. Let us call 



^» For a three-dimensional crystal having the external shape of a cube, the three- 

 dimensional wave function has an integral number of half wave-lengths along lines 

 parallel to each edge of the crystal. This condition is illustrated in a simplified 

 form by the wave patterns for the two-dimensional drum head shown in Fig. 7; 

 for each normal mode, there is an integral number of half wave-lengths parallel to 

 each boundary of the membrane, and, in fact, the values of these numbers are given 

 by p and q. 



