1000 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



ferent phases in adjoining unit cells, and thus to correspond to a finite wave 

 length, the frequency drops. 



The opposed motions are referred to as "optical modes" by analogy 

 with polar crystals. In a crystal of sodium chloride there is one Cl~ and one 

 Na+ per unit cell. In the opposed type of motion, ions of hke sign move 

 one way and opposite to those of the other sign. This relative motion of 

 charge polarizes the crystal and phonons of this type of vibration can ab- 

 sorb or emit light. Because of this optical activity the mode is termed 

 optical. 



The name "optical" is carried over to valence cr>^stals to describe the 

 opposed form of motion, although no polarization accompanies the dis- 

 placement in the latter case. 



3c. The Selection Rules 



We shall next consider the laws which govern the interchange of energy 

 between an electron and the phonons. There are two important laws, closely 

 analogous to the laws of conservation of energy and momentum for two 

 masses in colhsion. The quantity analogous to momentum for the phonon 

 is a vector, called Py , directed along the direction of propagation of the 

 phonon, and having a magnitude given by the relationship between mo- 

 mentum and wavelength - 



Py = h{l/X) = h/\ Py II propagation direction (3.3) 



In a transition in which an electron exchanges energy with the phonons, 

 and changes its momentum from Pi to P2 , so that diiy = dzl for one of 

 the modes, one selection rule requires that 



P2- Pl+ bUyPy = 0. (3.4) 



This is analogous to conservation of momentum; actually it is based on far 

 more subtle effects. The conservation of energy requires that 



82 + dityhvy = 81 (3.5) 



where 



82 = Pl/2m, 81 = Pl/lm (3.6) 



are the electron's energies before and after collision. The mass m need not 

 be the mass of an electron but may instead be the "effective mass," a mass- 

 like quantity of the same order as the electron mass which takes into ac- 

 count the influence of the periodic potential of the crystal structure upon 

 the electron wave packet. 

 The effective mass concept represents a simplification that may not 



