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Unformatted text preview: 16.3 Debye theory of the heat capacity of solids The atoms in a crystal consisting of N atoms do not vibrate independently of each other about fixed sites. Instead, they execute very complicated coupled vibrations. From classical mechanics, however, one knows that the oscillations of such a system can be described in terms of 3N independent normal modes of vibration of the whole crystal, each with its own characteristic frequency 1 , 2 , 3N . In terms of these normal modes , the lattice vibrations of the crystal are equivalent to 3 N independent harmonic oscillators with these frequencies. To illustrate the above statement, consider a system of two coupled one-dimensional oscillators described by the equations of motion: where q 1 and q 2 are the position coordinates, while the coupling factor ( =0 for Einstein model). Let Q 1 =q 1 +q 2 and Q 2 =q 1-q 2 , then above equation become Here Q 1 and Q 2 are called the normal coordinates. In terms of them, the system behaves like two independent linear oscillators with frequencies of ( 2 + ) 1/2 and ( 2- ) 1/2 , respectively....
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