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# phys documents (dragged) 70 - 64 Physics Formulary by ir...

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Unformatted text preview: 64 Physics Formulary by ir. J.C.A. Wevers 12.3.4 Thermal heat capacity The total energy of the crystal vibrations can be calculated by multiplying each mode with its energy and sum over all branches K and polarizations P : U = K P h n k,p = D ( ) h exp( h /kT )- 1 d for a given polarization . The thermal heat capacity is then: C lattice = U T = k D ( ) ( h /kT ) 2 exp( h /kT ) (exp( h /kT )- 1) 2 d The dispersion relation in one dimension is given by: D ( ) d = L dK d d = L d v g In three dimensions one applies periodic boundary conditions to a cube with N 3 primitive cells and a volume L 3 : exp( i ( K x x + K y y + K z z )) exp( i ( K x ( x + L ) + K y ( y + L ) + K z ( z + L ))) . Because exp(2 i ) = 1 this is only possible if: K x ,K y ,K z = 0; 2 L ; 4 L ; 6 L ; ... 2 N L So there is only one allowed value of K per volume (2 /L ) 3 in K-space, or: L 2 3 = V 8...
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