The Born-Karman model and dispersion curves

By crystal dynamics one usually understands the theory describing atomic oscillations around their equilibrium positions and those features of the properties that depend on these oscillations. Only harmonious oscillations occurring because of the action of quasi-elastic forces will be considered here (refer to Chapters 2.2 and 2.4). 

In a real three-dimensional crystal there are similar waves extending in all directions. Taking into account that in the crystal the existence of waves of three polarizations is possible (one being longitudinal L and two transverse wave Tj and refer to Section 2.8.2), 

Note that the acoustic branch corresponds to the oscillation of the crystal unit cells relative to each other, whereas the optical branches describe the oscillation of atoms relative to each other within the volume of a single unit cell. 

Since the elastic waves in aystals are caused by atomic oscillations and are interconnected, waves are developed as collective excitations. Hence, in modem terminology. 

Participation of electron excitations in heat capacity is limited by quantum statistics laws (refer to Section 9.2.2). As only a small part of the electrons takes part in excitation, the electronic heat capacity usually makes a very small addition in comparison with the lattice one. 

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