To understand the motion of atoms, we need first know the force:
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(55) |
For stable solids at some temperature, it is much more useful and informative to cast the expressions in terms of an expansion of the energy in powers of displacements
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(56) |
where Greek subscripts ,..., denote Cartesian components.
Within harmonic approximation, the vibrational modes at frequency are described by displacements
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(57) |
so that equation 55 becomes:
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(58) |
the solution of the equation is the set of independent oscillators, each with vibrational frequency ,
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(59) |
In a crystal, the atomic displacement eigenvectors obey the Bloch theorem, i.e., the vibrations are classified by k with the displacement. Therefore, the vibrations can be reduced to the first BZ. For a unit cell with N atoms, the phonons at different k yield dispersion curves, which are solution of the determinant equation:
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(60) |
There are three acoustic modes with for
and the other
modes are classified as opitc.