Atomic Motion and Normal Modes

Having identified our approximation to the total energy associated with small vibrations about equilibrium, we are now in a position to construct the classical equations of motion which dictate the ionic motions. If we make the simplifying assumption that we are concerned with an elemental material (i.e. all the masses are identical, etc.), thus surrendering some of the possible generality of our treatment, the equation of motion for the component of displacement associated with the 

This series of coupled linear equations in the unknowns Uia may be seen in a different light if written in direct form as the matrix equation, 

When our problem is recast in this form it is clear that we are reconsidering it as one of matrix diagonalization. Our task is to find that change of variables built around linear combinations of the UiaS that results in a diagonal K. Each such linear combination of atomic displacements (the eigenvector) will be seen to act as an independent harmonic oscillator of a particular frequency, and will be denoted as normal coordinates . 

Before undertaking the formal analysis of vibrations in solids, it is worthwhile to see the machinery described above in action. For concreteness and utter simplicity, we elect to begin with a two-degrees-of-freedom system such as that depicted in fig. 5.3. Displacements in all but the x-direction are forbidden, and we assume that both masses m are equal and that all three springs are characterized by a stiffness k. By inspection, the total potential energy in this case may be written as 

These equations can be rewritten in the more sympathetic matrix form as 

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