Materials Science and Engineering/Doctoral review questions/Crystallography

Tensor 1

Propagation of Waves Along One-Dimensional Crystal with Two Kinds of Atoms

• Force-balance equations
• Trial solution
• With a given value of k, there are two different waves of angular frequency that may be propagated

Interpretation of the Solutions

Solutions of Small k

Solutions with negative sign at k = 0

Acoustic and optic branch

Correspondence to the Identical-Atom Problem

The optical mode is not present in the case with one kind of atom.

Tensor 2

Wave Propagation in a Continuous One-Dimensional Medium

The wave velocity is independent of $\lambda$

 $v={\sqrt {\frac {c}{\rho }}}={\sqrt {\frac {\beta a^{2}}{M}}}$

Total Number of Vibrational Modes that May be Supported by the Crystal

The number of vibration modes that can be supported is equal to the number of atoms in the crystal

Tensor 3

Propagation of Elastic Waves in Crystals

Relation of stress and strain in a cubic crystal

Tensor 6

Cauchy Relation

1. Central Forces
2. Each atom at center of symmetry
3. No initial stress