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- Idea
- One-Dimensional Crystal
- Modes
- Dispersion Equation. Meaning of Phonons
- Heat Capacity for 1D Crystal
- 3D Crystals

We^{} represented each atom as a
harmonic oscillator. It did not work.

Let us represent our system as a set of 3*N* oscillators, but choose
them differently, *including* collective motion.

We have *N* particles with masses *m* on springs *k*:

(3) |

(4) |

(5) |

All *u*_{i} are entangled! To disentangle them--use a
trick^{}:

(6) |

(7) |

Why Debye model is better than Einstein model: since at , some phonons are excited even at low temperatures!

We obtained *N* independent oscillators (modes!) with frequencies

Each gives a contribution to the heat capacity. Total contribution:

If --integration instead of summation. We haveWhat's new for 3D?

- 1.
- is now
*vector*--instead of*dq*we must integrate over ,*q*=^{2}*q*_{x}^{2}+*q*_{y}^{2}+*q*_{z}^{2}. - 2.
- There are three waves for every : two transversal and one longitudinal

Instead of integrating by we integrate by . Number of modes:

Limits: lower limit , upper limitResult (for simple lattice!):

with(8) |

**High temperatures:**- for upper limit in the integral
in (8) is small (1/
*u*), and*C*=3*Nk*_{B} **Low temperatures:**- for integral is constant, and

© 1997

Wed Oct 1 00:45:35 EDT 1997