Subsections

# Debye Model of Crystals

## Idea

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

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

## One-Dimensional Crystal

We have N particles with masses m on springs k: Equation of motion: (3)
or (4)
Periodic boundary conditions: u1=uN+1. Sound velocity: (5)

## Modes

All ui are entangled! To disentangle them--use a trick : (6)
What can we say about q? We want u(Na)=u(0), or for all q. This means that ,or Physics: we represented deviation as sum of waves: These waves are called modes

## Dispersion Equation. Meaning of Phonons

Substitute (6) into (4): with For small q Solution: In real space we obtain sum of terms proportional to This describes sound waves propagating with velocity c (that is why we claimed that c is sound velocity)

Equation (7)
is called dispersion equation. Each q corresponds to one normal mode or one kind of phonons--in other words, to a sound wave.

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

## Heat Capacity for 1D Crystal

We obtained N independent oscillators (modes!) with frequencies Each gives a contribution to the heat capacity. Total contribution: If --integration instead of summation. We have N values for q in the interval --so ## 3D Crystals

What's new for 3D?

1. is now vector--instead of dq we must integrate over , q2=qx2+qy2+qz2.
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 limit should be , but we used instead of exact equation (7), so we correct this by taking instead: Result (for simple lattice!): with (8) High temperatures:
for upper limit in the integral in (8) is small (1/u), and C=3NkB
Low temperatures:
for integral is constant, and . This is an example of a scaling, made by a classic of physics as early as 1912!   Next: Quiz Up: Solid State Previous: Einstein Model of Crystals

© 1997 Boris Veytsman and Michael Kotelyanskii
Wed Oct 1 00:45:35 EDT 1997