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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