Real Solids - more than one atom per unit cell Molecular vibrations –Helpful to classify the different types of vibration Stretches; bends; frustrated rotations etc. Same is true of vibrations in solids –But to understand the possibilities need to look at a more complex model solid

A linear chain with 2 atoms per unit cell Must have 2N Avo vibrational modes per mole of substance (2R heat cap at hi T) Vibrations divide into two classes –Atoms in unit cell move in-phase; known as an acoustic mode (b) –Atoms in unit cell move in antiphase; known as an optical mode (a)

A three dimensional solid Get longitudinal and transverse waves.

The heat capacity For a solid with p atoms per primitve unit cell, there will be (per mole of primitive cells) –3N Avo p normal modes –3N Avo acoustic modes –3N Avo (p-1) optical modes And a hi T heat capacity of 3pR

Optical modes tend to be of a high frequency –Einstein model –Not excited at “low” T Acoustic modes vary in frequency from 0 to max. –Debye model –Contribute even at low T freq

Measurement of vibrations in solids Infra-red absorption –Excites optical modes where these give range to a change in dipole moment

Inelastic neutron scattering –Use thermal neutrons –Undergo energy loss/gain when they are scattered from a material –Energy exchange represents the phonon energy –More favourable selection rules than IR absorption

Thermal conduction Metals conduct heat via the conduction electrons, but some insulators are even better. Heat is carried by the phonons, which can travel unimpeded through a perfect crystal. Thermal resistance arises from –Scattering by imperfections –Phonon-phonon collisions

According to simple theory depends on the –heat cap. (C) –phonon vel. (v) –phonon mean free path (l) At low T, l= const=size of crystal. So K varies as T 3 (debye) At hi T, C= constant and l proportional to no.of phonons ie 1/T Diamond is a very good thermal conductor because of a. high sound velocity. b. high Debye T

The electronic heat capacity Peculiar observation in metals –Electrical conduction “a free electron gas” –Heat capacity - very small electronic heat capacity Arises because electrons are too light to follow Maxwell-Boltzmann laws Instead get a Fermi-Dirac distribution

At T=0, all the states up to E f are full. At T>0, only a small number of electrons close to E f can be excited. T f =E f /k=20,000 K typically. So at room T, C elec is about 0.01 of the expected classical value

At low T, lattice vibrations are small enough to see the electronic term