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 2NAvo 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)– 3NAvop normal modes

– 3NAvo acoustic modes

– 3NAvo (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 T3 (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

€

K = 13Cvl

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 Ef are full.

• At T>0, only a small number of electrons close to Ef can be excited.

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only a fracti on of electrons T/Tf take up energy

U thermal ≅ 32NkT × T

Tf (classical energy × fraction)

Cvelec ≅

dUdT

≅3RTTf

(exact = 12π

2R TTf

)

•Tf=Ef/k=20,000 K typically.•So at room T, Celec is about 0.01 of the expected classical value

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

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Cv = AT 3 + BTCvT

= AT 2 + B