Download Solids and Surfaces

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  1 3 Cv
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.
only a fract ion of electrons T/T f take up energy
U
thermal
T
 3 NkT 
(classical energy  fraction)
2
T
f
C elec

v
dU 3RT
T

(exact = 12  2 R )
dT
Tf
Tf
•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
Cv  AT 3  BT
Cv
2
 AT  B
T