Transcript Part II

Heat capacity of the lattice
General Theories
Thermodynamics
for
no numerical value
at finite T
Classical Statistical
Mechanics
Dulong-Petit law, numerical value
independent of T
For details see Mandl
Heat capacity: classical thermodynamics
We also get information about the heat capacity at T=0.
for a reversible process
and at constant volume
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In the limit of ever lower temperatures for T1, CV must
vanish to comply with the third law of thermodynamics.
Heat capacity: classical thermodynamics
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So classical thermodynamics does not tell us anything
about the heat capacity of solids at finite temperature but
we know that it must vanish at zero temperature.
Heat capacity: classical statistical mechanics
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The equipartition theorem states that every generalized
position or momentum co-ordinate which occurs only
squared in the Hamiltonian contributes a mean energy of
(1/2) kBT to the system.
This theorem is based on the classical partition function, an
integral in phase space. It is not valid if the real quantum
mechanical energy levels are separated by an energy
interval large compared to kBT.
The equipartition theorem allows us to make a quantitative
prediction of the heat capacity, even though this prediction
contradicts the vanishing heat capacity ot 0 K which was
obtained from a more general principle. At sufficiently high
temperatures, the prediction should still be ok.
Heat capacity: classical statistical mechanics
so for a one classical dimensional harmonic oscillator we have
so the mean energy is k T. For a three-dimensional oscillator
we have 3 k T. For one B
mole of ions we get
B
and so
This is called the Dulong-Petit law.
Comparison of the Dulong-Petit law to
experiment
-1
77 K (JK
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7
-1
)
273 K (JK
classical value
24.9
24.9
copper
12.5
24.3
aluminium
9.1
23.8
gold
19.1
25.2
lead
23.6
26.7
)
ironvalues of one mole
8.1 of substance24.8
20.4
27.6
At highsodium
temperatures the Dulong-Petit
law works
quite well.
silicon
5.8
21.8
At low temperatures,
it does not.
But we already know
from basic
principles that it wouldn’t.
Heat capacity of diamond
The Einstein model for the heat capacity
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correct Dulong-Petit value
for high temperatures
zero heat capacity at zero
temperature
The Einstein frequency /
temperature is an adjustable
parameter.
The Einstein model: low-temperature heat capacity
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but low temperature
behaviour of the
experiment not correctly
reproduced. T3 behaviour
in experiment, exponential
behaviour here
for very low T
10
Why does the Einstein model work at high T?
Why does it fail at low T?
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At high T the small spacing between
the energy levels is irrelevant.
At sufficiently low temperature, the
energy level separation is much
bigger than k T.
B
Eventually all the oscillators are
“frozen” in the ground state.
Increasing T a little does not change
this, i.e. it does not change the
energy.
total number of
states for a
maximum |k|
The Debye model for the heat capacity
substitute
low temperature,
large
for one mole
Comparison Debye model - experiment
Why does the Debye model work better at low
T than the Einstein model?
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The Debye model gives a better representation for the
very low energy vibrations.
At low temperatures, these vibrations matter most.
Limits of the Debye model