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Statistical Physics: PHYC30017
Dr. Andy Martin (Room 613)
Lecture 12Specific Heat of Solids:
The Einstein Solid
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Review of lecture 11Paramagnets
Ideal paramagnet B
B
Spinjparamagnet Ej = jgBB
Ej+1= (j 1)gBB
Ej1 = (j+ 1)gBBEj = jgBB
4 2 2 4
1.0
0.5
0.5
1.0
M/(N)
x
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Todays lectureReferences AV Chapters III-2.4 and IV-3
ZN = (Z1)
Breakdown of the Einstein model
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Specific heat of solids: introduction
ObservationIn the 19th century the value of CV/N, the specific heat per particle, ofmost solids was known to follow the Dulong-Petit law
CV = 3kN
Harmonic oscillator model
The energy of a single oscillator is
E=
p2
2m+
1
22
x
2
and there are thus six quadratic degrees of freedom, giving an energy perparticle of 6kBT/2, and a specific heat/particle of 3kB.
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Beyond Dulong-PetitBy the end of the century it became apparent that:
Some solids, e.g. diamond, had specific heats below the Dulong-Petitvalue.At lower temperatures the specific heat of all solids reduced, andseemed to go to zero as T !0.
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Einstein
From Quantum mechanics + Boltzmann's probability distribution, we
expect that the probability of an harmonic oscillator being excited to thefirst excited state is of order exp{-h"/(2#kBT)}, which goes to zero as T! 0, so we can understand the temperature dependence of the specificheat.
Einstein inverted this argument to show that the decrease of the specificheat of solids at low temperatures was an example of quantum mechanics
at work. This was the first example of the application of quantummechanics to matter.
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The Einstein solid
Einstein's model of a solid - each of the N ions is in a 3d harmonic
oscillator well with the frequency - all ions have the same frequency
ZN = (Z1)
Z1 is the partition function of a single harmonic oscillator in 3
dimensions, with energy levels:
n = nx+ny+nz+ 3 2
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The partition functionThe partition function, for one site
Z1 =
n=0
e(n+1/2)/(kT)
=
e
2kT
1 e
kT
= 12 sinh
2kT
3
The partition function, forNsites
Z= ZN = (Z1)N =
1
2sinh
2kT
3N
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The mean energy
U =
lnZ
=
1
Z
Z
= 3N1
2 coth1
2
Define the Einstein temperature: E = /kB
U= (3/2)NkBEcoth(E/(2T))
High temperature: T >> $E
U = 3NkBT limx0
cothx =x
+ x
3
...
Low temperature: T
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Specific heat
CV =
UT = 3Nk
BE
2T2
1sinh2 (E/ {2T})
High temperature: T >> $E
limx0
sinhx = x + x3
6 ...
CV = 3NkB
Low temperature: T
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Comparison with experiments
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Check entropy
F =
kBT lnZ = 3NkBT
ln 2 + ln sinhE
2T
S =
F
T
= 3NkB
ln 2 + ln sinh
E
2T
E
2T coth
E
2T
As T !0
S 3Nk
E
2T ln 2
E
2T + ln 2
0
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Einstein to Debye
The discrete energy levels of quantum mechanics, with the Boltzmann
factor determining probabilities, gives freezing of degrees of freedomand the third law of thermodynamics.
New experiments showed that the behaviour at low temperatures was notwell described by Einstein's theory, which gave an exponential decrease
to zero, much faster than observed. The data for insulators fit T3 law atlow temperatures.
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Debye theory
Next Lecture:)