Lecture 12 - Specific Heat of Solids; The Einstein Solid

7/24/2019 Lecture 12 - Specific Heat of Solids; The Einstein Solid

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Statistical Physics: PHYC30017

Dr. Andy Martin (Room 613)

Lecture 12Specific Heat of Solids:

The Einstein Solid


2/14

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


3/14

Todays lectureReferences AV Chapters III-2.4 and IV-3

ZN = (Z1)

Breakdown of the Einstein model


4/14

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.


7/14

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


13/14

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:)

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Lecture 12 - Specific Heat of Solids; The Einstein Solid