• And finally differentiate U w.r.t. to T to get the heat capacity.

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C = 3NAvohv( )−1

ehv

kt −1 ⎛ ⎝ ⎜ ⎞

⎠ ⎟2

−hv

kT 2ehv

kt

= 3NAvokhv

kT

⎛

⎝ ⎜

⎞

⎠ ⎟2

ehv

kt

ehv

kt −1 ⎛ ⎝ ⎜ ⎞

⎠ ⎟2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

= 3RθE

T

⎛

⎝ ⎜

⎞

⎠ ⎟2

eθ E

T

eθ E

T −1 ⎛

⎝ ⎜

⎞

⎠ ⎟

2 where θE = hv /k

Notes

• Qualitatively works quite well

• Hi T 3R (Dulong/Petit)Lo T 0

• Different crystals are reflected by differing Einstein T (masses and bond strengths)

• Neatly links up the heat capacity with other properties of solids which depend on the “stiffness” of bonds e.g. elastic constants, Tm

• But the theory isnt perfect.

Substance Einstein T/K

diamond 1300

Al 300

Pb 60

• Firstly, real monatomic solids can show a heat capacity at hi T which is greater than 3R.

• Any harmonic oscillator always has a limiting C of R.

• And we’ve counted up the oscillators correctly (3NAvo)

• So…. the vibrations can’t be perfectly harmonic.

• Also manifests itself in other ways e.g. thermal expansion of solids.

• Secondly, the heat capacity of real solids at low T is always greater than that predicted by Einstein

• A T3 dependence rather than an exponential

• This flags up a serious deficiency

• Vibrations in solids are much more complicated than the simplistic view of the Einstein model !

• Atoms don’t move independently - the displacement of one atom depends on the behaviour of neighbours!

• Consider a simple linear chain of atoms of mass m and and force constants k

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Hooke's Law

Force = −k × extension)( )

Fnn+1 = −k(un − un+1)

Fnn−1 = −k(un − un−11)

nett force on atom n = Fnn+1 + Fn

n−1

= −k(2un − un+1 − un−1)

• For situations where the atoms and neighbours are displaced similarly

• So the frequency will be very low for “in phase” motions

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nett force on atom n = Fnn+1 + Fn

n−1

= −k(2un − un+1 − un−1)

≈ 0

• For situations where the atoms and neighbours are displaced in opposite direction

• So the frequency will be very high for “out of phase” motions

€

nett force on atom n = Fnn+1 + Fn

n−1

= −k(2un − un+1 − un−1)

≈ −4kun

The nett result

• The linear chain will have a range of vibrational frequencies

€

vmin = 0 in phase

vmax =1

π

k

m out of phase

• So a real monatomic solid (one atom per unit cell) will have 3NAvo oscillators (As Einstein model).

• But they have a distribution of frequencies (opposite of the Einstein model)

• Each oscillator can be in the ground vibrational state…. Or can be excited to h, 2 h …n h

• Desrcribed by saying the oscillator mode is populated by n PHONONS

• How to get the specific heat?

• Look back at the Einstein derivation

€

Uvib = 3NAvo ×hv

ehv

kT −1

⎛

⎝ ⎜

⎞

⎠ ⎟

i.e. mean energy per oscillator

=hv

ehv

kT −1

⎛

⎝ ⎜

⎞

⎠ ⎟

so for a real monatomic solid

Uvib =i=1

3NAvo

∑ hv i

ehvi

kT −1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

• If we know all the vibrational frequencies we can calculate the thermal energy and the specific heat.

• Normally a job for a computer since have a complicated frequency distribution

Debye approximation

• Vibrations in the linear chain have a wavelength• High frequency modes have a wavelength of the order of

atomic dimensions (c)• But for the low frequency modes, the wavelength is

much,much greater (b)

• In the low frequency, long wavelength limit the atomic structure is not significant

• Solid is a continuum - oscillator frequency distribution is well-understood, in this regime.

€

no. of vib betwen ω and ω +dω

= Aw 2dω

ω = ang. freq = 2πv

E ph = hω = hv

• Debye assumption- the above distibution applies to all the 3Navovibrational modes, between 0 and a maximum frequency, D, which is chosen to get the correct number of vibrations.

• So a monatomic solid on the Debye model has 3NAvo oscillators…

• with a frequency distibution in the

range 0- D

• And a normalised spectral distribution

€

no. of vib betwen ω and ω +dω

=9NAvo

ωD3

ω2dω

€

Uvib =hω

ehω

kT −1

⎛

⎝ ⎜

⎞

⎠ ⎟

0

ωD

∫ 9NAvo

ωD3

ω2 ⎛

⎝ ⎜

⎞

⎠ ⎟dω

=9NAvoh

ωD3

0

ωD

∫ ω3

ehω

kT −1

⎛

⎝ ⎜

⎞

⎠ ⎟dω

• Finally we can differentiate w.r.t. to T to get the specific heat

€

C =dU

dT= 3R

3T 3

ΘD3

ΘD

T

⎛

⎝ ⎜

⎞

⎠ ⎟4

eΘD

T

eΘD

T −1 ⎛

⎝ ⎜

⎞

⎠ ⎟

2 dΘD

T

⎛

⎝ ⎜

⎞

⎠ ⎟

0

ΘD

T

∫

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

i.e it is a complicated function of a variable

x = ΘDT where ΘD =

hω

k≡ Debye Temperature of solid

• Understand the physical principles and logic behind the derivation - don’t memorise all the expressions!

• The term in square brackets tends to 1 at hi TIe C=3R as expected for harmonic oscillators.

• At low T, the integral tends to a constant, so C varies as T3. Fits the experimental observations much better.

• Physically, there are very low frequency oscillators which can still be excited, even when the higher frequency modes cannot.

Like the Einstein T, the Debye T is a measure of the vibrational frequency I.e. determined by bond strengths and atomic masse.