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• 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
• 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
€
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.