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

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, T m But the theory isnt perfect. SubstanceEinstein T/K diamond1300 Al300 Pb60

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 (3N Avo ) 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 T 3 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

For situations where the atoms and neighbours are displaced similarly So the frequency will be very low for “in phase” motions

For situations where the atoms and neighbours are displaced in opposite direction So the frequency will be very high for “out of phase” motions

The nett result The linear chain will have a range of vibrational frequencies

So a real monatomic solid (one atom per unit cell) will have 3N Avo 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

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.

Debye assumption- the above distibution applies to all the 3N avo vibrational 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 3N Avo oscillators… with a frequency distibution in the range 0-  D And a normalised spectral distribution

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

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 T Ie C=3R as expected for harmonic oscillators. At low T, the integral tends to a constant, so C varies as T 3. 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.