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

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, Tm • But the theory isnt perfect.

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

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 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

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 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

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 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.

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