Heat capacity of the lattice we ignore the heat capacity of the electrons in metals for the time being. Important issue for itself. Many electrons in metals, should give strong contribution to heat capacity but they don’t. We also ignore differences between CV and CP. Theorists like CV and experimentalists like CP. Explain why.

statistical mechanics result for one mole of substance. General Theories Thermodynamics for no numerical value at finite T Classical Statistical Mechanics statistical mechanics result for one mole of substance. but we ignore free electrons in metals for now! Dulong-Petit law, numerical value independent of T

Heat capacity: classical thermodynamics We also get information about the heat capacity at T=0. for a reversible process and at constant volume classical thermodynamics: very general theory which allows very general conclusions fundamental equations defining the heat capacity the integral is ln(T) from T1 to T2. for the limit of T1->0 CV has to go to 0 in order to keep the integral finite, otherwise its going to minus infinity. If not, the integral can be compensated by an infinite entropy at 0 but this does not work. In the limit of ever lower temperatures for T1, CV must vanish to comply with the third law of thermodynamics.

Heat capacity: classical thermodynamics So classical thermodynamics does not tell us anything about the heat capacity of solids at finite temperature but we know that it must vanish at zero temperature.

Heat capacity: classical statistical mechanics The equipartition theorem states that every generalized position or momentum co-ordinate which occurs only squared in the Hamiltonian contributes a mean energy of (1/2) kBT to the system. This theorem is based on the classical partition function, an integral in phase space. It is not valid if the real quantum mechanical energy levels are separated by an energy interval large compared to kBT. The equipartition theorem allows us to make a quantitative prediction of the heat capacity, even though this prediction contradicts the vanishing heat capacity ot 0 K which was obtained from a more general principle. At sufficiently high temperatures, the prediction should still be ok.

Heat capacity: classical statistical mechanics so for a one classical dimensional harmonic oscillator we have so the mean energy is kBT. For a three-dimensional oscillator we have 3 kBT. For one mole of ions we get and so independent of temperature independent of the force constant, mass and hence omega important is the number of oscillators always three times the number of atoms, no matter if single harmonic oscillators, finite chain, whatever this is 24.9 JK-1mol-1 This is called the Dulong-Petit law.

Comparison of the Dulong-Petit law to experiment values for one mole of substance 77 K (JK-1) 273 K (JK-1) classical value 24.9 copper 12.5 24.3 aluminium 9.1 23.8 gold 19.1 25.2 lead 23.6 26.7 iron 8.1 24.8 sodium 20.4 27.6 silicon 5.8 21.8 for some materials it still works quite well at LN2 but not for all. Need to understand this. It really does not seem to play a role if things are metals or not. In fact, things go particularly bad for Si which is a semiconductor in fact, at low T it really goes to zero At high temperatures the Dulong-Petit law works quite well. At low temperatures, it does not. But we already know from basic principles that it wouldn’t. 7

Heat capacity of diamond and it is even worse for diamond which is an insulator. this does indeed go to zero at zero temperature it seems double log scale: T increases one order of magnitude, C increases three orders of magnitude maybe on blackboard: log(C)=log(const*T^3)=log(const)+log(T^3)=log(const)+3*log(T) So: any microscopic theory should at least give two things: DP at high T, zero at low T, possibly also T^3 law

The Einstein model for the heat capacity correct Dulong-Petit value for high temperatures zero heat capacity at zero temperature The Einstein frequency / temperature is an adjustable parameter. pretty good. Classical value reached about Theta_E this has been of major importance in the development of quantum mechanics. For the first time, shown that classical theory fails while quantum does not this is because for high T x=hbar*oe/kT is really small so x^2*(1+x+1/2x^2+1/n!x^n.....)/(x+1/2x^2+.....)^2 also at low T it looks pretty good but not entirely

The Einstein model: low-temperature heat capacity but low temperature behaviour of the experiment not correctly reproduced. T3 behaviour in experiment, exponential behaviour here The Einstein model: low-temperature heat capacity so it’s not quite what we want for very low T

Why does the Einstein model work at high T? Why does it fail at low T? At high T the small spacing between the energy levels is irrelevant. At sufficiently low temperature, the energy level separation is much bigger than kBT. Eventually all the oscillators are “frozen” in the ground state. Increasing T a little does not change this, i.e. it does not change the energy. but it is also clear why it fails: you can’t get the oscillators out of the ground state or only by Boltzmann, which is exponential. something smarter than Einstein model needed to solve this problem

The Debye model for the heat capacity total number of states for a maximum |k| so with this we have the density of states

The Debye model for the heat capacity substitute low temperature, large at low T, x is big. We can integrate to infinity, then we differentiate and get the desired T^3 law. for one mole

Comparison Debye model - experiment there is again an important temperature, the Debye temperature which corresponds to the Debye angular frequency. It is different from the Einstein frequency. It is the highest frequency which would be reached, if we were to continue the acoustic dispersion. But it also gives a feeling for the vibrational frequencies. why diamond?

Why does the Debye model work better at low T than the Einstein model? The Debye model gives a better representation for the very low energy vibrations. At low temperatures, these vibrations matter most.

Limits of the Debye model phonon density of states from copper. Not very realistic but good at low T