By A M Guénault

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Thus S is obtained from a knowledge of the εj s (which depend on V ), of N and of T (as it enters the Boltzmann distribution). Method 2: Use the definition of Z There is a direct shortcut from the partition function to U . This is particularly useful if only U and perhaps dU /dT (= CV , the heat capacity at constant volume) are wanted. 27) Note that it is usually convenient to retain β as the variable here rather than to use T . Method 3: The ‘royal route’ This one never fails, so it is worth knowing!

However, the energy level spacing has reduced (from 2μB1 to 2μB2 ) and so the temperature must have also decreased in the same ratio, in order to keep μB/kkB T constant. 7) This is a nice example of the importance of entropy. In the first leg we reduce the entropy of the solid at a high temperature by an amount S, say, by extracting a quantity T1 S of heat. The adiabatic leg then transfers the entropy reduction to a lower temperature, and the spins can absorb heat of at least T2 S from their surroundings before warming up again to the starting point.

The normal modes are then travelling waves of the form ψ ∼ exp(ikkx x + ikky y + ikkz z) or ψ ∼ exp(ik·r) The values of k differ from the earlier ones in two respects, and indeed the microscopic picture is quite different in detail. The first difference is that the spacing of the k-values is doubled, since now the boundary condition requires fitting an integral number of full (and not half) waves into the box. 3) is no longer valid – another factor of ( 18 ) is needed on the right-hand side. The second difference is that the restriction on k to the positive octant is lifted.

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