Contributions to the specific heat

From formula (3.21), we see that the temperature at which the maximum of the magnetic contribution to the specific heat occurs is determined by the energy splitting AE of the levels ... 

An example of magnetic contributions to the specific heat is reported in Fig. 3.9 that shows the specific heat of FeCl24H20, drawn from data of ref. [35,36]. Here the Schottky anomaly, having its maximum at 3K, could be clearly resolved from the lattice specific heat as well as from the sharp peak at 1K, which is due to a transition to antiferromagnetic order (lambda peak). 

As an example, in Fig. 3.11, a schematic two-dimension representation of the structure of cristobalite (a crystalline form of Si02) and of vitreous Si02 is shown. A, B and C represent three cases of double possible equilibrium positions for the atoms of the material in the amorphous state [41]. Atoms can tunnel from one position to another. The thermal excitation of TLS is responsible for the linear contribution to the specific heat of amorphous solids. 

The overall specific heat of a polymer is given by a combination of the various contributions to the specific heat of longitudinal and transversal phonons. At temperatures below 1K, the linear contribution due to the TLS must be added. 

General expression for the fluctuation contribution to the specific heat is given by the first line (15) and can be resolved with the help of the self-consistent solution of (11), (19) (or (20), (21), (22) in the limiting cases). Assuming for rough estimates that fluctuations can be described pcrturbativcly and putting m2 — r/ vip u, from the second line of (15) we find for T <. 

Also in the CFL phase (T < 7).) there is a contribution to the specific heat of Goldstone-like excitations, cf. phonons in the ordinary condensed matter. For T nip.a, where mp.a is the mass of the pseudo-GoIdstone excitation, we get... 

Comparing the mean field (12) and the fluctuation (15) contributions to the specific heat (in the low and high temperature limiting cases one may use Eqs. (22), (24)) we may estimate the fluctuation temperature < Tc, at which the contribution of fluctuations of the order parameter becomes to be as important as the mean field one (so called Ginzburg - Levanyuk criterion),... 

Note that the full curve drawn in the figure corresponds to the specific heat calculated with no free parameters. While the agreement of this model to the specific heat is rather impressive, it is in fact the 3 inter-cluster degrees of freedom, i.e. motion of the whole cluster, which govern the phonon contribution to the specific heat at temperatures below about 15-20 K [99]. 

It should be pointed out here that the measured specific heat [25, 54,55] of AU55 showed no trace of a linear term which would normally indicate the presence of an electronic contribution to the specific heat, at least down to 60 mK. We will return to this point in Sect. 4.5. 

Once we have obtained the dispersion curves for the metal, wc may proceed to other properties just as we did with the covalent solids. In particular, we may quantize the vibrations as was done for the covalent solids and obtain the appropriate contribution to the specific heat. We shall not repeat that analysis now for the simple metals but shall wish to use the customary terminology by referring to the vibrations as phonons. 

Specific heat. In the Fermi liquid theory, the expression for the electronic contribution to the specific heat is linear in temperature CXT) = yT, where the Sommerfeld constant is... 

At high temperatures (i.e. large Tj ) this function tends to unity, and the contribution to the specific heat from the vibration tends to R per mole. At low temperatures the function tends to zero. 

We may, therefore, write the vibrational contribution to the specific heat as... 

The specific heat (C) is the amount of energy required, per unit mass or per mole, to raise the temperature of a substance by one degree. This is the derivative of its internal energy dU/dT, and since magnetic levels make a contribution to this their separations can in principle be measured from C(T) measurements. However, the magnetic contribution to the specific heat must be disentangled from that of lattice vibrational modes. 

It is quite evident at this stage, that although the interfacial term plays the fundamental role, it gives a constant contribution to the free energy per cation and may be dropped in thermodynamical calculations (with the restriction that Yi could depend, for instance, on T and give rise to a contribution to the specific heat). 

Thus, we can see that in (14.6.64) the dominant term with respect to S is the term proportional to z2h 3,il oc S[ln(S/s0)]3/11. This is just the term which gives the dominant contribution to the specific heat. 

The paramagnon contribution to the specific heat has been evaluated by Doniach and Engelsberg (1966) and Brinkman and Engelsberg (1968). It yields an enhancement of the linear electronic term, which can be conceived as an enhancement of the effective mass of the electrons. It can be written as... 

Fig. 3.23. (a) Temperature dependence of the electronic contribution to the specific heat of USn3 plotted as Ct. / T versus T2. The solid curve is a guide for the eye (Norman et al. 1986). (b) Temperature dependence of the electronic contribution to the specific heat of NpSn3 plotted as Cc/T versus T2. The solid curve is the mean-field prediction (Trainor et al. 1976b). 

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