Lattice specific heat

We notice from Fig. 3.4(b) that the lattice specific heat cph is not modified by the superconducting transition cph = /3T3 with the same of the normal state. 

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). 

Fig. 17. SpeciEic heat versus temperature for UAs (solid curve) and isostructural non-magnetic ThAs (point-dashed curve). The dashed line represents the estimate lattice specific heat of UAs and the small dashed line a law of short range magnetic ordering effect. (Blaise et al. )...

Debye characteristic temperature. A parameter relating to the lattice specific heat of a solid. The temperature at which the specific heat of a simple specific cubic crystal equals 5.67 calories per degree per mole. 

We have already shown that at low temperatures the lattice specific heat is only observed (C v T3)(1). By adiabatic method on a polycrystalline sample we have looked for some anomaly around 150 K (Figure 4) a reversible anomaly as in KCP (5) and in TTF-TCNQ (6),is found which indicates the presence of a phase transition. 

The lattice specific heats of solids at low temperatures are generally proportional to T. With metals, an electronic term, proportional to T, must also be taken into consideration. For a first approximation, the low-temperature specific heats of normal metals follow the two-term function of (2). In statistical analyses of experimental data, however, improvements in fit have often been obtained by extending that function as a Taylor expansion in odd powers of T, For example, Martin et al [ ] fitted the specific heat of high-purity Cu in the temperature range 0.3 to 30 K to a polynomial of the form Generally, such a procedure... 

Lattice specific heat, according to the Debye theory, is given by Ciatuce.r = 3R DF(xt) J/mole-deg, where R = 8.314 J/mole-deg, and the Debye function at temperature T, given by... 

Low-Temperature Specific Heat. Although no liquid-helium-temperature data exist for boron, the specific heat has been measured between 13 and 305 K by Johnston et al [ ]. As a result, a Debye temperature of 1219 K has been assigned to the temperature range of 60 to 150 K [ ]. This value is assumed to be applicable at low temperatures and is used in synthesizing a value for the lattice specific heat coefficient ( 3 or jS) of the boron/aluminum composite, on the assumption that the mixture principle is applicable, viz. ... 

Sundstrom, L.J., 1968, A Theoretical investigation of the lattice specific heat of gadolinium, dysprosium and erbium metals (Ann. Acad. Sci. Fennicae A VI, No. 280). Taylor, K.N.R. and M.I. Darby, 1972, Physics of rare earth solids (Chapman and Hall, London). 

Meulen et al. 1988a,b, Junod 1996). This indicates that some conduction electrons do not pair at low temperatures, most likely due to an anisotropic or d-wave gap (i.e., with nodes). The lattice specific heat at temperatures not exceeding 10% of the Del e temperature can be approximated by The Debye temperature for R123x lies between 350 and 450 K and /S = 0.3-0.5 mJmole (van der Meulen et al. 1988a,b, Junod 1996). In most of... 

Assuming that the phonon DOS does not depend on temperature the lattice specific heat per atom at constant volume is defined as... 

The lattice specific heat at constant pressure is given by the expression... 

Fig. 26. Gruneisen analysis for UPtj. The upper plot shows the temperature-dependent Gruneisen parameter Q T) determined from C(T ) and ii T) using eq. (1). The equation S2 T) = S2 C, ICT) + Q C JCt) is then solved for C (T) and CpiiCT) under the assumption Q =73 and flp, =2.35. These are shown in the middle plot (Cf solid circles Cp dotted curve) and compared to the theoretical prediction for a Kondo doublet with 7 k = 16K (solid lines) and to the lattice-specific heat deduced from phonon dispersion curves (dashed line). The bottom plot shows the Kondo and phonon contributions to the thermal expansion deduced from the same analysis. Data are from Franse et al. (1989).

The theory of lattice specific heat was basically solved by Einstein, who introduced the idea of quantized oscillation of the atoms. He pointed out that, because of the quantization of energy, the law of equipartition must break down at low temperatures. Improvements have since been made on this model, but all still include the quantization of energy. Einstein treated the solid as a system of simple harmonic oscillators of the same frequency. He assumed each oscillator to be independent. This is not really the case, but the results, even with this assumption, were remarkably good. All the atoms are assumed to vibrate, owing to their thermal motions, with a frequency v, and according to the quantum theory each of the three degrees of freedom has an associated energy of which replaces the kT as postulated by... 

Example 3.3. Determine the lattice specific heat of chromium at 20 K as given by the Debye function. 

Solution. Equation (3.17) may be used to determine the electron heat capacity, while Eq. (3.10) may be used to calculate the lattice specific heat provided T/6 is less than 1/12. Table 3.3 gives a Debye temperature of 310 K for copper, for copper from Table 3.5 is 0.011 J/kgK. The molecular weight of copper is 63.55. 

Note that the ratio of the electronic specific heat to the lattice specific heat is proportional to 1/T. ... 

The specific heat is the lattice specific heat of the solid. Its variation with temperature is plotted on the same temperature scale in Fig. 3.16b. The mean velocity v of the phonons is the mean of the velocity of sound and varies only slightly with temperature as shown in Fig. 3.16c. The phonon gas differs from a real gas in that the number of particles varies with the temperature, increasing in number as the temperature is increased. At high temperatures, the large number of phonons leads to more collisions between phonons. Thus, as the temperature increases, X decreases, as shown in Fig. 3.16d. 

Determine the lattice specific heat of copper at liquid nitrogen temperatures. The experimental value of is 0.0473 cal/g K. 

Determine the lattice specific heat of aluminum at liquid hydrogen temperatures. Recall that if 7/0 >< 12, a simple function may be used. 

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