Phonon Heat Capacity

The phonon heat capacity, Q, is defined as the derivative of the internal energy, Up stored within the vibration lattice. 

An expression for the phonon density of states is required to calculate the lattice heat capacity, Q. Two common models for the calculation of density of phonon states are (a) Debye model and (b) Einstein model. 

---

If all the energy Ej - Eo is initially input to the phonons only, then the initial phonon quasitemperature (0) = Ej - Eo)IC, ph will be much greater than the final temperature T/, because the phonon heat capacity Qp/, in Eq. (9) is much smaller than the total heat capacity of phonons plus vibrations, C T) [50]. For instance with a shock that heats naphthalene by 400K, the initial phonon quasitemperature dph(0) is nearly 2400K [51]. 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

In typical metals, both electrons and phonons contribute to the heat capacity at constant volume. The temperaPire-dependent expression... 

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

The thermal conductivity plateau has traditionally been considered by most workers as a separate issue from the TLS. In addition to the rapidly growing magnitude of phonon scattering at the plateau, an excess of density of states is observed in the form of the so-called bump in the heat capacity temperature dependence divided by T. The plateau is interesting from several perspectives. For one thing, it is nonuniversal if scaled by the elastic constants (say, co/)... 

Figure 11. Displayed are the TLS heat capacities as computed from Eq. (29) appropriate to the experiment time scales on the order of a few microseconds, seconds, and hours. A value of c = 0.1 was used here. If one makes an assumption on the specific value of Aq, it is possible to superimpose the Debye contribution on this graph, which would serve as the lowest bound on the total heat capacity. As checked for Aq = cod. the phonon contribution is negligible at these temperatures.

Section V, other quantum effects are indeed present in the theory and we will discuss how these contribute both to the deviation of the conductivity from the law and to the way the heat capacity differs from the strict linear dependence, both contributions being in the direction observed in experiment. Finally, when there is significant time dependence of cy, the kinematics of the thermal conductivity experiments are more complex and in need of attention. When the time-dependent effects are included, both phonons and two-level systems should ideally be treated by coupled kinetic equations. Such kinetic analysis, in the context of the time-dependent heat capacity, has been conducted before by other workers [102]. 

When a model for a CUORICINO detector (see Section 15.3.2) was formulated and the pulses simulated by the model were compared with those detected by the front-end electronics, it was evident that a large difference of about a factor 3 in the pulse rise time existed. This discrepancy was mainly attributed to the uncertainty in the values of carrier-phonon decoupling parameter. For the thermistor heat capacity, a linear dependence on temperature was assumed down to the lowest temperatures. As we shall see, this assumption was wrong. 

It can be observed that these thermal conductances G(7) are typical of phonon conduction between two solids at very low temperature, as already reported [45], The value of the heat capacity was calculated from equation C = r G, where the thermal time constant r is obtained from the fit to the exponential relaxation of the wafer temperature. 

Since in our temperature range, the Debye temperature of Ge is 370K [47], the phonon contribution to the heat capacity can be neglected. Hence, the heat capacity of our samples is expected to follow the equation ... 

Low-temperature thermometers are obtained by cutting a metallized wafer of NTD Ge into chips of small size (typically few mm3) and bonding the electrical contacts onto the metallized sides of the chip. These chips are seldom used as calibrated thermometers for temperatures below 30 mK, but find precious application as sensors for low-temperature bolometers [42,56], When the chips are used as thermometers, i.e. in quasi-steady applications, their heat capacity does not represent a problem. In dynamic applications and at very low temperatures T < 30 mK, the chip heat capacity, together with the carrier-to-phonon thermal conductance gc d, (Section 15.2.1.3), determines the rise time of the pulses of the bolometer. 

The HEM is a thermal model which represents a doped semiconductor thermistor (e.g. Ge NTD) as made up of two subsystems carriers (electrons or holes) and phonons. Each subsystem has its own heat capacity and is thermally linked to the other one through a thermal conductance which takes into account for the electron-phonon decoupling (see Fig. 15.2). 

Figure 15.8 shows the thermal scheme of one detector there are six lumped elements with three thermal nodes at Tu T2, r3, i.e. the temperatures of the electrons of Ge sensor, Te02 absorber and PTFE crystal supports respectively. C), C2 and C3 are the heat capacity of absorber, PTFE and NTD Ge sensor respectively. The resistors Rx and R2 take into account the contact resistances at the surfaces of PTFE supports and R3 represents the series contribution of contact and the electron-phonon decoupling resistances in the Ge thermistor (see Section 15.2.1.3). 

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

The original specific heat experiments on BaPb xB Og by Methfessel et al (60) immediately raised the prospect that an unusual mechanism was operative in this newly found system. Their finding of no heat capacity anomaly at Tc could actually have a number of possible interpretations, including an impurity phase giving rise to superconductivity, a non-phonon mechanism, or some new form of conductivity. 

This competition between electrons and the heat carriers in the lattice (phonons) is the key factor in determining not only whether a material is a good heat conductor or not, but also the temperature dependence of thermal conductivity. In fact, Eq. (4.40) can be written for either thermal conduction via electrons, k, or thermal conduction via phonons, kp, where the mean free path corresponds to either electrons or phonons, respectively. For pure metals, kg/kp 30, so that electronic conduction dominates. This is because the mean free path for electrons is 10 to 100 times higher than that of phonons, which more than compensates for the fact that C <, is only 10% of the total heat capacity at normal temperatures. In disordered metallic mixtures, such as alloys, the disorder limits the mean free path of both the electrons and the phonons, such that the two modes of thermal conductivity are more similar, and kg/kp 3. Similarly, in semiconductors, the density of free electrons is so low that heat transport by phonon conduction dominates. 

The discussion of the previous section would also lead us to believe that since most ceramics are poor electrical conductors (with a few notable exceptions) due to a lack of free electrons, electronic conduction would be negligible compared to lattice, or phonon, conduction. This is indeed the case, and we will see that structural effects such as complexity, defects, and impurity atoms have a profound effect on thermal conductivity due to phonon mean free path, even if heat capacity is relatively unchanged. 

As with other disordered materials, the thermal condnctivities of polymers are low due to phonon scattering. As a resnlt, even thongh polymers tend to have heat capacities of the same order of magnitude as metals (1.5 to 3.5 J/g K), their thermal conductivities (0.1 to 1.0 W/m K) are 1000 times lower than metals. Polymers, therefore, are generally good insulators, as long as their use temperature is below their thermal stability temperature. Few correlations for heat capacity and thermal conductivity of polymers... 

---