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# 物理代写|固体物理代写solid physics代考|The lattice heat capacity

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## 物理代写|固体物理代写solid physics代考|Historical background

The energy content of a physical system is thermodynamically accounted for by its internal energy $\mathcal{U}$ (see appendix $\mathrm{C}$ ) whose derivative with respect to temperature
$$\mathcal{C}{V}=\left.\frac{d \mathcal{U}}{d T}\right|{V},$$
is known as the heat capacity at constant volume $V$ : it represents the amount of heat we need to quasi-statically provide in order to increase the system temperature by one degree.

The first attempts to derive a microscopic theory for $\mathcal{C}{V}$ in crystalline solids were developed at the dawn of the XXth century. In many respects, we can consider these investigations as the beginning of quantum solid state physics [1]. Developing a microscopic theory was certainly worthy of effort since the classical theory of $\mathcal{C}{V}$ is contradicted by the experimental evidence. In order to outline this theory, outdated but still valuable for our pedagogical approach to the thermal properties, we preliminarily remark that there are three main contributions to the heat capacity of a crystal, respectively, deriving from lattice vibrations, conduction electrons, and magnetic ordering. In non-magnetic insulators the first one is by far the leading one and in this chapter we focus just on it ${ }^{1}$.

## 物理代写|固体物理代写solid physics代考|The Debye model for the heat capacity

The Einstein model is correct in treating atomic vibrations as quantum oscillators, but it fails in attributing the same frequency to all of them: simply, this is inconsistent with the knowledge of the dispersion relations we developed in chapter 3. We must therefore introduce in the theory the fundamental notion that atomic oscillators can vibrate at different frequencies. Within the Debye model this notion is developed in a simplified way which allows us to carry on a clean analytical calculation of the heat capacity.

According to Debye, all phonon dispersion relations are effectively described by only three effective acoustic branches whose extension in wavevector, however, exceeds the boundary of the $1 \mathrm{BZ}$. This is shown in figure $4.2$ : the low and high $q$-values of the effective branch, respectively, describe an acoustic and an optical vibration of the real crystal. Furthermore, since for any direction there are in fact three possible phonon polarisations, the linearisation of their dispersions must properly take care to distinguish between one effective longitudinal and two effective transverse branches with slope $v_{\mathrm{g}}^{(L)}$ and $v_{\mathrm{g}}^{(T)}$, respectively (see section 3.2.1). To this aim it is useful to introduce the effective speed of sound $v_{\text {eff }}$ defined as
$$\frac{3}{v_{\mathrm{eff}}^{3}}=\frac{1}{\left[v_{\mathrm{g}}^{(L)}\right]^{3}}+\frac{2}{\left[v_{\mathrm{g}}^{(T)}\right]^{3}}$$

## 物理代写|固体物理代写SOLID PHYSICS代考|The general quantum theory for the heat capacity

Although the Debye expression for the lattice heat capacity is rather accurate over a wide range of temperatures for most materials, deviations from laboratory measurements are nevertheless found. The most effective way to develop the comparison is to fit experimental data taken at different temperatures by means of equation (4.10), while keeping $T_{\mathrm{D}}$ as the only calibration parameter for the fitting. For many systems this procedure returns a Debye temperature varying within few tens of Kelvin degrees: this is the fingerprint of some failure of the interpolation scheme, which is conceptually based on the existence of a unique Debye temperature. Good for us, these deviations are small for many practical applications and, therefore, the Debye model can be used as a very good approximation.

If, however, a high degree of accuracy is needed, then there is no better solution than using a full quantum theory where the lattice contribution to the internal energy $\mathcal{U}$ is calculated according to equation (3.36) so that
$$\mathcal{U}=\mathcal{U}{0}+\sum{s \mathbf{q}}\left[n_{\mathrm{BE}}(s \mathbf{q}, T)+1 / 2\right] \hbar \omega_{s}(\mathbf{q})$$
where $\mathcal{U}{0}$ is the total energy content of the static lattice. We accordingly calculate ${ }^{4}$ \begin{aligned} \mathcal{C}{V}^{\text {quantum }}(T) &=\frac{\partial}{\partial T} \sum_{s \mathbf{q}} \frac{\hbar \omega_{s}(\mathbf{q})}{\exp \left[\hbar \omega_{s}(\mathbf{q}) / k_{\mathrm{B}} T\right]-1} \ &=\sum_{s \mathbf{q}} \hbar \omega_{s}(\mathbf{q}) \frac{\partial n_{\mathrm{BE}}(s \mathbf{q}, T)}{\partial T}=\sum_{s \mathbf{q}} \mathcal{C}{V, s \mathbf{q}}(T) \end{aligned} where we used equation (3.37) for the phonon population $n{\mathrm{BE}}(s \mathbf{q}, T)$ and we introduced the specific contributions $\mathcal{C}_{V, s q}(T)$ of each $(s, \mathbf{q})$ mode to the heat capacity.

## 物理代写|固体物理代写SOLID PHYSICS代考|HISTORICAL BACKGROUND

$$\mathcal{C} {V}=\left.\frac{d \mathcal{U}}{d T}\right| {V}，$$

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