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## 物理代写|固体物理代写Solid-state physics代考|The lattice heat capacity

The energy content of a physical system is thermodynamically accounted for by its internal energy $\mathcal{U}$ (see appendix $C$ ) 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 In order to outline this theory, outdated but still valuable for our pedagogical approach to the thermal properties, we preliminarily remark that there are three main contributions to the heat capacity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 ${ }^1$.

Classically the internal energy $\mathcal{U}$ of a crystal containing $N$ atoms corresponds to the vibrational energy of $3 N$ one-dimensional harmonic oscillators, as calculated by means of the equipartition theorem: if the crystal is in equilibrium at temperature $T$, an average energy $k_{\mathrm{B}} T / 2$ is attributed to each energy contribution which is quadratic either in general coordinates or momenta. Therefore, the average energy of each atomic oscillator is estimated to be $\langle u(T)\rangle=k_{\mathrm{B}} T$ so that
$$\mathcal{U}=3 N\langle u(T)\rangle=3 N k_{\mathrm{B}} T=3 R T,$$
where we have hereafter assumed that $N=\mathcal{N}_A$ (i.e. we have an Avogadro number of atoms in the crystal), while $R=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ is the universal gas constant. The corresponding classical prediction for the heat capacity $\mathcal{C}_V=3 R$ is known as the Dulong-Petit law. Contrary to this, experimental measurements provide evidence that $\mathcal{C}_V \rightarrow 0$ for $T \rightarrow 0$. More specifically, it is found that $\mathcal{C}_V \sim T^3$ in the range of vanishingly small temperatures. The measured $\mathcal{C}_V$ approaches the predicted value only at very high temperature. In conclusion, the classical theory is unable to justify the observed $\mathcal{C}_V=\mathcal{C}_V(T)$ trend over the full range of temperatures.

## 物理代写|固体物理代写Solid-state 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
$$\frac{3}{v_{\text {eff }}^3}=\frac{1}{\left[v_{\mathrm{g}}^{(L)}\right]^3}+\frac{2}{\left[v_{\mathrm{g}}^{(T)}\right]^3} .$$
We can now calculate the density of vibrational states in the Debye model $G_{\mathrm{D}}(\omega)$ by making use of equation (3.42) elaborated for a single branch so that
$$G_{\mathrm{D}}(\omega)=3 \frac{V}{2 \pi^2} q^2 \frac{1}{d \omega / d q}=3 \frac{V}{2 \pi^2} \frac{\omega^2}{v_{\text {eff }}^3}=\frac{V}{2 \pi^2}\left{\frac{1}{\left[v_{\mathrm{g}}^{(L)}\right]^3}+\frac{2}{\left[v_{\mathrm{g}}^{(T)}\right]^3}\right} \omega^2,$$
where we set $\omega=v_{\text {eff }} q$ consistently with the linearisation procedure; the factor 3 takes into account the three possible polarisations. Interesting enough, we find $a$ quadratic dependence of the VDOS upon the frequency: this feature is indeed found in real materials in the acoustic region of the vibrational spectrum, that is, exactly where the phonon dispersion relations are linear as supposed in the Debye model (see section 3.7).

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|The lattice heat capacity

$$\mathcal{C}V=\left.\frac{d \mathcal{U}}{d T}\right|_V,$$ 称为定容热容 $V$ ：它表示我们需要准静态提供的热量，以便将系统温度提高一度。 第一次尝试为 $\mathcal{C}_V$ 在 20 世纪初开发了结晶固体。在许多方面，我们可以将这些研究视为量子固态物理学的开端[1]。发展一个微观理论当然是值得努力的，因为经典 的理论 $\mathcal{C}_V$ 与实验证据相矛盾。为了概述这个已经过时但对我们对热性质的教学方法仍然有价值的理论，我们初步指出，对晶体的热容量有三个主要贡献，分别来 自晶格振动、传导电子和磁排序. 在非磁性绝缘体中，第一个是迄今为止领先的，在本章中我们只关注它 1 . 经典的内能 $\mathcal{U}$ 含有晶体的 $N$ 原子对应的振动能量 $3 N$ 通过均分定理计算的一维谐振子: 如果晶体在温度下处于平衡状态 $T$ ，平均能量 $k{\mathrm{B}} T / 2$ 归因于在一般坐标或动量 中是二次的每个能量贡献。因此，每个原子振荡器的平均能量估计为 $\langle u(T)\rangle=k_{\mathrm{B}} T$ 以便
$$\mathcal{U}=3 N\langle u(T)\rangle=3 N k_{\mathrm{B}} T=3 R T,$$

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

$$\frac{3}{v_{\mathrm{eff}}^3}=\frac{1}{\left[v_{\mathrm{g}}^{(L)}\right]^3}+\frac{2}{\left[v_{\mathrm{g}}^{(T)}\right]^3}$$

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