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## 物理代写|固体物理代写Solid-state 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{RF}}(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 $\mathcal{C}{V, s \mathbf{q}}(T)$ of each $(s, \mathbf{q})$ mode to the heat capacity. This expression is more easily handled in the limit of a very large crystal: the density of allowed $\mathbf{q}$ wavevectors in the reciprocal space is so high that we can treat them as a continuum $$\mathcal{C}_V^{\text {quantum }}(T)=\frac{V}{(2 \pi)^3} \frac{\partial}{\partial T} \sum_s \int \frac{\hbar \omega_s(\mathbf{q})}{\exp \left[\hbar \omega_s(\mathbf{q}) / k{\mathrm{B}} T\right]-1} d \mathbf{q},$$
where the integral is taken for $\mathbf{q} \in 1 \mathrm{BZ}$.

## 物理代写|固体物理代写Solid-state physics代考|Anharmonic effects

The crystal lattice dynamics has been so far described under the harmonic approximation which allowed us to understand many fundamental intrinsic properties of solids. It is, however, just an approximation, as emerged from the discussion developed in section $3.1$ where it has been presented as a convenient truncation of a Taylor expansion of the total ionic potential energy $U=U(\mathbf{R})$ (see equation (3.1) and relative discussion). Beyond this formal argument, robust experimental evidences suggest that a real system is in fact not purely harmonic; they are mostly related to thermal properties like:

• if $k_{\mathrm{B}} T / \hbar$ is much larger than typical phonon frequencies, deviations of the predicted heat capacity from the experimental data are actually observed: they are the onset of anharmonic effects, not yet explicitly included in the theory leading to equation (4.17);
• a crystalline solid differently resists to positive or negative strains of identical magnitude; since any volume variation reflects a change in the lattice spacing, this suggests that ions are confined nearby their equilibrium positions by a non-parabolic (that is, non harmonic) potential;
• real solids undergo thermal expansion; this would not be possible if the ions thermally oscillate under the action of a perfectly parabolic potential since the average ion-ion distance would not increase upon temperature;
• finally, a beam of phonons travelling along a given direction within an infinite defect-free crystal would propagate with no damping if anharmonic effects were not included (harmonic vibrational modes overlap without interference); this would imply an infinite lattice thermal conductivity.

In the following we are going to treat separately thermal expansion and thermal conduction in the next subsections.

# 固体物理代写

## 物理代写|固体物理代写Solid-state physics代考|The general quantum theory for the heat capacity

(4.10) 拟合在不同温度下取得的实验数据，同时保持 $T_{\mathrm{D}}$ 作为配件的唯一校准参数。对于许多系统，此过程返回在几十开尔文度内变化的德拜温度：这是揷值方 案的某些失败的指纹，它在概念上基于唯一德拜温度的存在。对我们有好处，这些偏差对于许多实际应用来说很小，因此，德拜模型可以用作非常好的近似值。

$$\mathcal{U}=\mathcal{U} 0+\sum s \mathbf{q}\left[n_{\mathrm{BE}}(s \mathbf{q}, T)+1 / 2\right] \hbar \omega_s(\mathbf{q}),$$

$$\mathcal{C}V^{\text {quantum }}(T)=\frac{\partial}{\partial T} \sum s \mathbf{q} \frac{\hbar \omega_s(\mathbf{q})}{\exp \left[\hbar \omega_s(\mathbf{q}) / k{\mathrm{B}} T\right]-1} \quad=\sum_{s \mathbf{q}} \hbar \omega_s(\mathbf{q}) \frac{\partial n_{\mathrm{RF}}(s \mathbf{q}, T)}{\partial T}=\sum_{s \mathbf{q}} \mathcal{C} V, s \mathbf{q}(T),$$

$$\mathcal{C}_V^{\text {quantum }}(T)=\frac{V}{(2 \pi)^3} \frac{\partial}{\partial T} \sum_s \int \frac{\hbar \omega_s(\mathbf{q})}{\exp \left[\hbar \omega_s(\mathbf{q}) / k \mathrm{~B} T\right]-1} d \mathbf{q},$$

## 物理代写|固体物理代写Solid-state physics代考|Anharmonic effects

• 如果ķ乙吨/ℏ远大于典型的声子频率，实际观察到预测的热容量与实验数据的偏差：它们是非谐波效应的开始，尚未明确包含在导致方程（4.17）的理论中；
• 结晶固体以不同的方式抵抗相同大小的正应变或负应变；由于任何体积变化都反映了晶格间距的变化，这表明离子被非抛物线（即非谐波）势限制在其平衡位置附近；
• 真实的固体经历热膨胀；如果离子在完美抛物线势的作用下发生热振荡，这是不可能的，因为平均离子-离子距离不会随温度增加；
• 最后，如果不包括非谐波效应（谐波振动模式重叠而无干扰），则在无限无缺陷晶体内沿给定方向行进的声子束将无阻尼地传播；这意味着无限的晶格热导率。

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## MATLAB代写

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