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# 物理代写|固体物理代写Solid Physics代考|PHY-558 Anharmonic effects

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## 物理代写|固体物理代写SOLID 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.

## 物理代写|固体物理代写SOLID PHYSICS代考|Thermal expansion

Thermal expansion is due to the dependence of vibrational frequencies on the crystal volume. To exploit this notion, we will make use of some fundamental thermodynamic definitions reported in appendix $\mathrm{C}$.

Our first goal is to work out an equation of state $P=P(V, T)$ relating the pressure $P$ acting on the system to its volume $V$ and temperature $T$. To this aim, we use the Helmholtz free energy $\mathcal{F}$, since we assume that our solid is coupled to a heat reservoir, that is $T=$ constant. We also understand that no matter is added to or removed from the system and, therefore, the numbers of moles of any chemical species are also constant. Under these assumptions, we can write (see equations (C.4) and (C.8))
$$d \mathcal{F}=d(\mathcal{U}-T S)=-P d V-S d T,$$
so that the equation of state for the pressure is cast in the form
$$P=-\left.\frac{\partial \mathcal{F}}{\partial V}\right|{T},$$ which is conveniently developed as follows \begin{aligned} P &=-\left.\frac{\partial(\mathcal{U}-T S)}{\partial V}\right|{T} \ &=-\frac{\partial}{\partial V}\left(\mathcal{U}-\left.T \int_{0}^{T} \frac{\partial S}{\partial T^{\prime}}\right|{V} d T^{\prime}\right){T} \ &=-\frac{\partial}{\partial V}\left(\mathcal{U}-\left.T \int_{0}^{T} \frac{1}{T^{\prime}} \frac{\partial \mathcal{U}}{\partial T^{\prime}}\right|{V} d T^{\prime}\right){T}, \end{aligned}

## 物理代写|固体物理代写SOLID PHYSICS代考|Thermal expansion

$$d \mathcal{F}=d(\mathcal{U}-T S)=-P d V-S d T$$

$$P=-\frac{\partial \mathcal{F}}{\partial V} \mid T$$

$$P=-\frac{\partial(\mathcal{U}-T S)}{\partial V} \mid T \quad=-\frac{\partial}{\partial V}\left(\mathcal{U}-T \int_{0}^{T} \frac{\partial S}{\partial T^{\prime}} \mid V d T^{\prime}\right) T=-\frac{\partial}{\partial V}\left(\mathcal{U}-T \int_{0}^{T} \frac{1}{T^{\prime}} \frac{\partial \mathcal{U}}{\partial T^{\prime}} \mid V d T^{\prime}\right) T$$

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