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# 物理代写|固体物理代写solid physics代考|Anharmonic effects

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## 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}
where we used the identity $T(\partial S / \partial T){V}=(\partial \mathcal{U} / \partial T){V}$. In order to proceed we need an explicit expression for the internal energy: by using equation (4.15) in the equation of state, after some non trivial algebra we obtain
\begin{aligned} P(V, T)=& \underbrace{-\left.\frac{\partial \mathcal{U}{0}}{\partial V}\right|{T}-\left.\frac{1}{2} \sum_{s \mathbf{q}} \frac{\partial \hbar \omega_{s}(\mathbf{q})}{\partial V}\right|{T}}{T=0 \text { contribution }} \ &+\underbrace{\sum_{s \mathbf{q}}\left[-\left.\frac{\partial \hbar \omega_{s}(\mathbf{q})}{\partial V}\right|{T} n{\mathrm{BE}}(s \mathbf{q}, T),\right.}_{T>0 \text { contribution }} \end{aligned}
indicating that the pressure depends on $V$ and $T$ through the variations of the phonon frequencies upon volume changes and through the Bose-Einstein phonon population, respectively.

## 物理代写|固体物理代写solid physics代考|Phonon–phonon interactions

The Grüneisen model has phenomenologically treated anharmonic effects in lattice dynamics through volume-dependent vibrational frequencies. The most fundamental description of anharmonicity is, however, based on quantum theory and exploits the phonon language: while in the harmonic approximation phonons are described as a gas of free pseudo-particles, in a most realistic anharmonic crystal they actually undergo mutual interactions.

Phonon-phonon interactions are not so strong to fully invalidate the harmonic picture: this is proved by the true existence of well-resolved peaks in neutron scattering spectra (see section 3.6), each peak being the fingerprint of a specific harmonic phonon mode. Therefore, anharmonicity can be treated as a perturbation on the quantum states of the harmonic crystal: while its energy spectrum remains basically unaffected by phonon-phonon interactions (that is, we can still speak about phonon frequencies and vibrational modes with different character $s$ and wavevector $\mathbf{q}$ ), anharmonicity causes transitions between different states of quantum harmonic oscillator.

The formal treatment of such a perturbation is non trivial and falls beyond the present level of discussion $[2,12,13]$, but we can assimilate the underlying physical concept by means of an analogy with atomic physics: the energy spectrum of, say, an isolated hydrogen atom remains unaffected by a low-intensity electromagnetic field ${ }^{9}$, whose perturbative effect is only to promote electronic transitions between the discrete stationary-state levels of the atom [14]. We can say that, for both absorption or emission transitions, the occupation of the initial and final stationary state has been varied by $-1$ and $+1$, respectively, while a photon has been annihilated (absorption) or created (emission). This is a three-particle event involving twoelectron and one-photon populations.

Similarly, any anharmonic term of the vibrational Hamiltonian appearing in equation (3.1) causes transitions among harmonic eigenstates, correspondingly affecting their phonon populations. The physical picture is simple: we can say that the $n$th order term (with $n \geqslant 3$ ) in the Taylor expansion of the lattice potential energy activates interactions among $n$ phonons, which we will refer to as $n$-phonon scattering events. Since the phonon number is not conserved, during a scattering event phonons of some harmonic mode are annihilated (their population is decreased), while other phonons of different modes are created (their population is increased).

## 物理代写|固体物理代写SOLID PHYSICS代考|THERMAL EXPANSION

dF=d(在−吨小号)=−磷d在−小号d吨,

$$d \mathcal{F}=d(\mathcal{U}-T S)=-P d V-S d T,$$
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}
where we used the identity $T(\partial S / \partial T){V}=(\partial \mathcal{U} / \partial T){V}$. In order to proceed we need an explicit expression for the internal energy: by using equation (4.15) in the equation of state, after some non trivial algebra we obtain
\begin{aligned} P(V, T)=& \underbrace{-\left.\frac{\partial \mathcal{U}{0}}{\partial V}\right|{T}-\left.\frac{1}{2} \sum_{s \mathbf{q}} \frac{\partial \hbar \omega_{s}(\mathbf{q})}{\partial V}\right|{T}}{T=0 \text { contribution }} \ &+\underbrace{\sum_{s \mathbf{q}}\left[-\left.\frac{\partial \hbar \omega_{s}(\mathbf{q})}{\partial V}\right|{T} n{\mathrm{BE}}(s \mathbf{q}, T),\right.}_{T>0 \text { contribution }} \end{aligned}

## 物理代写|固体物理代写SOLID PHYSICS代考|PHONON–PHONON INTERACTIONS

Grüneisen 模型通过与体积相关的振动频率对晶格动力学中的非谐效应进行了现象学处理。然而，非谐性的最基本描述是基于量子理论并利用声子语言：虽然在谐波近似中，声子被描述为自由伪粒子的气体，但在最现实的非谐晶体中，它们实际上经历了相互作用。

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