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物理代写|固体物理代写Solid Physics代考|PHYS440 Quantum theory of harmonic crystals

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物理代写|固体物理代写SOLID PHYSICS代考|Quantum theory of harmonic crystals

Moving to a quantum description, as simple as it may appear, represents a major conceptual step forward in our search for a truly fundamental description of lattice dynamics. To appreciate its relevance, we anticipate a result more extensively discussed in the next chapter. The specific heat of a crystal described as an assembly of classical harmonic oscillators is calculated to be independent of temperature (Dulong-Petit law). Contrary to this prediction, experimental measurements provide evidence that the specific heat becomes vanishingly small as $T \rightarrow 0$, thus proving that it is in fact temperature-dependent. Only a full quantum treatment is able to reconcile theoretical predictions to measurements.

Based on the theory developed in the previous section, we will agree to describe each classical (sq) vibrational mode as a quantum one-dimensional harmonic oscillator [1-3] whose energy is restricted to the values $\left(n_{s \mathbf{q}}+1 / 2\right) \hbar \omega_{s}(\mathbf{q})$ where $n_{s \mathbf{q}}=0,1,2, \ldots$ is the vibrational quantum number and $\omega_{s}(\mathbf{q})$ is obtained by diagonalising the dynamical matrix. Since the vibrational energy levels are equally spaced, we can look at the state with energy $\left(n_{s \mathbf{q}}+1 / 2\right) \hbar \omega_{s}(\mathbf{q})$ as a single $n_{s q}$ th excited state or, equivalently, as the state obtained by adding $n_{s q}$ identical energy quanta $\hbar \omega_{s}(\mathbf{q})$. We will adopt this second approach since it is especially effective in describing the dynamical and thermal characteristics of a crystal lattice through the properties of $a$ gas of pseudo-particles, hereafter named phonons. This choice introduces a corpuscular description of lattice dynamics, where phonons are the energy quanta of the ionic displacement field .

物理代写|固体物理代写SOLID PHYSICS代考|Experimental measurement of phonon dispersion relations

The experimental determination of the $\omega=\omega_{s}(\mathbf{q})$ dispersion relations over the entire $1 \mathrm{BZ}$ needs a probe fulfilling two conditions: (i) its wavelength must be comparable with the typical interatomic distances in the crystal structure and (ii) its energy must be of the same order of typical phonon quanta $\hbar \omega_{s}(\mathbf{q})$, which range mostly in the interval $\left[1,10^{2}\right] \mathrm{meV}$. Optical probes are unsuitable: $x$-rays have the right wavelength, but a too high energy of the order $\mathcal{O}\left(10^{4} \mathrm{eV}\right)$; other kinds of photons, instead, can only explore the $\mathbf{q} \sim 0$ region of the Brillouin zone, i.e. they can only detect (some) zone-centre phonons. A probe consisting in a flux of electrons is also impractical for a twofold reason: (i) their surface scattering is very strong and, therefore, they are unable to probe the bulk region of the crystal; (ii) multiple scattering is likely to occur in the case of electrons and this makes the analysis of the experiment a very challenging task. In contrast, both requirements of suitable wavelength and energy are guaranteed by a flux of thermal neutrons which have typical wavelengths of the order of just a few $\AA$ and energies in between a few and a few tens of meV. Accordingly, neutron spectroscopy is the most powerful technique for measuring the phonon dispersion relations $[16,17]$.

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