Abstract
One basic experimental method to probe condensed matter is to subject the system to small perturbing forces, using for example electromagnetic fields, and measure responses. In this chapter, we first develop a linear-response theory for analyzing the resulting data. We then use it to obtain theoretical formulas for ultrasonic attenuation and nuclear-spin relaxation in s-wave superconductors. It is thereby shown that changes in the excitation spectrum through the superconducting transition can be captured unambiguously by these experiments.
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- 1.
The density matrix (5.1) in equilibrium remains invariant under the change in definition of the ket and bra, because the additional phase factor \(\mathrm{e}^{-\mathrm{i}\mathcal{E}_{\nu }t/\hslash }\) of \(\vert \varPhi _{\nu }\rangle\) in the absence of \(\hat{\mathcal{H}}"\) is canceled by \(\mathrm{e}^{\mathrm{i}\mathcal{E}_{\nu }t/\hslash }\) of \(\langle \varPhi _{\nu }\vert\).
- 2.
Probabilities w ν are assumed to have no time dependence at all, which is justified when considering linear responses.
- 3.
We adopt a normalization for \(\hat{\mathcal{H}}_{\omega }\) different from (11.9) for the continuous spectrum.
- 4.
The energy of sound in the temperature scale is of the order of \(\varDelta T \equiv 2\pi \hslash f/k_{\mathrm{B}} \lesssim 2\pi \times 10^{-34} \times 10^{9}/10^{-23} \sim 0.1\) K, which is much smaller than T c in general. The corresponding wave number q is given in terms of the speed of sound s ∼ 103 m/s by \(q = 2\pi f/s \lesssim 2\pi \times 10^{9}/10^{3} \sim 10^{7}\) m−1, which is also much smaller than \(k_{\mathrm{F}} \sim a^{-1} \sim 10^{10}\) m−1 with \(a \sim 10^{-10}\) m denoting the lattice spacing of metals.
- 5.
- 6.
For electrons in solids, there appears another factor \(\langle \vert u_{\mathbf{k}}(\mathbf{0})\vert ^{2}\rangle _{\mathrm{F}}^{2}\) on the right-hand side of (11.57) [12], where \(\vert u_{\mathbf{k}}(\mathbf{0})\vert ^{2}\) is the relative density of electrons at the nuclear site with Bloch vector k, and \(\langle \cdots \,\rangle _{\mathrm{F}}\) denotes the Fermi-surface average.
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Kita, T. (2015). Responses to External Perturbations. In: Statistical Mechanics of Superconductivity. Graduate Texts in Physics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55405-9_11
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DOI: https://doi.org/10.1007/978-4-431-55405-9_11
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