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Unconventional Superconductivity in Heavy Fermion and Ruthenate Materials
Published in David A. Cardwell, David C. Larbalestier, I. Braginski Aleksander, Handbook of Superconductivity, 2023
The point at which a magnetic ordering temperature (or other critical temperature) falls to 0 K is called a ‘quantum critical point’. This term was introduced by Hertz [44] to denote the fact that, unlike critical points at non-zero temperature, there is no regime in which classical statistical mechanics can be applied to the critical fluctuations of the order parameter, so the theory of second-order phase transitions, famously developed by Kadanoff, Fisher, Wilson, and others, must be recast using quantum statistical mechanics.
Modelling Techniques to Enhance Precision in Metal Additive Manufacturing
Published in Richard Leach, Simone Carmignato, Precision Metal Additive Manufacturing, 2020
Sankhya Mohanty, Jesper H. Hattel
The ‘length scale’ of a phenomenon refers to the particular length or distance at which the phenomenon becomes significant. The concept is especially important when understanding a complex process (such as metal AM) as a physical phenomenon occurring at different length scales can be decoupled, that is, it is possible to make a justifiable assumption that the phenomena do not directly affect each other. For example, heat transfer in large bodies (say a few centimetres in size) can be described adequately using Fourier’s law and the heat conduction equations, while the heat transfer between a proton and an electron requires a different set of physics involving quantum statistical mechanics, particle dynamics and electromagnetic theory (see Marla et al. 2018 for different models relevant to laser processing). Nonetheless, the physical laws relevant at the lower length scale can always be used to describe the phenomena at the larger length scales, albeit often requiring rigorous handling from a mathematical/algorithmic perspective to ensure a solution can be found in a reasonable time. Analogously, ‘time scale’ refers to the shortest period of time over which the phenomena can be observed, and the property of decoupling is also applicable to phenomena at different time scales. Using the previous example of heat transfer in a large body, Fourier’s law governing heat transfer is valid when considering time periods much larger than 10–12 s to 10–14 s, below which relativistic heat conduction equations become more applicable (Molina et al. 2014).
The Theory of Bardeen, Cooper, and Schrieffer
Published in R. D. Parks, Superconductivity, 2018
The Hamiltonian [Eq. (21)] determines the properties of the system in terms of the numbers bk0, but we have yet to determine these numbers. This is where the requirement of self-consistency enters, for the numbers bk0 are the thermal and quantum averages of the operators c_k↓ck↑ when the system is described by the model Hamiltonian. It is shown in textbooks on quantum statistical mechanics that such an average is given by bk0=Tr[exp(−βℋM)c−k↓ck↓]/Trexp(−βℋM)
Thermodynamic properties for some diatomic molecules with the q-deformed hyperbolic barrier potential
Published in Molecular Physics, 2023
Ahmed Diaf, Mohammed Hachama, Mohamed M'hamed Ezzine
Quantum statistical mechanics have attracted many researchers over the years, especially to improve the understanding and interpretation of thermodynamic properties of various systems [1–16]. Usually, this requires finding the energy spectrum for the interaction potential by solving the Schrödinger equation [17–20] or within the path integral formulation [21–23]. One key step in studying the thermodynamic properties of a given system is the determination of the vibrational partition function [24–26], which can be calculated by direct summation over all possible vibrational energy levels ; and allows to derive all thermodynamic quantities such as the vibrational entropy , vibrational mean energy U, vibrational specific heat C and the free energy F.