Free electron model

Summary

In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the Drude–Sommerfeld model. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially

the Wiedemann–Franz law which relates electrical conductivity and thermal conductivity;
the temperature dependence of the electron heat capacity;
the shape of the electronic density of states;
the range of binding energy values;
electrical conductivities;
the Seebeck coefficient of the thermoelectric effect;
thermal electron emission and field electron emission from bulk metals.

The free electron model solved many of the inconsistencies related to the Drude model and gave insight into several other properties of metals. The free ele

Official source

https://en.wikipedia.org/wiki/Free_electron_model

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Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.

EPFL2007

Ultrafast Transverse Modulation of Free Electrons by Interaction with Shaped Optical Fields

Francesco Barantani, Fabrizio Carbone, Simone Gargiulo, Ido Kaminer, Veronica Leccese, Ivan Madan, Alexey Sapozhnik, Phoebe Marie Tengdin, Giovanni Maria Vanacore

Spatiotemporal electron-beam shaping is a bold frontier of electron microscopy. Over the past decade, shaping methods evolved from static phase plates to low-speed electrostatic and magnetostatic displays. Recently, a swift change of paradigm utilizing light to control free electrons has emerged. Here, we experimentally demonstrate arbitrary transverse modulation of electron beams without complicated electron-optics elements or material nanostructures, but rather using shaped light beams. On-demand spatial modulation of electron wavepackets is obtained via inelastic interaction with transversely shaped ultrafast light fields controlled by an external spatial light modulator. We illustrate this method for the cases of Hermite-Gaussian and Laguerre-Gaussian modulation and discuss their use in enhancing microscope sensitivity. Our approach dramatically widens the range of patterns that can be imprinted on the electron profile and greatly facilitates tailored electron-beam shaping.

AMER CHEMICAL SOC2022

Conformal Field Theories (CFTs) are crucial for our understanding of Quantum Field Theory (QFT). Because of their powerful symmetry properties, they play the role of signposts in the space of QFTs. Any method that gives us information about their structure, and lets us compute their observables, is therefore of great interest. In this thesis we explore the large quantum number sector of CFTs, by describing a semiclassical expansion approach. The idea is to describe the theory in terms of fluctuations around a classical background, which corresponds to a superfluid state of finite charge density. We detail the implementation of the method in the case of U (1)-invariant lagrangian CFTs defined in the epsilon-expansion. After introducing the method for generic correlators, we illustrate it by performing the computation of several observables.First, we compute the scaling dimension of the lowest operator having a given large charge n under the U (1) symmetry. We demonstrate how the semiclassical result in this case bridges the gap between the naive diagrammatic computation (which fails at too large n) and the general large-charge expansion of CFTs (which is only valid for n large enough).Second, we apply the method to the computation of 3- and 4-point functions involving the same operator. This lets us derive some of the OPE (Operator Product Expansion) coefficients.Finally, we consider the rest of the spectrum of charge-n operators, and propose a way to classify them by studying their free-theory equivalent. In the free theory, we construct the complete set of primary operators with number of derivatives bounded by the charge.We also find a mapping between the excited states of the superfluid and the vacuum states of standard quantization, which is valid when the spin of said states is bounded by the square root of the charge.

EPFL2022