Semiclassical methods in solid state physics : two examples

Energy spectrum of tight-binding electrons on a two-dimensional (2D) square lattice under a magnetic field perpendicular to the plane has an exquisite structure, known as Hofstadter butterfly. Spectra for 2D lattices with a variety of symmetry have been calculated.3–6 Experimental verification of the full Hofstadter butterfly spectra is a formidable task, although such an attempt using semiconductor lateral superlattice seems to attain a certain degree of success. Superconducting wire network constitutes an experimental system intimately related to the Hofstadter problem. It has been demonstrated that the Little-Parks oscillation of transition temperature Tc of a superconducting wire network exhibits fine structures at commensurate values of average flux per unit cell, and that ∆Tc(H) reproduces the edge shape of the Hofstadter spectrum for the corresponding 2D lattice. These and the majority of subsequent studies were conducted in a uniform magnetic field. In the present study, we address ourselves to the case of spatially varying magnetic field, extending our previous work. In the next Section, we calculate the spectra for a square lattice subjected to both a spatially varying and a uniform magnetic field. Section 3 describes experiment using superconductor/ferromagnet hybrid systems. A brief summary is given in Section 4.

The energy spectrum of electrons on a square lattice in an applied magnetic field composes the famous Hofstadter butterfly with a recursive internal subband structure. An effective method for calculating the Green function for such a system is proposed. The standard approach requires an explicit knowledge of the eigenstates and eigenenergies of the system; here we derive a Harper‐like equation, that allows us to calculate the Green function for the lattice electrons in the field directly. The method is particularly useful in the weak‐field regime, where the standard calculations are cumbersome.

Nonlinear d-dimensional vector σ models, such as O(N), SU(N), and CPN, are considered in the limit of an infinite number of components N. It is shown that the equation for the two-point correlation function in these models is similar to the Schrodinger equation for a quantum particle moving in a δ-function potential well (−T)δ(x), where T is the temperature. This equation adequately describes the systems under study both above and below the Curie point. Within this approach, the critical behavior of the SU(N)-invariant Ginzburg-Landau model in an external uniform magnetic field is determined in the vicinity of the upper critical magnetic field. The critical indices in this case are the same as in the spherical model in a random magnetic field. An exact equation describing the Hc2(T) curve of continuous phase transitions is derived, which allows one to determine the asymptotes of this curve in strong and weak fields. The relation between the one-particle Schrodinger equation and critical phenomena is analyzed, and applications of this method to various models in solid state physics and statistical mechanics are discussed.

Corrigendum to "Intercomparison of dose enhancement ratio and secondary electron spectra for gold nanoparticles irradiated by X-rays calculated using multiple Monte Carlo simulation codes" [Phys. Med. 69 (2020) 147-163

Purpose: In a radiological examination, low-energy X-radiation is used (<100 keV). For other radiological procedures, the energy used is several MeV. ICRP in publication 103 has currently considered that photons irrespective of their energy have the same radiation weighting factor. Nevertheless, there are topological differences at the nanoscale of X-ray energy deposition as a function of its energy spectrum, meaning that the different interactions with living matter could vary in biological efficacy.Materials and methods: To study these differences, we characterized our irradiation conditions in terms of initial photon energies, but especially in terms of energy spectra of secondary electrons at the cell nucleus level, using Monte Carlo simulations. We evaluated signaling of DNA damage by monitoring a large number of γH2A.X foci after exposure of G0/G1-phase synchronized human primary endothelial cells from 0.25 to 5 Gy at 40 kV, 220 kV and 4 MV X-rays. Number and spatial distribution of γH2A.X foci were explored. In parallel, we investigated cell behavior through cell death and ability of a mother cell to produce two daughter cells. We also studied the missegregation rate after cell division.Results: We report a higher number of DNA double-strand breaks signaled by γH2A.X for 40 kVp and/or 220 kVp compared to 4 MVp for the highest tested doses of 2 and 5 Gy. We observed no difference between the biological endpoint studies with 40 kVp and 220 kVp X-ray spectra. This lack of difference could be explained by the relative similarity of the calculated energy spectra of secondary electrons at the cell monolayer.Conclusion: The energy spectrum of secondary electrons seems to be more closely related to the level of DNA damage measured by γH2A.X than the initial spectrum of photon energy or voltage settings. Our results indicate that as the energy spectrum of secondary electrons increases, the DNA damage signaled by γH2A.X decreases and this effect is observable beyond 220 kVp.

Scattering of light by an electron embedded simultaneously in a uniform magnetic field and in a microwave field is first analysed within the framework of classical electrodynamics. It is shown that the intensities of the different components of the scattered light can be increased by the presence of the magnetic field. Then coherent quantum states of the electron in the same field configuration are constructed and are applied to the description of the scattering process using semiclassical methods. The results of the classical and semiclassical calculations are compared and discussed and various limiting cases are considered to show the connection with previously published work.

The study of the behavior of Bloch electrons in an uniform external electric field is as old as the quantum theory of solids. Analysis of the motion of electrons in such external fields turned out, perhaps rather surprisingly, to be quit complicated and even at present the subject is very much alive. The source of the difficulties of this problem is that no matter how small the electron field strength is, for sufficiently large distances, the perturbed potential becomes arbitraly strong. In fact the perturbation created by electric field is singular from the spectral theoretic point of view. In consequence, strightforward application of the naive perturbation method is dangerous and rigorous results are hard to come by. For a better understanding of the difficulties, let us remined here the case of the Stark effect in atomic physics : although the Stark effects were the first example of quantum mechanical perturbation theory, it needed half a centry to develop a satisfactory mathematical description. Actually the existence of the Stark-Wannier resonance states, a quantum mechanical concept proposed by Wannier in solid state physics about 40 years ago, was strongly debated until recently.

We discuss some new developments in the theory of functional integration and some applications in nonrelativistic quantum theory, quantum field theory, statistical mechanics, solid state physics and hydrodynamics.

In this chapter, the problem under consideration is the existence of the thermo dynamic limit for the energy in Thomas-Fermi-van Weizsiicker type molecular models. We again consider a set of IAI nuclei of unit charges located on a set A of points of integer coordinates in R3. Denoting byh the energy of the neutral molecular system and by PA its electronic ground state, we investigate the behaviour of bothh and PA as the set A goes to the periodic lattice Z3. For the (quite substantial!) physical background of this mathematical problem, we refer the reader to Chapter 1 and to the references mentioned therein from solid state physics, thermodynamics, and statistical mechanics.

A method for deriving the electron and photon energy spectra from electronbeam central axis percentage depth dose (PDD) curves has been investigated. ThePDD curves of 6, 12 and 20 MeV electron beams obtained from the Monte Carlofull phase space simulations of the Varian linear accelerator treatment headhave been used to test the method. We have employed a `random creep' algorithmto determine the energy spectra of electrons and photons in a clinicalelectron beam. The fitted electron and photon energy spectra have beencompared with the corresponding spectra obtained from the Monte Carlo fullphase space simulations. Our fitted energy spectra are in good agreement withthe Monte Carlo simulated spectra in terms of peak location, peak width,amplitude and smoothness of the spectrum. In addition, the derived depth dosecurves of head-generated photons agree well in both shape and amplitude withthose calculated using the full phase space data. The central axis depth dosecurves and dose profiles at various depths have been compared using anautomated electron beam commissioning procedure. The comparison hasdemonstrated that our method is capable of deriving the energy spectra for theVarian accelerator electron beams investigated. We have implemented thismethod in the electron beam commissioning procedure for Monte Carlo electronbeam dose calculations.

Using translational properties of the Hamiltonian, a standard form is derived for the wave functions describing conduction electrons moving in a uniform magnetic field. This is used to obtain a solution of the Schrödinger equation in order to examine the magnetic energy levels. For non-overlapping bands the notion of effective mass is shown adequate (neglecting level broadening) but in an overlap region neighbouring bands interact strongly and distort the single band levels. A brief application to the diamagnetism of metals is considered.

Book reviews

This paper is devoted to problems coming from statistical mechanics. The transfer matrix (or transfer operator) approach consists in reducing the analysis of asymptotic properties of statistical systems to the analysis of the spectral properties of their transfer operator. Sometimes the new problem appears to have a semi-classical nature. Although the problem is similar to the semiclassical study of Kac's operator presented in our paper with M. Brunaud [BruHe] which was devoted to the study of \(\) for h small, new features appear for the model \(\). Our first results concern semi-classical analysis of the ground state for this second operator. We then analyze the two models in the large dimension situation. One basic technique is Sjostrand's formalism of the 0-standard functions as introduced in [Sj5]. The one-dimensional case was presented in [He11].

Semiclassical Analysis, Witten Laplacians, and Statistical Mechanics

Intensity enhancements of the ring current electrons associated with the VLF emissions during the geomagnetic storms and substorms, which have been observed by the equatorially orbiting S³‐A satellite (Explorer 45), are so far limited only to electrons of energies below the order of 10 keV. Furthermore, onsets of the enhancement are detected in the lowest observable energy channel first, then shifting toward higher energies with time, a feature, which is opposite to the well‐known energy dependence expected from the drift motion of electrons in the earth's dipole magnetic field. In order to explain these findings on the ring current electron enhancements, trajectories of electrons injected into the nightside of the magnetosphere from the geomagnetic tail are calculated by modifying Ejiri's calculations which were used for the interpretation of the nose events of the ring current protons observed also by the S³‐A satellite. The results indicate that the electron intensity enhancements are limited to electrons below the order of 10 keV simply because of the location of the observations which are initially confined to the dusk‐midnight sector outside the plasmasphere. If the observations are made in the morning sector, where many VLF emissions such as chorus are observed, the enhancement should not be limited only to electrons below the order of 10 keV but should extend to electrons up to the order of 100 keV. The apparent inverse dispersion, i.e., the appearance of the enhancement at lower energies first, is also due to the location of the observations. Namely, if the observations are made in the morning sector outside the plasmasphere, the normal dispersion with the first enhancement by high‐energy electrons should be seen. Since the adiabatic energy increase of charged particles during cross‐magnetic field inward motion in the dipole field is largest for the 90° pitch angle particles, the energy at the source is lowest for 90° pitch angle electrons for the same energy at the observing location. The smaller enhancement of the small pitch angle electron intensity can therefore be explained by the energy spectra of source electrons in the geomagnetic tail.