Electron Momentum Distribution Function Research Articles

Fast nonlinear scattering of runaway electron beams through resonant interactions with plasma waves

Abstract The resonant interaction between a runaway electron (RE) beam and a reactor-grade background plasma is investigated through two-dimensional particle-in-cell simulations, employing a simplified model of the system. The temporal evolutions of the electron momentum distribution function for two separate initial beam energies (1 and 10 MeV) are tracked, revealing the occurrence of plasma wave growth concomitant with pitch angle scattering or momentum distribution diffusion within the RE beam. Notably, we identify and confirm the dependence of the dominant resonance condition on the initial kinetic energy of the RE beam. Furthermore, we quantify the effect of particle-wave interactions on the RE momentum distribution diffusion by assessing the average kinetic energy flux from the RE distribution function.

Momentum-space-resolved measurements using oblique electron cyclotron emission for the validation of the quasi-linear theory of electron cyclotron current drive

Electron cyclotron resonance heating (ECRH) can drive large current densities through electron cyclotron current drive (ECCD). ECCD is expected to be crucial for high-performance plasmas in future fusion reactors like ITER and DEMO, making the current drive efficiency of ECCD a critical design parameter for future reactors. In present-day devices, good agreement between measured and predicted current drive efficiency has been found. However, to ensure the reliability in future machines, a direct validation of the electron momentum distribution function is needed.As a first step towards this goal, we present in this paper oblique electron cyclotron emission (ECE) measurements of a low-density plasma in the ASDEX Upgrade tokamak. Two oblique ECE diagnostics are used to allow the simultaneous measurements of electrons streaming co- and counter-directionally with the plasma current. Predictions for the distribution function are computed with the bounce-averaged Fokker-Planck code RELAX. 9 9E Westerhof et al, Rijnhuizen report,1992. To allow direct comparison with the measurements, synthetic radiation temperatures are computed with the code ECRad. 10 10S Denk et al, Computer Physics Communications, p. 107175, 2020. Good agreement is found if radial transport occurring predominantly at low electron energies is included.We demonstrate that oblique ECE diagnostics measure the electron distribution function directly at the ECRH deposition site in phase space. Furthermore, they are sensitive to the abundance of pitch-angle scattered electrons that reduce the ECCD efficiency. Limitations and uncertainties of the measurements and the modeling are discussed.

Cherenkov instability of a magnetized beam-plasma system with allowance for a momentum spread of beam electrons

Cherenkov instability is considered in a completely magnetized homogeneous beam-plasma system featuring a thermal momentum spread of beam electrons. The thermal spread in the beam is described in the scope of both the hydrodynamic approach and the kinetic equation method by giving the electron momentum distribution function in the form of theMaxwellian and semi-Maxwellian distributions. It is shown that two beam-plasma instability regimes, the single-particle and collective Cherenkov effects (Compton and Raman regimes) differing by the physical mechanism and the increments, are possible in a system (waveguide) with homogeneous transverse beam and plasma density profiles. Solutions to dispersion equations for these and a more general regime are obtained and analyzed.

Fokker-Planck simulation of runaway electron generation in disruptions with the hot-tail effect

To study runaway electron generation in disruptions, we have extended the three-dimensional (two-dimensional in momentum space; one-dimensional in the radial direction) Fokker-Planck code, which describes the evolution of the relativistic momentum distribution function of electrons and the induced toroidal electric field in a self-consistent manner. A particular focus is placed on the hot-tail effect in two-dimensional momentum space. The effect appears if the drop of the background plasma temperature is sufficiently rapid compared with the electron-electron slowing down time for a few times of the pre-quench thermal velocity. It contributes to not only the enhancement of the primary runaway electron generation but also the broadening of the runaway electron distribution in the pitch angle direction. If the thermal energy loss during the major disruption is assumed to be isotropic, there are hot-tail electrons that have sufficiently large perpendicular momentum, and the runaway electron distribution becomes broader in the pitch angle direction. In addition, the pitch angle scattering also yields the broadening. Since the electric field is reduced due to the burst of runaway electron generation, the time required for accelerating electrons to the runaway region becomes longer. The longer acceleration period makes the pitch-angle scattering more effective.

Interaction quench in the Holstein model: Thermalization crossover from electron- to phonon-dominated relaxation

We study the relaxation of the Holstein model after a sudden switch-on of the interaction by means of the nonequilibrium dynamical mean field theory, with the self-consistent Migdal approximation as an impurity solver. We show that there exists a qualitative change in the thermalization dynamics as the interaction is varied in the weak-coupling regime. On the weaker interaction side of this crossover, the phonon oscillations are damped more rapidly than the electron thermalization timescale, as determined from the relaxation of the electron momentum distribution function. On the stronger interaction side, the relaxation of the electrons becomes faster than the phonon damping. In this regime, despite long-lived phonon oscillations, a thermalized momentum distribution is realized temporarily. The origin of the thermalization crossover found here is traced back to different behaviors of the electron and phonon self-energies as a function of the electron-phonon coupling. In addition, the importance of the phonon dynamics is demonstrated by comparing the self-consistent Migdal results with those obtained with a simpler Hatree-Fock impurity solver that neglects the phonon self-energy. The latter scheme does not properly describe the evolution and thermalization of isolated electron-phonon systems.

Exact distribution function of a one-dimensional Fro¨hlich–Toyozawa–Luttinger liquid

An exact analytical expression for the electron momentum distribution function of a one-dimensional Frohlich–Toyozawa–Luttinger (FTL) liquid is obtained. It is explicitly shown that the residual Fermi surface disappears when the electron–electron interaction is increased. It is furthermore shown that the electron–phonon interactions also flatten the distribution function, the optical phonons having a larger effect than the acoustic phonons. However, it is found that the nature of the distribution function of an FTL liquid is qualitatively different from that of the corresponding Luttinger liquid above kF.

Optical momentum alignment and spin orientation of hot electrons in GaAs/AlAs superlattices

Recent studies of dimensionality effects on hot electron photoluminescence (HPL) polarization in GaAs/AlAs superlattices (SLs) are reviewed. The effect of miniband formation in semiconductor superlattices on selection rules for optical transitions has been studied by means of polarized HPL. In the SL parameters two regimes can be distinguished which are characterized by their own selection rules. The peculiarities of selection rules for each of the regimes reveal themselves in linear and circular polarization of photoluminescence, momentum distribution function of photoexcited electrons and their behavior in a magnetic field. It is also demonstrated that the polarized HPL can be successively used in the study of ferromagnetic diluted magnetic semiconductors.

NUMERICAL STUDY OF HOLE CORRELATION IN THE t-J MODEL WITH FOUR HOLES ON A 32-SITE LATTICE

We solve the t-J model with four holes on a 32-site lattice using exact diagonalization. This is the largest scale calculation ever reported for this model. The size of the matrix involved in this problem is about 3×109 by 3×109. We find that at the realistic parameter J/t = 0.3, holes in the ground state of this system are unbound. They have short range repulsion due to lowering of kinetic energy. The electron momentum distribution function resembles hole pockets. We show that in case antiferromagnetic order is strong, holes form d-wave bound pairs and there is mutual repulsion among the holes. This presumably will occur at low doping level.

Hole correlation and antiferromagnetic order in thet−Jmodel

We study the t-J model with four holes on a 32-site square lattice using exact diagonalization. This system corresponds to doping level x=1/8. At the ``realistic'' parameter J/t=0.3, holes in the ground state of this system are unbound. They have short range repulsion due to lowering of kinetic energy. There is no antiferromagnetic spin order and the electron momentum distribution function resembles hole pockets. Furthermore, we show evidence that in case antiferromagnetic order exists, holes form d-wave bound pairs and there is mutual repulsion among hole pairs. This presumably will occur at low doping level. This scenario is compatible with a checkerboard-type charge density state proposed to explain the ``1/8 anomaly'' in the LSCO family, except that it is the ground state only when the system possesses strong antiferromagnetic order.

A Direct Method for Measuring Heat Conductivity in Intracluster Medium

The inverse Compton scattering of the cosmic microwave background (CMB) radiation with electrons in the intracluster medium which has a temperature gradient, was examined by the third-order perturbation theory of the Compton scattering. A new type of the spectrum distortion of the CMB was found and named as gradient T Sunyaev-Zel'dovich effect (gradT SZE). The spectrum has an universal shape. The spectrum crosses over zero at 326GHz. The sign of the spectrum depends on the relative direction of the line-of-sight to the direction of the temperature gradient. This unique spectrum shape can be used to detect the gradT SZE signal by broad-band or multi-frequency observations of the SZE. The amplitude of the spectrum distortion does not depend on the electron density and is proportional to the heat conductivity. Therefore, the gradT SZE provides an unique opportunity to measure thermally nonequilibrium electron momentum distribution function when the ICM has a temperature gradient and the heat conductivity in the ICM. However, the expected amplitude of the signal is very small. The modifications to the thermal SZE spectrum due to variety of known effects, such as relativistic correction etc., can become problematic when using multi-frequency separation techniques to detect the gradT SZE signal.

A GRADIENT-T SZE

The inverse Compton scattering of the cosmic microwave background (CMB) radiation with electrons in the intracluster medium which has a temperature gradient, was examined by the third-order perturbation theory of the Compton scattering. A new type of the spectrum distortion of the CMB was found and named as gradient T Sunyaev-Zel'dovich effect (gradT SZE). The spectrum has an universal shape. There is a zero distortion point, the cross over frequency, at 326GHz. When the hotter region locates closer to an observer, the intensity becomes brighter than the CMB in the frequency region lower than the cross over frequency and fainter than the CMB in the frequency region higher than the cross over frequency. When the cooler region locates closer to an observer, the distorted part of the spectrum has an opposite sign to the above case. The amplitude of the spectrum distortion does not de-pend on the electron density and depends on the heat conductivity and the total temperature variation along a line of sight. Therefore, the gradT SZE provides an unique opportunity to measure thermally nonequilibrium electron momentum distribution function in the ICM and combined with the X-ray measurements of the electron temperature distribution provides an opportunity of direct measurement of the heat conductivity in the ICM.

Low-energy states with different symmetries in thet−Jmodel with two holes on a 32-site lattice

We study the low energy states of the t-J model with two holes on a 32-site lattice with periodic boundary conditions. In contrary to common belief, we find that the state with d_{x^2-y^2} symmetry is not always the ground state in the realistic parameter range 0.2\le J/t\le 0.4. There exist low-lying finite-momentum p-states whose energies are lower than the d_{x^2-y^2} state when J/t is small enough. We compare various properties of these low energy states at J/t=0.3 where they are almost degenerate, and find that those properties associated with the holes (such as the hole-hole correlation and the electron momentum distribution function) are very different between the d_{x^2-y^2} and p states, while their spin properties are very similar. Finally, we demonstrate that by adding ``realistic'' terms to the t-J model Hamiltonian, we can easily destroy the d_{x^2-y^2} ground state. This casts doubt on the robustness of the d_{x^2-y^2} state as the ground state in a microscopic model for the high temperature superconductors.

Dynamic evolution of relativistic electrons in the radiation belts

Geomagnetic storms are responsible for the large increase of trapped electron fluxes in the magnetosphere resulting in the formation of radiation belts. Such belts can also be artificially produced by injection of electron beams. The dynamic evolution of the electron fluxes is very important for satellite protection purpose. Up to now most of the existing studies have been restricted to the stationary description of natural radiation belts. Recently, great effort has been made for their dynamic description [Bourdane et al., 1996]. We have developed a numerical code, which is able to follow the time behavior of the electron population under influence of collisions with tenuous atmosphere and resonant scattering by plasma waves. The code solves the phase‐averaged Fokker‐Planck equation of the electron momentum distribution function on magnetic shells within the Earth's inner magnetosphere. The physical arguments incorporated in the code include phase‐averaged, approximate collision operators and quasi‐linear diffusion due to resonant interaction with whistler plasma waves. The magnetic field is supposed to be tilted and decentered. Numerical experiments to simulate the decay and particles' loss in an artificial Van Allen belt are presented, and comparisons with available published data are discussed to validate the code [Abel and Thorne, 1998].

Electron momentum distribution of a single mobile hole in the t- J model

We investigate the electron momentum distribution function (EMDF) for the two-dimensional t- J model. The results are based on the self-consistent Born approximation (SCBA) for the self-energy and the wave function. In the Ising limit of the model we give the results in a closed form, in the Heisenberg limit the results are obtained numerically. An anomalous momentum dependence of EMDF is found and the anomaly is in the lowest order in number of magnous expressed analytically. We interpret the anomaly as a fingerprint of an emerging large Fermi surface coexisting with hole pockets.

Electron momentum distribution function in the t– t′– J model

We study the electron momentum distribution function (EMDF) for the two-dimensional t– t′– J model doped with one hole on finite clusters by the method of twisted boundary conditions. The results quantitatively agree with our analytical results for a single hole in the antiferromagnetic background, based on the self-consistent Born approximation (SCBA). Moreover, within the SCBA an anomalous momentum dependence of EMDF is found, pointing to an emerging large Fermi surface. The analysis shows that the presence of next-nearest-neighbor (NNN) hopping terms changes EMDF only quantitatively if the ground state (GS) momentum is at (π/2,π2) and qualitatively if the GS momentum is shifted to (π,0).

Electron momentum distribution in underdoped cuprates

We investigate the electron momentum distribution function (EMD) in a weakly doped two-dimensional quantum antiferromagnet (AFM) as described by the $t\ensuremath{-}J$ model. Our analytical results for a single hole in an AFM based on the self-consistent Born approximation (SCBA) indicate an anomalous momentum dependence of EMD showing ``hole pockets'' coexisting with a signature of an emerging large Fermi surface. The position of the incipient Fermi surface and the structure of the EMD is determined by the momentum of the ground state. Our analysis shows that this result remains robust in the presence of next-nearest neighbor hopping terms in the model. Exact diagonalization results for small clusters are with the SCBA reproduced quantitatively.

Electron Distribution Function in a Nonideal Plasma in Strong Electric and Laser Fields

AbstractUsing a two‐term approximation to the kinetic equation, the generalized electron energy and momentum distribution which governs the electron transport and rate properties in nonideal plasma is analyzed. A class of general solutions of the equation is found in the model case which is widely used in semiconductor plasma. An equation that generalizes the well‐known Zeldovich‐Raizer quantum kinetic equation in the case of nonideal plasma is deduced. Explicit nonequilibrium solutions of this equation for the electron energy and momentum distribution function in a strong laser field are constructed.

Single-integral expressions for electron cyclotron damping, relativistically and with an arbitrary distribution function

The relativistic expressions for the dielectric-tensor elements can be expressed as a double integral over both parallel and perpendicular momenta. The anti-Hermitian (damping) parts can be integrated over a delta function for either one of these two momenta, leaving a single integral over the other momentum variable. We review these expressions for either integration option, and compare them with statements in the literature. We give the limits on the remaining integral explicitly, for all positive or negative values of the parallel refractive index, whose magnitude can be less or greater than one. This single-integral form can be applied to electron cyclotron harmonic damping for any distorted electron momentum distribution function. As an example, we recover the results for a Maxwellian distribution. We also provide approximate results when the distribution function can be taken as a Maxwellian only in the perpendicular-momentum direction. A computer program will be written for a distribution function subjected to distortion by both electron cyclotron and lower-hybrid waves.

Comprehensive numerical and analytical study of two holes doped into the two-dimensionalt−Jmodel

We report on a detailed examination of numerical results and analytical calculations devoted to a study of two holes doped into a two-dimensional, square lattice described by the t-J model. Our exact diagonalization numerical results represent the first solution of the exact ground state of 2 holes in a 32-site lattice. Using this wave function, we have calculated several important correlation functions, notably the electron momentum distribution function and the hole-hole spatial correlation function. Further, by studying similar quantities on smaller lattices, we have managed to perform a finite-size scaling analysis. We have augmented this work by endeavouring to compare these results to the predictions of analytical work for two holes moving in an infinite lattice. This analysis relies on the canonical transformation approach formulated recently for the t-J model. From this comparison we find excellent correspondence between our numerical data and our analytical calculations. We believe that this agreement is an important step helping to justify the quasiparticle Hamiltonian, and in particular, the quasiparticle interactions, that result from the canonical transformation approach. Also, the analytical work allows us to critique the finite-size scaling ansatzes used in our analysis of the numerical data. One important feature that we can infer from this successful comparison involves the role of higher harmonics in the two-particle, d-wave symmetry bound state -- the conventional (\cos(k_x) - \cos(k_y)) term is only one of many important contributions to the d-wave symmetry pair wave function.

A NUMERICAL AND ANALYTICAL STUDY OF TWO HOLES DOPED INTO THE 2D t– J MODEL

Exact diagonalization numerical results are presented for a 32-site square cluster, with two holes propagating in an antiferromagnetic background described by the t– J model. We characterize the wave function of the lowest energy bound state found in this calculation, which has d x 2− y 2 symmetry. Analytical work is presented, based on a Lang–Firsov-type canonical transformation derived quasiparticle Hamiltonian, that accurately agrees with numerically determined values for the electron momentum distribution function and the pair correlation function. We interpret this agreement as strong support for the validity of this description of the hole quasiparticles.