Robert L. Singleton Jr

Charged particle motion in a highly ionized plasma

Abstract A recently introduced method utilizing dimensional continuation is employed to compute the energy loss rate for a non-relativistic particle moving through a highly ionized plasma. No restriction is made on the charge, mass, or speed of this particle. It is, however, assumed that the plasma is not strongly coupled in the sense that the dimensionless plasma coupling parameter g = e 2 κ D / 4 π T is small, where κ D is the Debye wave number of the plasma. To leading and next-to-leading order in this coupling, d E / d x is of the generic form g 2 ln [ Cg 2 ] . The precise numerical coefficient out in front of the logarithm is well known. We compute the constant C under the logarithm exactly for arbitrary particle speeds. Our exact results differ from approximations given in the literature. The differences are in the range of 20% for cases relevant to inertial confinement fusion experiments. The same method is also employed to compute the rate of momentum loss for a projectile moving in a plasma, and the rate at which two plasmas at different temperatures come into thermal equilibrium. Again these calculations are done precisely to the order given above. The loss rates of energy and momentum uniquely define a Fokker–Planck equation that describes particle motion in the plasma. The coefficients determined in this way are thus well-defined, contain no arbitrary parameters or cutoffs, and are accurate to the order described. This Fokker–Planck equation describes the straggling—the spreading in the longitudinal position of a group of particles with a common initial velocity and position—and the transverse diffusion of a beam of particles. It should be emphasized that our work does not involve a model, but rather it is a precisely defined evaluation of the leading terms in a well-defined perturbation theory.

Temperature equilibration rate with Fermi-Dirac statistics

We calculate analytically the electron-ion temperature equilibration rate in a fully ionized, weakly to moderately coupled plasma, using an exact treatment of the Fermi-Dirac electrons. The temperature is sufficiently high so that the quantum-mechanical Born approximation to the scattering is valid. It should be emphasized that we do not build a model of the energy exchange mechanism, but rather, we perform a systematic first principles calculation of the energy exchange. At the heart of this calculation lies the method of dimensional continuation, a technique that we borrow from quantum field theory and use in a different fashion to regulate the kinetic equations in a consistent manner. We can then perform a systematic perturbation expansion and thereby obtain a finite first-principles result to leading and next-to-leading order. Unlike model building, this systematic calculation yields an estimate of its own error and thus prescribes its domain of applicability. The calculational error is small for a weakly to moderately coupled plasma, for which our result is nearly exact. It should also be emphasized that our calculation becomes unreliable for a strongly coupled plasma, where the perturbative expansion that we employ breaks down, and one must then utilize model building and computer simulations. Besides providing different and potentially useful results, we use this calculation as an opportunity to explain the method of dimensional continuation in a pedagogical fashion. Interestingly, in the regime of relevance for many inertial confinement fusion experiments, the degeneracy corrections are comparable in size to the subleading quantum correction below the Born approximation. For consistency, we therefore present this subleading quantum-to-classical transition correction in addition to the degeneracy correction.

Effective theory for quenched lattice QCD and the Aoki phase

We discuss the symmetries of quenched QCD with Wilson fermions, starting from its Lagrangian formulation, taking into account the constraints needed for convergence of the ghost-quark functional integral. We construct the corresponding chiral effective Lagrangian, including terms linear and quadratic in the lattice spacing. This allows us to study the phase structure of the quenched theory, and compare it to that in the unquenched theory. In particular we study whether there may be an Aoki phase (with parity and flavor spontaneously broken) or a first-order transition line (with no symmetry breaking but meson masses proportional to the lattice spacing), which are the two possibilities in the unquenched theory. The presence of such phase structure, and the concomitant long-range correlations, has important implications for numerical studies using both quenched and dynamical overlap and domain-wall fermions. We argue that the phase structure is qualitatively the same as in the unquenched theory, with the choice between the two possibilities depending on the sign of a parameter in the low-energy effective theory.

Charged Particle Stopping Power Effects on Ignition: Some Results from an Exact Calculation

A completely rigorous first-principles calculation of the charged particle stopping power has recently been performed by Brown, Preston, and Singleton (BPS). This calculation is exact to leading and next-to-leading order in the plasma number density, including an exact treatment of two-body quantum scattering. The BPS calculation is therefore extremely accurate in the plasma regime realized during the ignition and burn of an inertial confinement fusion capsule. For deuterium-tritium fusion, the 3.5MeV alpha particle range tends to be 20%–30% longer than most models in the literature have predicted, and the energy deposition into the ions tends to be smaller. Preliminary numerical simulations indicate that this increases the ρR required to achieve ignition.

Temperature equilibration in a fully ionized plasma: Electron-ion mass ratio effects.

Brown, Preston, and Singleton (BPS) produced an analytic calculation for energy exchange processes for a weakly to moderately coupled plasma: the electron-ion temperature equilibration rate and the charged particle stopping power. These precise calculations are accurate to leading and next-to-leading order in the plasma coupling parameter and to all orders for two-body quantum scattering within the plasma. Classical molecular dynamics can provide another approach that can be rigorously implemented. It is therefore useful to compare the predictions from these two methods, particularly since the former is theoretically based and the latter numerically. An agreement would provide both confidence in our theoretical machinery and in the reliability of the computer simulations. The comparisons can be made cleanly in the purely classical regime, thereby avoiding the arbitrariness associated with constructing effective potentials to mock up quantum effects. We present here the classical limit of the general result for the temperature equilibration rate presented in BPS. In particular, we examine the validity of the m(electron)/m(ion)-->0 limit used in BPS to obtain a very simple analytic evaluation of the long-distance collective effects in the background plasma.

Is there an Aoki phase in quenched QCD

We argue that quenched QCD has non-trivial phase structure for negative quark mass, including the possibility of a parity-flavor breaking Aoki phase. This has implications for simulations with domain-wall or overlap fermions.

Charged Particle Motion in a Plasma: Electron-Ion Energy Partition

A fast charged particle traversing a plasma loses its energy to both the electrons and the ions in the plasma. We compute the energy partition, the fractions Ee/E0 and EI/E0 of the initial energy E0 of this ‘impurity particle’ that are deposited into the electrons and ions when it has slowed down into an equilibrium distribution that we shall determine. We do this using a well-defined FokkerPlanck equation for the phase space distribution of the charged impurity particles in a weakly to moderately coupled plasma. The Fokker-Planck equation holds to first sub-leading order in the dimensionless plasma coupling constant, which translates to computing to order nlnn (leading) and n (sub-leading) in the plasma density n. Previously, the order n terms have been estimated, not calculated. Since the charged particle does not come to rest, but rather comes into a statistical distribution, the energy loss obtained by a simple integration of a dE/dx has an ambiguity on the order of the plasma temperature. Our Fokker-Planck formulation provides an unambiguous, precise definition of the energy fractions. For equal electron and ion temperatures, we find that our precise results agree well with a fit obtained by Fraley, Linnebur, Mason, and Morse. The case with differing electron and ion temperatures, a case of great importance for nuclear fusion, will be investigated in detail in the present paper. The energy partitions for this general case, partitions that have not been obtained before, will be presented. We find that now the proper solution of the Fokker-Planck equation yields a quasi-static equilibrium distribution to which fast particles relax that has neither the electron nor the ion temperature. This “schizophrenic” final ensemble of slowed particles gives a new mechanism to bring the electron and ion temperatures together. The rate at which this new mechanism brings the electrons and ions in the plasma into thermal equilibrium will be computed.

Extension of the planar Noh problem to aluminum, iron, copper, and tungsten

The classic Noh verification test problem is extended beyond the traditional ideal gas and applied to shock compression of condensed matter. Using the stiff-gas equation of state (EOS), which admits an exact analytical solution for the planar Noh problem, we examine the shock compression of Al, Fe, Cu, and W. Analytical EOS predictions for the jump in density and the location of the shock are compared to numerical results obtained using the same EOS within Los Alamos compressible-flow codes Flag and xRage. Excellent agreement between the numerical and exact results is observed. Both codes exhibit first-order spatial convergence with increasing mesh resolution.

Effects of alpha stopping power modelling on the ignition threshold in a directly-driven inertial confinement fusion capsule

Abstract The alpha-particle energy deposition mechanism modifies the ignition conditions of the thermonuclear Deuterium-Tritium fusion reactions, and constitutes a key issue in achieving high gain in Inertial Confinement Fusion implosions. One-dimensional hydrodynamic calculations have been performed with the code Multi-IFE [R. Ramis, J. Meyer-ter-Vehn, Comput. Phys. Commun. 203, 226 (2016)] to simulate the implosion of a capsule directly irradiated by a laser beam. The diffusion approximation for the alpha energy deposition has been used to optimize three laser profiles corresponding to different implosion velocities. A Monte-Carlo package has been included in Multi-IFE to calculate the alpha energy transport, and in this case the energy deposition uses both the LP [C.K. Li, R.D. Petrasso, Phys. Rev. Lett. 70, 3059 (1993)] and the BPS [L.S. Brown, D.L. Preston, R.L. Singleton Jr., Phys. Rep. 410, 237 (2005)] stopping power models. Homothetic transformations that maintain a constant implosion velocity have been used to map out the transition region between marginally-igniting and high-gain configurations. The results provided by the two models have been compared and it is found that – close to the ignition threshold – in order to produce the same fusion energy, the calculations performed with the BPS model require about 10% more invested energy with respect to the LP model. Graphical abstract

The energy partitioning of non-thermal particles in a plasma: the Coulomb logarithm revisited

The charged particle stopping power in a highly ionized and weakly to moderately coupled plasma has been calculated exactly to leading and next-to-leading accuracy in the plasma density by Brown, Preston and Singleton (BPS). Since the calculational techniques of BPS might be unfamiliar to some, and since the same methodology can also be used for other energy transport phenomena, we will review the main ideas behind the calculation. BPS used their stopping power calculation to derive a Fokker–Planck equation, also accurate to leading and next-to-leading orders, and we will also review this. We use this Fokker–Planck equation to compute the electron–ion energy partitioning of a charged particle traversing a plasma. The motivation for this application is ignition for inertial confinement fusion—more energy delivered to the ions means a better chance of ignition, and conversely. It is therefore important to calculate the fractional energy loss to electrons and ions as accurately as possible. One method by which one calculates the electron–ion energy splitting of a charged particle traversing a plasma involves integrating the stopping power dE/dx. However, as the charged particle slows down and becomes thermalized into the background plasma, this method of calculating the electron–ion energy splitting breaks down. As a result, it suffers a systematic error that may be as large as T/E0, where T is the plasma temperature and E0 is the initial energy of the charged particle. The formalism presented here is designed to account for the thermalization process and it provides results that are near-exact.