Energy loss of heavy ions in dense plasma. I. Linear and nonlinear Vlasov theory for the stopping power

Correct calculation of nitrogen charge state passing through highly ionized carbon plasmas

In the present work, we reanalyze the energy loss experimental data from Cayzac et al. [Nat. Commun. 8, 15693 (2017)10.1038/ncomms15693] using our successful ion charge state theoretical model. We predict lower nitrogen charge values, from 3.5+ to 5.0+, than the ones calculated by Cayzac et al., fitting better to their data. For energy loss estimations, we use the same stopping model, so our predictions agree better with the experimental data only due to our charge state model. Different projectile electron loss and capture processes are taken into account to estimate the projectile charge state. The projectile electron loss, or ionization, with plasma ions and free electrons are considered. On the other hand, the projectile electron capture, or recombination, with plasma free or bound electrons are also considered. The projectile ionization with plasma ions is shown as the main factor that modifies the mean charge of the projectile. Here, the new Kaganovich fitting formula for this projectile ionization is used because it seems to be more accurate than Gryzinsky's fitting in the low energy range. Our charge state model fits better with experimental data than any other model in the bibliography. Thus, it should be considered in any charge state and any energy loss estimation to obtain reliable results in future work.

Observation of a high degree of stopping for laser-accelerated intense proton beams in dense ionized matter

Intense particle beams generated from the interaction of ultrahigh intensity lasers with sample foils provide options in radiography, high-yield neutron sources, high-energy-density-matter generation, and ion fast ignition. An accurate understanding of beam transportation behavior in dense matter is crucial for all these applications. Here we report the experimental evidence on one order of magnitude enhancement of intense laser-accelerated proton beam stopping in dense ionized matter, in comparison with the current-widely used models describing individual ion stopping in matter. Supported by particle-in-cell (PIC) simulations, we attribute the enhancement to the strong decelerating electric field approaching 1 GV/m that can be created by the beam-driven return current. This collective effect plays the dominant role in the stopping of laser-accelerated intense proton beams in dense ionized matter. This finding is essential for the optimum design of ion driven fast ignition and inertial confinement fusion.

A detailed understanding of particle stopping in matter is essential for nuclear fusion and high energy density science. Here, the authors report one order of magnitude enhancement of intense laser-accelerated proton beam stopping in dense ionized matter in comparison with currently used models describing ion stopping in matter.

Reduction of the Coulomb logarithm due to electron-neutral collisions

The frictional force (stopping power) acting on a test electron moving through an ideal electron gas is calculated by taking into account electron-neutral atom collisions using the linear plasma response formalism. This allows us to elucidate how the effective Coulomb logarithm is affected by electron-neutral collisions. In agreement with a recent investigation by Hagelaar, Donko, and Dyatko [Phys. Rev. Lett. 123, 025004 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.025004], we observe that the effective Coulomb logarithm decreases considerably due to electron-neutral collisions and becomes inversely proportional to the collision frequency in the highly collisional limit.

Experimental Validation of Low-Z Ion-Stopping Formalisms around the Bragg Peak in High-Energy-Density Plasmas

We report on the first accurate validation of low-Z ion-stopping formalisms in the regime ranging from low-velocity ion stopping-through the Bragg peak-to high-velocity ion stopping in well-characterized high-energy-density plasmas. These measurements were executed at electron temperatures and number densities in the range of 1.4-2.8 keV and 4×10^{23}-8×10^{23}  cm^{-3}, respectively. For these conditions, it is experimentally demonstrated that the Brown-Preston-Singleton formalism provides a better description of the ion stopping than other formalisms around the Bragg peak, except for the ion stopping at v_{i}∼0.3v_{th}, where the Brown-Preston-Singleton formalism significantly underpredicts the observation. It is postulated that the inclusion of nuclear-elastic scattering, and possibly coupled modes of the plasma ions, in the modeling of the ion-ion interaction may explain the discrepancy of ∼20% at this velocity, which would have an impact on our understanding of the alpha energy deposition and heating of the fuel ions, and thus reduce the ignition threshold in an ignition experiment.

Stopping power of dense plasmas: The collisional method and limitations of the dielectric formalism

We present a study of the stopping power of plasmas using two main approaches: the collisional (scattering theory) and the dielectric formalisms. In the former case, we use a semiclassical method based on quantum scattering theory. In the latter case, we use the full description given by the extension of the Lindhard dielectric function for plasmas of all degeneracies. We compare these two theories and show that the dielectric formalism has limitations when it is used for slow heavy ions or atoms in dense plasmas. We present a study of these limitations and show the regimes where the dielectric formalism can be used, with appropriate corrections to include the usual quantum and classical limits. On the other hand, the semiclassical method shows the correct behavior for all plasma conditions and projectile velocity and charge. We consider different models for the ion charge distributions, including bare and dressed ions as well as neutral atoms.

Experimental discrimination of ion stopping models near the Bragg peak in highly ionized matter

The energy deposition of ions in dense plasmas is a key process in inertial confinement fusion that determines the α-particle heating expected to trigger a burn wave in the hydrogen pellet and resulting in high thermonuclear gain. However, measurements of ion stopping in plasmas are scarce and mostly restricted to high ion velocities where theory agrees with the data. Here, we report experimental data at low projectile velocities near the Bragg peak, where the stopping force reaches its maximum. This parameter range features the largest theoretical uncertainties and conclusive data are missing until today. The precision of our measurements, combined with a reliable knowledge of the plasma parameters, allows to disprove several standard models for the stopping power for beam velocities typically encountered in inertial fusion. On the other hand, our data support theories that include a detailed treatment of strong ion-electron collisions.

The energy loss of ions in plasma is a challenging issue in inertial confinement fusion and many theoretical models exist on ion-stopping power. Here, the authors use laser-generated plasma probed by accelerator-produced ions in experiments to discriminate various ion stopping models near the Bragg peak.

Measurements of Ion Stopping Around the Bragg Peak in High-Energy-Density Plasmas

For the first time, quantitative measurements of ion stopping at energies around the Bragg peak (or peak ion stopping, which occurs at an ion velocity comparable to the average thermal electron velocity), and its dependence on electron temperature (T(e)) and electron number density (n(e)) in the range of 0.5-4.0 keV and 3×10(22) to 3×10(23) cm(-3) have been conducted, respectively. It is experimentally demonstrated that the position and amplitude of the Bragg peak varies strongly with T(e) with n(e). The importance of including quantum diffraction is also demonstrated in the stopping-power modeling of high-energy-density plasmas.

Predictions for the energy loss of light ions in laser-generated plasmas at low and medium velocities

The energy loss of light ions in dense plasmas is investigated with special focus on low to medium projectile energies, i.e., at velocities where the maximum of the stopping power occurs. In this region, exceptionally large theoretical uncertainties remain and no conclusive experimental data are available. We perform simulations of beam-plasma configurations well suited for an experimental test of ion energy loss in highly ionized, laser-generated carbon plasmas. The plasma parameters are extracted from two-dimensional hydrodynamic simulations, and a Monte Carlo calculation of the charge-state distribution of the projectile ion beam determines the dynamics of the ion charge state over the whole plasma profile. We show that the discrepancies in the energy loss predicted by different theoretical models are as high as 20-30%, making these theories well distinguishable in suitable experiments.

Self-Consistent Simulation of Transport and Energy Deposition of Intense Laser-Accelerated Proton Beams in Solid-Density Matter

The first self-consistent hybrid particle-in-cell (PIC) simulation of intense proton beam transport and energy deposition in solid-density matter is presented. Both the individual proton slowing-down and the collective beam-plasma interaction effects are taken into account with a new dynamic proton stopping power module that has been added to a hybrid PIC code. In this module, the target local stopping power can be updated at each time step based on its thermodynamic state. For intense proton beams, the reduction of target stopping power from the cold condition due to continuous proton heating eventually leads to broadening of the particle range and energy deposition far beyond the Bragg peak. For tightly focused beams, large magnetic field growth in collective interactions results in self-focusing of the beam and much stronger localized heating of the target.

Molecular dynamics simulations of classical stopping power

Molecular dynamics can provide very accurate tests of classical kinetic theory; for example, unambiguous comparisons can be made for classical particles interacting via a repulsive 1/r potential. The plasma stopping power problem, of great interest in its own right, provides an especially stringent test of a velocity-dependent transport property. We have performed large-scale (~10(4)-10(6) particles) molecular dynamics simulations of charged-particle stopping in a classical electron gas that span the weak to moderately strong intratarget coupling regimes. Projectile-target coupling is varied with projectile charge and velocity. Comparisons are made with disparate kinetic theories (both Boltzmann and Lenard-Balescu classes) and fully convergent theories to establish regimes of validity. We extend these various stopping models to improve agreement with the MD data and provide a useful fit to our results.

Energy loss and Z oscillations of atomic beams in plasmas

We apply a semiclassical partial-wave-scattering method based on the Wentzel-Kramers-Brillouin approximation to study the transport cross section and the energy loss of neutral or ionized atomic beams in plasmas. This approach reproduces the exact quantum result in a satisfactory manner, even in several extreme conditions of plasma densities and temperatures, and agrees with the results of linear or perturbative calculations for bare ions in the appropriate limits. We pay special attention to low projectile speeds where strong oscillations in the transport cross section and energy loss-as a function of projectile's atomic number-are observed. We study these oscillatory phenomena varying the projectile speed and its ionization degree and the plasma temperature and density. We analyze in physical terms these effects and present a diagram of plasma conditions showing the regions where these oscillations may occur for both neutral and ionized beams.

Time evolution and energy deposition for ion clusters injected into magnetized two-component plasmas

A two-dimensional particle-in-cell simulation model is proposed to study the time evolution and energy deposition for ion clusters injected into magnetized two-component plasmas. The injection of an isolated ion cluster is studied in the case of weak and strong magnetic fields. For strong magnetic fields, the ions tend to deposit their energy smoothly along the trajectory of the cluster, due to the confinement by the strong magnetic fields. However, in the case of weak magnetic fields, a large amount of energy is deposited by the ions near the initial cluster injection position, where the cluster density is expected to be largest. We attribute these to the influences of interference effects between the cluster ions, which have close relations to the distances between the ions. Furthermore, the influences of various magnetic fields, injection angles, and injection velocities on the time evolution and energy deposition of a beam pulse, which contains several similar ion clusters, are investigated in detail. The influences of different magnetic fields on the beam pulse show similar to that of a single ion cluster. For increasing injection angles, the beam velocity perpendicular to the magnetic field increases, leading to increasing oscillations in the beam trajectory and energy deposition profile. Besides, the amount of energy that transferred from the beam pulse to the plasma increases as the beam injection velocity approaches the electron thermal velocity.

Energy loss of ions by electric-field fluctuations in a magnetized plasma

The results of a theoretical investigation of the energy loss of charged particles in a magnetized classical plasma due to the electric-field fluctuations are reported. The energy loss for a test particle is calculated through the linear-response theory. At vanishing magnetic field, the electric-field fluctuations lead to an energy gain of the charged particle for all velocities. It has been shown that in the presence of strong magnetic field, this effect occurs only at low velocities. In the case of high velocities, the test particle systematically loses its energy due to the interaction with a stochastic electric field. The net effect of the fluctuations is the systematic reduction of the total energy loss (i.e., the sum of the polarization and stochastic energy losses) at vanishing magnetic field and reduction or enhancement at strong field, depending on the velocity of the particle. It is found that the energy loss of the slow heavy ion contains an anomalous term that depends logarithmically on the projectile mass. The physical origin of this anomalous term is the coupling between the cyclotron motion of the plasma electrons and the long-wavelength, low-frequency fluctuations produced by the projectile ion. This effect may strongly enhance the stochastic energy gain of the particle.

Number-conserving linear-response study of low-velocity ion stopping in a collisional magnetized classical plasma

The results of a theoretical investigation of the low-velocity stopping power of ions in a magnetized collisional and classical plasma are reported. The stopping power for an ion is calculated through the linear-response (LR) theory. The collisions, which lead to a damping of the excitations in the plasma, are taken into account through a number-conserving relaxation time approximation in the LR function. In order to highlight the effects of collisions and magnetic field, we present a comparison of our analytical and numerical results obtained for nonzero damping or magnetic field with those for vanishing damping or magnetic field. It is shown that the collisions remove the anomalous friction obtained previously [Nersisyan et al., Phys. Rev. E 61, 7022 (2000)] for the collisionless magnetized plasmas at low ion velocities. One of the major objectives of this paper is to compare and to contrast our theoretical results with those obtained through a diffusion coefficient formulation based on the Dufty-Berkovsky relation evaluated for a magnetized one-component plasma modeled with target ions and electrons.

Wake effect and stopping power for a charged ion moving in magnetized two-component plasmas: two-dimensional particle-in-cell simulation

A two-dimensional particle-in-cell (PIC) model is proposed to study the wake field and stopping power induced by a nonrelativistic charged particle moving perpendicular to the external magnetic field in two-component plasmas. The effects of the magnetic field on the wake potential and the stopping due to the polarization of both the plasma ions and electrons are discussed. The velocity fields of plasma ions and electrons are investigated, respectively, in the weak and strong magnetic field cases. Our simulation results show that in the case of weak magnetic field and high ion velocity, the wakes exhibit typical V-shaped cone structures and the opening cone angles decrease with the increasing ion velocity. As the magnetic field becomes strong, the wakes lose their typical V-shaped structures and become highly asymmetrical. Similar results can be obtained in the case of low ion velocity and strong magnetic field. In addition, stopping power is calculated and compared with previous one-dimensional and full three-dimensional PIC results.

Shielding of a moving test charge in a quantum plasma

The linearized potential of a moving test charge in a one-component fully degenerate fermion plasma is studied using the Lindhard dielectric function. The motion is found to greatly enhance the Friedel oscillations behind the charge, especially for velocities larger than half of the Fermi velocity, in which case the asymptotic behavior of their amplitude changes from 1/r3 to 1/r2.5. In the absence of the quantum recoil (tunneling) the potential reduces to a form similar to that in a classical Maxwellian plasma, with a difference being that the plasma oscillations behind the charge at velocities larger than the Fermi velocity are not Landau damped.

Interaction of fast charged projectiles with two-dimensional electron gas: Interaction and collisional-damping effects

The results of a theoretical investigation on the stopping power of ions moving in a two-dimensional degenerate electron gas are presented. The stopping power for an ion is calculated employing linear-response theory using the dielectric function approach. The collisions, which lead to a damping of plasmons and quasiparticles in the electron gas, is taken into account through a relaxation-time approximation in the linear-response function. The stopping power for an ion is calculated in both the low- and high-velocity limits. In order to highlight the effects of damping, we present a comparison of our analytical and numerical results, in the case of pointlike ions, obtained for a nonzero damping with those for a vanishing damping. It is shown that the equipartition sum rule first formulated by Lindhard and Winther for three-dimensional degenerate electron gas does not necessarily hold in two dimensions. We have generalized this rule introducing an effective dielectric function. In addition, some results for two-dimensional interacting electron gas have been obtained. In this case, the exchange-correlation interactions of electrons are considered via local-field corrected dielectric function.

Dynamic polarization and energy dissipation for charged particles moving in magnetized two-component plasmas

Energy losses of test particles in magnetized two-component plasmas are investigated within the framework of the linearized Vlasov-Poisson theory, taking into account the dynamic polarization effects of both the plasma ions and electrons. General expressions of the potential and stopping power are obtained and calculations are performed for protons in a magnetized hydrogen plasma. The influences of the magnetic field, the angle between the proton velocity and magnetic field, and certain plasma parameters on the stopping power are studied. Numerical results show that for low particle velocities and strong magnetic field the dynamic polarization effects of the plasma ions become obvious and contribute mainly to the stopping power.

Low ion-velocity slowing down in a strongly magnetized plasma target

An ion projectile stopping at a velocity smaller than the target electron thermal velocity in a strong magnetic field is investigated within a different diffusion formulation, based on Green-Kubo integrands evaluated in magnetized one component plasma models, respectively framed on target ions and electrons. Analytic expressions are reported for slowing down orthogonal and parallel to an arbitrary large magnetic field, which are free from the usual uncertainties plaguing the standard perturbative derivations. Magnetic and target temperature dependences of the low velocity slowing down are thoroughly detailed for dense plasmas of fast ignition concern and ultracold plasmas envisioned for ion beam cooling, as well.

Influence of damping on proton energy loss in plasmas of all degeneracies

The purpose of the present paper is to describe the effects of electron-electron collisions on the stopping power of plasmas of any degeneracy. Plasma targets are considered fully ionized so electronic stopping is only due to the free electrons. We focus our analysis on plasmas which electronic density is around solid values n(e) approximately = 10(23) cm(-3) and which temperature is around T approximately = 10 eV ; these plasmas are in the limit of weakly coupled plasmas. This type of plasma has not been studied extensively though it is very important for inertial confinement fusion. The electronic stopping is obtained from an exact quantum mechanical evaluation, which takes into account the degeneracy of the target plasma, and later it is compared with common classical and degenerate approximations. Differences are around 30% in some cases which can produce bigger mistakes in further energy deposition and projectile range studies. Then we consider electron-electron collisions in the exact quantum mechanical electronic stopping calculation. Now the maximum stopping occurs at velocities smaller than for the calculations without considering collisions for all kinds of plasmas analyzed. The energy loss enhances for velocities smaller than the velocity at maximum while decreases for higher velocities. Latter effects are magnified with increasing collision frequency. Differences with the same results for the case of not taking into account collisions are around 20% in the analyzed cases.

Effects of target plasma electron-electron collisions on correlated motion of fragmented protons

The objective of the present work is to examined the effects of plasma target electron-electron collisions on H2 + protons traversing it. Specifically, the target is deuterium in a plasma state with temperature Te=10 eV and density n=10(23) cm(-3), and proton velocities are vp=vth, vp=2vth, and vp=3vth, where vth is the electron thermal velocity of the target plasma. Proton interactions with plasma electrons are treated by means of the dielectric formalism. The interactions among close protons through plasma electronic medium are called vicinage forces. It is checked that these forces always screen the Coulomb explosions of the two fragmented protons from the same H2 + ion decreasing their relative distance. They also align the interproton vector along the motion direction, and increase the energy loss of the two protons at early dwell times while for longer times the energy loss tends to the value of two isolated protons. Nevertheless, vicinage forces and effects are modified by the target electron collisions. These collisions enhance the calculated self-stopping and vicinage forces over the collisionless results. Regarding proton correlated motion, when these collisions are included, the interproton vector along the motion direction overaligns at slower proton velocities (vp=vth) and misaligns for faster ones (vp=2vth, vp=3vth). They also contribute to a great extend to increase the energy loss of the fragmented H2 + ion. This later effect is more significant in reducing projectile velocity.

Energy loss of ions in a magnetized plasma: conformity between linear response and binary collision treatments

The energy loss of a heavy ion moving in a magnetized electron plasma is considered within the linear response (LR) and binary collision (BC) treatments with the purpose to look for a connection between these two models. These two complementary approaches yield close results if no magnetic field is present, but there develop discrepancies with growing magnetic field at ion velocities that are lower than, or comparable with, the thermal velocity of the electrons. We show that this is a peculiarity of the Coulomb interaction which requires cutoff procedures to account for its singularity at the origin and its infinite range. The cutoff procedures in the LR and BC treatments are different as the order of integrations in velocity and in ordinary (Fourier) spaces is reversed in both treatments. While BC involves a velocity average of Coulomb logarithms, there appear in LR Coulomb logarithms of velocity averaged cutoffs. The discrepancies between LR and BC vanish, except for small contributions of collective modes, for smoothened potentials that require no cutoffs. This is shown explicitly with the help of an improved BC in which the velocity transfer is treated up to second order in the interaction in Fourier space.

Stopping power of charged particles due to ion wave excitations

Stopping power due to ion wave excitations is derived for a charged particle moving in a two-component plasma. Unlike previous theories based on ion-acoustic-wave approximation (IAWA), the excitation of short-wavelength ion waves is taken into account. The obtained stopping power has a magnitude larger than that of IAWA. Stopping power at subsonic velocities, where stopping power in IAWA disappears, is even larger than that of supersonic velocities.

Time-dependent electron-ion collision frequency at arbitrary laser intensity-temperature ratio

In superintense laser beams collisional absorption exhibits a large amplitude modulation over a laser cycle. In this paper formulas for the time-dependent electron-ion collision frequency nu(ei)(t) are presented. On the basis of a ballistic interaction model we deduce an expression for nu(ei)(t) which holds for an arbitrary isotropic distribution function and arbitrary anharmonic oscillatory electron motion [Eq. (4)]. For a Maxwellian we present compact formulas for the various ratios v(os)(t)/v(th). It is shown that the strong time dependence over one laser cycle leads to the generation of intense odd harmonics. The cycle-averaged collision frequency nu(ei);(t) is compared with expressions derived from the more complex dielectric model. It is shown that the correct choice of cutoffs as a consequence of dynamical screening is essential.

Stopping power of ions in a magnetized two-temperature plasma

Using the dielectric theory for a weakly coupled plasma, we investigate the stopping power of an ion in an anisotropic two-temperature electron plasma in the presence of a magnetic field. The analysis is based on the assumption that the energy variation of the ion is much less than its kinetic energy. A general expression for the stopping power is analyzed for weak and strong magnetic fields (i.e., for the electron cyclotron frequency less than and greater than the plasma frequency), and for low and high ion velocities. It is found that the usually velocity independent friction coefficient contains an anomalous term which diverges logarithmically as the projectile velocity approaches zero. The physical origin of this anomalous term is the coupling between the cyclotron motion of the electrons and the long-wavelength, low-frequency fluctuations produced by the projectile ion.

Beam-plasma coupling effects on the stopping power of dense plasmas

The stopping power for ion beams in dense plasmas is investigated on the basis of quantum kinetic equations. Strong correlations between the beam ions and the plasma particles which occur for high ion charge numbers and strongly coupled plasmas are treated on the level of the statically screened T-matrix (binary collision) approximation. Dynamic screening effects are included using a combined scheme which considers both close collisions and collective effects. Applying this approach, the ion charge number dependence of the stopping power is determined. The result is a modification of the Z(2)(b) scaling law. In particular, the stopping power is reduced for strong beam-plasma coupling. Good agreement is found between T-matrix results and simulation data (particle-in-cell and molecular dynamics) for low beam velocities.