Effect of soft-core potentials on inverse bremsstrahlung heating during laser matter interactions

Inverse bremsstrahlung heating (IBH) is the process in which an electron absorbs a photon while colliding with an ion or with another electron.¹ It is an essential mechanism for coupling laser energy to matter.² Thus, optimizing and modeling the processes of laser absorption is very important. It is often the case that theoretical models need to be solved computationally in order to obtain results under specific conditions, which tend to be in the non-linear regime of light-matter interactions. For instance, IBH is largely studied in the literature using different approaches.

Seely and Harris derived a kinetic equation and used it to calculate the change in kinetic energy of the electrons.³ Inverse bremsstrahlung absorption and evolution of the electron distribution function (EDF) were studied in Refs. 4 and 5. They found that the electron-electron collisions tend to enhance the inverse bremsstrahlung absorption; the contribution ratio of electron-electron collisions to the inverse bremsstrahlung absorption rate increases with increasing ion charge state Z_i in the high laser frequency regime. In another study, collisional absorption was investigated with the quantum statistical methods as well as molecular dynamics simulations. It was found that the energy absorption not only depends on the electron temperature but also on the temperature of the heavy ion component.⁶ Pfalzner and Gibbon numerically studied inverse bremsstrahlung absorption in strongly coupled plasmas produced by high intensity lasers. They verified the Langdon effect in a direct microscopic particle simulation.^7–9 

The quantum statistical expressions for the absorption rate in terms of the Lindhard dielectric function were derived for high frequency laser fields by Bornath et al.¹⁰ Stallcop and Billman developed a simple analytical expression for calculating the electron-ion, free-free Gaunt factor for high electron temperatures and long wavelengths and the values of the Gaunt factor are displayed in a graphical form.¹¹ Wierling et al. reproduced a free-free absorption coefficient for radiation in hot, weakly coupled plasmas by using the Gaunt factor.¹² A numerical one-dimensional Lagrangian hydrodynamics model was used to show that for long pulses, at low intensities and short wavelengths, inverse bremsstrahlung absorption is favored.¹³ 

Absorption using kinetic theory, in many reports, has been taken into account.^14–28 However, to date, no one has established the relationship between the potential depth (or any parameter of a soft-core potential) and the energy absorption rate via IBH in plasmas or nano-plasmas. Models which cannot use the singular Coulomb potential (due to the finite numerics or for performance reasons) must soften the singularity. A wide-spread model is the soft-core model in which the Coulombic potential is replaced by a smooth finite function at small distances^23,29,36 for the numerical stability, not the correct statistical behavior of the plasma, since in nano-plasmas all properties are transient. Thus, the soft-core potential is a numerical technique to enable high speed molecular dynamics. How the parameters of the soft-core potential affect the energy absorption of the electrons due to IBH remained unknown, despite the model's prevalence.

In molecular dynamics, Newton's equations are numerically integrated by breaking up the motion into small time segments given by $Δt$⁠. The time segment must be small enough to resolve the motion of the electron through the potential. The Coulomb potential cannot be used due to the singularity. However, this gives rise to the question: How does the depth parameter affect the results obtained using molecular dynamics when large angle scattering (small impact parameter collisions) is the driving mechanism behind IBH? Typically, the depth of the soft-core potential is taken to be just above the ionization potential of the singly charged ion.^30,31 A theoretical model is needed in order to understand how the previous results would change due to the potential depth as well as to guide future work in choosing correct parameters.

To determine the regimes of plasmas as either classical, degenerate or strongly coupled, we have to define the plasma parameters such as the Wigner-Seitz (WS) radius, coulomb coupling parameter, degeneracy, thermal de Broglie wavelength, and Bohr radius. The discussion using plasma parameters is in Refs. 32–34. The Wigner-Seitz (WS) radius is

where n is the density. For particles of mass m and charge e, the Coulomb coupling parameter is

where T is the temperature of electrons and k_B is Boltzmann's constant. The degeneracy is

where $ℏ$ is Planck's constant. The thermal de Broglie wavelength is

When the Wigner-Seitz (WS) radius is large in comparison with the Bohr radius (⁠$aB=ℏ2me2$⁠), i.e., if $a≥aB$ the plasma behaves classically. Otherwise, the plasma is degenerate. If the Wigner-Seitz (WS) radius is comparable to the de Broglie wavelength, quantum mechanical effects play a significant role. For moderate coupling, the coupling parameter is very close to unity. For strong coupling, the coupling parameter is much greater than unity. In our case, at the highest laser intensity of 10¹³ W/cm² and with a wavelength of 100 nm, the Wigner-Seitz (WS) radius (a $≈ 2.25×10−8$ cm) is large in comparison with the Bohr radius (⁠$aB=0.529×10−8$ cm). Other plasma parameters also lie in the classical regime. Thus, we are dealing with classical plasmas in this article. In the classical regime, the width parameter w is similar or equal to thermal de Broglie wavelength. So, the potential depth is

Where $z2e$ is the charge of ions

In this paper, we present, for the first time, an analytical expression for the potential depth dependence of the rate of inverse bremsstrahlung heating produced by electrons in laser-matter interactions. The differential cross-section is derived by using scattering theory. In Sec. II, a differential cross-section is derived and the rate of energy absorption by IBH is estimated. In Sec. III, absorption calculations are discussed. In Sec. IV, we present our results and discussion. Finally, we end with the conclusions.

In inverse bremsstrahlung heating, an electron absorbs energy from the laser pulse during a collision with a charged particle. The equation of motion of an electron in an oscillatory electric field is

where m is the mass, $v$ is the velocity, and e is the charge of electrons, $Ep$ is the electric field due to other particles and $EL$ is the external laser field. $Ep$ is given by Eq. (15). The laser field, E_L, is related to the intensity of the laser pulse by $EL=2cμ0I$ where I is the laser intensity.

The average rate of energy absorption by the electron from the electric field is the product of the average kinetic energy of electrons and the effective collisional frequency. The effective collisional frequency is derived from the scattering theory. In scattering theory, for an electron of mass m and charge e that interacts with an ion of mass M and charge e, the differential cross-section is the square of scattering amplitude

where $dσ$ is the differential cross-section, $dΩ$ is the element of solid angle, and $f(θ,ϕ)$ is the scattering amplitude. The Schrödinger wave equation for this scattering is

where $ψ(r)$ is the wave function, μ is the reduced mass, $V(r)$ is the potential, and E is the energy. A solution to the Schrödinger equation is sought out using partial-wave analysis. The ansatz wave function is thus written as the sum of the incident plane wave and the scattering part

where $ϕinc(r)$ is the incident plane wave.

For a weak potential $V(r′)$⁠, the scattered part will be only slightly distorted. Then, the first Born approximation consists of approximating the scattered wave function $ψ(r)$ by a plane wave. The scattering amplitude in the first Born approximation is

where $q=|k0−k|=k02+k2−2k0k cos θ, ℏq$ is the momentum transfer, $ℏk0$ is the linear momentum of the incident particle, and $ℏk$ is the linear momentum of the scattered particle.

For a spherical symmetric scatterer, we obtain the standard form of the Born approximation for the scattering amplitude

The soft-core potential is given by

where $z1e$ and $z2e$ are the charges of electrons and ions, respectively, erf is the error function, and w is the width parameter which is related to the potential depth, D, by

Figure 1 shows the plots of the soft-core potential from Eq. (13) with width parameter $w=0.12$ (long dashed, red), $w=0.22$ (dotted, green), and Coulombic potential (solid, blue). Twice the widths of the potentials are shown as straight lines to illustrate the meaning of the w-parameter.

The electric field is the negative potential gradient. So, the particle's electric field is

Figure 2 shows the plots of the electric field from Eq. (15) with width parameter $w=0.12$ (long dashed, red), $w=0.22$ (dotted, green), and coulombic potential (solid, blue). The plot shows how the soft-core potential results in a maximum electric field and thus a maximum acceleration which determines the minimum time step that can be used to solve Newton's equations of motion. The goal of the following is to determine how the screening of the Coulomb singularity affects the rate of energy absorption in a laser field.

Substituting the value of $V(r′)$ from Eq. (13) to Eq. (12)

The direct integration of Eq. (16) does not converge. A factor, $limd→∞ exp (−r′2/d)$⁠, is introduced into Eq. (16) and integrated giving

Here, d is a parameter having dimension the same as the dimension of $r′2$⁠. This convergence factor was obtained by consideration of the integral representation of the error function which is also quadratic in the exponential.

Substituting the value of $f(θ)$ into Eq. (8) we get

Now Eq. (18) can be expressed as a cross section of scattering per unit interval in momentum transfer Q

where $Q=ℏq, Q2=ℏ2(k02+k2−2k0k cos θ), 2QdQ=2ℏ2k0k sin (θ)dθ, dΩ=sin(θ)dθdϕ=−Q dQ dϕ/(ℏ2k0k)$⁠.

It is now possible to calculate the total cross-section $(σ)$⁠, using the differential cross-section for the soft-core potential in Eq. (13). The total cross-section with soft-core potential integrated over momentum transfers is

where Ei is the exponential integral, and Q_max and Q_min are maximum and minimum values of momentum transfers. The values of Q_max and Q_min are³⁵ 

where E and v are the energy and velocity of the electron, and $ω′$ is the laser frequency.

The total cross-section with Coulomb potential is known to be

The collisional frequency between the ion and electron is

where n_i is the ion density. With the collisional frequency, we can then calculate the rate of energy absorption via IBH. Taking the product of the collisional frequency and the time average kinetic energy of the electron, the average rate of energy absorption by the electron from the electric field is

Here, $EL=2cμ0I$ is the slowly varying amplitude of the laser field.

Single particle calculations of the energy absorbed by an electron in the presence of a short (10 fs) laser pulse are now presented. The electron is of mass m and charge e interacting with an ion of mass M and charge Ze. The laser pulse's irradiance is varied from 10¹² W/cm² to $5×1013$ W/cm², with a wavelength fixed at 100 nm, 800 nm, 1600 nm, and 2400 nm. During the interaction, an electron absorbs laser energy and transfers some of the absorbed energy to the ions. The density at which the plasma frequency equals the frequency of an electromagnetic electron wave is called the critical density of the plasma. It depends on the wavelength of the laser light which is calculated by

where c is the speed of light, λ is the laser wavelength, and ϵ₀ is the permittivity of free space. For an 800 nm laser pulse, the value of the critical density is $1.7446×1021[1/cm3]$⁠. To estimate the value of the total cross-section, the collision frequency and the rate of energy absorption via IBH, the ion density, k, k₀, and velocity of electrons are needed. The density of condensed argon is 1.3954 g/cm³.³⁷ The value of the critical density is $1.1166×1023[1/cm3]$ for an 100 nm laser pulse and the ion density is $ni=2.089×1022cm−3$⁠. k is estimated from $k=2μEℏ2$ and $k0=2μE0ℏ2$⁠, where E is the energy of the scattered particle and E₀ is the initial energy of particle. The collisional frequency is obtained from Eq. (25), where the average velocity of the electron is $|eEL/mω′|$⁠.

The rate of energy absorption via IBH is then calculated from the product of the average kinetic energy of the electrons and the collisional frequency.

The total cross-section using the soft-core potential from Eq. (21) as well as the result using a Coulomb potential [c.f. Eq. (24)] is plotted in Fig. 3(a) as a function of the potential depth, D. The different intensities are shown as the dotted, medium-dashed, large-dashed, and dotted-dashed lines for intensities of 10¹², $5×1012$⁠, 10¹³, and $5×1013$ W/cm², respectively, for a λ = 100 nm pulse with a density of ions equal to $0.18709nc$⁠, and an average charge state of ions equal to 1. The solid line represents the total cross-section for a Coulomb potential at laser intensity $5×1013$ W/cm². Figure 3(a) shows a direct relationship between the intensity and total cross-section. However, the relationship is nonlinear. Further, the total cross-section increases rapidly with the increase in the potential depth and quickly becomes saturated. Additionally, lower intensities saturate at a much larger potential depth.

Figure 3(b) shows the rate of energy absorption as a function of the potential depth (the dotted, medium-dashed, large-dashed, and dotted-dashed lines represent the laser intensities of 10¹², $5×1012$⁠, 10¹³, and $5×1013$ W/cm², respectively, for a λ = 100 nm pulse with a density of ions equal to $0.18709nc$⁠, and an average charge state of ions equal to 1. The solid line represents the rate of energy absorption for a Coulomb potential at laser intensity $5×1013$ W/cm²). The rate of energy absorption also first increases rapidly with increases in the potential depth and then quickly becomes saturated. Additionally, the rate of energy absorption via IBH saturates faster as laser intensity increases with respect to the potential depth. The rate of energy absorption for a soft-core potential is almost equal to that of a Coulomb potential at laser intensity $5×1013$ W/cm² when the depth is greater than around 100 eV.

Again we see that the rate of energy absorption is non-linear with respect to the laser intensity. This non-linearity is due to the non-linear relationship between the velocity and collisional frequency of the electrons in a nano-plasma; when slow electrons are present, the nano-plasma is depleted when heated by a laser beam (which includes weibel instability).¹⁸ Similarly, the total cross-section and rate of energy absorption are plotted with respect to potential depth for laser wavelength λ = 800 nm in Figs. 4(a) and 4(b), respectively. Here, we see that the total cross-section and rate of energy absorption are also non-linear with respect to the laser's intensity. Again, the rate of energy absorption via IBH with a soft-core potential saturates faster (for a small potential depth) as laser intensity increases.

Figures 5(a) and 5(b) compare the total cross-section and rate of energy absorption as a function of the potential depth at a laser intensity of 10¹² W/cm² with different values of laser wavelength (dotted, medium-dashed, dotted-dashed, and solid lines represent that of soft-core potential at laser wavelengths, λ, of 800 nm, 1600 nm, 2400 nm, and that of a Coulomb potential for a wavelength of 1600 nm, respectively). Here, we see that the longer the laser wavelength, the higher the total cross-section as well as the rate of energy absorption via IBH. Additionally, they are non-linear with respect to the laser's wavelength. The rate of energy absorption via IBH saturates faster as wavelength increases.

In conclusion, the effect of a soft-core model for the electrostatic potential of particles in laser-matter interaction calculations which include inverse bremsstrahlung heating (IBH) was studied using scattering theory. A new scaling for the total cross-section and the rate of energy absorption via IBH is derived. The total cross-section as well as the rate of energy absorption via IBH increases rapidly with an increase in the potential depth and then becomes (mostly) saturated for larger potential depths. The rate of energy absorption via IBH with respect to the potential depth saturates faster as laser intensity increases. The rate of energy absorption via IBH is found to change non-linearly with increases in laser intensities, independent of the potential depth. As laser wavelength increases, the total cross-section as well as rate of energy absorption of IBH also increases. Rate of energy absorption via IBH saturates faster as laser wavelength increases. Lower intensities are found to saturate at much larger potential depths at all wavelengths.

E.A. would like to thank D. Kaplan for insightful discussions. This work was supported by U.S. AFOSR FA9550-14-1-0247.