Comparison of the effect of soft-core potentials and Coulombic potentials on bremsstrahlung during laser matter interaction

Electromagnetic radiation produced by the acceleration or deceleration of a charged particle, such as an electron deflected (scattered) by another charged particle, is known as bremsstrahlung. The accelerated electron loses kinetic energy and emits radiation. Bremsstrahlung radiation becomes a dominant mechanism of emission at high temperatures in low density steady-state plasmas. Current driven plasmas are already a significant source of bremsstrahlung, as the energy deposition into the system is high.

The wide availability of high intensity, ultrashort pulse lasers has led to an explosion of research in ultrafast laser-matter interactions. Bremsstrahlung radiation from plasmas has been studied extensively, often providing an essential diagnostic tool, and has been used, for example, to diagnose hot electron populations in inertial confinement fusion experiments,^1,2 provide information about star formation in starburst galaxies,^3,4 and extract radiation temperatures in discharges.⁵ Bremsstrahlung emission in the range of 100 keV has been measured from plasmas produced by irradiation of solid targets with ps laser pulses of up to $7×1017$ W/cm².⁶ Experimental measurements of the fast electron beam created by the interaction of an intense, relativistic, frequency-doubled laser light with planar solid targets and its subsequent transport within the target are presented and compared with those of a similar experiment using the laser's fundamental frequency as well as the measured bremsstrahlung radiation.⁷ Highly energetic electrons produced by intense laser-matter interaction subsequently penetrate the solid thick and thin gold targets and produce bremsstrahlung photons, generating an x-ray source which can affect the electron recirculation.⁸ Meadowcroft et al. presented the design, the characterization, and the modeling of two target diagnostics to measure the high-energy (100 keV–2 MeV) bremsstrahlung emission from hot electrons in laser-plasma experiments.⁹ Galy et al. reported a dedicated series of calculations on the generated bremsstrahlung distributions from two experimental electron spectra measured using the giant pulse VULCAN laser and a gas jet target.¹⁰ The scaling of the intensity, angular and material dependence of bremsstrahlung radiation from an intense laser solid interaction was characterized, and the bremsstrahlung spectra were observed to have a two effective temperature energy distribution depending on the laser intensity and the observation angle.^11,12 Also, the hot electron velocity distributions are usually nonisotropic and non-Maxwellian.¹³ The energy distribution of a high-power, short-pulse laser produced fast electrons was experimentally and numerically studied using high-energy bremsstrahlung x-rays and compared with the result using a 2D particle-in-cell code, PICLS, which includes a newly developed radiation transport module.¹⁴ The generation of bremsstrahlung radiation from intense laser produced plasmas has been previously reported.^15–24 This generation is caused by the ponderomotive force accelerating electrons along the target normal direction for normal incidence.²⁵ 

Atomic clusters are groups of neutral atoms held together by van der Waals forces. Their properties bridge the gap between the solid and gas phases of matter. When an intense laser pulse irradiates the cluster, it causes inverse bremsstrahlung heating.²⁶ If the laser pulse is of sufficient intensity, a warm dense nano-plasma with energetic electrons in the cluster is formed. As these electrons undergo scattering with ions, they emit bremsstrahlung radiation. The production of bremsstrahlung radiation (hard X-rays) in the 1–5 keV range is observed when rare gas clusters of nanometer sizes are heated by strong optical fields (⁠$F>109$ V/cm).²⁷ The general results of the theory of polarizational bremsstrahlung and the results of calculations of the cross-sections of this process for a number of targets (clusters) are discussed.²⁸ This occurs when the frequency of the emitted photon is comparable with the energy of a plasmon resonance in a cluster.²⁹ Absolute doubly differential bremsstrahlung cross-sections of Xe, Kr, Ar, and Ne have been measured for electron bombarding energies of 28 and 50 keV.³⁰ Standard Molecular Dynamics (MD) is limited to classical systems. Nevertheless, it can be extended to weakly degenerate plasmas by replacing the Coulomb potential by effective potentials, such as those proposed by Kelbg,^31–33 Deutsch,³⁴ Rogers,³⁵ Gombert,³⁶ Perrot³⁷ and Filinov.³⁸ Therefore, the soft-core potential containing an “erf” function is used as a representative of a class of widely used potentials. One common approach to solve for the dynamics of laser-cluster interactions is to treat the motion of the particles as classical and the ionization/recombination through cross-sections, using the so-called quantum classical hybrid model. Newton's equations are numerically integrated by breaking up the motion into time segments small enough to resolve the motion of the electrons and ions through the potential in molecular dynamics. Due to the singularity, the Coulombic potential is replaced by the soft-core potential.^39–43 How does the width (or its inverse, the depth) parameter affect the bremsstrahlung radiation using molecular dynamics? Usually, the depth of the soft-core potential is taken to be just above the ionization potential of the singly charged ions.^40,44 A theoretical model is needed in order to understand how the bremsstrahlung emission would change due to the well-depth. To our knowledge, such a theory is yet to be presented.

In this paper, we present, for the first time, a simple analytic expression for the potential depth dependence of the radiation cross-section and the radiation power of bremsstrahlung produced by electrons in laser-matter interactions and for comparing the effect of soft-core potentials and Coulombic potentials on bremsstrahlung for any momentum transfer. In Sec. II, the radiation cross-section and the radiation power via bremsstrahlung are derived. In Sec. III, emission calculations are discussed. In Sec. IV, we present our results and discussion. Finally, we end with the conclusions.

The soft-core potential is given by the expression

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

The electric field is the negative potential gradient. So, the particle electric field with the soft-core potential is expressed as

During the scattering of the electron with ions, the scattering amplitude for the soft-core potential is²⁶ 

where $q=|k0−k|=k02+k2−2k0k cos θ, ℏq$ is the momentum transfer, θ is the scattering angle from the direction of the incident particle, $ℏk0$ is the linear momentum of the incident particle and $ℏk$ is the linear momentum of the scattered particle. Here, it is noted that there is no limitation placed on θ.

The scattering amplitude for Coulombic potential is

The ratio of the scattering amplitude for the soft-core potential and the scattering amplitude for the Coulombic potential is given by

The ratio of the scattering amplitude for the soft-core potential and that for the Coulombic potential decreases exponentially with an increase in momentum transfer q as shown in Eq. (6).

Using the soft-core potential model, the differential cross-section is the square of the scattering amplitude

Equation (7) can be expressed as a cross-section of scattering per unit interval in momentum transfer Q

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

In a Coulomb collision with momentum transfer Q, the incident particle is accelerated and emits radiation. The differential radiation cross-section is defined as⁴⁵ 

where $dI(ω′,Q)/dω′$ is the energy radiated per unit frequency interval in a collision with momentum transfer Q. The low-frequency radiation spectrum is given by

The radiation cross-section integrated over momentum transfer with a soft-core potential is

or

where Ei is the exponential integral and Q_max and Q_min are the maximum and minimum values of momentum transfers, respectively. $dχs dω′$ has dimensions of (area × energy/frequency). The values of Q_max and Q_min are⁴⁵ 

where E and v are the energy and the velocity of the electron and $ℏω′$ is the radiated energy.

The radiation cross-section via bremsstrahlung using the Coulomb potential is⁴⁵ 

$dχc dω′$ has dimensions of (area × energy/frequency). The total energy of bremsstrahlung radiation is the product of the radiation cross-section, the number density of ions, and the velocity of light. The power of the bremsstrahlung radiation for the soft-core potential is

while the power of the bremsstrahlung radiation for the Coulomb potential is

where n_i is the number density of ions. $Ps(ω′)$ and $Pc(ω′)$ both have dimensions of (energy × frequency). $dχs dℏω′$ and $dχc dℏω′$ both have dimensions of (area).

The power of the bremsstrahlung radiation for the soft-core potential [Eq. (16)] and the power of the bremsstrahlung radiation for the Coulomb potential [Eq. (17)] are plotted with respect to the frequency at a laser intensity of 10¹⁶ W/cm², $z1=1,z2=1$ (Fig. 1). The dashed line represents a soft-core potential at the potential depth $D=50 eV$⁠, while the solid line represents a Coulomb potential.

When an intense, short laser pulse is incident on rare-gas clusters, nano-plasmas with energetic electrons are produced. Then, electrons undergo bremsstrahlung collisions with ions. During the bremsstrahlung process, an electron emits energy. Single particle calculations of the bremsstrahlung radiation emitted by an electron are now presented. The electron is of mass m, has a charge e interacting with an ion of mass M and charge Ze. The laser pulse's irradiance is set to 10¹⁶ W/cm² with the wavelength fixed at $100 nm, 800 nm$ and $1600 nm$⁠. The critical density of the plasma depends on the wavelength of the laser light, which is calculated as $nc=4π2ϵ0mc2 e2λ2,$ where c is the speed of light, λ is the laser wavelength, and $ϵ0$ is the permittivity of free space. For a 100 nm laser pulse, the value of the critical density is $1.1166×1023[1/cm3]$⁠. To estimate the value of the radiation cross-section and the power radiated via bremsstrahlung, the ion density, k, and k₀ are needed. The density of condensed argon is 1.3954 g/cm³.⁴⁶ The ion density is estimated from the formula, $ni=ρi/mi=0.187nc$⁠. This is the ion density before cluster expansion. k is estimated from $k=2μEℏ2$ and $k0=2μE0ℏ2$⁠, where E is the energy of the scattered electron and E₀ is the initial energy of the electron. The radiation cross-section via bremsstrahlung for a soft-core potential is calculated using Eq. (12), and for the Coulomb potential, it is calculated using Eq. (15). The power radiated via bremsstrahlung for the soft-core potential is calculated using Eq. (16) and that for the Coulomb potential is calculated using Eq. (17). To emit a bremsstrahlung photon, the electron's energy should be greater than the emitted photon's energy, $ℏω$⁠. The maximum energy of the emitted bremsstrahlung photon is the maximum energy of the electron that undergoes the bremsstrahlung collision.

The radiation cross-section (left) and the power radiated via bremsstrahlung (right) determined using the soft-core potential from Eqs. (12) and (16), as well as the results obtained using the Coulomb potential [c.f. Eqs. (15) and (17)] are plotted in Fig. 2(a) as a function of the potential depth, D, for a λ = 100 nm pulse at 10¹⁶ W/cm² laser intensity with the density of ions equal to $0.187nc$ and the average charge state of ions equal to 1. These equations do not directly depend on laser intensity, but the linear momentum parameters k and k₀ depend on the electron kinetic energy through the ponderomotive energy gained from the laser field. The solid line represents the soft-core potential result, while the dashed line represents the Coulomb potential result. Figure 2(a) shows that the radiation cross-section as well as the power radiated via Bremsstrahlung increases rapidly with an increase in the potential depth of up to 200 eV, and then asymptotically approaches the Coulombic values. The soft-core potential reaches the Coulomb result asymptotically. Additionally, the radiation power for the soft-core potential is almost equal to that of the Coulomb potential at 10¹⁶ W/cm² laser intensity when the depth is greater than around 200 eV.

Figure 2(b) also shows the radiation cross-section and the power radiated via bremsstrahlung as a function of the potential depth parameter D for a λ = 800 nm pulse at 10¹⁶ W/cm² laser intensity with the density of ions equal to $11.973nc$⁠, and the average charge state of ions equal to 1. The solid line represents a soft-core potential, while the dashed line represents a Coulomb potential. The radiation cross-section as well as the radiation power for a soft-core potential is almost equal to that of the Coulomb potential at 10¹⁶ W/cm² laser intensity when the depth is greater than around 200 eV.

Figure 2(c) also shows the radiation cross-section and the power radiated via bremsstrahlung against the potential depth parameter D, respectively, for a λ = 1600 nm pulse at 10¹⁶ W/cm² laser intensity with the density of ions equal to $47.895nc$ and the average charge state of ions equal to 1. The solid line represents the soft-core potential, while the dashed line represents the Coulomb potential. The radiation cross-section as well as the radiation power for the soft-core potential is almost equal to that of the Coulomb potential at 10¹⁶ W/cm² laser intensity when the depth is greater than around 200 eV. Also, the radiation cross-section as well as the power radiated via bremsstrahlung first increases rapidly with an increase in the potential depth and then quickly becomes saturated.

In conclusion, the effect of a soft-core model on the electrostatic potential of particles in laser-matter interaction calculations which include bremsstrahlung radiation was studied and compared with the effect of the Coulomb potential using scattering theory. A new scaling for the radiation cross-section and the radiation power via bremsstrahlung is derived. The radiation cross-section and the radiation power via bremsstrahlung increase rapidly with an increase in the potential depth of up to around 200 eV and then become (mostly) saturated. The radiation cross-section and the radiation power via bremsstrahlung are also non-linear with respect to the laser's wavelength. The ratio of the scattering amplitude for a soft-core potential and that for a Coulombic potential decreases exponentially with an increase in momentum transfer.

This work was supported by U.S. AFOSR FA9550-14-1-0247.