Mean force emission theory for classical bremsstrahlung in strongly coupled plasmas Open Access

Bremsstrahlung emission plays an important role in radiation transport in a variety of dense plasmas. These include plasmas in nuclear fusion^1–3 and astrophysical objects like white dwarfs,^4,5 neutron stars,⁶ and our sun.⁷ Even in low-temperature industrial applications, such as EUV lithography, the bremsstrahlung emission process is important in modeling energy loss.^8,9 Importantly, many of these systems include strongly coupled plasmas. This paper presents mean force emission theory, which extends the description of classical bremsstrahlung into the strongly coupled regime. It also tests this theory by comparing with classical molecular dynamics (MD) simulations of a model plasma that can be simulated in a first-principles manner.

Traditional descriptions of the classical bremsstrahlung spectrum apply in the high-frequency limit, which means they treat only the timescale associated with individual electron–ion collisions. A consequence is that they do not capture the low-frequency limit associated with timescales longer than the Coulomb collision time (⁠ $τ e i )$⁠. Furthermore, early treatments model electron–ion interactions as occurring through the Coulomb potential,^10–12 which leads to a logarithmic divergence of the emission spectrum in the low-frequency limit. Later models explored the impact of screening by cutting off the interaction distance at the Debye length,¹³ or using a linear-dielectric approach that includes Debye–Hückel screening.¹⁴ Accounting for screening resolves the divergence, predicting a constant value for the emission coefficient in the low-frequency limit, but does not capture the physical decrease that is expected as the timescale transitions to the fluid-scale associated with multiple collisions. Some authors have looked at bremsstrahlung in strongly coupled plasmas and included the effects of strong correlations through using effective interaction potentials to model electron–ion collisions. These effective potentials include an ion-sphere potential,¹⁵ and a soft-core potential with a Yukawa screening term.¹⁶ The high-frequency limit of mean force emission theory presented here also addresses strong coupling through an effective interaction potential, but is based on the potential of mean force, which has a rigorous basis in kinetic theory.¹⁷ 

The bremsstrahlung process has also been approached from a quantum perspective. The seminal work in this regard is Sommerfeld's calculation of non-relativistic quantum electron scattering off a Coulomb potential.¹⁸ Subsequent work has looked at the effects of relativity, screening, and electron degeneracy.^19–22 In the quantum formalism, strong coupling has been looked at by applying an effective potential to a Kubo–Greenwood approach.²³ Quantum effects are conveniently expressed in terms of a single quantity, the Gaunt factor.²⁴ The quantum and classical approaches also agree in that they give similar results for the Gaunt factor when applied to small-angle Coulomb scattering (see Sec. 3.5 in Ref. 12). More importantly for this paper, whether classical or quantum mechanical, any model must account for multiple collisions to capture the low-frequency regime. A popular correction used in quantum calculations of optical transport quantities is a “Drude” factor that includes an effective collision frequency.^23,25–27 Mean force emission theory suggests that the form of the Drude factor does not capture strong correlation effects that occur in the neighborhood of the electron plasma frequency.

The mean force emission model is based on mean force kinetic theory,¹⁷ which generalizes how the many-body effect of screening should be treated in kinetic theory at strongly coupled conditions. It shows that binary interactions should be modeled via the potential of mean force, rather than the Coulomb or Debye–Hückel potential.²⁸ This has been shown to successfully extend plasma kinetic theory up to Coulomb coupling strengths of $Γ ≈ 10$⁠.^29–32 However, kinetic theory only models timescales longer than the Coulomb collision time (τ_ei). Here, we apply the mean force concept to a broad range of timescales in order to capture the full frequency spectrum of bremsstrahlung emission. At high frequencies, the theory looks similar to traditional models, but where electrons and ions interact via the potential of mean force. Under the molecular chaos approximation, these individual collisions are summed up to calculate the emission coefficient. The bremsstrahlung problem can also be reframed in terms of autocorrelation functions in a manner similar to previous work.¹⁶ In the low-frequency limit, the velocity autocorrelation is modeled using a Langevin approximation with a temporally local friction force computed from the mean force kinetic theory. This returns a Drude correction factor that captures the asymptotic low frequency decay of the emission spectrum, but misses an enhanced peak that is observed near the plasma frequency. To capture the enhanced peak, a model for the velocity autocorrelation is developed that includes oscillatory behavior associated with strong correlations that the Drude factor misses.

These predictions are benchmarked using classical MD simulations. MD simulations provide first-principles calculations that are commonly utilized to benchmark theories for a variety of physical processes in strongly coupled plasmas, including but not limited to: diffusion,^30,33 temperature relaxation,³⁴ thermal conductivity,³¹ friction,³⁵ and bremsstrahlung absorption.^36–38 However, MD simulations present significant complexities when used to simulate multicomponent plasmas with attractive interactions.³⁹ For one, the use of classical point particles and attractive interactions leads to “Coulomb collapse,” where ions and electrons recombine to form classical bound states.⁴⁰ Even in a weakly coupled system, close interactions between electrons and ions leads to unphysical amounts of radiation. In order to properly account for attractive interactions one needs to include quantum effects at short distances. In order to benchmark the classical theory against first principles classical MD simulations, this work will consider a fully ionized hydrogen plasma with positively charged electrons and ions. While this is a simplified picture, it still provides important information that is applicable to the broader problem at hand. Accurately capturing the surrounding plasma environment in strongly coupled systems and the effect of multiple collisions are details that are universally present in a theoretical description of bremsstrahlung emission, regardless of whether the problem is approached through repulsive interactions and classical physics.

This paper is organized as follows: Sec. II introduces the MD simulations and calculation of the bremsstrahlung emission coefficient. This section also presents a relation between the bremsstrahlung emission and correlation functions. Section III discusses mean force emission theory. Section IV discusses the successes and limitations of mean force emission theory and compares with relevant traditional models. Section V discusses conclusions based on these results.

Molecular dynamics simulations were run using LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator).⁴¹ These provided a first-principles calculation of the classical dynamics of a collection of charged particles interacting through the electrostatic Coulomb force. For numerical efficiency, the particle–particle particle–mesh (P3M) method was used.⁴² In P3M, interparticle forces are calculated directly within a specified distance (r_c), while outside of this distance charges are interpolated to a grid and the force on a particle is advanced from the electric field computed at the grid cells. To ensure that the computation remains first-principles, r_c was chosen so as to not impact the results. In this regard, it was found that $r c = 5 a$ was sufficient for our purpose.

Simulations were run using a two species plasma consisting of positively charged electrons and ions (Z = 1) at a fixed temperature (⁠ $T e = T i = T$⁠). Because the charge of each species was the same, the simulations can be characterized in dimensionless units entirely by Γ and the mass ratio $m i / m e ≈ 1836$⁠. Each simulation used a three dimensional box with periodic boundary conditions containing 10⁴ particles. Prior to data gathering, each plasma was equilibrated to a fixed Γ value using a Nose–Hoover thermostat.⁴² Once the particles settled into their equilibrium spatial and velocity distributions the thermostat was turned off, and the total energy in the simulation was fixed. The time step was chosen to be $10 − 3 ω p e − 1$ and the total runtime $10 4 ω p e − 1$⁠, except for the case of the weakest coupling explored (⁠ $Γ = 0.01$⁠), where the time step was set to $10 − 4 ω p e − 1$ in order to fully resolve all collisions. Here $ω p e = n e q 2 / ( m e ϵ 0 )$ is the electron plasma frequency.

In order to relate the calculation of the emission spectrum from Eq. (2) to the forthcoming theoretical work, one can divide by the total time of the simulation (⁠ $T$⁠) and the simulation volume (V) to calculate the emission coefficient which defines the power density per unit frequency per sterradian $j ( ω ) = W ( ω ) / ( T V )$⁠. Since the relation between the total spectrum and the emission coefficient is only dependent on parameters $T$ and V, this paper will henceforth refer to results displaying $j ( ω )$ as the spectrum and the emission coefficient interchangeably.

Spectra calculated from the MD simulations are shown in Fig. 1. In order to reduce high-frequency noise and isolate the general trends, a third-degree Savitzky–Golay filter was applied in post-processing the data.⁴⁴ Qualitative changes are observed as a function of coupling strength. Most notably, the spectrum is nearly constant over a wide frequency range at weak coupling, but becomes much more narrow and peaked near the plasma frequency at strong coupling. These breaks in the spectrum can be understood by considering the timescales associated with close interactions, and the transition to macroscopic fluid dynamics.

As the coupling strength increases toward unity, the emission spectrum is observed to become less flat and more peaked near the electron plasma frequency. This may be expected from the timescale arguments of Eqs. (5) and (7), since both $ω max$ and $ω min$ merge toward ω_pe as Γ approaches 1 (see Fig. 2). Equations (5) and (7) do not apply for values of $Γ ≳ 1$⁠. Accordingly, Fig. 2 uses the expression $ln Λ ≈ 0.65 ln ( 1 + 2.15 / ( 3 Γ 3 / 2 ) )$ from Ref. 33, which extends the electron–ion collision frequency to $Γ ≳ 1$⁠. This expression and previous work have observed that the electron–ion collision frequency plateaus to a value of approximately $0.2 ω p e$ for $Γ ≳ 10$ (see Fig. 3 of Ref. 29). This appears to agree with the characteristic value of the low-frequency limit in Fig. 1.

The merging of $τ min$ and τ_ei is a characteristic of strong coupling that can also be visualized by examining the time-dependent forces experienced by particles. Each panel in Fig. 3 shows three components of the force on an individual electron in an MD simulation. For the weakest coupling case (⁠ $Γ = 0.1$⁠), the collisions appear as distinct, random spikes as the electron experiences a strong force during the close encounter with another charged particle. At strong coupling (⁠ $Γ = 1 , 10$⁠), collisions are not well separated in time. Strong forces do not appear randomly but are a part of a more correlated electron motion through the plasma. In Sec. II C, it is shown that this correlated motion and its impact on the bremsstrahlung emission coefficient can be captured using autocorrelation functions.

The force and velocity autocorrelation functions calculated from MD are shown in Fig. 4 for different values of Γ. The characteristic trends in the weakly coupled cases are monotonically decaying correlation functions, whereas the strongly correlated cases exhibit oscillations near the plasma frequency. These oscillations cause the Fourier transforms in Eqs. (14) and (18), and therefore the emission spectrum, to be peaked near the plasma frequency.

The emission coefficient calculated from the force autocorrelations in Eq. (14) is compared to that computed directly from the general formula of Eq. (2) in Fig. 5. The comparison shows that Eq. (14) matches Eq. (2) at all frequencies. Thus, the correlation function approach gives a description of bremsstrahlung emission in the low and mid frequency ranges that is inaccessible to the typical high-frequency approach. In Secs. III C and III D, it is shown that modeling the velocity autocorrelation analytically allows for simple expressions that capture the bremsstrahlung spectrum in the low and mid frequency ranges.

Here, we develop this generalization in steps. For high frequencies (⁠ $ω ≳ ω p e$⁠), the spectrum is calculated by solving for the individual trajectories directly from Eq. (19). These are summed to obtain the total emission spectrum in this limit; see Sec. III B. In the low and mid frequency range (⁠ $ω ≲ ω p e$⁠), the spectrum is calculated by summing many trajectories to evaluate the velocity autocorrelation function. This leads to a direct application of mean force kinetic theory to compute an electron–ion collision rate. Then, the spectrum is computed using Eq. (18); see Secs. III C and III D. A simple comprehensive model is then developed that bridges these limits, providing a mean force emission theory that is valid across all frequencies.

The time-dependent force on an electron, $F e ( t )$⁠, in Eq. (24) is computed by directly solving for the electron trajectory using Eq. (19). Only two coordinates are needed since the collision takes place on a plane. Due to its large mass, the ion is taken to be fixed at the origin and the electron trajectory is started at some distance (d) considered to be asymptotically far away. Asymptotic in this sense refers to a distance larger than the screening length of the potential. Thus, for a weakly coupled plasma, we use $d ≫ λ D$ and in a strongly coupled plasma use $d ≫ a$⁠.

Figure 7 illustrates an example of the general workflow for computing the high-frequency emission spectrum in Eq. (24). Equation (19) was solved using a Runge–Kutta integrator to give $r e ( t )$ and $v e ( t ) = d r e / d t$ during a collision. The force felt by the electron was then calculated from the velocity $m e d v e / d t$ and input into Eq. (24) to give the single particle emission spectrum. Finally, the single particle emission spectrum was integrated over impact parameter, b, to find the differential emissivity, Eq. (25), and over pre-collision speed, v, to find the resulting emission coefficient, Eq. (26).

Figure 8 shows the results of this calculation in comparison with the MD data. For frequencies (⁠ $ω > ω p e$⁠) the calculated emission coefficient agrees well with MD simulations up to $Γ ∼ 30$⁠. At very strong coupling (Γ = 100), the results are in fair agreement, but begin to depart from MD. This is expected because the high-frequency solution makes the molecular chaos approximation, which becomes increasingly invalid at strong coupling.^17,29 For $ω < ω p e$⁠, the high-frequency solution predicts a constant $j ( ω )$ as $ω → 0$ and departs from the results given by the MD simulations. This is expected because the high-frequency solution treats only the electron–ion binary collision timescale. In order to capture the low-frequency limit of the spectrum, one needs to include the timescales associated with multiple collisions.

While Eq. (33) agrees with MD at low frequencies, it falls short of the emerging peak around $ω ∼ ω p e$ in the emission coefficient at strong coupling. Particularly in the case of strong coupling, it is clear from Fig. 4 that $Z e i ( t )$ is not exponential and there are distinct oscillations at intermediate timescales. In order to better capture the emission coefficient in the mid frequency range, the model of $Z e i ( t )$ given by Eq. (28) can be modified to account for correlations on intermediate timescales.

Figure 9 shows that Eq. (36) can effectively capture the peak for $Γ = 1 − 30$⁠. It is important to note that the model for $Z e i ( t )$ in Eq. (34) is not derived from a Langevin-type equation like the previous exponential model. It is possible to more rigorously derive an approximation of $Z e i ( t )$ by assuming that the frictional force in Eq. (27) is temporally non-local and depends on the past velocity history of the particle; for example, see Hansen and McDonald.⁵³ Although this formalism provides a more rigorous derivation for $Z e i ( t )$ at strong coupling, the resulting emission coefficient does not approach a constant for high ω and is not convenient for forming a comprehensive solution with the high-frequency model from Sec. III B. The functional form of the velocity autocorrelation model in Eq. (34) is simply motivated by the increasingly oscillatory behavior in $Z e i ( t )$ at strong coupling. Section III E presents comprehensive solutions for the emission coefficient that apply across all frequencies.

A comprehensive model can be obtained by combining the high-frequency result from Sec. III B with the low- and mid-frequency models presented in Secs. III C and III D. To do this, it is important to note that the high-frequency limit of Eqs. (33) and (36) is constant; see Figs. 8 and 9. Thus, one can simply multiply a frequency-dependent factor onto the high-frequency solution in order to obtain a solution that is valid across all frequencies.

Figure 11 shows the normalized power computed from MD simulations data vs the prediction using mean force emission theory. The theory predictions agree well with the simulation data up to $Γ ≈ 30$⁠. Furthermore, the emission coefficient from Eq. (38) leads to better agreement with the MD data than the emission coefficient from Eq. (37). Using Eq. (37) to calculate the power leads to an under prediction in the $Γ = 1 − 30$ range where the bremsstrahlung peak appears near $ω ∼ ω p e$⁠. The theory also under-predicts the total radiated power for the Γ = 100 data point, as expected from the emission spectrum. Next, we discuss a comparison of the mean force emission theory and previous models.

Figure 12 shows both the general numerical solution and the small-angle approximation. At high frequencies, the two solutions disagree. Physically, this is because the small-angle approximation ignores large-angle collisions that correspond to high-frequency radiation. Both solutions also diverge logarithmically as $ω → 0$ while the MD results do not. This low-frequency divergence is a direct result of using the Coulomb potential to model binary collisions. The Coulomb potential is long-range, and thus the electron–ion interaction is not negligible when the particles are infinitely far away from each other.

Figure 13 further illustrates this point by showing numerical solutions for $j ( ω )$ obtained by varying the initial distance between the electron and ion when solving for the electron trajectory under a Coulomb potential, then plugging the resultant $F e ( t )$ into Eqs. (24)–(26). In the low-frequency limit, the solution obtained using a Coulomb potential depends on the initial position. Using a closer initial position is the same as limiting the electron–ion interaction and resolves the low frequency divergence. Increasing the initial position returns the familiar logarithmic divergence as $ω → 0$⁠. In order to resolve this divergence self-consistently, traditional models have considered screening in binary interactions.

Figure 12 also shows $j ( ω )$ obtained using the high-frequency limit from Eq. (26), but based on the Debye–Hückel potential of Eq. (22). Because the Debye–Hückel potential is the $Γ ≪ 1$ limit of the potential of mean force, it matches the MD calculations well at weak coupling and high frequencies. The Debye–Hückel potential is also short-range and thus does not diverge in the low-frequency limit. While the Debye–Hückel potential accurately captures the plateau in $j ( ω )$ at weak coupling, at strong coupling the simple screening provided by a Debye–Hückel potential fails to accurately predict the results given by MD; see Fig. 14. Indeed, this is where mean force emission theory provides an accurate description by using the potential of mean force.

It is important to note that these traditional approaches do not account for multiple collisions, and are thus limited to high frequencies. Next, we consider a common traditional approach to incorporating multiple collisions.

One common approach used to capture the low-frequency limit of the bremsstrahlung spectrum comes from quantum calculations of the conductivity and absorption coefficient.^23,25–27 In this approach, a Kubo–Greenwood solution is corrected at low frequencies by multiplying the solution by a Drude factor $ω 2 / ( ν 2 + ω 2 )$ where ν is a collision frequency obtained using the conductivity sum rule. This factor comes from the Drude derivation of conductivity, which models a frequency-dependent transport quantity in the low-frequency limit. In local thermodynamic equilibrium, the emission and absorption coefficients are simply related by the Blackbody spectrum, so we expect this multiple collision correction to impact the emission coefficient as well.

Of course, the Drude factor is the solution provided by Eq. (33), derived from the exponential model of the velocity autocorrelation function. Thus, the mean force emission theory justifies the use of a Drude correction in radiation transport calculations for weakly coupled plasmas. However, in strongly correlated plasmas the Drude approach misses the effects of oscillations in $Z e i ( t )$ that impact the mid frequency range around $ω ∼ ω p e$⁠. In this way, mean force emission theory provides justification for a more complete correction factor that goes beyond the Drude model and captures important effects around the plasma frequency at strong coupling. Although this work is purely classical, it still provides important insight into techniques used in quantum calculations.

The essence of mean force emission theory is that binary electron–ion interactions occur through the potential of mean force. Here, it has been shown that this concept successfully extends the classical theory of bremsstrahlung emission into the strongly coupled regime, showing agreement with MD simulations across the entire frequency range for $Γ ≲ 30$⁠. In terms of coupling strength, it is primarily limited by its application of the molecular chaos approximation, which assumes particles start from an uncorrelated state as an initial condition of their interaction. This is similar to the range of applicability of the previous mean force kinetic theory applied to transport coefficients,^17,29,49 and can be considered a generalization of this to a broader range of timescales. In the high-frequency limit, individual binary collisions are summed to predict the emission coefficient. In the low-frequency limit, the theory is recast using autocorrelation functions to capture the effect of multiple successive collisions and this is used to find analytic expressions for the bremsstrahlung emission coefficient. Importantly, the autocorrelation approach shows that in the mid frequency range (⁠ $ω ∼ ω p e$⁠) a peak in emission occurs at strong coupling. This work provides analytic correction factors for the low and mid frequency range that are combined with the high-frequency model to produce a model that applies to all frequencies.

In addition to the intrinsic utility of developing a new theory, this work also gives insight into the physics of common approaches to bremsstrahlung emission in the literature. It emphasizes the importance of screening, even in weakly coupled plasmas, as screening keeps the emission coefficient in traditional models from diverging as $ω → 0$⁠. It also justifies the use of a Drude correction factor in weakly coupled plasmas in order to account for multiple collisions. However, it also suggests that this simple correction may not be enough in strongly coupled plasmas, because strongly correlated motion leads to a peak in emission around the plasma frequency that is not captured by the Drude formula.

Finally, the work here considers a fully ionized hydrogen plasma with positively charged electrons and ions. The inclusion of attractive interactions between electrons and ions is expected to influence the emission coefficient and will be the subject of future work. For one, the electronic trajectories under a repulsive and attractive potential will be different and lead to differences in the high-frequency solution. In the low-frequency limit, the electron–ion collision frequency calculated using mean force kinetic theory is expected to differ between the attractive and repulsive cases. This is known as the Barkas Effect and has been studied rigorously through molecular dynamics simulations and theory.⁵⁴ However, while the work here considers a simplified model with a repulsively interacting electron–ion plasmas, this development of mean force emission theory lays the groundwork for future calculations that incorporate attractive collisions and quantum effects at short distances.

The authors thank Dr. Charles Starrett for helpful conversations on this work. This work was funded by the NNSA Stockpile Stewardship Academic Alliances under Grant No. DE-NA0004100. Additionally, this research was supported in part through computational resources and services provided by Advanced Research Computing (ARC), a division of Information and Technology Services (ITS) at the University of Michigan, Ann Arbor.

The authors have no conflicts to disclose.

J. P. Kinney: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). H. J. Lefevre: Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (equal). C. C. Kuranz: Conceptualization (equal); Funding acquisition (lead); Writing – review & editing (equal). S. D. Baalrud: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.