X-ray absorption via electron-ion bremsstrahlung in Maxwellian plasma at the exact consideration of Coulomb potential

TL;DR

This paper investigates X-ray absorption in high-temperature plasma considering exact Coulomb interactions, revealing conditions for efficient absorption of hard X-ray photons beyond the Born approximation.

Contribution

It provides a numerical analysis of electron-ion bremsstrahlung absorption accounting for Coulomb potential without relying on the Born approximation.

Findings

Efficient absorption coefficients are achievable in high-temperature plasma.

Exact Coulomb interaction significantly affects X-ray absorption calculations.

Results are relevant for high nuclear charge plasmas.

Abstract

Based on the results of exact consideration of electron-ion Coulomb interaction, by numerical simulations for ultimate absorption coefficient, we study the practically more interesting case of absorption of hard x-ray quanta of frequency when photon energy is of the order of plasma temperature, in high temperature plasma, within the high nuclear charges as well (at which the Born approximation is not applicable), constituting the temperature dependence of one-photon absorption coefficient. It is shown that one can achieve the efficient absorption coefficient in these cases.

Equations

$$\alpha =\frac{\hslash \omega }{c{E}_0^2/8\pi }\int \mathbf{f}(\mathbf{k})[w_a(\mathbf{k})-w_e(\mathbf{k})]\frac{d^3\mathbf{k}}{(2\pi {)}^3}$$

$$\int \mathbf{f}(\mathbf{k})d^3\mathbf{k}=(2\pi {)}^3{N}_e,$$

$$\mathbf{f}(k)=N_e{\left(\frac{2\pi {\hslash }^2}{\mu \kappa T}\right)}^{3/2}\exp \left(-\frac{{\hslash }^2{\kappa }^2}{2\mu \kappa T}\right)$$

$$a/k_{\omega }\ll 1.$$

$$\alpha ={\alpha }_B\sqrt{\pi }{2}^3\left(\frac{a}{k_{\omega }}\right){\chi }^{3/2}I\left(\chi ,\frac{a}{k_{\omega }}\right),$$

$$I\left(\chi ,\frac{a}{k_{\omega }}\right)=\underset{0}{\overset{\infty }{\int }}\exp \left(-\chi t^2\right)\frac{\ln \left(t+\sqrt{1+t^2}\right)}{\exp \left(\frac{2\pi a}{k_{\omega }t}\right)-1}$$

$$\times \left[1-\exp \left(-\chi \right)\exp \left(\frac{2\pi a}{k_{\omega }t}\right)\right]dt,$$

$${\alpha }_B=\frac{32{\pi }^2{N}_e{N}_i}{3}\frac{Z^2{e}^6}{\mu c{\hslash }^2{\omega }^3{k}_{\omega }},$$

$$\alpha ={\alpha }_{B}4\pi \left(\frac{a}{k_{\omega }}\right)\frac{\hslash \omega }{\kappa T}.$$

$$\alpha ={\alpha }_{B}4{\pi }^{3/2}\left(\frac{a}{k_{\omega }}\right){\left(\frac{\hslash \omega }{\kappa T}\right)}^{1/2}\frac{id}{d{x}_0}{\left|F\left(1-i\frac{a}{k_{\omega }},1,x_0\right)\right|}^2$$

$$\times {\left(1-\exp (-2\pi a/k_{\omega })\right)}^{-1}{\left(\exp (-2\pi a/k_T)-1\right)}^{-1},$$

Full text

I Introduction

With the advent of contemporary intense x-ray free electron lasers (FELs) 1 ; 2 ; 3 , interest has grown to x-ray stimulated bremsstrahlung (SB) absorption, because for the short laser pulses it may become the dominant mechanism of the absorption of strong laser radiation in plasma 4 . Further, in the field of intense x-ray laser an electron in the SB process may acquire the essential energy, which makes it effective mechanism for laser-plasma heating, absorbing even one-two photons 5 , 6 . The theoretical investigation of the SB process in gas or plasma medium in the strong electromagnetic (EM) radiation field has been dedicated many papers in various approximations over the scattering potential or EM wave field using nonrelativistic 4 , 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 ; 32 ; 33a ; 34 ; 35 or relativistic descriptions 36 ; 37 ; 38 ; 39 ; 40 . What concerns the multiphoton absorption of plasma at high radiation intensities, the relativistic theory of the latter has been succeeded in the Born 41 ; 42 ; 43 , eikonal 38 , and generalized eikonal 44 approximations over the scattering potential. Beyond this approximation for the infrared and optical lasers in the multiphoton interaction regime, when one could been applied classical theory and its main usable low frequency approximation is considered in Refs. 4 , 30 , and it have been generalized for the relativistic intensities of EM wave in Refs. 39 , 40 . It was shown in Ref. 45 , where the interaction of strong lasers with thin plasma targets of solid densities was investigated via particle-in-cell simulations, that SB is dominant for electron densities above 1021cm*-3*. The x-ray multiphoton absorption via inverse bremsstrahlung absorption in such dense classical as well as quantum plasmas in the present of strong wave field is investigated using quantum consideration in Ref. 5 in the Born approximation by the scattering potential of a plasma ion, which demonstrates that for the relativistic laser intensities the SB rate exhibits a tenuous dependence on plasma temperature. So for middle wave intensities the consideration of dependence on plasma temperature is of interest.

Meanwhile, for the optical or infrared lasers the quantum effects are small because of the smallness of the photon energy. However for intense x-ray radiation the nonlinear over the field quantum effects will be considerable. What concerns the multiphoton absorption of plasma at high (and superhigh) radiation intensities, the relativistic theory of the latter has been succeeded major in the regime of small angles scattering for electrons SB 5 , 40 . In spite of this, and to find out the SB absorption rates dependence versus the electrons temperature, in the present paper the inverse-bremsstrahlung absorption of an x-ray laser field in the classical Maxwellian plasma is considered by the quantum mechanical theory taking into account a scattering potential field exactly, while the x-ray radiation is treated by perturbation theory.

The earlier investigations of the one-photon absorption of high frequency electromagnetic radiation by plasma via the stimulated bremsstrahlung of electrons on the ion scattering potentials with the exact consideration of Coulomb potential have been succeeded analytically only in two boundary cases of photon frequency: for $\hbar\omega\ll\kappa T$ and $\hbar\omega\gg\kappa T$ ($\kappa$ is the Boltzmann constant, $\omega$ is the EM wave frequency) 35 . Taking into account the significance of the problem involving, specifically, the laser heating of plasma by current x-ray FELs (Free Electron Laser) and its diagnostics, in the present paper, based on mentioned results of exact consideration of electron-ion Coulomb interaction, by further numerical simulations for ultimate absorption coefficient, we study the practically more interesting case of absorption of hard x-ray quanta of frequencies $\hbar\omega\sim kT$ in high temperature plasma, within the high nuclear charges $Z$ as well (at which the Born approximation is not applicable), constituting also the temperature dependence of one-photon absorption coefficient.

The organization of this paper is as follows. In Sec. II the quantum dynamics of x-ray absorption via electron-ion SB in Maxwellian plasma is presented with analytical results for inverse-bremsstrahlung absorption coefficient at the exact consideration of Coulomb field. In Sec. III, the analytic formulas are considered numerically. Finally, conclusions are given in Sec. IV.

II Basic theory of SB taking Coulomb potential exactly

Let us investigate the inverse-absorption of x-ray radiation considering the Coulomb potential exactly, while EM wave field is treated by perturbation theory. The absorption coefficient $\alpha$ of the weak wave in the one-photon approximation determined by the following formula:

[TABLE]

where $E_{0}$ is the amplitude of the electric field strength $\mathbf{E}(t)=\widehat{\mathbf{x}}E_{0}\sin\omega t$ of a linearly polarized EM wave in dipole approximation with \widehat{\mathbf{x}}\is the unit vector, $cE_{0}^{2}/8\pi$ is the intensity of the EM wave in linear polarization case, $w_{a}$ and $w_{e}$ total probabilities of one-photon absorption and emission respectively, $\mathbf{f}(k)$ is the distribution function of the electrons over the wavevectors $\mathbf{k=p/\hbar}$ , which is normalized on the electron number density $N_{e}$ as follows:

[TABLE]

$\varphi(r)=-Ze/r$ is the the Coulomb attractive potential, $e$ is the electron charge, $Ze$ is the nuclear charge. We using the results obtained according to earlier analytic calculations 35 for the isotropic plasma with Maxwellian distribution function:

[TABLE]

($T$ is the temperature of electrons in plasma) in the case of hard radiation

[TABLE]

Here $a=Ze^{2}\mu/h^{2}$, $k_{\omega}=\left(2\mu\omega/\hbar\right)^{1/2}$. The absorption coefficient 35 in lowest order of $a/k_{\omega}$ can been reduced to the form:

[TABLE]

where $\chi=\hbar\omega/\kappa T$ is the electron-wave interaction parameter, $N_{i}$ is the ion number density, the both fields-dependent function $I\left(\chi,\frac{a}{k_{\omega}}\right)$ is

[TABLE]

[TABLE]

where

[TABLE]

is the absorption coefficient in the Born approximation, which is the same in the cases for anisotropic electron distribution - monochromatic beam 35 , as well for isotropic plasma with arbitrary momentum distribution of electrons in the corresponding hard quantum case ($a/k_{\omega}\ll 1$) 4 .

Previous analytical studies were successful in two boundary cases of photon frequency. The first, for high-temperature plasma, $\kappa T\gg\hbar\omega\gg\hbar^{2}a^{2}/2\mu$, by expanding exponent $\exp\left(-\chi\right)$ in power series and performing integration in (5) one was obtained the formula:

[TABLE]

The second, for the case of $\hbar\omega\gg\kappa T$, when the absorption coefficient in Maxwellian plasma was expressed by the following formula:

[TABLE]

[TABLE]

where $k_{T}=\sqrt{2\mu\kappa T}/\hbar$, $x_{0}=i4a/k_{\omega}$, $F\left(z,1,x_{0}\right)$ is the confluent hypergeometric function, $x_{0}=i4a/k_{\omega}$. In the Born approximation by the scattering potential, when $a/k_{\omega}\ll 1$ and $a/k_{T}\ll 1$, the expression (8) coincides with the expression (6).

Further we continue our study by numerical calculations for x-ray radiation one-photon inverse-bremsstrahlung absorption coefficient.

III Numerical results for one-photon SB absorption coefficient of hard

x-ray

To investigate the absorption coefficient $\alpha$ by numerical simulations we will utilize Eq. (4). We study the dependence of integral function $I\left(\chi,\frac{a}{k_{\omega}}\right)$ (5) versus parameter $\chi$. Besides, as follows from (4), when Coulomb potential is taken into account exactly the absorption coefficient in the limit of low temperatures $a/k_{T}\gg 1$ decreases exponentially $\alpha\sim\exp\left(-2\pi a/k_{T}\right)$.(according to the formula (8)).

The magnitude of the electron momentum change during the scattering process of low-temperature electrons in the Coulomb potential is much less than is necessary for real absorption of quantum with energy many times greater than the electron’s energy $\kappa T$.

We study the practically more interesting case of absorption of hard x-ray quanta, when $a/k_{\omega}\ll 1$, making integration by codes using Gauss’ rule. To show the dependence of the one-photon inverse-bremsstrahlung rates on the photon energy, in the Fig. 1 the absorption coefficient $\alpha$ scaled to $\alpha_{B}$ (6) of linearly polarized wave in Maxwellian plasma versus ${\chi}$ for various photon energies is shown ($a/k_{\omega}=10^{-5}$). As seen from this figure, the SB coefficient is considerably suppressed with the increase of the wave quanta energy, and in the case of hard radiation in high temperature plasma with the high nuclear charges $Z$ (beyond the Born approximation) the inverse-bremsstrahlung absorption coefficient dependence on ${\chi}$ (and plasma temperature) has the characteristic maximum after which it decreases with increasing of the parameter $\chi$. In Fig. 2 the absorption coefficient $\alpha$ scaled to $\alpha_{B}$ dependence versus parameter $\chi$ is showed for the frequencies $\omega\sim\kappa T/\hbar$ ($T_{n}\equiv T/\hbar\omega\sim 1$), which demonstrated that as the energy of the quantum of the wave increases, the probability of SB absorption reaches the maximal value when $\hbar\omega=\kappa T$.

To compare with the case when the electron-ion interaction is considered in the Born approximation 4 , in Fig. 3 the dependence of the one-photon hard radiation inverse-bremsstrahlung absorption coefficient $\alpha/\alpha_{B}$ scaled to $1/T_{n}$ versus the plasma temperature is shown for various laser radiation frequencies and for small $a/k_{\omega}$ (for linearly polarized wave), and Fig. 4 shows the same in the vicinity of $\hbar\omega=\kappa T$. Fig. 3 demonstrates that when the electron-ion interaction is taken into account exactly the absorption coefficient of high temperature plasma at $\kappa T\gg\hbar\omega\gg\hbar^{2}a^{2}/2\mu$ (${\chi\ll 1}$) dependence versus the plasma temperature $T_{n}$ is suppressed. Thus, now obtained results revealed a different dependence of absorption coefficient on the plasma temperature $T$, in contrast to the known results obtained in the Born approximation (6). If the Coulomb potential is taken into account exactly the absorption coefficient of hard radiation depends on plasma electron temperature as $\alpha\sim 1/T$, it decreases more slowly with plasma temperature, than in the Born approximation by the electron-ion interaction ($\alpha\sim\ln T/T^{3/2}$) 4 . As it is showed in Fig. 4, for hard radiation in the case of frequencies $\hbar\omega\sim\kappa T$ in high temperature plasma with the high nuclear charges $Z$ one-photon absorption coefficient dependence versus $T$ has the maximum. The numerical simulations give the characteristic curve for the dependence of $\alpha/\alpha_{b}$ versus plasma temperature.

IV Conclusion

We have presented the numerical investigation of x-ray radiation one-photon SB absorption coefficient under the limits on the main characteristic coefficients of the process, such as parameter ${\chi}$ and plasma temperature, based on earlier results of the exact quantum mechanical consideration of the dynamic of electron-ion Coulomb interaction in the presence of a transverse EM wave, when Coulomb potential is taken exactly, while the EM wave is treated by perturbation theory. The coefficient of SB absorption has been calculated considering the classical Maxwellian distribution. We study the more interesting case of absorption of hard x-rays ($a/k_{T}\ll 1$, ${\chi\sim 1}$) in high temperature plasma with high nuclear charges as well, at which the Born approximation by scattering potential is broken. The obtained results demonstrate that one can achieve the efficient absorption coefficient in this case, and the SB rate is suppressed with increase of the parameter ${\chi}$ and plasma temperature, and wave quanta energy. For large values of $T$, the absorption coefficient $\alpha$ decreases as $1/T$ in contrast to the case, when the scattering potential considering as perturbation where one has the dependence $\ln T/T^{3/2}$ 4 . In the limit of low plasma electron temperatures, $a/k_{T}\gg 1$, one-photon SB absorption coefficient exponentially decreases because of the classical nature of the electron interaction with the Coulomb potential, whereas it interaction with EM wave has quantum nature (one-photon inverse bremsstrahlung). The problem is significant connected with the laser heating of plasma by current x-ray FELs and its diagnostics.

Acknowledgements.

The author is deeply grateful to the Prof. Hamlet K. Avetissian for permanent discussions during the work on the present paper, for valuable comments and recommendations. This work was supported by the RA MES State Committee of Science.