X-ray generation by electron photo-recombination in charged atomic clusters formed in intense femtosecond laser pulses

The interaction of intense femtosecond laser pulses with large atomic clusters consisting of a large number of noble gas atoms is accompanied by efficient transformation of laser energy into hard x-rays. These x-rays are produced at the radiative recombination of electrons into atomic internal shells. The experimentally observed photon energy of hard x-rays from K-shell is of the order of some keV. Highly charged atomic ions inside the clusters are produced at the collisional inner ionization by heated electrons in the process of induced inverse bremsstrahlung. The review of various effects at the irradiation of large clusters by intense femtosecond laser pulses has been given in [1–3].

Above-barrier inner field ionization and subsequent collisional ionization of the cluster atoms and atomic ions generate multicharged ions and electrons. When the cluster size is increasing, more and more of the electrons remain trapped inside the cluster. X-rays are generated in the interaction of the trapped electrons with the cluster ions due to collisional–radiative recombination [4–12]. Three such processes are considered: direct photo-recombination of free electrons and ions, dielectronic recombination and excitation of ions by electron impact followed by collisional recombination. The main contribution to x-ray emission comes from electron impact excitation, while dielectronic recombination is less important, and photo-recombination is negligibly small. The excitation is created before the cluster explodes, whereas the emission of x-rays continues due to the laser pulse, or thereafter, depending on the laser intensity. Argon is a good choice for a cluster target because K-shell emission is readily obtained and is more suitable for detailed spectroscopic analysis than krypton or xenon, which would yield more complex L- and M-shell spectra.

In this paper, we suggest the new mechanism of the generation of hard x-ray radiation. We develop the theory of the photo-recombination for electrons which first leave atomic clusters due to the outer ionization and then go into the ground quantum state of the neighbouring charged cluster. Hot electrons leave the cluster, and then they can be captured by another positively charged cluster. The value of cluster charge depends on the cluster size. For example, deuterium clusters with a diameter of 5 nm (a few thousand atoms per cluster) in the laser field with the intensity of 10¹⁷ W cm⁻² are quickly transformed into a positive spherical ball consisting only of deuterons [13]. Due to Coulomb explosion the cluster diameter increases up to 10 nm only in 30 fs.

A free electron can be captured into the ground state of the Coulomb potential well of the charged cluster with the ejection of the spontaneous high-energy x-ray photon. The depth of the potential well is equal to several keV depending on the cluster parameters and parameters of the laser pulse. Outer ionization (i.e. ejection of free electrons from the cluster) can be considered using two different assumptions for the remaining electron distribution within the cluster: (1) the remaining electrons concentrate in the central region of the cluster: this leads to a neutral cluster core; a positive charged surface region consists only of positive atomic ions; this assumption corresponds to a statistical equilibrium of the electron cloud; (2) the remaining electrons are distributed uniformly within the whole charged cluster ball because of their high velocities (non-equilibrium distribution). In [14, 15], we considered photo-recombination of ejected electrons into the charged cluster, considering using as a quantum-mechanical potential well, the second assumption about uniform electron distribution inside the cluster.

Here we focus on the case of a surface cluster charge (the first assumption). This assumption describes an equilibrium state in electron dynamics when the time for establishment of the equilibrium Maxwell distribution determined by electron–electron collisions inside the cluster is less than the laser pulse duration. At the front of the laser pulse, the multiple step-wise inner ionization occurs quickly up to some final value of the charge multiplicity of atomic ions. The time between electron–electron collisions is of the order of 1–10 fs depending on the laser intensity, final ion charge and electron temperature, i.e. it is less than the pulse duration.

Potential electron energy in the field of the surface cluster charge can be approximated as (figure 1)

$$\begin{equation} U(r) = \left\{ {\begin{array}{@{}ll@{}} {-} Q/r,& r \ge R;\\ {-} Q/R,& r < R. \\ \end{array}} \right. \end{equation} \tag{ 1 }$$

Here R is the cluster radius; Q is the cluster charge after outer ionization. We assume here that the surface region of the cluster charge is sufficiently thin so that the whole cluster charge Q is found on the cluster surface. This assumption is realized for sufficiently large atomic clusters with weak outer ionization compared to the inner ionization (10³–10⁵ atoms in the cluster).

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Let us consider the wavefunction of the ground state in this potential well, equation (1), with the orbital quantum number l = 0. The Schrödinger equation for the radial function ϕ(r) is of the form (here and thereafter the atomic system of units is used, e = m_e = ℏ = 1)

$$\begin{equation} \frac{1}{{r^2 }}\frac{{\rm d}}{{{\rm d}r}}\left( {r^2 \frac{{{\rm d}\phi }}{{{\rm d}r}}} \right) + 2[E - U(r)]\phi = 0. \end{equation} \tag{ 2 }$$

Here, E < 0 is the electron bound energy.

The regular normalized solution of equation (2) for the ground state can be approximated as

$$\begin{equation} \phi (r) = \left\{ \begin{array}{@{}ll@{}} \dsty\frac{1}{r}\sqrt {\dsty\frac{2}{R}} \sin \left( {\dsty\frac{{\pi r}}{R}} \right),& r < R \\\ms 0,& r > R.\\ \end{array} \right. \end{equation} \tag{ 3 }$$

Indeed, let us consider, as an example, the large atomic cluster consisting of Ar, Kr or Xe atoms with the radius R = 5 nm and with the typical number density of plasma electrons n_e = 2 ×10²² cm⁻³ after irradiation by intense femtosecond laser pulse with an intensity of 10¹⁷ W cm⁻². This cluster contains N = n_e(4πR³/3) = 10⁴ atoms. When the final charge multiplicity of atomic ions is Z = 15, the irradiated cluster contains 1.5 × 10⁵ electrons. According to various derivations [1], typically approximately 20% of electrons are ejected by the laser field from the cluster due to outer ionization. Hence, the charge of the irradiated cluster is Q ≈ 3 × 10⁵. On the cluster surface r = R, the wavefunction is extremely small. Hence, we can take ϕ = 0 in the region r > R. Thus, we obtain a simple expression for the energy of the ground state:

$$\begin{equation} E = \frac{{\pi ^2 }}{{2R^2 }} - \frac{Q}{R} < 0; \qquad \frac{{\pi ^2 }}{{2R^2 }} \ll \frac{Q}{R}. \end{equation} \tag{ 4 }$$

In order to derive the photo-recombination cross section, we first calculate the cross section of the inverse process, i.e. the photo-ionization cross section. The differential photo-ionization cross section is

$$\begin{equation} {\rm d}\sigma _{{\rm ion}} = \frac{p}{{2\pi \omega c}}|M_{fi} |^2\, {\rm d}\Omega _{\bf p} . \end{equation} \tag{ 5 }$$

Here p is the electron momentum in the final continuum state, ω is the frequency of the absorbed photon, M_fi is the matrix element for bound-free electron transition and c is the speed of light. According to the energy conservation law, we have ω = –E + p²/2. It should be noted that the electron energy in the final continuum state p²/2 ≪|E|, since when p²/2 > |E|, the cross section of photo-ionization is negligibly small. Then, ω = |E| ≈ Q/R. The final electron state f is described by the WKB wave $C\exp \left( {{\mathop{\rm i}\nolimits} {\bf pr}} \right)$. This is correct since the wavefunctions of the initial and final electron states overlap only at r < R. In this region, the potential of the charged cluster, equation (1), is constant. The WKB continuum wavefunction of the final electron state is the plane wave in this region with the coefficient

$$\begin{equation*} C = \sqrt {\frac{p}{{\left( {2|E|} \right)^{1/2} }}} \approx \sqrt {\frac{p}{{\left( {2Q/R} \right)^{1/2} }}} . \end{equation*}$$

The transition matrix element M_{f i} in equation (5) is of the form

$$\begin{equation} M_{fi} = C({\bf ep})\int \exp ({-} {\rm i} {\bf pr} + {\rm i} {\bf kr})\phi (r)\, {\rm d} {\bf r} . \end{equation} \tag{ 6 }$$

Here e is the unit polarization vector of the absorbed photon, k is the photon wave vector, k = ω/c = = Q/Rc. It should be noted that the dipole approximation is inapplicable for photon absorption and emission.

In order to calculate this matrix element, let us introduce the notation θ for the angle between vectors r and (k − p). Integrating equation (6) over θ one obtains

$$\begin{equation} M_{fi} = \frac{{4\pi C({\bf ep})}}{{|{\bf p} - {\bf k}|}}\int\nolimits_0^R {\sin (|{\bf p} - {\bf k}|r)\phi (r)r\,{\rm d}r} . \end{equation} \tag{ 7 }$$

Let us further introduce the angle φ between vectors k and p. The vector k can be chosen along the axis Z of the spherical coordinate system, the polarization vector e can be chosen along the axis X. Let us introduce the angle ψ between the projection of the vector p to the plane (X, Y) and the axis X. Then (ep) = psin φcos ψ. Equation (7) can be rewritten in the form

$$\begin{eqnarray} &&\fl M_{fi} = \frac{{4\sqrt 2 \pi pC\sin \varphi \cos \psi }}{{\beta \sqrt R }}\int\nolimits_0^R {\sin \left( {\beta r} \right)\sin \left( {\frac{{\pi r}}{R}} \right)\;{\rm d}r} ; \nonumber\\ &&\fl \beta = \sqrt {p^2 + k^2 - 2pk\cos \varphi } \nonumber\\ &&\hphantom{\fl \beta } = \sqrt {p^2 + ( {Q/Rc} )^2 - ( {2pQ/Rc})\cos \varphi } . \end{eqnarray} \tag{ 8 }$$

Integral

$$\begin{equation} I = \int\nolimits_0^R {\sin ( {\beta r} )\sin \left( {\frac{{\pi r}}{R}} \right)\;{\rm d}r} = \frac{{\pi R\sin (\beta R)}}{{\pi ^2 - \beta ^2 R^2 }}. \end{equation} \tag{ 9 }$$

The differential cross section, equation (5), takes the form (after integration over angle ψ)

$$\begin{equation} {\rm d}\sigma _{{\rm ion}} = \frac{{16\pi ^2 p^4 I^2 \sin ^3 \varphi }}{{\omega c\beta ^2 \left( {2QR} \right)^{1/2} }}\,{\rm d}\varphi . \end{equation} \tag{ 10 }$$

and

$$\begin{eqnarray} &&\fl \sigma _{{\rm ion}} = \frac{{16p^4 R^{5/2} }}{{c\sqrt {2Q^3 } }}\int\nolimits_{ - 1}^1 {\left[ {\frac{{\sin \beta R}}{{\beta ( {1 - ( {\beta R/\pi })^2 })}}} \right]^2 } (1 - t^2 )\,{\rm d}t,\nonumber\\ &&t = \cos \varphi . \end{eqnarray} \tag{ 11 }$$

Let us introduce the dimensionless electron momentum $P = \frac{{pRc}}{Q}$. Then, $\beta = \frac{Q}{{Rc}}\sqrt {P^2 + 1 - 2Pt}$ and

$$\begin{eqnarray} &&\fl \sigma _{{\rm ion}} = \frac{{8P^4 \sqrt {2QR} }}{{c^3 }}\nonumber\\ &&\times\,\int\nolimits_{ - 1}^1 {\frac{{\sin ^2 \big( {\frac{Q}{c}\sqrt {P^2 + 1 - 2Pt} } \big)(1 - t^2 )\,{\rm d}t}}{{( {P^2 + 1 - 2Pt})\big( {1 - \big( {\frac{Q}{{\pi c}}} \big)^2 ( {P^2 + 1 - 2Pt})} \big)^2 }}} \end{eqnarray} \tag{ 12 }$$

The cross section of photo-recombination can be found using the principle of detail equilibrium for direct and inverse processes:

$$\begin{equation} \sigma _{{\rm rec}} = \frac{{2\omega ^2 }}{{c^2 p^2 }}\sigma _{{\rm ion}} = \frac{2}{{P^2 }}\sigma _{{\rm ion}} . \end{equation} \tag{ 13 }$$

The final result of the recombination cross section is

$$\begin{eqnarray} &&\fl \sigma _{{\rm rec}} (P) = \frac{{16P^2 \sqrt {2QR} }}{{c^3 }}\nonumber\\ &&\times\int\nolimits_{ - 1}^1 {\frac{{\sin ^2 \big( {\frac{Q}{c}\sqrt {P^2 + 1 - 2Pt} } \big)(1 - t^2 ){\rm d}t}}{{( {P^2 + 1 - 2Pt})\big( {1 - \big( {\frac{Q}{{\pi c}}} \big)^2 ( {P^2 + 1 - 2Pt})}\big)^2 }}} .\nonumber\\ \end{eqnarray} \tag{ 14 }$$

The rate of photo-recombination can be obtained by multiplying by electron velocity and dividing by cluster volume.

We can see that the cross section has a sharp peak at P = 1, i.e. at p = Q/Rc. The maximum value of the cross section at P = 1 is equal to

$$\begin{equation*} \sigma _{{\rm rec}} (P = 1) = \frac{{8\sqrt {2QR} }}{{c^3 }}\int\nolimits_{ - 1}^1 {\frac{{\sin ^2 \big( {\frac{Q}{c}\sqrt {2 - 2t} } \big)(1 + t)\,{\rm d}t}}{{\big( {1 - 2\big( {\frac{Q}{{\pi c}}} \big)^2 ( {1 - t})} \big)^2 }}} . \end{equation*}$$

Changing the integration variable

$$\begin{equation*} x = \frac{Q}{c}\sqrt {2 - 2t} , \end{equation*}$$

and taking into account the fact that for large clusters Q/c ≫ 1, one obtains

$$\begin{equation*} \sigma _{{\rm rec}} (P = 1) = \frac{{16\pi ^4 \sqrt {2R} }}{{cQ^{3/2} }}\int\nolimits_0^\infty {\frac{{\sin ^2 ( x )x\,{\rm d}x}}{{( {\pi ^2 - x^2 })^2 }}} . \end{equation*}$$

Numerical derivation of the integral (it does not contain singularity at x = π) allows us to find the maximum cross section of photo-recombination

$$\begin{equation} \sigma _{{\rm rec}} (P = 1) = \frac{{490R^2 }}{{c\left( {QR} \right)^{3/2} }} = \frac{{3.63R^2 }}{{\left( {QR/a_B } \right)^{3/2} }} \ll \pi R^2 . \end{equation} \tag{ 15 }$$

Here we restored the usual units, and the Bohr radius is a_B = ℏ²/me².

Oppositely, when c/Q ≪ (P − 1) ≪ 1, then the argument of sine in equation (14) oscillates fast, sin ²x → 1/2 and values of t → 1 are significant in the integrand of equation (14). One obtains, changing the integration variable 1 − t = x ≪ 1:

$$\begin{eqnarray} &&\fl \sigma _{{\rm rec}} (P) = \frac{{16\pi ^4 c\sqrt {2QR} }}{{Q^4 }}\int\limits_0^\infty {\frac{{x\,{\rm d}x}}{{( {( {P - 1})^2 + 2x} )^3 }}}\nonumber\\ &&\hphantom{\fl \sigma _{{\rm rec}} (P) } = \frac{{2\pi ^4 \sqrt 2 }}{{( {P - 1})^2 }}\frac{{cR^2 }}{{Q^{7/2}( {R/a_B })^{3/2} }}. \end{eqnarray} \tag{ 16 }$$

We can unify equations (15) and (16):

$$\begin{equation} \sigma _{{\rm rec}} (P) = \frac{{3.63R^2 c^2 }}{{Q^{7/2}( {R/a_B })^{3/2} [ {1.80( {P - 1})^2 + ( {c/Q})^2 }]}}. \end{equation} \tag{ 17 }$$

In the typical case (see above) R = 100 au and Q = 3 × 10⁵, we have Q/Rc = 30 (since e²/ℏc = 1/137). Thus, the electron momentum p = 30 au in maximum (the electron energy is about of 12 keV). The dependence, equation (14), of the recombination cross section on the dimensionless electron momentum P is shown in figure 2 for the case Q = 3 × 10⁵. It is in agreement with a simple expression, equation (17).

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According to our results [14, 15], for a uniformly charged cluster the maximum photo-recombination cross section is

$$\begin{equation} \sigma _{{\rm rec}} (P = 1,{\rm uniform}) = \frac{{0.063R^2 }}{{\left( {QR/a_B } \right)^{5/4} }}. \end{equation} \tag{ 18 }$$

The ratio of equations (15) and (18) is

$$\begin{equation*} \frac{{\sigma _{{\rm rec}} (P = 1)}}{{\sigma _{{\rm rec}} (P = 1,{\rm uniform})}} = \frac{{57.6}}{{( {QR/a_B } )^{1/4} }}. \end{equation*}$$

In the above example, R = 100 au and Q = 3 × 10⁵ both cross sections are of the same order of magnitude.

Of course, the photo-recombination cross section on charged clusters is much less than the photo-recombination cross section on the separate multicharged ions inside the cluster. However, the photon energy ℏω = Qe²/R is greater. At R = 100 au and Q = 3 × 10⁵ it is equal to 3000 au = 80 keV. The photon energy of the x-ray even for the Ar¹⁷⁺ ion is only 3.1 keV. A decay channel of relevance for diagnostics and possible applications of laser-cluster interactions is the emission of energetic photons particularly in the soft x-ray domain. The x-ray spectra contain line emission that reflects the recombination of electrons in weakly bound atomic levels with core-level vacancies.

In conclusion, let us estimate the number N of x-ray photons emitted according to our model. Using equation (15), one obtains in the maximum of the photo-recombination cross section:

$$\begin{equation*} N = N_e \sigma _{{\rm rec}} (P = 1)v_e n_e \tau = \frac{{3.6Z^2 n_i^2 \tau }}{{c\sqrt {QR} }}\left( {\frac{R}{L}} \right)^3 V. \end{equation*}$$

Here v_e is the electron velocity, τ is the laser pulse duration, n_e is the number density of ejected electrons in the electron plasma between irradiated clusters, Z is the ion charge, L = 10R is the distance between the neighbouring clusters, n_i is the atomic number density inside the cluster, V is the laser focal volume. Substituting the above-cited typical values of parameters, one obtains N = 10⁷ x-ray photons. Thus, we can conclude that these photons can be observed in experiments.

This work was supported by the Russian Foundation for Basic Research (project N 13-02-00072).