Many-electron effects on the x-ray Rayleigh scattering by highly charged He-like ions

Many-electron effects on the x-ray Rayleigh scattering by highly charged He-like ions

A. V. Volotka, V. A. Yerokhin, A. Surzhykov, Th. Stöhlker, and S. Fritzsche Helmholtz-Institut Jena, D-07743 Jena, Germany
Department of Physics, St. Petersburg State University, 198504 St. Petersburg, Russia
Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
GSI Helmholtzzentrum für Schwerionenforschung, D-64291 Darmstadt, Germany
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, D-07743 Jena, Germany
Abstract

The Rayleigh scattering of x-rays by many-electron highly charged ions is studied theoretically. The many-electron perturbation theory, based on a rigorous quantum electrodynamics approach, is developed and implemented for the case of the elastic scattering of (high-energetic) photons by helium-like ion. Using this elaborate approach, we here investigate the many-electron effects beyond the independent-particle approximation (IPA) as conventionally employed for describing the Rayleigh scattering. The total and angle-differential cross sections are evaluated for the x-ray scattering by helium-like Ni, Xe, and Au ions in their ground state. The obtained results show that, for high-energetic photons, the effects beyond the IPA do not exceed 2% for the scattering by a closed -shell.

pacs:
32.80.Wr, 31.15.V-, 31.30.J-

I Introduction

The elastic scattering of a light by bound electrons is commonly known as Rayleigh scattering. It has been found a powerful and versatile tool for investigating the structure and dynamics of bound electrons as well as for probing the atomic environment. Apart from the fundamental interest, a quantitative understanding of the Rayleigh scattering is needed in various fields, including the study of the solid-state, complex molecules or even nano-objects Sfeir et al. (2004); Kampel et al. (2012); Kulik et al. (2012); Wu et al. (2015), astrophysics Maeda et al. (2012); The and Burrows (2014) as well as in medical diagnostics Zhang et al. (2012).

Experimentally, recent progress in exploring the Rayleigh scattering of hard x rays by atomic or solid-state targets has been achieved by a major improvement of the detection techniques as well as the quality of light sources. Indeed, novel solid-state photon detectors Fritzsche et al. (2005); Weber et al. (2010) together with advances in the available synchrotron sources Bilderback et al. (2005) have paved the way towards new generations of experiments. For example, a measurement of the linear polarization of the Rayleigh scattered light has been recently performed with the help of the segmented solid-state detectors at the PETRA III synchrotron at DESY K.-H. Blumenhagen et al., in preparation. (). The advances in the Rayleigh scattering experiments nowadays also demand further and accurate predictions at the side of theory.

The theoretical investigations on the elastic scattering of photon by bound electrons dates back to the mid-1930s Franz (1936). While initially quite simple approximations were applied, based on the atomic form factors, the rigorous quantum electrodynamical (QED) approach has later been developed by using the relativistic second-order -matrix amplitude; we refer the reader to Ref. Kissel and Pratt (1985) for a comprehensive historical overview. This latter approach has now become the standard for treating the Rayleigh scattering. These developments were triggered specially by the pioneering work of Brown, Peierls, and Woodward Brown et al. (1954), who developed the method for the calculation of the second-order transition amplitudes. Within this QED approach, quite a number of calculations were carried out for different atoms and photon energies Brenner et al. (1954); Brown and Mayers (1956, 1957); Johnson and Cheng (1976); Kissel et al. (1980); Kane et al. (1986); Roy et al. (1986); we refer the reader to Ref. Roy et al. (1999) for a comprehensive review on this approach. In recent years Costescu et al. (2011); Safari et al. (2012); Surzhykov et al. (2013, 2015); Safari et al. (2015) the angular and polarization correlation between the incident and outgoing photons have also been investigated. Unfortunately, however, the formalism of the second-order -matrix theory does not enable one to investigate systematically the many-electron effects on the Rayleigh scattering. This is caused by the electron-electron interaction that is treated only approximately in this formalism by means of a central screening potential, and which is known as screening potential or independent-particle approximation (IPA). Until the present, many-electron effects beyond the IPA were only investigated for the helium atom in Ref. Lin et al. (1975). It was found that the interelectronic-interaction corrections beyond the IPA significantly modify the Rayleigh cross section for the low-energy photons and disappear for higher energies. The main aim of the present study is to investigate the many-electron effects for highly charged ions.

In this paper, we present a rigorous QED treatment of the Rayleigh scattering of light by highly charged helium-like ions. The QED perturbation expansion with regard to the interelectronic interaction is applied up to the first-order for the Rayleigh scattering as an important light-matter interaction process at medium and high photon energies. In particular, formulas have been derived for the zeroth- and first-order interelectronic-interaction corrections to the scattering amplitude. This framework enables one to systematically investigate the many-electron effects beyond the IPA. In Sec. II, the details of this formalism are presented, while the computational techniques and methods are discussed in Sec. III. The total and angle-differential cross sections are evaluated for the scattering of x rays on the ground state of helium-like Ni, Xe, and Au ions. We shall here consider especially two experimental scenarios in which the incoming light is either unpolarized or completely linearly polarized. In Sec. IV, the obtained numerical results are presented and discussed. Attention is paid to the comparison between the IPA and the many-electron description. For the Rayleigh scattering of x rays by helium-like ions in their ground state, we find that the many-electron effects beyond the IPA treatment do not exceed 2 %. In Sec. V, we briefly summarize our results and give a brief outlook.

Relativistic units () and the Heaviside charge unit [, ] are used throughout the paper.

Ii Theoretical background

The process of elastic photon scattering is characterized by the energy conservation of the incident and outgoing photons in the center-of-mass frame of the overall scattering system. This means that no energy transfer is possible between the photon and the target with its internal degree of freedom. However, since the energy of the scattered photon is typically much smaller than the atom’s rest-mass energy, we therefore consider in the following the scattering of a photon in the rest frame of the atom, so that the incoming and outgoing photon energies simply remain the same. For the theoretical description of (quite) heavy atoms, moreover, it is naturally to utilize the Furry picture, in which the (infinitely heavy) nucleus is taken as the source of the classical Coulomb field and where the interaction of electrons with this field is then treated exactly by just solving the Dirac equation in the nuclear Coulomb potential.

According to the basic principles of QED Berestetsky et al. (1982), the differential cross section for the scattering of a photon by an atom is given by

(1)

where the initial and final state of the photon are characterized by the four-momentum and and polarization vectors and , respectively. Here, the zero and spatial components of the four-vector define the photon frequency and the photon wave vector . Moreover, the total energy of the bound electrons are and for the initial and final state of the atom. The shorthand notations and stand for a unique specification of the bound-electron states and , where and are the total angular momenta, and their corresponding projections, and where and denote all additional quantum numbers that are needed for a unique specification of the states. The energy conservation clearly shows that no energy transfer occurs to the atom and that the moduli of the wave vectors are the same for the incoming and outgoing photons, . Thus, the angle-differential cross section for the elastic scattering in a solid angle takes the form

(2)

The scattering amplitude can be related to the scattering -matrix element by following expression

(3)

and where and denote the scattering and identity operators. The scattering -matrix element contains two types of processes: the scattering by the Coulomb potential of the nucleus as well as the scattering by the bound electrons. The first type corresponds to the Delbrück scattering amplitude, while the second one is usually defined as the Rayleigh scattering by the bound electrons. Here, we restrict ourselves to the Rayleigh scattering only. In order to evaluate the corresponding Rayleigh -matrix element, which we denote as , one has to employ the bound-electron QED perturbation theory. For this purpose, we here utilize the (so-called) two-time Green-function method as developed in Refs. Shabaev (1990a, b, 2002), where the perturbation theory is formulated for the two-time Green functions. The Rayleigh -matrix element can be generally related to the two-time Green functions by the equation Shabaev (2002)

(4)

where the contour encloses the pole corresponding to the bound-electron states with the energy . This contour also excludes all further singularities of the Green functions , , and , which are defined in a similar way as in Ref. Shabaev (2002). The factor is a renormalization constant for the absorbed and emitted photons lines. The Green function describes the scattering of a photon by bound electrons, while the Green functions and characterize the initial and final bound-electron states. Since the Rayleigh -matrix element is expressed in terms of the Green functions, it can be calculated order-by-order by applying the QED perturbation theory with regard to the radiation-matter interaction.

In the following, we shall consider in further detail the non-resonant Rayleigh scattering of light by helium-like ions, i.e. for photon energies which are not close to possible excitations of any quasi-stationary bound state. The zeroth-order two-electron wave functions and are constructed as linear combinations of Slater determinants, and as

(5)

where is a shorthand notation for the summation over the Clebsch-Gordan coefficients

(6)

the one-electron total angular momentum and its projection, is the permutation operator, giving rise to the sign of the permutation for any permutation of the electron coordinates. The same notations hold for the final state . The one-electron wave functions and are found by solving the Dirac equation with the Coulomb potential of the nucleus.

ii.1 Zeroth-order approximation

In order to calculate the -matrix element of the Rayleigh scattering according to Eq. (4), we decompose the two-time Green functions in a perturbation series with an expansion parameter and group the terms of the same order together. In zeroth-order approximation, the corresponding Feynman diagrams are depicted in Fig. 1. Then, by employing Eqs. (3) and (4) we obtain the zeroth-order Rayleigh scattering amplitude in a following form

(7)

where the superscript “” indicates the order of the perturbation expansion. According to the Feynman rules, the zeroth-order Green function can be written as

(8)

and are the emission and absorption operators, is the vector of the Dirac -matrices, and where the zeroth-order energy of the bound electrons is equal to the sum of the one-electron Dirac energies . Furthermore, the factor in expression (8) preserves the proper treatment of poles of the electron propagators. In the case of the non-resonant scattering, the expressions in the curly brackets of Eq. (8) are regular functions of and as long as and . By substituting Eq. (8) into Eq. (7) and by integrating over and , one thus easily obtains

(9)

Before we shall proceed further, let us discuss this expression for the zeroth-order Rayleigh scattering amplitude . Obviously, this amplitude splits into two pieces as indicated by the round brackets. These pieces correspond to either the scattering by just the (first round brackets) or (second round brackets) electrons. In the zeroth-order approximation, hence, the obtained Rayleigh scattering amplitude corresponds to the IPA formulas as they are widely used for the theoretical description of the Rayleigh scattering, see, e.g., Ref. Roy et al. (1999).

Figure 1: Feynman diagrams of the Rayleigh scattering in zeroth-order approximation. The double line indicates the electron propagators in the Coulomb field of the nucleus, while the photon absorption and emission are depicted by the wavy line with incoming and outgoing arrows, respectively.

Using the (zeroth-order) Rayleigh scattering amplitude from above, the angle-differential Rayleigh scattering cross section defined by Eq. (2) is given in zeroth-order approximation by

(10)

while the corresponding zeroth-order total cross section gives rise to

(11)

In the following, we shall go beyond the zeroth-order approximation and investigate the effects due to the electron-electron correlations. To do so, we need to account for the interelectronic-interaction correction to the Rayleigh scattering amplitude. In the next subsection we now present the corresponding formulas for the first-order interelectronic-interaction effects.

Figure 2: Feynman diagrams to represent the first-order interelectronic-interaction corrections to the Rayleigh scattering process. The internal wavy line stands for the photon propagator. The rest notations are the same as in Fig. 1.

ii.2 First-order interelectronic-interaction correction

In order to obtain the expression for the first-order interelectronic-interaction correction to the Rayleigh -matrix element, we have to collect all first-order terms in Eq. (4) as they arise from the perturbation expansion of the Green functions. Using Eq. (3), the corresponding interelectronic-interaction correction to the scattering amplitude is given by

(12)

where the first-order interelectronic-interaction correction to the scattering Green function can be represented by Feynman diagrams as displayed in Fig. 2. The corresponding corrections to the Green functions and are defined by the first-order interelectronic-interaction diagram shown in Fig. 3.

Figure 3: One-photon exchange Feynman diagram. Notations are the same as in Figs. 1 and 2.

In addition, by making use of the Feynman rules for the two-time Green functions one can derive the final expressions for the scattering amplitude correction . This derivation is technically very similar to those as performed in Refs. Indelicato et al. (2004); Volotka et al. (2011), where the interelectronic-interaction effects were investigated for one- and two-photon bound-bound transitions in helium-like ions. For the sake of brevity, we therefore omit here the cumbersome expressions and go on to the final form of the first-order interelectronic-interaction correction which can be written as a following sum

(13)

Here, the different contributions are distinguished by the superscripts , , and and correspond to the so-called irreducible parts of the diagrams as shown in Fig. 2 (A), (B), (C), and (D), respectively. These terms are given by the expressions:

(14)
(15)
(16)
(17)

and where refers to the interelectronic-interaction operator with the photon propagator . The diagrams displayed in Fig. 2(A) and (C) contain also the so-called reducible parts, while this is not the case for the diagrams shown in Fig. 2(B) and (D). Thus, the last term in Eq. (13) is the total reducible contribution, which arises from the second term in the square brackets of Eq. (12) and from the reducible parts of the diagrams (A) and (C) in Fig. 2. The total reducible term can be written as

(18)

where

(19)

is the one-photon exchange correction and

(20)

is the first-order derivative of the one-photon exchange correction.

Finally, the angle-differential Rayleigh scattering cross section up to the first-order in the interelectronic interaction is given by

(21)

while the corresponding total cross section takes the form

(22)

These Rayleigh scattering cross sections take rigorously into account the many-electron effects on a level of the one-photon exchange. These calculations enable us to analyze systematically the importance of the many-electron effects upon the elastic scattering cross sections. However, before we continue with this analysis let us consider a different approach for including the many-electron effects in the next subsection.

ii.3 Screening potential approximation

Apart from the rigorous approach above, the zeroth-order or independent-particle approximation of Eq. (9) also facilitates a partial account of the interelectronic-interaction effects in the Rayleigh cross sections. To this end, we shall start from an extended Furry picture in which a central screening potential is incorporated into the zeroth-order Hamiltonian. In this case, the formula (9) for the zeroth-order Rayleigh amplitude remains formally the same, while the initial, intermediate, and final one-electron wave functions are now generated within a mean-field potential by including, in addition to the Coulomb field, also some screening potential. In practice, however, this approach only includes some (major) parts of the many-electron effects for highly charged ions. Here, we shall separate this part from the complete first-order result in Eq. (13) by restricting ourselves to the static Coulomb part in the photon propagator as well as to the spherical terms in its multipole expansion. Below, we shall refer to this approximation by the superscript ”scr” in the Rayleigh scattering amplitude

(23)

Let us mention here that the contribution of the diagrams (D) in Fig. 2 should also be excluded in this case since these diagrams are of inherent many-electron character. In the leading order this approximation is equivalent to the IPA with the Dirac-Hartree-Fock potential. The difference arisen from the higher-order terms can be neglected for highly charged ions.

With these ”screening” corrections to the Rayleigh scattering amplitude, the angle-differential and total cross sections take now the form

(24)

and

(25)

respectively. Since this approximation can be obtained from Eq. (9) by just making use of a screening potential in solving the Dirac equation, we shall refer to it as the IPA. — In the next section, we now discuss the numerical procedure and the methods employed in the computation of the cross sections.

Iii Computations

The formulas (9), (13), and (23) for the transition amplitudes, as obtained in the previous section, require further simplifications to make detailed computations feasible. For instance, in order to perform the angular integrations in the one- and two-electron matrix elements, we utilize the well-known multipole expansion technique. Indeed, this angular integration can be carried out analytically by expanding the transition operators and as well as the photon propagator in multipole series. For the sake of brevity, we do not recall the corresponding expressions here and just refer the reader to the literature for further details Varshalovich et al. (1988). The infinite multipole summations over the incoming and outgoing photon multipoles are further restricted by analyzing the convergence and more often than not we summed up to 10 multipoles.

Numerically most demanding in the computations is the infinite summation over the complete Dirac spectrum and , which not only contain the bound states but also the positive- and negative-energy Dirac continuum. In order to perform such a summation several independent approaches were employed previously in the consideration of the Rayleigh scattering. One method formulated in Ref. Brown et al. (1954) is based on a solution of an inhomogeneous Dirac equation, a method that was found quite successful and was utilized in a good number of calculations of the elastic scattering cross sections Brenner et al. (1954); Brown and Mayers (1956, 1957); Johnson and Cheng (1976); Kissel et al. (1980); Kane et al. (1986); Roy et al. (1986). Another approach is known as the finite basis-set method. This technique enables one to replace an infinite summation in the spectral representation of the electron propagator by a summation over a finite basis set and was utilized for calculating the Rayleigh scattering in Refs. Safari et al. (2012, 2015). Still another approach is based on the exact Dirac-Coulomb Green’s function which can be represented in a closed form as a superposition of the regular and irregular solutions of the Dirac equation. In the case of a point-like Coulomb potential, for example, it can be expressed analytically Wichmann and Kroll (1956). In Refs. Surzhykov et al. (2013, 2015) this method was also utilized in the calculation of the Rayleigh scattering cross sections.

Figure 4: (Color online) Total cross section for the Rayleigh scattering of x rays by helium-like Ni (), Xe () and Au () ions in their ground state and as function of the photon energy. Predictions are compared for three different approximations in the evaluation of the scattering amplitudes: A pure Coulomb potential (red dashed line), use of a screening potential (blue dash-dotted line) as well as by applying the full many-electron procedure in (black solid line). The lower panel displays the differences (in %) between the Coulomb (red dashed line) and the screening cross sections (blue dash-dotted line), each relative to the many-electron computations.

In the present work, we made use of the two latter approaches: the finite basis-set method and the Dirac-Coulomb Green’s function. The finite basis set was constructed from -splines Sapirstein and Johnson (1996) by employing the dual-kinetic-balance approach Shabaev et al. (2004). In addition, the analytic Dirac-Coulomb Green’s function for a point-like Coulomb potential was employed in terms of the Whittaker functions Yerokhin and Shabaev (1999). The finite basis set method enables us to considerably reduce the numerical effort in the calculations. The reason for this is the separation of the radial variables and the subsequent integrations of the single radial integrals. However, when the energies of the incident photons are larger than the ionization threshold, the application of the finite basis-set technique is critically hampered in some of the terms (diagrams). For example, this is the case for the first term of Eq. (9) with the denominator . This term has a pole in the energy continuum (spectrum) at that cannot be treated accurately in any finite basis-set method due to the discretized spectrum and the summation over just a finite number of basis functions. In contrast, the use of the Dirac-Coulomb Green’s function is free of such difficulties as it represents the exact electron propagator. Therefore, all the electron propagators with energies or were treated by means of the Dirac-Coulomb Green’s function :