^{1,a)}

### Abstract

X-ray diffraction combined with conventional spectroscopy could provide a powerful means to characterize electronically excited atoms and molecules. We demonstrate theoretically how x-ray diffraction from laser excited atoms can be used to determine electronic structure, including angular momentum composition, principal quantum numbers, and channel populations. A theoretical formalism appropriate for highly excited atoms, and easily extended to molecules, is presented together with numerical results for Xe and H atoms.

The author gratefully acknowledges helpful discussions with Christian Jungen, Niels E. Henriksen, Peter M. Weber, Stephen T. Pratt, and staff and postdoc's at ITAMP, in particular Mikhail Lemeshko. This work was supported by the European Union by grant COCOSPEC (FP7-IEF) and the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory. The author is grateful for the hospitality of the African Institute for Mathematical Sciences in Cape Town during early stages of this work.

I. INTRODUCTION

II. THEORY

A. Theoretical description of highly excited states

B. Electron density from Rydbergwave functions

C. X-ray diffraction

III. NUMERICAL CALCULATION OF RYDBERGWAVE FUNCTIONS

IV. RESULTS

A. Diffraction from H atoms

B. Diffraction from Xe atoms

V. EXPERIMENTAL CONSIDERATIONS

VI. CONCLUSIONS

### Key Topics

- X-ray diffraction
- 118.0
- Rydberg states
- 90.0
- Wave functions
- 60.0
- Angular momentum
- 23.0
- X-rays
- 21.0

## Figures

Schematic of the proposed experiment. The sample atoms are excited by a laser with planar polarization along the -axis. The direction of the incoming x-ray beam is **k** _{0}, which defines the -axis, and the diffracted x-rays are measured along **k**. The direction of **k** is expressed in spherical angles θ, ϕ relative the incoming x-ray beam **k** _{0}.

Schematic of the proposed experiment. The sample atoms are excited by a laser with planar polarization along the -axis. The direction of the incoming x-ray beam is **k** _{0}, which defines the -axis, and the diffracted x-rays are measured along **k**. The direction of **k** is expressed in spherical angles θ, ϕ relative the incoming x-ray beam **k** _{0}.

Radial Rydberg electron wave functions *R* _{ j }(*r*) for H and Xe atoms. The classical turning point, *r* _{ tp }, for the wave functions scales as *r* _{ tp } = 2*n* ^{2}. The wave functions are bound-state normalized. (a) (H atom) Radial wave functions for *L* = 0 (*M* = 0) and principal quantum numbers *n* = 1 (top), *n* = 10 (middle), and *n* = 20 (bottom). (b) (Xe atom) Radial electronic Rydberg wave functions for states *E* _{1} and *E* _{2} in channels *j* = 1–5 (see Table I). The core is contained inside the volume *r* _{ c } < 5 a.u. The principal quantum numbers in the channels *j* = 1–3 are *n* ≈ 20, and *n* ≈ 5 in *j* = 4–5.

Radial Rydberg electron wave functions *R* _{ j }(*r*) for H and Xe atoms. The classical turning point, *r* _{ tp }, for the wave functions scales as *r* _{ tp } = 2*n* ^{2}. The wave functions are bound-state normalized. (a) (H atom) Radial wave functions for *L* = 0 (*M* = 0) and principal quantum numbers *n* = 1 (top), *n* = 10 (middle), and *n* = 20 (bottom). (b) (Xe atom) Radial electronic Rydberg wave functions for states *E* _{1} and *E* _{2} in channels *j* = 1–5 (see Table I). The core is contained inside the volume *r* _{ c } < 5 a.u. The principal quantum numbers in the channels *j* = 1–3 are *n* ≈ 20, and *n* ≈ 5 in *j* = 4–5.

The Rydberg electron density in the *yz*-plane for state *E* _{1} in Xe (see Table I). (a)–(c) show channels *j* = 1, 3, and 5, respectively. The density is isotropic in channels *j* = 1, 4, and anisotropic in the other channels. The electron density has been multiplied by the radius *r*, and re-normalized so that the maximum value is 1 in each channel in order to aid visualization. (a) Channel *j* = 1. (b) Channel *j* = 3 (and channel *j* = 2 by a 90° rotation around the perpendicular -axis). (c) Channel *j* = 5 (note that *j* = 4 is similar in size and nodal pattern, but is isotropic like *j* = 1).

The Rydberg electron density in the *yz*-plane for state *E* _{1} in Xe (see Table I). (a)–(c) show channels *j* = 1, 3, and 5, respectively. The density is isotropic in channels *j* = 1, 4, and anisotropic in the other channels. The electron density has been multiplied by the radius *r*, and re-normalized so that the maximum value is 1 in each channel in order to aid visualization. (a) Channel *j* = 1. (b) Channel *j* = 3 (and channel *j* = 2 by a 90° rotation around the perpendicular -axis). (c) Channel *j* = 5 (note that *j* = 4 is similar in size and nodal pattern, but is isotropic like *j* = 1).

Diffraction patterns from states in H atoms with different angular momentum *L*, but otherwise identical quantum numbers *n* = 10 and *M* = 0. The angular momenta *L* = 0 (top), *L* = 1 (middle), and *L* = 2 (bottom) correspond to *s*, *p*, and *d* orbitals. The righthand column shows the electron density for each state in the *yz*-plane (see Figure 3 for technical details) and the lefthand column shows contour plots of the diffraction, | *f*(θ, ϕ)|^{2}, along the radial θ and angular ϕ coordinates (see Figure 1). The x-ray wavelength is λ = 150 a.u. (156 eV) and the incoming x-ray is aligned with the -axis. Note that background diffraction from, e.g., ground state atoms is not included.

Diffraction patterns from states in H atoms with different angular momentum *L*, but otherwise identical quantum numbers *n* = 10 and *M* = 0. The angular momenta *L* = 0 (top), *L* = 1 (middle), and *L* = 2 (bottom) correspond to *s*, *p*, and *d* orbitals. The righthand column shows the electron density for each state in the *yz*-plane (see Figure 3 for technical details) and the lefthand column shows contour plots of the diffraction, | *f*(θ, ϕ)|^{2}, along the radial θ and angular ϕ coordinates (see Figure 1). The x-ray wavelength is λ = 150 a.u. (156 eV) and the incoming x-ray is aligned with the -axis. Note that background diffraction from, e.g., ground state atoms is not included.

Diffraction signal as function of the radial angle θ for different x-ray probe wavelengths λ from H atoms with angular momentum *L* = *M* = 0 and principal quantum number *n* = 10 (classical turning point at 200 a.u.). The probe wavelength λ needs to be matched to the spatial dimensions of the radial wave function to avoid loss of information. The range λ = 5–600 a.u. corresponds to photon energies 4.7 keV–39 eV. Details of the diffraction pattern are shown in the insert.

Diffraction signal as function of the radial angle θ for different x-ray probe wavelengths λ from H atoms with angular momentum *L* = *M* = 0 and principal quantum number *n* = 10 (classical turning point at 200 a.u.). The probe wavelength λ needs to be matched to the spatial dimensions of the radial wave function to avoid loss of information. The range λ = 5–600 a.u. corresponds to photon energies 4.7 keV–39 eV. Details of the diffraction pattern are shown in the insert.

The radial diffraction signal as a function of principal quantum number *n*. The greatest difference is between *n* = 1 and *n* = 2 (top), but it remains significant for *n* = 10 and *n* = 11 (middle) and even for *n* = 20 and *n* = 21 (bottom). In each case, the x-ray wavelength λ is chosen to match the size of the electronic state (λ = 6 a.u. at the top, λ = 150 a.u. in the middle and λ = 250 a.u. at the bottom, corresponding to photon energies 3.9 keV, 156 eV, and 94 eV, respectively). Angular quantum numbers are *L* = *M* = 0, and the difference signal △_{ n,n+1} is included as a dotted line in each plot.

The radial diffraction signal as a function of principal quantum number *n*. The greatest difference is between *n* = 1 and *n* = 2 (top), but it remains significant for *n* = 10 and *n* = 11 (middle) and even for *n* = 20 and *n* = 21 (bottom). In each case, the x-ray wavelength λ is chosen to match the size of the electronic state (λ = 6 a.u. at the top, λ = 150 a.u. in the middle and λ = 250 a.u. at the bottom, corresponding to photon energies 3.9 keV, 156 eV, and 94 eV, respectively). Angular quantum numbers are *L* = *M* = 0, and the difference signal △_{ n,n+1} is included as a dotted line in each plot.

Contour plots of the diffraction, | *f*(θ, ϕ)|^{2}, from the Rydberg and the core electrons in Xe atom states *E* _{1} and *E* _{2} (see Table I and Figure 2(b)) for x-ray probe wavelength λ = 5 a.u. (4.7 keV) and λ = 300 a.u. (78 eV). The diffraction angles θ, ϕ are defined in Figure 1. The short wavelength λ = 5 a.u. (bottom row) predominantly probes the core, while λ = 300 a.u. (top row) is more sensitive to the Rydberg electron. The diffraction pattern for state *E* _{1} (left column) is quite different from the pattern for state *E* _{2} (right column), due to the large change in channel populations between the two states, and associated change in spatial angular distribution. (a) State *E* _{1}, λ = 300 a.u. (b) State *E* _{2}, λ = 300 a.u. (c) State *E* _{1}, λ = 5 a.u. (d) State *E* _{2}, λ = 5 a.u.

Contour plots of the diffraction, | *f*(θ, ϕ)|^{2}, from the Rydberg and the core electrons in Xe atom states *E* _{1} and *E* _{2} (see Table I and Figure 2(b)) for x-ray probe wavelength λ = 5 a.u. (4.7 keV) and λ = 300 a.u. (78 eV). The diffraction angles θ, ϕ are defined in Figure 1. The short wavelength λ = 5 a.u. (bottom row) predominantly probes the core, while λ = 300 a.u. (top row) is more sensitive to the Rydberg electron. The diffraction pattern for state *E* _{1} (left column) is quite different from the pattern for state *E* _{2} (right column), due to the large change in channel populations between the two states, and associated change in spatial angular distribution. (a) State *E* _{1}, λ = 300 a.u. (b) State *E* _{2}, λ = 300 a.u. (c) State *E* _{1}, λ = 5 a.u. (d) State *E* _{2}, λ = 5 a.u.

## Tables

The calculated character of the two bound Xe states *E* _{1} and *E* _{2} given as the % population in each channel. The three channels *j* = 1–3 correspond to the ground electronic state of the core and *j* = 4–5 to the first excited state. Core states are given as , where *J* _{ c } is the total, *L* is the orbital and *S* is the spin angular momentum of the core. The Rydberg electron is given as , where *l* is the electron orbital and *j* _{ e } is the total electron angular momentum, including the electron spin.

The calculated character of the two bound Xe states *E* _{1} and *E* _{2} given as the % population in each channel. The three channels *j* = 1–3 correspond to the ground electronic state of the core and *j* = 4–5 to the first excited state. Core states are given as , where *J* _{ c } is the total, *L* is the orbital and *S* is the spin angular momentum of the core. The Rydberg electron is given as , where *l* is the electron orbital and *j* _{ e } is the total electron angular momentum, including the electron spin.

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