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 = 2n 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).
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.
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.
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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