Atomic form-factor measurements in the low-momentum transfer region for Li, Be, and Al by inelastic x-ray scattering

We report measurements of the atomic form factor of lithium, beryllium, and aluminum single crystals at low-momentum transfers $(Q=1.6–50{\mathrm{nm}}^{-1})$ from the intensity of phonons observed by inelastic x-ray scattering. Comparing to Hartree-Fock calculations, the form factor deviates significantly in the case of lithium and beryllium around $k_F$. These deviations can be mostly understood on the basis of electron redistribution by a pseudopotential. The influence of multiple scattering due to coherent phonon scattering and possible deviations from the adiabatic approximation are also discussed.

Figure 1

Schematic view of the spectrometer developed at beamline 3-ID. The beam comes from an undulator (A) and premonochromator (C) and then passes through the high-resolution monochromator (D) and focusing mirror (E) before it illuminates the sample (G). The scattered intensity is focused by an analyzer (I) into the detector (J). The ionization chamber (F) is used to monitor the incident flux on the sample. The slit systems that determine the source size are (B) and (H). The momentum resolution is determined by a slit (K) in front of the analyzer that has a slit opening of $x$ in the real space. In the reciprocal space, half width of the momentum resolution is represented by $\Delta Q_h$. The momentum-transfer vector and the incident beam wave vector are represented by $\mathbf{Q}$ and ${\mathbf{k}}_i$, respectively. The scattering angle is represented by ${\theta }_s$ and the deviation of the scattering angle is shown by $\Delta {\theta }_s$.

Figure 2

Energy scans for lithium along the $\left[0\zeta \zeta \right]$ direction for longitudinal modes.

Figure 3

Energy scans for beryllium along the $\left[00\zeta \right]$ direction for longitudinal modes.

Figure 4

Energy scans for aluminum along the $\left[00\zeta \right]$ direction for longitudinal modes. The elastic scattering at $\zeta =0.6$ is due to air scattering.

Figure 5

(a) Some of the energy scans for different momentum positions in terms of the reduced wave unit $\zeta$ are shown with circles, and fits to these data are shown with thick gray lines. (b) Energy scans for beryllium at $\zeta =0.15$ $(Q=2.63{\mathrm{cm}}^{-1})$ for two different tilt angle positions rotated around incident x-ray direction showing an increase in the intensity due to coherent phonon scattering caused by the additional wave field inside the crystal induced by the (0 1 3) Bragg reflection.

Figure 6

Dispersion relations for lithium along the $\left[0\zeta \zeta \right]$ direction, and for beryllium and aluminum along the $\left[00\zeta \right]$ direction. Circles are present measurements. Dotted lines are results from inelastic neutron scattering experiments (Refs. 24, 25, 26). The lower pictures are the full width of half maximum of the phonon peaks deconvoluted from the spectrometer’s overall energy resolution. Thus, $\Gamma$ projects to zero at the zone boundary where there should be no contribution from dispersion. Solid lines are the theoretical calculations of the widths due to finite momentum resolution that contributes to measured width. It is calculated from sine function of a dispersion relation (Ref. 27) as explained in the text.

Figure 7

Adjusted measured integrated intensities for lithium, beryllium, and aluminum. The conversion of integrated intensity to electron units include as adjustable parameter the analyzer efficiency, the accuracy of which is estimated to be around 20%, as explained in the text. Circles represent the data from the pseudo-Voigt model for beryllium, and squares represent the data from the DHO model for lithium and aluminum. Solid lines are the theoretical calculations.

Figure 8

Comparison of measured atomic form factor (circles) with the free-atom Hartree-Fock calculation (solid line) and the metallic-atom form factor calculated by the QHNC approximation (dashed line) for a given direction of lithium, beryllium and aluminum. The dotted line is the ionic form factor from Hartree-Fock calculation. Triangles show the experimental data determined from Bragg reflections (Refs. 13, 16). The arrow indicates minimum $Q$ values corresponding to lowest order Bragg diffraction. On the right side of the figure, deviations of the measured form factor from free-atom form factor, $\Delta$, in terms of a fraction of the valence electron are plotted together with the deviations of the theoretical calculations (dashed line).