Momentum density spectroscopy of Pd: Comparison of 2D-ACAR and Compton scattering using a 1D-to-2D reconstruction method

Two-dimensional (2D) angular correlation of annihilation radiation and Compton scattering are both powerful techniques to investigate the bulk electronic structure of crystalline solids through the momentum density of the electrons. Here we apply both methods to a single crystal of Pd to study the electron momentum density and the occupancy in the first Brillouin zone and to point out the complementary nature of the two techniques. To retrieve the 2D spectra from one-dimensional Compton profiles, a direct inversion method is implemented and benchmarked against the well-established Cormack's method. The comparison of experimental spectra with first-principles density functional theory calculations of the electron momentum density and the two photon momentum density clearly reveals the importance of positron probing effects on the determination of the electronic structure. While the calculations are in good agreement with the experimental data, our results highlight some significant discrepancies.

Figure 1

(a) $\mathrm{\Gamma }$-centered electron sheet and $X$-hole pockets of the Fermi surface of Pd in the first Brillouin zone. The $\mathrm{\Gamma }$, X, K, and L high-symmetry points are shown in red (with $\mathrm{\Gamma }$ at the zone center). (b) Open-hole sheet and $L$-hole pockets of the Fermi surface. Additionally, the dHvA $\alpha$ and $\epsilon$ orbit is shown. The $\epsilon$ orbit lies in the (001) plane through the center of the Brillouin zone (plotted in XCrySDen [36]).

Figure 2

2D radial anisotropy of the five measured ACAR spectra (left half of each subplot) and the corresponding theoretical calculation, with (bottom right) and without Drummond (top right) enhancement. All calculations have been convolved with the experimental resolution, and all experimental spectra were symmetrized to the according crystal symmetry. The angle $\mathrm{\Phi }$ denotes the angle measured from the [100] direction within the (011) plane.

Figure 3

Directional differences of experimental (blue) and theoretical Compton profiles (red). The labeled angles are measured from the $\mathrm{\Gamma }$-K direction towards the $\mathrm{\Gamma }$-X direction of the fcc Brillouin zone. The calculated profiles are convolved with a one-dimensional Gaussian accounting for the experimental resolution. For every third experimental data point, an error bar showing the statistical error of one standard deviation is plotted. The plots have been offset by $0.4\mathrm{el}/\mathrm{a}.\mathrm{u}.$ from one another for clarity.

Figure 4

(a) 2D radial anisotropy of the projected EMD calculated by DFT (left), reconstructed from ten 1D Compton profiles by DIM (top right) and by Cormack (bottom right). The theoretical spectrum was convolved with a two-dimensional Gaussian accounting for the experimental resolution. (b) Cut through the 2D radial anisotropy along the red path ($1\to 2\to 3$) shown in (a).

Figure 5

2D LCW of the experimental and theoretical 2D-EMD: Reconstructed experimental spectrum by DIM (top left) and reconstructed experimental data by Cormack (bottom left); theoretical EMD (bottom right); reconstruction from theoretical 1D Compton profiles by Cormack (top right, T.C.) and DIM (top right, T.D.). The theoretical data in the right half of the plot was convolved with a two-dimensional Gaussian accounting for the experimental resolution, before back-folding.

Figure 6

2D radial anisotropy of the EMD reconstructed (DIM) from ten 1D Compton spectra (left) and the corresponding 2D projection obtained by 2D-ACAR (right). Both spectra were normalized to the corresponding electron density of the EMD before calculating the radial anisotropy. The positions of the projected high-symmetry $\mathrm{\Gamma }$, K, and X points are indicated.

Figure 7

2D-LCW of the reconstructed (DIM) Compton spectrum (top left) and the 2D-ACAR spectrum (top right). 2D-LCW calculated from the radial anisotropies of the reconstructed (DIM) Compton spectrum (bottom left) and the 2D-ACAR spectrum (bottom right).

Figure 8

Difference between the TPMD calculation including enhancement (Drummond model [51]) and the IPM calculation in the reduced zone scheme.

Figure 9

Directional differences between 1D projections along $\mathrm{\Gamma }$-X and $\mathrm{\Gamma }$-K of the Compton and 2D-ACAR experiments (purple and red points, respectively), together with the calculated (lines) EMD (blue) and TPMD, calculated with (brown) and without (orange) positron enhancement. The calculated profiles are convolved with Gaussian functions accounting for their respective experimental resolutions. For every fourth experimental data point, error bars show the statistical error of one standard deviation.