Orientation-dependent NMR study of the giant-unit-cell intermetallics $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$, Bergman-phase ${\mathrm{Mg}}_{32}{(\mathrm{Al},\mathrm{Zn})}_{49}$, and ${\xi }^{\prime }\text{−}{\mathrm{Al}}_{74}{\mathrm{Pd}}_{22}{\mathrm{Mn}}_4$

We present a ${}^{27}\mathrm{Al}$ NMR study of three giant-unit-cell complex metallic compounds, $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$, the “Bergman-phase” ${\mathrm{Mg}}_{32}{(\mathrm{Al},\mathrm{Zn})}_{49}$, and ${\xi }^{\prime }\text{−}\mathrm{Al}\text{−}\mathrm{Pd}\text{−}\mathrm{Mn}$, which contain some hundreds up to more than a thousand atoms in the unit cell. The NMR spectra of monocrystalline samples are strongly inhomogeneously broadened by the electric quadrupole interaction and the line shapes are featureless and powderlike, but still exhibit significant orientation-dependent variation of the intensity on the satellite part of the spectrum in the magnetic field. Measuring orientation-dependent satellite intensity in appropriate frequency windows yields rotation patterns that can be related to the structure and symmetry of the giant unit cells. For a theoretical reproduction of the rotation patterns, we derived a distribution of the electric-field-gradient (EFG) tensors for each of the investigated compounds from existing structural models using point-charge and ab initio calculations. The EFG distribution yields important structural information on the manifold of different local atomic environments in the unit cell and distinguishes crystallographically inequivalent lattice sites from the equivalent ones. The distribution of the EFGs for the 1168-atom unit cell of $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$ was successfully determined by a point-charge calculation, whereas the ab initio approach was successful for the 160-atom unit cell of the ${\mathrm{Mg}}_{32}{(\mathrm{Al},\mathrm{Zn})}_{49}$ Bergman phase. For the 258-atom unit cell of ${\xi }^{\prime }\text{−}\mathrm{Al}\text{−}\mathrm{Pd}\text{−}\mathrm{Mn}$, the experimental rotation patterns revealed a pseudotenfold symmetry, whereas the theoretical point-charge calculation revealed predominant twofold symmetry with traces of tenfold symmetry, so that no quantitative matching between the theory and experiment could be obtained.

Figure 1

(a) ${}^{27}\mathrm{Al}$ field-swept NMR spectrum (solid line) of the $\beta \text{−}{\mathrm{Al}}_{3.26}{\mathrm{Mg}}_2$ monocrystal, measured at a constant irradiation frequency $\nu \left({}^{27}\mathrm{Al}\right)=26.053\mathrm{MHz}$ and a frequency-swept spectrum (open circles), measured in a constant field $6.34\mathrm{T}$ at $T=80\mathrm{K}$ and [1 0 0] axis parallel to the field. The spectra are normalized to the same integral. The origin of the frequency axis is taken at the ${}^{27}\mathrm{Al}$ resonance of the $\mathrm{Al}{\mathrm{Cl}}_3$ aqueous solution and the uniform magnetic shift $\Delta {\nu }_m$ (the isotropic Knight shift) has been subtracted. (b) Central lines in 2.35 and $4.7\mathrm{T}$ on an expanded frequency scale. The lines are obtained by a Fourier-transform technique and are scaled to the same height. (c) Temperature-dependent frequency shift of the central line in $4.7\mathrm{T}$.

Figure 2

(a) Two ${}^{27}\mathrm{Al}$ frequency-swept NMR spectra of $\beta \text{−}{\mathrm{Al}}_{3.26}{\mathrm{Mg}}_2$ (normalized to the same integral) in a field $6.34\mathrm{T}$ and $T=80\mathrm{K}$ at orientations $\beta =0$ (field parallel to [1 0 0]) and $\beta =45°$ (field along [1 1 0]). Vertical dashed lines indicate the frequencies at which the window-intensity rotation patterns were recorded. (b) Theoretical satellite parts of the spectra at $\beta =0$ and 45° obtained by a point-charge calculation on the Samson structural model[11] of $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$. (c) Rotation pattern around [0 0 1] of the window intensity at $70.70\mathrm{MHz}$ in one full rotation $0<\beta <360°$. (d) Rotation patterns around [0 0 1] of the window intensity in the interval $0<\beta <90°$ at the four frequencies indicated by vertical dashed lines in (a). Solid lines are theoretical fits described in the text.

Figure 3

Two arrangements [(a) mode 1 and (b) mode 2] of the pentagonal prisms and tetrahedra within the unit cell of $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$. While the tetrahedra can have no orientational disorder within a given mode, each pentagonal prism can assume six different orientations with equal probability, so that their sublattice contains inherent configurational disorder. In (c), one complete unit cell of $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$ (Samson model) is shown.

Figure 4

Marginal distributions (a) $f\left(V_{ZZ}\right)$ and (b) $r\left(\eta \right)$ of the ${}^{27}\mathrm{Al}$ EFG-tensor eigenvalues in the unit cell of $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$, calculated from the Samson model and used for the theoretical reproduction of the orientation-dependent NMR window intensities [solid lines in Fig. 2d].

Figure 5

The distribution of the EFG PAS orientations in the crystal-fixed frame of $\beta \text{−}{\mathrm{Al}}_3{\mathrm{Mg}}_2$ represented in a spherical plot, where the orientations of the $V_{ZZ}$ principal direction are shown on a sphere. In (a), the [1 0 0] crystalline axis points out of the graph and the number (intensity) of the EFGs is denoted by a gray scale, whereas in (b), the same distribution is viewed along [1 1 1]. In (c) and (d), the orientational distributions of the $V_{YY}$ and $V_{XX}$ principal directions are shown, respectively, where the [1 0 0] direction points out of the graphs.

Figure 6

Theoretical rotation patterns of the window intensity at $70.70\mathrm{MHz}$ [corresponding to Fig. 2c] for two cases: (a) the model of configurationally disordered pentagonal prisms and (b) the ordered model, where all prisms assume the same orientation.

Figure 7

(a) Six successive shells of the Bergman icosahedral cluster (see text). (b) The structure of the Bergman phase is obtained by a body-centered packing of these clusters by sharing a hexagonal plane of the fourth shell (the soccer ball).

Figure 8

(a) ${}^{27}\mathrm{Al}$ frequency-swept NMR spectra of monocrystalline ${\mathrm{Mg}}_{29.4}{(\mathrm{Al},\mathrm{Zn})}_{51.6}$ Bergman-phase sample at $80\mathrm{K}$ and orientations $\beta =0$, 111°, and 147°. The dashed vertical line indicates the frequency at which the window-intensity rotation patterns were recorded. (b) Theoretical reproduction of the same spectra (the satellite parts are shown only) by the ab initio calculation (see text).

Figure 9

(a) Central lines of the ${}^{27}\mathrm{Al}$ NMR spectra of ${\mathrm{Mg}}_{29.4}{(\mathrm{Al},\mathrm{Zn})}_{51.6}$ for the $\beta =0$ orientation at $80\mathrm{K}$ in two magnetic fields 2.35 and $4.7\mathrm{T}$. The lines are obtained by a Fourier-transform technique and are scaled to the same height. (b) Temperature-dependent frequency shift of the central line in $4.7\mathrm{T}$.

Figure 10

(a) Experimental rotation patterns of the ${}^{27}\mathrm{Al}$ window intensity at the frequency $70.25\mathrm{MHz}$ [marked by the vertical dashed line in Fig. 8a] for seven different axes of rotation within the $(y,z)$ plane of the crystal-fixed frame. $\phi$ denotes the angle between the rotation axis and the [0 0 1] direction, so that $\phi =0$ corresponds to rotation about [0 0 1] and $\phi =90°$ to rotation about [0 1 0]. In (b), the theoretical rotation patterns, obtained by ab initio calculation on an Al-Mg-Al unit cell, are displayed side by side with the experimental ones.

Figure 11

Marginal distributions (a) $f\left(V_{ZZ}\right)$ and (b) $r\left(\eta \right)$ of the ${}^{27}\mathrm{Al}$ EFG-tensor eigenvalues for an Al-Mg-Al unit cell of the Bergman phase, calculated from the Sun model[17] and used for the theoretical reproduction of the orientation-dependent window intensities in Fig. 10b. In (c), the distribution of orientations of the $V_{ZZ}$ principal direction in the crystal-fixed frame is shown in a spherical plot as viewed along the [1 0 0] crystalline direction, whereas in (d), the same distribution is viewed along [1 1 1].

Figure 12

Comparison of (a) the experimental window-intensity rotation pattern for the $\phi =0$ rotation with the theoretical patterns calculated by (b) the ab initio method and (c) the point-charge calculation.

Figure 13

(a) A view of the ${\xi }^{\prime }\text{−}\mathrm{Al}\text{−}\mathrm{Pd}\text{−}\mathrm{Mn}$ skeleton structure[4, 34] along the [0 1 0] direction. Mn atoms form a planar flattened-hexagon lattice and are located in the centers of pseudo-Mackay icosahedra. The two interpenetrating polyhedra that form the outer shell of the pseudo-Mackay cluster [a 12-atom Pd icosahedron (white atoms), and a 30-atom Al icosidodecahedron (black atoms)] are shown. The orthothombic unit cell is also shown and one of the flattened hexagons is divided into a Penrose tiling by dashed lines. (b) The same structure viewed along [0 0 1]. There are two planes of flattened hexagons at $y=0$ and $1∕2$ in the unit cell.

Figure 14

(a) ${}^{27}\mathrm{Al}$ frequency-swept NMR spectra of monocrystalline ${\xi }^{\prime }\text{−}{\mathrm{Al}}_{73}{\mathrm{Pd}}_{22.9}{\mathrm{Mn}}_{4.1}$ at $T=80\mathrm{K}$ in a magnetic field $6.34\mathrm{T}$ and angles $\beta =18°$ and 36°. Vertical dashed lines indicate the frequencies at which the window-intensity rotation patterns were recorded. (b) Rotation patterns of the window intensity in the rotation about the [0 1 0] pseudotenfold direction at the selected frequencies 70.8, 70.9, and $71.0\mathrm{MHz}$.

Figure 15

(a) Theoretical distribution of the orientations of the $V_{ZZ}$ principal axis on the sphere viewed along the [0 1 0] crystalline direction, calculated for the structural model of ${\xi }^{\prime }\text{−}\mathrm{Al}\text{−}\mathrm{Pd}\text{−}\mathrm{Mn}$ by Boudard.[4] A twofold-symmetric pattern is observed with traces of tenfold symmetry (ten equally spaced radial directions of the intensity can be noticed). The corresponding theoretical marginal distributions (b) $f\left(V_{ZZ}\right)$ and (c) $r\left(\eta \right)$ are also shown.