Direct atomic structure determination by the inspection of structural phase

Abstract

A century has passed since Bragg solved the first atomic structure using diffraction. As with this first structure, all atomic structures to date have been deduced from the measurement of many diffracted intensities using iterative and statistical methods. We show that centrosymmetric atomic structures can be determined without the need to measure or even record a diffracted intensity. Instead, atomic structures can be determined directly and quickly from the observation of crystallographic phases in electron diffraction patterns. Furthermore, only a few phases are required to achieve high resolution. This represents a paradigm shift in structure determination methods, which we demonstrate with the moderately complex α-Al₂O₃. We show that the observation of just nine phases enables the location of all atoms with a resolution of better than 0.1 Å. This level of certainty previously required the measurement of thousands of diffracted intensities.

Determining the atomic structure of a material is a vital step in the fundamental understanding of its properties. Crystal structures can be described by a Fourier series whose coefficients are called “structure factors” (1–3). The structure factor magnitude indicates how much matter there is, and its phase indicates where it is. As with the first structure determined by Bragg (4), all atomic structures to date have, in essence, had to be deduced indirectly from measurements of magnitude only. This is because phase information is either entirely lost in the intensities of X-ray and neutron diffraction patterns or difficult to extract analytically from electron diffraction patterns (the notorious phase problem) (1, 5–8). The absence of phase information, and hence explicit information on the location of matter, has necessitated an indirect approach to structure determination that has prevailed since Bragg’s original discovery. The structure must be deduced by refining structural models iteratively until they reproduce the diffracted intensities measured in experiments (1, 3, 9–19). To achieve reliability, thousands of diffracted intensities must be measured. This is time-consuming, sometimes inaccurate, and uniqueness cannot be guaranteed if the phases are inaccessible. As bemoaned by many a crystallographer, the quantity we should ideally measure is phase (5, 7, 8).

Here we describe a method for the determination of structure from the direct experimental observation of phase, without recourse to structure refinement by iterative pattern matching (1, 3, 9–19). The method identifies distinctive features arising from dynamical scattering in convergent beam electron diffraction (CBED) patterns taken in a three-beam condition to determine individual phases by inspection. It depends on qualitative interpretation of the intensity distribution alone with all of the necessary information obtained purely by inspection. Using the observed phases, the position of matter within the unit cell is rapidly and unequivocally identified. This method requires no knowledge of the crystal’s unit cell dimensions nor how many atoms are in the cell and is independent of specimen thickness. It does not even assume that atoms exist (although it is well known that they do). This is a paradigm shift in atomic structure determination. We explain the method here, step-by-step, using the moderately complex hexagonal centrosymmetric structure of α-Al₂O₃ (30 atoms in its unit cell) as an example structure. The outline structure is evident after the observation of just five structure factor phases and is determined to better than 0.1 Å resolution after nine phases, sufficient to identify the subtle off-axis shifts of the oxygen atoms. Using conventional structure determination methods, the same structure required the measurement of several thousand magnitudes to deliver a comparable level of accuracy (20, 21).

We make no assumptions about the structure of α-Al₂O_3. The first step in the structure determination is to determine the space group. This is readily determined by inspecting the symmetries of appropriately oriented CBED patterns, using long-established methods (22). No measurement of intensities is required. From this we find that α-Al₂O₃ belongs to space group 167, R-3c.

Having established the space group, one can index the reflections present in any CBED pattern from the same material by relating CBED patterns from different zones via common reciprocal lattice vectors.

With the space group and pattern indexing in hand, we now consider the special geometry of three-beam CBED patterns, as illustrated in Fig. 1. Three-beam patterns involve the simultaneous satisfaction of the Bragg condition for two different reflections, say g and h, where the scattering vectors, g and h, are not collinear, meaning the Bragg condition loci for each reflection intersect (ζ_g = 0 and ζ_h = 0, where ζ is the excitation error or, in other words, deviation from the Bragg condition). This region of intersection is “three-beam” in nature if and only if no other Bragg loci pass through it. Previous work established that such diffraction geometries in electron scattering allow the magnitudes of the structure factor triplet, V_g, V_h, and V_g–h (where g–h is the vector that couples discs g and h), to be measured directly from the position of features in the intensity distributions of the discs (23–26). More importantly, the position, Γ₃, of one of these features gives the three-phase invariant (23–26):

where σ is the interaction constant for electrons of a particular energy and is always positive. [It is of interest that the quantity, Γ₃, which arises naturally in the analysis of three-beam dynamical scattering, is the same as that introduced by Hauptman, who called it the three-phase invariant and derived it from an entirely different approach (11).]

Fig. 1.

An example of the determination of the phases of V_g, V_h, and V_g–h by inspection. A is a schematic representation of the geometry of a three-beam CBED pattern. The vectors g, h, and g–h and excitation error coordinate system (ζ_g, ζ_h) define the scattering geometry. Discs g and h, corresponding to reciprocal lattice vectors g and h, respectively, satisfy the Bragg condition (ζ_g = 0 and ζ_h = 0). The Bragg loci divide the origin disk into quadrants based on the sign of the excitation error in each of discs g and h. B and C are schematic diagrams of the intensity distributions in disk g that one can expect for the cases of (B) positive and (C) negative three-phase invariants. An experimental three-beam CBED pattern taken with 200 keV electrons about [16 –10 –1] is shown in D. The deflections of the intensity fringes about locus C (perpendicular to g–h and shown by dotted lines in both 1 1 6 and 0 –1 10) are away from 0 0 0, indicating that the three-phase invariant is positive. In E, the 0 0 0 disk (dotted square in D) is shown with the contrast level set to that observed on the fluorescent screen of the transmission electron microscope. It shows that the + – quadrant is darkest, indicating a difference in phase between V_g, V_h, and V_g–h.

The sign of Γ₃ gives the sign of the three-phase invariant and can be determined easily by inspection of the direction of the displacement, Γ₃, from the origin of the disk. The simplest and canonical approach to this inspection is to consider a locus perpendicular to g–h (locus C) (25, 26) and look for the direction of the deflection (indicated by arrows in Fig. 1 B–D) of the intensity “fringes” in the diffracted discs about locus C. If this is away from the origin of the pattern (central beam disk), then the three-phase invariant is positive (Fig. 1B), and if it is toward the origin, then the three-phase invariant is negative (Fig. 1C).

This is illustrated in the experimental example of Fig. 1D, taken about [16 –10 –1] with 200 keV electrons. The deflections in both disk g and disk h (about the dotted loci) indicate the three-phase invariant for this triplet of structure factors is positive.

The sign of the three-phase invariant alone is, however, not enough to determine the individual phases of the structure factors. For some centrosymmetric space groups, it will suffice to determine as many three-phase invariants as are accessible by three-beam CBED, as long as some of them have opposing signs and one or more common structure factors. This allows the individual phases to be deduced and fed back into the other triplets, which allow the other individual phases to be deduced in turn. For other centrosymmetric space groups, additional information is necessary.

The intersection of two noncollinear loci satisfying Bragg conditions divides the disk at the origin of the pattern into quadrants as marked in Fig. 1A. These have been marked according to the signs of the excitation error axes, ζ_g and ζ_h, in each of the quadrants. Goodman showed that the Borrmann effect (27) [a phenomenon arising from interference (28)] that manifests quite strongly in two-, three-, and four-beam patterns can be used to assess the relative phase of the structure factors relevant to a particular few-beam CBED pattern (29). Specifically, if there is no difference in phase between the complex components (i.e., absorption terms) of each structure factor, then the “–” quadrant of 0 0 0 should show the strongest absorption and appear the darkest of the quadrants. This implies that the real parts of the structure factors will also all have the same sign (phase) as the real and imaginary parts almost always have opposite signs. The only situation in which this may not always be true is in the case of very weak structure factors, but these are in any case not optimal for use in the present type of three-beam structure determination, as they are not likely to represent densely packed atomic planes. If, on the other hand, the “+ –” or “– +” quadrants appear darker, then a difference in phase of the structure factors is implied.

In the experimental pattern from α-Al₂O₃ shown in Fig. 1D, a closer examination of the 0 0 0 disk in Fig. 1E shows the + – quadrant to be darkest. When this is coupled with the observation above that the three-phase invariant is positive (i.e., V_g.V_h.V_g–h is positive), then it is evident that two of the structure factors must have negative signs because of the relative phase difference between the individual structure factors implied by the dark + – quadrant.

For α-Al₂O₃, six different three-beam triplets were examined by CBED, and it rapidly became evident that all three-phase invariants in this structure are positive. This alone would not allow the individual phases of the structure factors to be determined, but in combination with observations of the Borrmann effect in the 0 0 0 disk, as described above, all individual structure factor phases could be assigned easily. These observations are summarized in Fig. 2.

Fig. 2.

Phase determination from the inspection of six experimental three-beam CBED triplets in α-Al₂O₃. Columns from left to right show the indices of the minor zone axes at which the inspections were made; the direction of the deflection of the intensity fringes along locus C in discs g and h; the sign of the three-phase invariant deduced from the deflections; the quadrant in the 0 0 0 disk that is the darkest; the indices of g, h, and g–h; and the absolute sign of each structure factor deduced directly from the information in this table. All of the above information was obtained by inspection only. The right-hand side of the table condenses these results into a set of 11 symmetry-inequivalent structure factors with their signs, as determined by inspection.

It is worth noting that the instrumental requirements for obtaining the necessary experimental data are modest. A routine, thermal emission gun transmission electron microscope with flexible condensing optics, as was used here, is more than adequate.

Fig. 2 presents the 11 structure factor phases determined by the inspection of intensity distribution in three-beam CBED patterns. The next step is to use these 11 phases to determine the atomic structure itself. It should be emphasized that at no stage do we measure or use any structure factor magnitudes in this determination.

Fig. 3 shows the process of structure determination, which proceeds based on the concept of plane intersections introduced by Lonsdale in solving the planar structure of the benzene ring in hexamethylbenzene (30). In contrast to Lonsdale, who plotted planes as thin lines in a 2D representation of the unit cell, we plot “slabs” of finite width in the 3D cell. We refer to this method from here as the method of intersecting slabs. The slab width is set at half the interplanar spacing, centered on the plane itself, and thus corresponding to the range ±π/2 in the argument of the cosine of the Fourier series expansion:

More specifically, we allocate the value V_g to the region that falls within a distance of d_hkl/4 (d_hkl is the interplanar spacing for the set of planes with Miller indices h, k, and l) from the plane and –V_g to the region between planes where d_hkl/4 < d ≤ d_hkl/2. In all cases, |V_g| is set to 1, as the present method relies only on the phases of the structure factors and not their magnitudes. This process effectively mimics the Fourier series with the exception that the cosine function is essentially replaced by a square wave function with an amplitude of unity. Therefore, even in a crystal structure where all of the structure factor phases are positive (e.g., CsCl; SI Text) or all of them are negative, this method always plots an equal number of positive and negative slabs, each with thickness d_hkl/2 and magnitude of unity. The structure factor phases dictate where those positive and negative slabs are located with respect to the corresponding sets of crystal planes. Next we will show that fewer than 10 structure factor phases are necessary for an unequivocal atomic structure determination of α-Al₂O₃ (and CsCl; see Fig. S1 and SI Text).

Fig. 3.

Direct three-dimensional structure determination by three-beam CBED and the method of intersecting slabs is illustrated for α-Al₂O₃. Slabs coplanar to (hkl) are given a finite thickness of half the interplanar spacing, centered on each (hkl) plane. Plots involve assigning the region d ≤ d_hkl/4 about a plane the value V_hkl and the region d_hkl/4 < d ≤ d_hkl/2 the value –V_hkl (|V_hkl| = 1, so only the sign of V_hkl plays any role). A shows the case for the (0 3 0) set of planes, and in B, all symmetry equivalents of (0 3 0) are added, hence {0 3 0}. The other sets of planes for which we have determined phases are subsequently added in C–I in order of the total number of phase inspections made under different experimental conditions (Fig. 2). Iso-surfaces are drawn to reveal the bimodality of the maxima. In J, the structure previously determined by conventional single crystal X-ray diffraction measurements of 2,000 structure factor magnitudes (21) is overlaid on I (blue atoms, Al; red atoms, O). An animated version of this figure is provided as Movie S1.

Fig. 3A shows the positive slabs corresponding to (0 3 0) planes, and Fig. 3B follows this by adding all symmetry-equivalent slabs. At this point, the positive iso-surface has the form of rods of intersection of the {0 3 0} slabs. Addition of successive sets of slabs proceeds in the order listed in Fig. 2. This process has been animated and is provided as Movie S1, which makes the evolution of the slab intersection maxima easier to follow visually. By Fig. 3E, after using only four symmetry-inequivalent sets of slabs, the outline structure appears, which reproduces the well-established structure (21) of α-Al₂O₃ to within 0.12 Å. Addition of more sets of slabs, as shown in Fig. 3 F–I, further improves the resolution. After a total of eight sets of slabs (Fig. 3I), the resolution is <0.1 Å, as quantified in Fig. 4.

Fig. 4.

Quantitative comparison of the structure of α-Al₂O₃ determined from the inspection of nine phases, with the structure determined from 2,000 magnitude measurements by X-ray diffraction (21). The blob weight is listed in the first column and is defined as the mean intersection density within the volume enclosed by its iso-surface, multiplied by that volume. This is followed by the fractional coordinates of the centroid of each of the blobs. The bimodal values in column 1 allow the atom types (Al and O, colored blue and red, respectively) to be assigned. The fractional coordinates of the atom positions previously determined by conventional methods (21) are given in the next columns followed by the mismatch between the structure determined with the current method. It is only in the calibration of the mismatch in Å that lattice parameters for α-Al₂O₃ must be used.

In Fig. 3J, we compare the structure determined using this method (Fig. 3I) with the established structure determined by conventional methods [from about 2,000 structure factor magnitude measurements by X-ray diffraction (21)]. A quantitative comparison is made in Fig. 4, which summarizes the locations of the centroids of each “blob” seen in Fig. 3I and compares them with the previously published atomic positions for α-Al₂O₃ as determined by X-ray diffraction (21). We choose to use the word blob here as no assumption has been made so far about the existence of atoms in the probed material. We know, however, that the composition of the material is two aluminum atoms for every three oxygen atoms, and we know that aluminum is much heavier than oxygen. It is therefore trivial to assign an atomic species to each blob in Fig. 3I, as the blob weights are clearly bimodal. This is done in Fig. 4, and it clearly shows that the structure of α-Al₂O₃ determined with this new approach is in full agreement with the structure previously determined by single crystal X-ray diffraction (21). A visual comparison is made in Fig. 3J by superimposing the structure determined here (Fig. 3I) on the established structure (21) (also provided in Movie S1). The close match is evident.

To gauge the actual distances involved in any atomic position mismatch between the structure determined in the present work and the conventional determination by X-rays (21), we need knowledge of the lattice parameters. It is notable that nowhere have we yet depended on any knowledge of the lattice parameters of α-Al₂O₃, as the positions of the maxima are expressed in fractional coordinates, as is also customary in all conventional structure analysis. When we apply the known lattice constants, the last column in Fig. 4 shows that the disagreements in atomic positions across all 30 atoms in the hexagonal cell are at the level of just a few picometres. Notably, this method of atom location by phase inspection even resolves the very small (0.07 Å) off-axis shifts of the oxygen atoms viewed along [0 0 1].

Conclusions

In summary, we have shown a method for the direct determination of a centrosymmetric crystal structure to picometre resolution using phases determined directly by experiment and no other information. This represents a paradigm shift in structure determination because phases, not magnitudes, of structure factors are used. No knowledge or measurements of magnitudes, lattice parameters, or the number of atoms in the unit cell were required. We showed that individual structure factor phases in centrosymmetric crystals can be observed directly and simply from the inspection of features within three-beam CBED patterns. These features are readily identified in a qualitative assessment of CBED patterns. No quantitative intensity measurements are needed. Armed with direct knowledge of only nine structure factor phases, we have shown the determination of a moderately complex structure (α-Al₂O₃) to within a few picometres per atom relative to the structure determined using conventional single crystal X-ray diffraction, which required 2,000 structure factor magnitude measurements (21). This approach succeeds even for structures where all structure factors have the same sign (SI Text).