ZnO studied via CXDI

The key feature of the CXDI technique is the ability to map displacements within the crystal. The phase maps shown in Figure 6.1 demonstrate scalar cross sections of direct space phase through a translucent isosurface of the amplitude of a ZnO rod reconstructed

Figure 6.1: Phase map cross sections taken at 80nm intervals along the length of the rod (a)150nm → (f)550nm. The (101) Q vector direction is also shown.

in Section 5.2.4. The cross sections were taken at 80nm intervals along its length parallel to the Q vector.

Close inspection of the phase maps shown in Figure 6.1 along an extrapolated (101) Q vector highlight a phase modulation ranging from +π/4 to -π/4 to +π/4, a non-quadratic relationship, which is present in all phase maps along the length of a ZnO rod shown in Figure 6.1. A misalignment of the diffraction data due to pixelation can be ruled out as the origin of this feature as this would generate a linear phase ramp, previously discussed in Section 5.1.2. In this case a linear phase ramp was not evident however a linear feature can underly the observed phase modulation. A suitable method for implementing this correction has not been considered to date, however the centering process should limit its

impact on the result. This particular measurement was completed using KB focussing, hence, if a quadratic phase structure were to be generated the curvature of the wavefront would need to be considered as a possible source. The quadratic phase structure would see a smooth quadratic modulation in the phase from facet to facet along the sampled Q-vector, this was not observed. Any complicated phase modulation is likely to correspond to a strain in the crystal. The scalar cut planes slice through the phase perpendicular to the c-axis, hence of the six equivalent hexagonal 100 facets; the two along and against the Q-vector respectively exhibit the same sign, the remaining four which are less coupled to Q also exhibit modulation in the phase.

6.1.1 Interpreting the phase in direct space

In Section 2.2 we introduced CXDI experiments, the interpretation of the generated phase maps will be reiterated. In direct space, the calculated phase corresponds to the displace-ment along the Q vector relative to an underlying equilibrium crystal lattice, φ = Q.u(r).

In the positive Q direction a positive phase corresponds to an expansion, a negative phase corresponds to a compression. In negative Q, a positive phase corresponds to a compres-sion, a negative phase corresponds to an expansion. Fitting to the phase modulation will provide an insight into their physical origins.

The phase modulation seen in Figure 6.1 shows a positive phase on both 100 facets along and against the Q-vector. This represents an expansion in the positive Q direction and a compression in the opposite direction, the origins of which are undetermined. We ex-pect the presence of oxygen and zinc vacancies, if they were to be uniformly distributed

throughout the centre of the rod the atomic spacing in the crystal would be constant throughout the crystal thus the Bragg peak would simply shift. If however a sample was placed in vacuum an increase in oxygen vacancies at the surface would lead to a con-traction near the facet relative to the rest of the crystal [164]. Equally a concon-traction on the six hexagonal facets would not be expected as they are non polar and atomically flat with equal numbers of cations and anions in the surface plane. A contraction is predicted on the zinc face however the scale of contraction, 0.4˚A, is likely below the resolution of the data, close to 50nm, so would not be observed. It is clear strains of some form have been observed and are considerably larger than the resolution of the direct space phase (∼ 0.15rad) but their origins are not obvious.

6.1.2 How reliable is a reconstruction?

This is the primary cause for concern. The nature of the ‘Phase problem’, a problem with N unknowns and N knowns leads to not only the multiple ambiguous solutions discussed in Section 4.2.4, but a significant number of different solutions (local minima in solution space) if the constraints are insufficient. Two approaches exist to overcome these prob-lems, first, ambiguous solution identification [20,213] and second, averaging [35,37,168]. A combination of the two will be implemented here as they are both advantageous for differ-ent reasons. The first makes the second applicable and averaging highlights reproducible phase features and averages out the erroneous phase features.

6.1.3 Phasing variables

The phasing variables discussed in section 4 provide the constraints for the phasing oper-ation. The constraint with the largest impact on the solution is the support constraint.

The shrinkwrap method introduced by Marchesini et al. has proved very successful in forward scattering phase retrieval experiments [35,36,123]. The algorithm begins with the autocorrelation function as the first estimate of the illuminated object, several iterations of HIO and ER are run and a support tailored to the current solution via a convolution operation with a 3x3x3 voxel cube and smoothing with a gaussian function. Further it-erations of HIO and ER follow, the support is updated after each set of itit-erations. The algorithm converges when the support is found to be self consistent between algorithm iterations (<1% variance in shape). For a detailed overview of the modified Shrinkwrap algorithm see Appendix G.2.

Although a valid method, the shrinkwrap method outlined relies on the initial solution of pre-existing algorithms with a very loose support constraint, an enlarged copy of the autocorrelation function. In solution space a number of solutions are available with loose constraints, and are subsequently optimised by shrinking the support around them. The shrinkwrap method has optimised the solution based on the support constraint, it does not definitively identify the global minimum. It simply identifies a solution based on a set of loose constraints and optimises it. From random starting points we would expect and observe a large variation in reconstructions but their supports are optimised. From this position we can apply the two approaches mentioned previously and find a more reliable average. In Section 5.2.2 an argument was put forward for using an object with a flat

phase variation as the starting point, thus a comparison will be made here.

6.1.4 Identifying ambiguous solutions

The random starting point leads to multiple different solutions, in order to combine them the average phase needs to be set to zero, solutions need to be overlaid, ambiguous solutions identified and made equivalent (i.e enantiomorphs) and finally averaged together. To identify ambiguous solutions we refer to Section 4.2.4. The solutions are now equivalent and can be averaged accordingly. The average solution must then be normalised before it is used to seed further phasing iterations.