Refining short-range order parameters from the three-dimensional diffuse scattering in single-crystal electron diffraction data

This study compares, for the first time, short-range order parameters refined from the diffuse scattering in single-crystal X-ray and single-crystal electron diffraction data.


S2.1. Long-range order model
For the long-range order model, a cell with a size of 6 × 6 × 6 NbCoSb unit cells (cell parameter a = 5.89864(3) Å, space group F4 ̅ 3m (Zeier et al., 2017)) was created, and 1/6 of the Nb atoms were replaced by vacancies to form the B1 structure as defined in (Roth et al., 2020).Each Sb atom was moved by 0.148 Å towards its neighbouring vacancy, and each Co atom was moved by 0.128 Å away from its neighbouring vacancy (displacements refined from the Bragg reflections in the single-crystal X-ray diffraction data of the slowly cooled sample Nb0.81CoSb (SC-0.81))(Roth et al., 2021).The Sb atoms were moved along the cubic <100> directions, while the Co atoms were moved along the cubic <111> directions (Fig. 1(b)).The resulting B1 cell (Fig. S1) has cell parameter a = 35.3918(2)Å and space group P1.A structure with a size of 6×6×6 B1 cells was created in DISCUS.
In (Roth et al., 2021), the BD structure was used instead of the B1 structure.The BD structure is a combination of the B1 structure and the A2 structure.The BD, B1 and A2 structures are defined in (Roth et al., 2020).Because we noticed that the diffraction patterns calculated from the B1 structure agree better with the experimental diffraction patterns than the ones calculated from the BD structure, we used the B1 structure instead of the BD structure.

S2.2. Short-range order model
For the short-range order model, a starting structure with a size of 25 × 25 × 25 NbCoSb unit cells (cell parameter a = 5.89864(3) Å, space group F4 ̅ 3m (Zeier et al., 2017)) was created.1/6 of the Nb atoms were randomly selected and replaced by vacancies.Periodic boundary conditions were imposed to avoid edge effects.
A Monte Carlo simulation in DISCUS is used to minimize the energy E of the crystal until the target correlations between nearest neighbour vacancy pairs ( (1/2,1/2,0) ) and next-nearest neighbour vacancy pairs ( (1,0,0) ) are achieved.The energy E of the crystal is defined as (Roth et al., 2020): ) , vac

𝑖=1
(1) with  vac the number of vacancies in the crystal.The summation in the first term is over all 12 nearest neighbour (NN) vacancy sites j of vacancy i, whereas the summation in the second term is over all six next-nearest neighbour (NNN) vacancy sites j' of vacancy i.   = 1 if site j is occupied by a vacancy and   = 0 if site j is occupied by a Nb atom.Similarly,  ′ = 1 if site j' is occupied by a vacancy and  ′ = 0 if site j' is occupied by a Nb atom. 1 is the energy assigned to a nearest neighbour vacancy pair, and  2 is the energy assigned to a next-nearest neighbour vacancy pair.
Each Monte Carlo step, two randomly selected Nb atoms/vacancies are switched positions.When the new configuration has a lower energy E, then it is always accepted.When the new configuration has a higher energy E, then it is only accepted when the transition probability P, given by is less than a random number η, chosen uniformly in the range [0,1].∆E is the energy difference between the new and the old configuration, T is the temperature, and k is Boltzmann's constant.The temperature T controls the proportion of accepted modifications that lead to a higher energy E. If T = 0, only changes that decrease the energy E will be accepted.The higher the temperature T, the more moves will be accepted that lead to a higher energy  & Proffen, 2008).The number of Monte Carlo cycles was chosen equal to 1000 times the number of atoms within the crystal.
The short-range order model in DISCUS was calculated for kT = 0.001 (Equation 2).If T = 0, only changes that decrease the energy E of the crystal will be accepted.The higher the temperature T, the more moves will be accepted that lead to a higher energy E. Fig. S2 shows the diffuse scattering in the h0l plane calculated for different values of kT.Differences in the sharpness of the diffuse scattering can be explained by differences between the target and the achieved correlation coefficients.The diffuse scattering was calculated for a target correlation between nearest neighbour vacancies of  (1/2,1/2,0) = −0.20 and a target correlation between next-nearest neighbour vacancies of  (1,0,0) = −0.10.For kT = 0.001, the achieved correlations are  (1/2,1/2,0) = −0.18 and  (1,0,0) = −0.09,while for kT = 1, the achieved correlations are  (1/2,1/2,0) = −0.16 and  (1,0,0) = −0.07.The achieved correlations are thus lower for higher values of kT, which explains the differences in the calculated diffuse scattering.For each crystal, the diffuse scattering was also averaged over 50 lots with a size of 12 × 12 × 12 unit cells.The diffuse scattering was calculated for a target correlation between nearest neighbour vacancies of  (1/2,1/2,0) = −0.20 and a target correlation between next-nearest neighbour vacancies of  (1,0,0) = −0.10.

S2.3. Monte Carlo refinement
A Monte Carlo refinement in DISCUS was used to refine the target correlation between next-nearest neighbour vacancy pairs ( (1,0,0) ), the target distance between a vacancy i and a neighbouring Sb atom k (  ), and the target distance between a vacancy i and a neighbouring Co atom k' ( ′ ).Each refinement cycle, the short-range order parameters ( (1,0,0) ,   and  ′ ) are adjusted, and the model crystal is recalculated.The diffuse scattering is calculated and compared with the observed diffuse scattering.This process is repeated until the best agreement between calculated and observed diffuse scattering intensities is obtained.
The differential evolutionary algorithm (Price et al., 2005) mimics the changes in a plant or animal population according to the Darwinian principle of natural evolution.The algorithm starts with a group of M members (parents).Each member represents a set of N short-range order parameters.
Next, the algorithm creates a new group of M members (children) by adjusting the short-range order parameters of their parents.The parents and the children with the lowest R-values (Equation 4) survive and will be the parents of the new generation (survival of the fittest).This procedure is repeated for a number of refinement cycles (generations) until the R-value converges to its minimum (Neder & Proffen, 2008).
The sum is over all measured reciprocal lattice points Qi, Iobs and Icalc are the observed and calculated diffuse scattering intensities.The weights   were set to unity so that all data points i contribute equally to the summation.
S3. Dynamical refinement of the average crystal structure

S6. The 3D-∆PDF
The x0z plane of the experimental 3D-∆PDF in Fig. 5 is almost identical for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-0.81).A positive peak is found at the origin since the distance of an atom to itself is always zero.Strong negative features are visible at interatomic vectors (0.5,0,0.5) and (1,0,0 finding two nearest or two next-nearest neighbour vacancies is also lower in the real structure than in the average structure.Fig. S19 shows the x0z plane of the 3D-∆PDF for longer interatomic distances. Strong positive features are visible at interatomic vectors (1.5,0,1.5),(2,0,0) and (3,0,0), showing the preferred distances between the vacancies.Besides, the magnitudes of the 3D-∆PDF features decrease more quickly for the thermally quenched sample than for the slowly cooled sample, which means that the correlation length of the local Nb-vacancy order is longer for the slowly cooled sample than for the thermally quenched sample.

Figure S1
Figure S1 B1 structure, as defined by (Roth et al., 2020), showing the Nb-vacancy ordering in the long-range order model.Sb and Co atoms are omitted for clarity.

Figure S2
Figure S2 Diffuse scattering in the h0l plane calculated for a) kT = 0.001, b) kT = 0.5, c) kT = 1 and d) kT = 5.The diffuse scattering was averaged over ten crystals with a size of 25 × 25 × 25 unit cells.

Figure S3
Figure S3 Comparison of the h0l plane from single-crystal X-ray and single-crystal electron diffraction, both for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-0.81).The top row shows the experimental diffuse scattering; the bottom row shows the diffuse scattering calculated in Scatty from the structure models calculated in DISCUS.The experimental single-crystal X-ray diffraction data were previously reported by (Roth et al., 2021).

Figure S4
Figure S4Comparison of the h0.5l plane from single-crystal X-ray and single-crystal electron diffraction, both for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-

Figure S5
Figure S5 Comparison of the hhl plane from single-crystal X-ray and single-crystal electron diffraction, both for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-0.81).The top row shows the experimental diffuse scattering; the bottom row shows the diffuse scattering calculated in Scatty from the structure models calculated in DISCUS.The experimental single-crystal X-ray diffraction data were previously reported by (Roth et al., 2021).Angular broadening of the Bragg reflections is mainly due to crystal mosaicity.

Figure
Figure S6 (a) X-ray and (b) electron atomic form factors of Co, Nb and Sb as a function of d*=2sin()/λ.With d the distance between the lattice planes,  the scattering angle and λ the wavelength.

Figure S7
Figure S7 Symmetry with Laue class m3 ̅ m was applied to the three-dimensional diffuse scattering calculated in Scatty.The effect of the vacancy distribution on the h0l plane is shown in Fig. S8.The h0l plane of a perfectly ordered NbCoSb crystal without vacancies shows sharp Bragg reflections at integer hkl values [Fig.S8(a)].In Nb0.84CoSb, 1/6 of the Nb sites are occupied by vacancies.When the vacancy distribution is random and when there are no displacements of Sb and Co atoms, a broad diffuse background will be visible [Fig.S8(b)], which is called monotonic diffuse Laue scattering (Warren et al., 1951).The short-range Nb-vacancy order in Fig. S8(c) results in highly structured diffuse scattering between the Bragg reflections, whereas the long-range Nb-vacancy order in Fig. S8(d) results in sharp satellite reflections.

Figure S8
Figure S8 Structure models and their corresponding calculated single crystal X-ray diffraction patterns.(a) The calculated h0l plane of a perfectly ordered NbCoSb crystal without vacancies shows sharp Bragg reflections at integer hkl values.(b) Nb0.84CoSb structure with a random vacancy distribution and without displacements of Sb and Co atoms.The random vacancy distribution gives rise to monotonic diffuse Laue scattering.(c) Nb0.84CoSb structure with correlations between nearest and next-nearest neighbour vacancies.Displacements of Sb and Co atoms are indicated by arrows.The short-range Nb-vacancy order in (c) results in highly structured diffuse scattering between the Bragg reflections, whereas the long-range Nb-vacancy order in (d) results in sharp satellite reflections.Fig. S9(a) shows the diffuse scattering calculated for a Nb0.84CoSb crystal with only occupational disorder (correlations between nearest and next-nearest neighbour vacancies).The intensity of the diffuse scattering for a crystal with only occupational disorder decreases with increasing scattering angle.Fig. S9(b) shows the diffuse scattering calculated for a Nb0.84CoSb crystal with only displacive disorder (displacements of Sb and Co atoms around the vacancies).The diffuse scattering for a crystal with only displacive disorder shows asymmetries with respect to the Bragg reflections.It should be noted that the diffuse scattering calculated for the crystal with only displacive disorder looks different from the one reported in the Supporting Information of (Roth et al., 2021), which was calculated using a custom Python script.The observed diffuse scattering in the h0l plane in Fig. 3 is thus due to both occupational and displacive disorder (Fig. S9(c)).

Figure S9
Figure S9 Structure models and their corresponding calculated single crystal X-ray diffraction patterns.(a) The calculated h0l plane for a Nb0.84CoSb crystal with only occupational disorder.Correlations between nearest and next-nearest neighbour vacancies give rise to the observed diffuse scattering.(b) The calculated h0l plane for a Nb0.84CoSb crystal with only displacive disorder.Displacements of Sb and Co atoms give rise to the observed diffuse scattering.(c) The highly structured diffuse scattering in the h0l plane is due to both occupational and displacive disorder.

Figure S14
Figure S14 Monte Carlo refinement applied to the diffuse scattering in the h0l plane from singlecrystal X-ray diffraction data of the thermally quenched sample (Q-0.84 #2).Evolution of (a) the Rvalue, (b) the target correlation between next-nearest neighbour vacancies ( (1,0,0) ), (c) the target distance between a vacancy i and a neighbouring Sb atom k (  ), and (d) the target distance between a vacancy i and a neighbouring Co atom k' ( ′ ).The figure shows the average value (blue) and the smallest and highest value (red) at each refinement cycle.The value with the lowest R-value at each refinement cycle is shown in black.

Figure S15
Figure S15 Monte Carlo refinement applied to the diffuse scattering in the h0l plane from threedimensional electron diffraction (3D ED) data of the thermally quenched sample (Q-0.84 #2).Evolution of (a) the R-value, (b) the target correlation between next-nearest neighbour vacancies ( (1,0,0) ), (c) the target distance between a vacancy i and a neighbouring Sb atom k (  ), and (d) the target distance between a vacancy i and a neighbouring Co atom k' ( ′ ).The figure shows the average value (blue) and the smallest and highest value (red) at each refinement cycle.The value with the lowest R-value at each refinement cycle is shown in black.

Figure
Figure S16 (a) h0l plane reconstructed from three-dimensional electron diffraction (3D ED) data acquired on the slowly cooled sample (SC-0.81).(b) h0l plane after removing the Bragg reflections.(c) x0z plane of the three-dimensional difference pair distribution function (3D-∆PDF).Positive 3D-∆PDF features are red and negative features are blue.

Figure S17
Figure S17 Comparison of the x0.27z plane of the X-ray and electron three-dimensional difference pair distribution function (3D-∆PDF), both for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-0.81).The 3D-∆PDF was reconstructed from the three-dimensional diffuse scattering data of which the h0l plane is shown in Fig. 3.The top row shows the 3D-∆PDF of the

Figure S18
Figure S18 Comparison of the h0l plane reconstructed from single-crystal X-ray diffraction and three-dimensional electron diffraction (3D ED) after removing the Bragg reflections, both for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-0.81).

Figure S19
Figure S19 Comparison of the x0z plane of the X-ray and electron three-dimensional difference pair distribution function (3D-∆PDF), both for the thermally quenched sample (Q-0.84 #2) and the slowly cooled sample (SC-0.81).The 3D-∆PDF was reconstructed from the three-dimensional diffuse scattering data of which the h0l plane is shown in Fig. 3.The top row shows the 3D-∆PDF of the

Figure S20
Figure S20 Comparison of the x0z plane of the simulated electron three-dimensional difference pair distribution function (3D-∆PDF) for two different Q-ranges (-8 ≤ h,k,l ≤ 8 and -20 ≤ h,k,l ≤ 20).The 3D-∆PDF maps were calculated from the simulated three-dimensional reciprocal lattice of the slowly cooled sample (SC-0.81).Positive 3D-∆PDF features are red and negative features are blue.
During the Monte Carlo simulation, the values of the distances   and  ′ in Equation3are adjusted.When the target distances   and  ′ are achieved, the Lennard-Jones potential energy will achieve its minimum.Each Monte Carlo step, one Sb atom is moved towards its neighbouring vacancy and one Co atom is moved away from its neighbouring vacancy.When the new configuration has a lower energy E, then it is always accepted.When the new configuration has a higher energy E, then it is only accepted when the transition probability P in Equation 2 is less than a random number η, chosen uniformly in the range [0,1] (Neder

Table S1
Average structure refinement for the thermally quenched sample (Q-0.84 #2).The dynamical refinement from the Bragg reflections in three-dimensional electron diffraction (3D ED) data acquired on three different crystals is compared with the reference refinement from the Bragg reflections in single-crystal X-ray diffraction data(Roth et al., 2021).Refined atomic displacement parameters for Sb, Co and Nb (split model in Table1).