research papers
A phase retrieval algorithm for triply periodic minimal surface like structures
^{a}Department of Physics, Faculty of Science, Nanomaterials Research Division, Research Institute of Electronics, Shizuoka University, Shizuoka, 4228529, Japan
^{*}Correspondence email: oka.toshihiko@shizuoka.ac.jp
A method to solve the crystallographic Acta Cryst. A78, 430–436]. Using this feature, a new iterative phase retrieval algorithm for was developed. The algorithm modifies electron densities outside the upper and lower thresholds in the iterative Fourier transformation process with fixed amplitudes for the structure factors, and efficiently searches for the structure with the smallest difference between the maximum and minimum electron densities. The proper structure was determined by this algorithm for all tested data for lyotropic bicontinuous cubic phases and mesoporous silicas. Although some cases required constraints such as the for more than half could be determined without any constraints, including space groups.
of materials with triply periodic minimal surface like structures, such as lyotropic bicontinuous cubic phases, is reported. In triply periodic minimal surface like structures, the difference between the maximum and minimum electron densities tends to be the smallest for the true phase combination among the possible combinations [Oka (2022).Keywords: crystallographic phase retrieval; lyotropic liquid crystals; mesoporous silica; triply periodic minimal surfaces.
1. Introduction
Crystallography serves as a fundamental method to offer structural information for the understanding of materials. This statement applies to both typical, i.e. well ordered, crystals, and highly disordered systems such as liquid crystals. However, the application of crystallographic methods to the latter is very difficult because of the limited number of reflections available. The bicontinuous cubic phase of lyotropic liquid crystals (LLC) has a triply periodic minimal surface (TPMS) like structure and threedimensional periodicity with natural beauty (Hyde et al., 1996). Thus, these systems are good targets for structural studies. The structure of the LLC bicontinuous cubic phase was first determined by the pioneering work of Luzzati et al. using Xray powder diffraction (Luzzati et al., 1988; Mariani et al., 1988). Recently, the present author established a singlecrystallization method for the LLC bicontinuous cubic phase (Oka & Hojo, 2014), and performed structural analyses of the LLC bicontinuous cubic phases while considering model structures (Oka, 2017; Oka et al., 2018, 2020). The essential difficulty is the socalled which is still unsolved.
Numerous researchers have tackled the ). and variants are routinely used for it is impossible to imagine the current practice of crystallography without them (Giacovazzo, 2001). These methods are based on the general properties of crystals. The chargeflipping method, which has a simple iterative algorithm for structural determination, utilizes the fact that the positive electron density of atoms is concentrated in a small region, while the remaining regions have zero electron density (Oszlányi & Sütő, 2004, 2008; Palatinus, 2013). Its success implies the possibility of using the structural features of highly disordered systems if we can identify suitable expressions for the features.
(Sayre, 2015In a previous paper (Oka, 2022), the author proposed two indicators reflecting the plausibility of phase combinations of experimental data for the LLC bicontinuous cubic phase. The indicators are based on the structural features of materials: the electron density tends to be constant in the direction in which molecules diffuse. This property suggests that the continuity of the density is a good indicator. The difference density between the maximum and minimum (I_{ρ}) seems to be a good and simple indicator. Another indicator (I_{K}), which utilizes the Hessian matrix of the electron density, is also acceptable. In the previous paper, the electron density and indicators were calculated for all possible phase combinations for the test data with centrosymmetric space groups. The result showed that the two indicators work well. Although the potential utility of the method based on the two indicators was confirmed for LLC bicontinuous structures, testing all phase combinations becomes impractical with an increase in the number of independent reflections. In addition, the method is only applicable to centrosymmetric space groups.
In this paper, an iterative phase retrieval algorithm for ; Palatinus, 2013). It is emphasized that the iterative algorithm opens the possibility of the application to structures without centrosymmetry, which tremendously widens the search space of phase combinations. The method was tested for LLC bicontinuous cubic phases and mesoporous silicas, and structures were successfully determined for all tested data, although additional constraints were necessary in some cases. Notably, the method converged to the proper structures without information on the space group.
to overcome these difficulties is proposed. The algorithm was developed with reference to the chargeflipping method (Oszlányi & Sütő, 20082. Phase retrieval method
The algorithm was designed to find the structure or structurefactor phase with the smallest difference between the maximum and minimum electron densities in a
The electron density was calculated in a of 32 × 32 × 32 voxels. All calculations were performed using a homemade script in python3.A finite number of structurefactor amplitudes are observed in crystal diffraction experiments. Unobserved amplitudes, including , were set to zero in the iterative process. cannot be determined in principle because no zeroelectrondensity region is observed in the target sample. Since is not included, the electron density, , in the
has positive and negative values and its mean is 0.In the iterative calculations, the initial phases were given random values in the range of −π to π which satisfy the Friedel law. When the was used as a constraint on the initial phases, the phases were assumed to be random to the extent that these satisfy the phase relations in (Shmueli et al., 2010). The initial is as follows:
where H_{obs} is the set of reciprocallattice vectors h for which the was observed.
The following calculation steps from (i) to (iv) are repeated:
(i) is calculated by determining the Fourier transform of ,
(ii) is modified as follows to obtain :
where k_{f} is the flipping parameter and is the magnitude of the electrondensity shift. When the of the positive region v_{p} is not set, 0. t_{+} and t_{−} are the upper and lower thresholds. in the equation is the amount above or below the threshold. The flipping parameter k_{f} is often set between 0 and 1. When k_{f} = 0, the density outside the upper and lower thresholds is replaced by the threshold value: . When k_{f} = 1, it is replaced by the threshold minus the amount above or below the threshold: = = . The upper and lower threshold values are and . When , , where σ is the standard deviation of . When , is the rootmeansquare of in the region , and is the rootmeansquare of in the region . only when v_{p} is set. v_{p} is the in the where and takes values between 0 and 1. The threshold parameter k_{t} is often set from 0.2 to 1.3.
(iii) is Fourier transformed to obtain the
= .(iv) A new
is obtained as follows:When k_{t} = 1 and , the upper and lower threshold values t_{+} and t_{−} are σ, the standard deviation of . Thus, most final structures have electron densities outside the upper and lower thresholds. When the parameters k_{f} and k_{p} were fixed in the calculation, the rate of obtaining the correct solution was low. Therefore, during the calculation cycle, they were changed. In the jth cycle, the parameters were as follows: , where , and n are the mean, width and period of a parameter k, respectively. Different values were set for n_{f} and n_{p} so that the periods of parameter changes of k_{f} and k_{p} do not coincide.
If the structure is known to have centrosymmetry, it is possible to add the constraint that the
be real. In this case, the in step (iv) becomeswhere Re() is a function that extracts the real part of the value in the parentheses. The progress of the iterative calculation can be monitored by the difference between the maximum and minimum electron densities in the ). Calculation steps are repeated a set number of iterations, and the final structure is the one with the minimum during the calculation. Multiple independent calculations may produce different results. Also, the originshifted structures are outputted even if the structures are completely equivalent.
= (Oka, 2022The output results were evaluated using and I_{K}. I_{K}, an indicator based on the convexity of the electron density, is described in detail in the previous paper (Oka, 2022) and briefly here. In materials with TPMSlike structures, convex regions, i.e. regions with closed isoelectrondensity surfaces, are considered to be small. The convex regions can be determined by the eigenvalues of the Hessian matrix of the electron density, :
where the subscripts indicate partial derivatives. If the signs of the eigenvalues of the Hessian matrix are all the same, then the region is strictly convex (Rockafellar & Wets, 2010). Let C be the electrondensity regions that are strictly convex. Then, the indicator I_{K} is defined by . Both indicators have been shown to be useful in the of LLC bicontinuous cubic phases (Oka, 2022). When both of these indicators are small and close values are obtained in several independent calculations, it can be presumed that the proper structure is found.
The phases of the outputted structure can be compared with that of the true structure by the following R_{p} value (Oka, 2022):
where = [ ]. The origin shift r_{shift} of the outputted structure was obtained by minimization of . R_{p} approaches 0 as the phase agreement becomes higher. According to Babinet's principle, when the electron density of a sample is inverted, the diffraction intensity is equivalent to that before the inversion. For this reason, a structure and a densityinversed structure are equivalent, and the smaller R_{p} of the two structures is adopted as the R_{p} of the structure. Thus, R_{p} is a value between 0 and 1.
3. examples
3.1. Phase retrieval with a centrosymmetric spacegroup constraint
Table 1 lists the 11 data sets used in the The six LLC bicontinuous cubic phases correspond to Xray diffraction data from single crystals we have measured previously (Oka, 2017; Oka et al., 2018, 2020). These are considered to be accurate with regard to the phase of the structure factors, which is obtained by optimizing the model to the Xray diffraction data. As examples other than the LLC bicontinuous cubic phases, the structures of mesoporous silicas were determined. Four of the mesoporous silica data sets were obtained using highresolution (Sakamoto et al., 2004; Gao et al., 2006; Zhang et al., 2011; Cao et al., 2016), and the structure factors, including phase, are considered reliable. The data for MCM48 were obtained by powder Xray diffraction at a synchrotron radiation facility, and the phase of the was obtained by model optimization (Solovyov et al., 2005). Therefore, the phases are reliable.
‡Volume fraction of silica on one gyroid network region. The other gyroid network region (including TPMS) is void. 
Materials with TPMSlike structures generally have large structural disorder, and the spatial resolution of the data obtained is lower than for solid crystals. Also, due to their high symmetry, the number of independent reflections is not large (Table 1). In the typeI LLC bicontinuous cubic phase, polar regions with high electron density gather on the TPMS and nonpolar regions with low electron density gather on the network side; in type II, the positions of polar and nonpolar regions are opposite to those in type I (Hyde et al., 1996). In all mesoporous silicas in Table 1, with one exception, the silica walls are located on the TPMS and the network sides are vacant. Only PEO_{117}bPS_{77}PtBA_{179} templated mesoporous silica has a singlegyroid structure with silica located on only one of the two srs nets, the other being vacant (Cao et al., 2016). Therefore, this structure is chiral and not centrosymmetric, whereas all the other structures are centrosymmetric. The of the TPMS side of each sample is shown in Table 1.
First, the structural determination was tried using the constraint of centrosymmetry and using as few other constraints as possible. The initial phases were set to random values within the range satisfying the phase relationship in each centrosymmetric et al., 2010). Since the space groups are centrosymmetric, the constraint that the be real in the iterative processes was used. A structure lacking centrosymmetry was not tried here because of the complexity of the initial constraints derived from the The results of the structure determinations are summarized in Table 2. For eight data sets, out of 100 independent calculations without additional constraints, structures with sufficiently small R_{p} were obtained for all of them. The R_{p} were calculated between the obtained structures and the previously determined true structures.
(Shmueli

Fig. 1(a) shows the changes in the indicator I_{ρ} and the parameters k_{t} and k_{f} during the iterative calculation process of in the data of phytantriol . I_{ρ} was useful as an indicator to show the progress of since I_{ρ} appeared to be minimized in the iterative process. The top part of the figure shows the change in I_{ρ} over 100 independent calculations. The I_{ρ} values for those 100 independent calculations converge to two value traces when the number of iterations is less than 10, and to a single trace at about 20 iterations. Thereafter, the value of I_{ρ} may increase temporarily in response to changes in the parameters k_{f} and k_{t}, but it remains close to the lowest value in many regions. Although the value of I_{ρ} fluctuated during the iterative calculations, the structure with minimum I_{ρ} was adopted as the final solution. The lower part of the figure shows changes in the parameters k_{f} and k_{t}, which were changed periodically in the iterative process. The reason for the periodic changes of both parameters was to perform calculations with different combinations of parameters and to avoid staying in local minima. Fixing k_{f} and k_{t} in the iterative process often resulted in convergence to several different local minima, which frequently did not result in the proper structure.
The I_{ρ} and I_{K} values for the structures obtained from the phytantriol and data are shown in Figs. 2(a) and 2(b). The true and obtained structures have very close or coincident I_{ρ} and I_{K} values. The previous study showed that the I_{ρ} and I_{K} values of the true structure have values close to the minimum (Oka, 2022). The structure obtained in this study [Fig. 2(a)] is consistent with the structure with the minimum values of both I_{ρ} and I_{K} previously obtained for phytantriol . The two obtained structures in phytantriol [Fig. 2(b)] also agree with the first and second minimum structures previously obtained.
Each data set for monoolein in three space groups showed a single final structure (Table 2). For and , the R_{p} value was 0, which is in perfect agreement with the true structure. On the other hand, had R_{p} = 0.080, the highest minimum R_{p} among the data used in this study. This is probably due to the fact that the true structure in is I_{K}minimal but not I_{ρ}minimal, as shown in a previous paper (Oka, 2022). The mesoporous silica, except for MCM48, yielded a final solution close to the true structure (Table 2).
For C_{12}EO_{6} and MCM48, the proper structure could not be obtained without additional constraints. The proper structure was obtained when the parameter v_{p} was set. When v_{p} = 0.75 was used in C_{12}EO_{6}, structures close to the true structure were obtained 19 times out of 100 independent calculations. Fig. 2(c) shows the distribution of I_{ρ} and I_{K} for the 100 structures obtained from the C_{12}EO_{6} calculation; the 19 indicator points [overlapped by a single point in Fig. 2(c)] almost overlap with the point of the true structure. However, the I_{ρ} and I_{K} of the other points are widely distributed, but many of them have smaller I_{ρ} than the I_{ρ} of the true structure. Many points are also distributed around the indicator values of the structure determined without using the v_{p} constraint. This result is consistent with a previous study showing that the I_{ρ} of the true structure is not minimal in C_{12}EO_{6}. Therefore, I_{ρ} cannot be used as an indicator of the true structure in C_{12}EO_{6}. On the other hand, as shown in previous studies, I_{K} is valid as an indicator, and the structure with the minimum I_{K} among the obtained structures is close to the true structure. The true structure was always obtained when v_{p} = 0.25 in MCM48. The I_{ρ} of the obtained structure was larger than that when v_{p} was not set, but I_{K} was smaller (data not shown). It has been previously shown that when the of the TPMS side is above 0.7, I_{ρ} is less valid as an indicator of structure and I_{K} is superior to I_{ρ} (Oka, 2022). These examples suggest that if the of the high (or low) electrondensity region is far from 0.5, it is better to set an additional constraint v_{p}. It is also better to focus on I_{K} rather than I_{ρ} in determining the final structure when v_{p} needs to be set. A reminder about v_{p} here: v_{p} determines the zero position of the electron density when calculating the standard deviation, and is one of the parameters similar to k_{f} and k_{t}. It is recommended to set a value close to the real but it is not necessary to set it to the same value. It would be better to adjust v_{p} based on the results of iterative calculations.
3.2. Phase retrieval without a spacegroup constraint
Next, the structure was determined without using the spacegroup constraint and with as few other constraints as possible. The results are summarized in Table 3. Under the condition of no constraints, proper structures with R_{p} < 0.1 were obtained with more than half probability for six data sets. To obtain the proper structures in the other data, constraints were necessary.
‡Sum of two chiral (mirror image) structures. 
An example of data for which a proper structure was obtained under completely unconstrained conditions is shown for phytantriol . The change in I_{ρ} during the iterative process is shown in Fig. 1(b). The value traces of I_{ρ} repeatedly increase and decrease, but some of them reach a minimum value around I_{ρ} = 14 after a large increase in I_{ρ}. Calculations with fixed parameters k_{f} and k_{p} were also tried: I_{ρ} was often trapped at a certain value and I_{ρ} did not decrease further even if the number of calculations was increased. On the other hand, when k_{f} and k_{p} were changed periodically, the search for a solution was efficient. In addition, the calculation time was acceptable for the current condition. Therefore, the parameters k_{f} and k_{p} were changed periodically. The data of phytantriol yielded 56 structures with R_{p} < 0.1 in 100 independent calculations over 700 iterations (Table 3). Scatter plots of I_{ρ} and I_{K} are shown in Fig. 2(a). The values of the obtained structures are distributed around that of the true structure. The distribution of the obtained values appears to be in the process of converging to the value of the structure using the as a constraint. When the number of iterations was set to 2800, the number of structures with R_{p} < 0.1 increased to 92 out of 100 calculations. If the number of iterations is set to a larger value, the distribution will probably converge to one or a few points. For phytantriol , 80 of 100 independent calculations resulted in R_{p} < 0.1, with a minimum R_{p} of 0.021 (Table 3). The scatter plot, Fig. 2(b), shows that the I_{ρ} and I_{K} distributions for the final structure are concentrated near the true structure.
For the monoolein and data, the
had to be constrained to a real number because without the constraint the proper structure could not be obtained. Compared with the other data, the result of monoolein may have been affected by the small number of independent reflections. On the other hand, monoolein , with the smallest number of independent reflections, converged to a nearly true structure without constraints. The ease of convergence may differ for each space group.For the _{12}EO_{6} has the largest value of 0.72 (Table 1). A constraint was necessary for C_{12}EO_{6}. When calculated with v_{p} = 0.75, the distribution of I_{ρ} and I_{K} for the obtained structure is shown in Fig. 2(c). The values of the obtained structures are distributed around that of the true structure. Also, under this condition, all calculations yielded structures with R_{p} < 0.1. In the calculation with v_{p} = 0.72, the structure with R_{p} < 0.1 was obtained in 60 out of 100 independent calculations (Table 3). Better results were obtained for C_{12}EO_{6} by changing v_{p} slightly from the actual volume fraction.
of the LLC bicontinuous cubic phase, CThe three mesoporous silicas, AMS10, EO_{20}PO_{70}EO_{20} templated and IBN9, yielded final structures close to the true structure without constraints. The volume fractions of silica in these structures ranged from 0.283 to 0.412, relatively close to 0.5 compared with the value of 0.25 for MCM48 (Table 1). MCM48 required two constraints: a real number for the and a set When the number of iterations was 700, only five out of 100 independent calculations yielded a structure with R_{p} < 0.1. Then, when the number of iterations was set to 7000, the number increased to 72 (Table 3). It seems that as iterations are increased, the number of final structures that are close to the true structure increases. For PEO_{117}bPS_{77}PtBA_{179} templated, 100 independent calculations yielded 96 structures with R_{p} < 0.1. Since the structure is chiral, this value includes the mirror image structure; excluding the mirror image, the value is 45. This calculation required the constraint of v_{p}; the best I_{K} values for the final structures were obtained when v_{p} = 0.40. The reason for the necessity of the v_{p} constraint is probably due to the large number of independent reflections.
The obtained electron densities of phytantriol/water () and C_{12}EO_{6} are shown in Fig. 3. The electron densities are the translationally shifted ones obtained by phase retrieval. These are in good agreement with the previously determined electron densities (Figs. S1 and S2 in the supporting information), although the was not used.
Without the use of space groups, nearly true structures were obtained without any constraints in more than half of the cases. Even when iteration did not work without constraints, the constraints of a real v_{p} yielded a nearly true structure. Judging from this result, this method can be used for a TPMSlike structure even when the is not known.
or setIn this paper, the electron density was calculated in a d_{min}/2 (Oszlányi & Sütő, 2004), where d_{min} is the minimum interplanar distance. Thus, the minimum required number of divisions per one edge of the is d_{min}/2a, where a is a lattice constant. In the data used in this paper, the minimum required number is 10–22. Since a small number of divisions results in a poor representation of the electron density and affects the values of the indicators I_{ρ} and I_{K}, 32 divisions were used.
of 32 × 32 × 32 voxels. The minimum grid spacing required for the chargeflipping method is4. Conclusion
An algorithm for determination of TPMSlike structures from diffraction data was developed and shown to be effective. Although additional constraints were necessary in some cases, R_{p}, which indicates phase difference, was close to zero. The algorithm appears to efficiently find the structure with the smallest I_{ρ} by cutting or inverting the electron density outside the upper and lower thresholds. Constraints on v_{p} were necessary when the was far from 0.5 and when the number of independent reflections was large. For some parameter settings, several different structures were obtained, but by using I_{K} in conjunction with I_{ρ}, the proper structure could be determined. Previous studies have shown that I_{K} is superior to I_{ρ} in determining TPMSlike structures. Therefore, when multiple structure candidates are obtained, I_{K} should also be used as a reference.
was achieved for all tested data sets. The obtained structures were close to the true structure, andThe determination was more efficient when information on centrosymmetric space groups was used. Therefore, when using this algorithm to determine unresolved structures, it appears better to use known information on space groups. On the other hand, this is not essential in
Thus, it is possible to determine a structure for which the is unknown. The number of independent reflections in the data sets for the was 8 for monoolein and 39 for MCM48, so the relative spatial resolution was low for monoolein and high for MCM48. Thus, the method appears to be applicable independently of spatial resolution and the number of independent reflections. The previously reported method requires a long computation time when the number of independent reflections is large, so the iterative method reported here should be used in such cases.Although the LLC bicontinuous cubic phase and mesoporous silica were used here, this method could be applied to TPMSlike structures such as thermotropic liquid crystals and polymers. If there exists a threedimensional periodic structure with bicontinuous or polycontinuous regions different from the TPMSlike structure, the method may be applicable to such a system as well.
Supporting information
Supporting information. DOI: https://doi.org/10.1107/S2053273322010786/ik5006sup1.pdf
Acknowledgements
I thank Kazuya Saito for useful discussions and helpful comments on the manuscript.
Funding information
The following funding is acknowledged: JSPS KAKENHI (grant No. JP20H04634).
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