Mathematical tools that connect different indexing 2 analyses

: As mathematical tools that can be commonly used for indexing analyses from different 8 types of experimental patterns, we have recently developed (i) rules on forbidden hkl ʹ s that can be 9 used even when the space group and setting are unknown, (ii) algorithm for error ‐ stable Bravais 10 lattice determination, (iii) generalization of the de Wolff figure of merit for powder diffraction (1D 11 data) to data in higher ‐ dimensions such as Kikuchi patterns (2D data) by electron backscatter 12 diffraction (EBSD). In particular, (ii) could be used in a variety of situations, not just for indexing. It 13 is explained how (i)—(iii) are used in the mathematical framework of our indexing algorithms. The 14 developed software is now available on the web.


Introduction
Mathematical tools that can be commonly used in ab-initio indexing analyses are introduced herein.They were originally invented for powder diffraction [1], and subsequently applied to indexing of Kikuchi bands in electron backscatter diffraction (EBSD) patterns [2]."Ab-initio" means that the indexing is carried out without any prior information on the parameters and Bravais type of the unit cell.
In the case of powder diffraction, the values of d-spacings (hence, lengths of reciprocal-lattice vectors) are obtained from positions of diffraction peaks.In the EBSD case, the orientations of reciprocal-lattice vectors are provided from the positions of Kikuchi bands.Our indexing algorithms for them use a common mathematical framework shown in Figure 1.First, the parameters of the primitive cell are determined, because (i) simple rules of systematic absence are available, if only basis vectors of a primitive lattice are considered.Subsequently, (ii) Bravais-type (and centering) determination is carried out.This process can be error-stable enough to deal with unit-cell parameters containing large errors due to zero-point shifts (powder [3]) or projection-center shifts (EBSD [4]).We also (iii) generalized the idea of the de Wolff figure of merit Mn [5], which has been the most efficient indicator in powder indexing.The generalized one presents similar properties for EBSD patterns [2].
In what follows, the mathematics used for (ii) is mainly discussed.Due to the limitation of the space, (i), (iii) are only mentioned, referring to published papers.The author believes that these theoretical results will be also useful in different analyses of crystallography.( ) , , , where ( )  is a vector, and S is the symmetric matrix with i j v v ⋅ in the (i,j)-entry.S is also the Gramian (or metric tensor [6]) of L. The stabilizer of S is defined as the following subgroup of ( ) N GL  (= the group of integral matrices with the determinant ±1): The Gramians S1, S2 belong to the same Bravais type, if , there exists ( ) As the boundaries of  are prone to complications, overlaps of the boundary : in ¶ =     ( in  : set of interior points of  ) are frequently allowed.In such a case,  should satisfies (i) and the following (ii)' and (iii)': for only finitely many ( ) It is straightforward to see that any S in in  satisfies

( )
Stab { 1} S =  .Thus, all the S with non-triclinic Bravais types belong to the boundary of  .The following are the definitions of Venkov [8] and Delaunay reductions used in Section 3; for any fixed 0 From the definition, Venkov-reduced with regard to 0 .S In particular, S is Selling-reduced, if S belongs to , where N A is the symmetric matrix with 2 in the diagonal entries and 1 in the other entries.

Determination of the primitive lattice
For some types of SA, forbidden reflections are not exceptional, but occur considerably high rate.
The rules of SA stated in International Tables depend on the space group and setting of atomic positions.Simpler rules of SA are required for developing algorithms that generally work.
In order to obtain such simple rules, only basis vectors of the primitive lattice are considered herein.
[Theorem 2, [9]] Regardless of the type of SA, there are infinitely many primitive sets , , 2 ,2 l l l l l l + + corresponds to an extinct reflection due to the SA.
Furthermore, there exist infinitely many 2D sublattices , l l contained in a basis of L*.In the powder case, the inner product Similarly, in the EBSD case, the direction * * l l of the reciprocal-lattice vector l are obtained from the coordinates of Kikuchi bands.Therefore, the vector-length ratio In both of Eq.( 5), ( 6), the lengths (or directions) of , , 2 l l l l + are sufficient to obtain the matrix (or the ratio of its components) in Eq.( 7).The remaining length (or direction) of 2l l + can be used to remove unlikely solutions quickly.
Theorem 2 is a 3D version of Theorem 1.

Theorem 2.
[Theorem 4 in [9]] Regardless of the type of SA, there are infinitely many bases , l l are assigned to various combinations of observed reflections.See Figure 5 of [2] for the EBSD case.

theoretical Background
After the parameters of the primitive cell are obtained in the indexing process, it is necessary to convert them into parameters of the conventional cell.For a Gramian matrix obs S extracted from observed data, how can one estimate the Bravais type of the unknown true value Ŝ of obs S ?In practice, the error can be observational errors or rounding errors of floating-point numbers [11].

Ŝtab( ) S . It is common in libraries developed
by mathematicians (e.g., Magma [13]) that the parameters of a lattice cannot be entered in floating-type numbers.For this reason, error-stable methods have been investigated in mathematical crystallography.This determination can be done by step 1 & 2 in Table 1 by using a finite set H0 with the following property, where  is a domain that fulfills (i), (ii)', (iii)' in Section 1.1.
is nearly reduced (i.e., close to  ) for some reduced obs S (i.e., obs S that belongs to  ).

Prepared sets in codes
1.For a domain  that fulfills (i), (ii)', (iii)' in Section 1.1, and its topological closure ,  let 0 G be the finite set consisting of all ( ) Step 2 Output the stored g, S after removing duplicates. 1 The same calculation can be done, even if  is replaced by a union of finitely many [ ] g  such as the Venkov reduced domain 0 S  .The only difference is that Lk may not be in the boundary of  .
If  is the Niggli-reduced domain[Chap.9.2.2, 6], G0 in Table 1 consists of 168 elements.The number m of lattice characters Lk is 42, after two triclinic cases are excluded [Table 9.2.5.1,6].H0 must contain G0, because all the non-triclinic S belong to the boundary of the Niggli-reduced domain.Hence, the computation time of the method of Table 1 is roughly estimated as |H0|×m≧168×42 = 7056.This is a little time consuming, if it is applied to multiple primitive cells generated in the indexing process.
The methods of Andrews & Bernstein [14,15] are basically same as this Niggli-reduced case, although it is not assured that their heuristics can always generate all the necessary operations (in their method, 25 operations in [16] are used to generate the elements of H0).
Use of the Delaunay reduced domain was proposed Burzlaff & Zimmermann [17,18].This reduces the number of lattice characters from 44 to 30.However, H0 is set to { 1 } in their method, so it can basically handle only the exact case.
Thus, the following are the problems, in order to develop a faster and more reliable Bravais-lattice determination method.
Question 1: Which reduction method minimizes the computation time for Table 1?  as  in Table 1.
・ S0 = A3 in Eq.( 4): if  is the Delaunay-reduced domain, ( ) By the choice of S0, 0 S  can include non-triclinic Lk in the interior distant from its boundary In such a case, if Ŝ is in Lk and Venkov-reduced with regard to 0 S , ôbs S S » is also in the interior of 0 .S  As a result, it is not necessary to consider the nearly-reduced.H0 may be set to { 1 } if the Venkov reduction is used.
Based on this idea, the author proved that error-stable determination is possible, under the following condition C on the error size of obs S [19] (This is an answer to Question 2): C: for any 3-by-3 symmetric matrix T and 0 Namely, C excludes the case: (If L is the crystal lattice with the Gramian Ŝ , the half of the squared-length of any non-zero vector in L is observed as a positive value).Hence, C only assumes that the error of obs S is not extraordinary large.Under this condition, the following is proved: Theorem 3. [Theorem 1--4 in [19]] For a given 0 obs S S Î  , assume that Ŝ belongs to the Bravais type B, in addition to C. In this case, Ŝ belongs to the VB, a union of finitely many linear subspaces in Table 2.There are only 5 Bravais types in Table 2, because it is straightforward to classify unit cells after the centering determination (i.e., conventional unit cells) into higher-symmetric Bravais types.
In Table 2, if H0 ={ 1 }, the number of operations required for the error-stable determination is same as the exact case.Therefore, contrary to our intuition, it is possible to output S with obs S S » and Ŝtab( ) Stab( ) S S = very generally, without increasing the computation time at all.However, the error of obs S affects the distance between the output S and its true value ˆ. S

Computation results
The implemented program is used in our indexing software [1,2].It was also used to build a database of quadratic forms [20].In [2], indexing analysis was carried out for EBSD patterns with projection-centers shifted as follows (z: the camera length).
, , 0, 0.005, 0.01, 0.02 The software succeeded in indexing of orthorhombic-cubic cells in most of the cases.Among them, there were only a few failures due to errors in Bravais-lattice determination (see Tables 4,5 and Figure 9 in [2] for more details).

Discussion
The theorems presented in Sections 2, 3 hold true for any symmetry types the crystal structures can have.Our error-stable Bravais-type determination is probably the first method where the result is mathematically guaranteed, even for parameters with large error.As I explained, the number of operations |H0|×m≧168×42 = 7056 cannot be decreased as long as the Niggli reduction is used, although it can be reduced from 7056 to 154 (58, conditionally) by using the Venkov reduction for I3 and A3.
However, there might be other reduction methods (or S0) that provide a faster method.No studies have been reported for lattices of dimensions more than 3.
Prior to such theoretical results, software developers of indexing analysis had to develop caseby-case algorithms or heuristics to deal with the symmetries by themselves.As ab-initio indexing software for powder diffraction patterns, ITO [21,22], TREOR [23], and DICVOL [24] are well known.
EBSD ab-initio indexing have been also studied in [25--27], although more accurate methods for band extraction and projection center identification are also needed for this indexing analysis.
From a theoretical point of view, the two indexing analyses have much in common.This suggests that updating the mathematical crystallography is effective to obtain reliable and efficient analytical methods in short time.
Funding: This study was financially supported by the PREST (JPMJPR14E6) and JST Mirai (JPMJMI18GD)

Figure 1 .
Figure 1.Common mathematical framework of our indexing algorithms

.
consisting of N-by-N symmetric matrices, an inner product is defined by the metric space (the distance between S and T equals The subset of N  consisting of all the positive definite symmetric matrices is denoted by N +  ．The action of The following is an overview of the lattice-basis reduction theory that discusses methods to provide the representatives for the orbits ( ) \ N N GL +   .Namely, N + Ì   is the subset that fulfills the following (i), (ii):

Question 2 :
Under which assumption on the error size of obs S , is it possible to output all the S with Question 1 was to use the following Venkov-reduced domain 0 S

Table 2 . B, S 0 , V B in Theorem 1.
This is yes, if the software user only needs the S closest to 1As for the face-centered case, our method simply uses the fact that obs S has the face-centered symmetry if and only if the inverse of obs S has the body-centered symmetry.2obs S among those with the Bravais type B.