Lattice Row Distance and Its Application in Row-Indexing

In this paper, we propose six general formulae to describe those relationships between the lattice row distance, the lattice parameters and the Miller indices h, k and l for all crystal systems along with any direction. This finally establishes the foundation of the row-indexing, a new method for deriving Miller indices from the lattice row distance. Triclinic talc is used as an example for row-indexing. This new indexing method is especially useful for beam-sensitive materials.


Introduction
Electron diffractometry is a powerful tool for determining crystal structure and microstructures [1].However, there are many factors that cause "defective electron diffraction patterns" which create difficulty in structural studies and can even render these "patterns" useless.These factors include defects and local disorder in both crystal chemistry and structure [2] such as variation in thickness of the crystal [3], and fast amorphization by electron radiation damage.This phenomenon is widely found in beam-sensitive materials and "soft matter" in geology [4][5][6], biology [7,8], physics [9,10], inorganic chemistry [11,12] and other fields [13][14][15][16].
Li et al. [17] proposed a row-indexing method to overcome the difficulty in indexing those "defective electron diffraction patterns".However, it is only available for crystal systems other than triclinic and for those electron diffraction patterns obtained along [uv0].According to the equation of row-row distance (X) for the monoclinic crystal system X = [(kb*) 2 +(ha*) 2 (sinβ*) 2 ] 0.5 [17], the row-row distance X is a function of Miller indices h, k and lattice parameters a*, b*, β* only, and no relation with l and c*.Therefore, the row-indexing method is useless to those electron diffraction patterns with Miler index l = 0.This limitation is also applied to cubic, tetragonal, orthorhombic, trigonal and hexagonal crystal systems [17].In addition, it is not easy to obtain special selected-area electron diffractions (SAED) along [uv0].The row-indexing along [uv0] is difficult to determine.Meanwhile, large amount of images of "defective electron diffraction patterns" along [uvw] are presented that contain important structural information.However, no parallel row-indexing work has been made since 2012.In this paper, we propose six formulae to describe these relationships between Miller indices h,k,l and row-row distance for all crystal systems in any orientation and finally present the foundation of "row-indexing".

Materials and Methods
Specimen talc (HW-1179) was collected from Shimen county, Hunan province, China.Natural talc powder was ground from the raw sample with a FD-4 hammer mill and random preparation was made for X-ray diffraction (XRD) analysis.The instrument used for XRD analysis is an X'Pert Crystals 2019, 9, 62 2 of 8 PRO MPD (PANalytical) diffractometer with a super detector (X'Celerator).The detection parameters of the instrument were as follows: Cu Kα radiation and working at 40 kV and 40 mA, a 0.017 • 2θ step size, a 20 s scan time per step and a 3-70 • 2θ scanning range in continuous scanning mode.After measurement, software X'Pert Highscore Plus version 4.6 was used to subtract background, strip Kα2 component, search peaks and match with the diffraction pattern from ICDD 2005 [18].Lattice parameters were refined by Unitcell after indexing.A transmission electron microscope (JEM-2100F) with energy dispersive spectrometer was used and working at 200kV to analyze lattice fringes.Sample preparation for transmission electron microscope (TEM) test was made by pipetting suspensions of ground powder onto the carbon-coated copper mesh.CrystalMaker version X was utilized for simulating the electron diffraction pattern.

Formulae of Lattice Row Distance
Equation (1) (see Appendix A for derivation) describes the relationship between the lattice row distance (X), the Miller indices (h,k,l) and the lattice parameters (a*, b*, c*, α*, β*, γ*) for the triclinic system (a For the cubic system (a = b = c, α = β = γ = 90 • ) the formula of lattice row distance is given as: For the tetragonal system (a = b = c, α = β = γ = 90 • ) it is given as: For the trigonal and hexagonal systems (a = b = c, α = β = 90 • , γ = 120 • ) given as: For the orthorhombic system (a = b = c, α = β = γ = 90 • ) as: and for the monoclinic system (a = b = c, α = γ = 90 • , β = 90 • ) as: From above equations, it can be seen that Miller indices can be uniquely determined from the known lattice parameters and measured lattice row distance(s).The method of finding Miller indices from lattice row distances is termed Row-Indexing.

Procedures of Row Indexing (Known Lattice Parameters)
Row-Indexing is a five-step indexing process: (1) Measure the distance(s), X mea (s), between two parallel reciprocal lattice rows in electron diffraction pattern (one of them passes through the original point).From the derivation (see Appendix A), it is clear that spot h 1 k 1 l 1 (or h 2 k 2 l 2 ) can represent a row and its direction (000→h 1 k 1 l 1 for instance), and that the "row distance" (000→h 1 k 1 l 1 ) means the perpendicular distance between this spot (and any others along the same row) and the parallel row passing through the origin.There exist many rows in many directions in a good ED pattern (Figure 1).One can measure any "row distance" in any direction.However, it should be noted that, (i) as described in Equations ( 1)-( 6), only for a correct choice of h 1 k 1 l 1 can the X mea s be (approximately) equal to the X cal s; (ii) only one obvious set of rows is visible in a typical defective patterns (Figure 2a for instance), so that the number of X mea (s) is small; (iii) from (ii) all rows not parallel to the actual row will produce non-matching X cal s and X mea (s) and will be rejected in step (3), regardless of the incident beam direction (see Table 1).
(2) Apply one of these Equations (( 1)-( 6), depending upon the crystal system) to the reciprocal lattice parameters and build a series of (h,k,l)s and corresponding X cal s; Note in step (1) one can build a series of (h,k,l)s in the way increasing the summation of h,k,l and calculate their corresponding X cal s.
(3) Compare X mea (s) with X cal s and find h, k and l from the closest match of X mea and X cal ; Note that only for rows passing through this h 1 k 1 l 1 spot can X mea be close to X cal and the match is accepted, see Table 1 for detail.
(4) Index whole pattern based on these known h, k and l; (5) Check indices according to crystallographic rules such as parallelogram law, interplanar spacing and angle formulae.Authors should discuss the results and how they can be interpreted in the perspective of previous studies and of the working hypotheses.The findings and their implications should be discussed in the broadest context possible.Future research directions may also be highlighted.
Crystals 2019, 9, x FOR PEER REVIEW 3 of 8 (1) Measure the distance(s), Xmea(s), between two parallel reciprocal lattice rows in electron diffraction pattern (one of them passes through the original point).From the derivation (see appendix), it is clear that spot h1k1l1 (or h2k2l2) can represent a row and its direction (000h1k1l1 for instance), and that the "row distance" (000h1k1l1) means the perpendicular distance between this spot (and any others along the same row) and the parallel row passing through the origin.There exist many rows in many directions in a good ED pattern (Figure 1).One can measure any "row distance" in any direction.However, it should be noted that, (i) as described in equations 1-6, only for a correct choice of h1k1l1 can the Xmeas be (approximately) equal to the Xcals; (ii) only one obvious set of rows is visible in a typical defective patterns (Figure 2a for instance), so that the number of Xmea(s) is small; (iii) from (ii) all rows not parallel to the actual row will produce non-matching Xcals and Xmea(s) and will be rejected in step (3), regardless of the incident beam direction (see Table 1).
(2) Apply one of these equations (1-6, depending upon the crystal system) to the reciprocal lattice parameters and build a series of (h,k,l)s and corresponding Xcals; Note in step (1) one can build a series of (h,k,l)s in the way increasing the summation of h,k,l and calculate their corresponding Xcals.
(3) Compare Xmea(s) with Xcals and find h, k and l from the closest match of Xmea and Xcal; Note that only for rows passing through this h1k1l1 spot can Xmea be close to Xcal and the match is accepted, see Table 1 for detail.
(4) Index whole pattern based on these known h, k and l; (5) Check indices according to crystallographic rules such as parallelogram law, interplanar spacing and angle formulae.Authors should discuss the results and how they can be interpreted in the perspective of previous studies and of the working hypotheses.The findings and their implications should be discussed in the broadest context possible.Future research directions may also be highlighted.Table 1.Calculated and measured distances between rows and their discrepancies in electron diffraction patterns of talc (Figure 2a).

Indices
Reciprocal Space Real Space  2b and c) and obtain the correct Miller indices (see Table 1).The row-indexing is also verified by published data (see Table 2).
Table 2. Calculated and measured distances between rows and their discrepancies in electron diffraction patterns of ZnPc [16] and Octacalcium phosphate (OCP) [19].

Indices Reciprocal Space
Real Space  Table 1.Calculated and measured distances between rows and their discrepancies in electron diffraction patterns of talc (Figure 2a).

Indices Reciprocal Space
Real Space Note: subscripts mea and cal stand for the measured and the calculated respectively and the unit of 1/X mea and 1/X cal is Å.

Validating By Electron Diffraction Patterns
Clay minerals are typical beam-sensitive materials and develop abundant defects and dislocations in their structures.Such defects and dislocations in structure and their beam-sensitive properties easily result in defective SAED patterns and cause considerable difficulty for indexing."Row-Indexing" can effectively overcome these difficulties.
Step 1: Measure the distances X mea s between [00L] and [HnKnL] (we use capital letters H, K and L with [ ] to represent reciprocal lattice row(s) to distinguish from the Miller indices h, k and l) in SAED patterns of talc (Figure 2a).Step 2: Compare them with a series of X cal s calculated by Equation (1).Step 3: Find out that those with the least difference between X mea and X cal are from the incident direction  2b,c) and obtain the correct Miller indices (see Table 1).
The row-indexing is also verified by published data (see Table 2).
Table 2. Calculated and measured distances between rows and their discrepancies in electron diffraction patterns of ZnPc [16] and Octacalcium phosphate (OCP) [19].Note: subscripts mea and cal stand for the measured and the calculated respectively and the unit of 1/X mea and 1/X cal is Å. *: reject (err >3%), **: reject (spot not on row).

Conclusions
Lattice row distance describes the distance from a row to the origin in reciprocal space.Any electron diffraction pattern consists of many but limited groups of rows in different orientation.A group of rows comprises a series of parallel rows in the same orientation and in a special interval of lattice row distance.The weakness, deformation and tailing of diffraction spots produced from beam-sensitive materials and "soft matter" make rows clear and outstanding.
The proposed equations well describe the relationships between the lattice row distances, the Miller indices and the reciprocal lattice parameters.Clear rows allow row-indexing to overcome the difficulties caused by spot weakness and deformation and make it possible to index those defective or weak electron diffraction patterns.The Row-indexing can be also applied to the FFT of HRTEM images.

Conclusions
Lattice row distance describes the distance from a row to the origin in reciprocal space.Any electron diffraction pattern consists of many but limited groups of rows in different orientation.A group of rows comprises a series of parallel rows in the same orientation and in a special interval of lattice row distance.The weakness, deformation and tailing of diffraction spots produced from beam-sensitive materials and "soft matter" make rows clear and outstanding.
The proposed equations well describe the relationships between the lattice row distances, the Miller indices and the reciprocal lattice parameters.Clear rows allow row-indexing to overcome the difficulties caused by spot weakness and deformation and make it possible to index those defective or weak electron diffraction patterns.The Row-indexing can be also applied to the FFT of HRTEM images.
Funding: This research was funded by THE NATIONAL NATURAL SCIENCE FOUNDATION OF CHINA, grant number 41872048, 40972038, 40872034.

Figure 1 .
Figure 1.A good electron diffraction pattern.Numbers 1-7 mark seven lattice rows in seven different directions (black lines).Four row distances (rows 1-4) are shown as (red) lines normal to their directions.

Figure 1 .
Figure 1.A good electron diffraction pattern.Numbers 1-7 mark seven lattice rows in seven different directions (black lines).Four row distances (rows 1-4) are shown as (red) lines normal to their directions.

Figure 2 .
Figure 2. (a) Electron diffraction pattern obtained from the defective domain of talc.(b) The simulated electron diffraction pattern of talc incidence along [010].(c) The simulated electron diffraction pattern of talc incidence along [310].Note the angle marked by white lines in (a) is very similar to that in (b) and is noticeably different from the one in (c).

Figure 2 .
Figure 2. (a) Electron diffraction pattern obtained from the defective domain of talc.(b) The simulated electron diffraction pattern of talc incidence along [010].(c) The simulated electron diffraction pattern of talc incidence along [310].Note the angle marked by white lines in (a) is very similar to that in (b) and is noticeably different from the one in (c).