Nobuo Ishizawa and Yumi Inagaki
セラミックス基盤工学研究センター年報 (2008). Vol. 8, 35-49

Review

A Guide to Discriminating the Rhombohedral Cellfrom the Face-Centred Pseudo Cubic Cell
Nobuo Ishizawa † and Yumi Inagaki ¶† Ceramics Research Laboratory, Nagoya Institute of Technology,

Asahigaoka 10-6-29, Tajimi, Gifu 507-0071 JAPAN¶ Graduate School of Nagoya Institute of Technology, Gokiso-cho, Nagoya, Aichi 466-8555

The crystal in cubic system contains four 3-fold axes with different orientations,whereas the rhombohedral modification with the face-centred pseudo cubic cell
contains only one ‘true’ 3-fold axis. The discrimination of the ‘true’ 3-fold axis fromthe other three ‘pseudo’ 3-fold axes is not difficult when the rhombohedral distortion is large. However, the difficulty increases as the distortion becomes small andthe cell dimensions become very close to values of ‘metrically cubic’. This note firstsummarizes the geometrical relation among the face-centred pseudo cubic, primitive rhombohedral, and triple-hexagonal cells. Then it provides a procedure tochoose the ‘true’ 3-fold axis, and accordingly the ‘true’ rhombohedral cell, from theexperimentally-obtained face-centred pseudo cubic cell. In the description of thisguide, a special attention has been paid for the Smart Apex II single-crystal X-ray
diffractometer users.

面心擬似立方単位胞から菱面体単位胞を識別するための手引
石澤伸夫†・稲垣友美¶†

名古屋工業大学セラミックス基盤工学研究センター 〒507-0071 岐阜県多治見市旭ヶ丘 10-6-29¶

名古屋工業大学大学院工学研究科 〒466-8555 愛知県名古屋市昭和区御器所町

立方晶系の結晶は異なった方位をもつ四種類の三回回転軸を含むが，面心擬似立方単位胞をもつ
菱面体晶系の多形における真の三回回転軸はこのうちの一種類のみである。菱面体歪みが大きい
時には，真の三回回転軸を他の三種類の擬三回回転軸から識別するのは難しくない。しかし菱面
体歪みが小さくなり，単位胞の寸法が数値的に立方体に極めて近くなるにつれて，識別に困難さが増す。本解説では，まず，面心擬似立方単位胞，菱面体単位胞，および三重六方単位胞の幾何学的関係を概説し，次に，実験的に得られた面心擬似立方単位胞から真の三回回転軸を見いだす
方法，従って真の菱面体単位胞を見いだす方法を述べる。この手引の記述に際しては，Smart Apex
II 単結晶 X 線回折計利用者のために特別な配慮を払った。

1. Introduction

having a doubled cube edge compared with the
ABO3 perovskite aristotype (͵ത).
Among the hettotype structures exist rhombohedral modifications which are classified into twogroups, one is the ‘͵ത group’ consisting of ͵ത,R3, R32, R3m and ͵ത, and the other is the ‘͵തgroup’ consisting of ͵ത and R3c. These two groups
can be discriminated by the systematic reflection

The perovskite-type compounds crystallize invarious symmetries. In the double perovskite definedas A2B’B”O6, where B’ and B” occupy the octahedral sites in the rock salt-type arrangement, thesymmetry of the aristotype (the ideal structure withthe highest symmetry) is ͵ത (for example, see
Howard et al., Acta Cryst. B59, 463-471, 2003),― 35 ―

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell

conditions as will be explained in Sections 2 and 6.
The rhombohedral cell can be alternatively
represented by the triple-hexagonal cells in obverseor reverse settings.
The cubic crystal contains four 3-fold axes withdifferent orientations, whereas the rhombohedral
modification contains only one ‘true’ 3-fold axis.
The discrimination of the ‘true’ 3-fold axis from theother three ‘pseudo’ 3-fold axes is not difficult when
the rhombohedral distortion is large. However it
becomes quite difficult when the distortion becomes
small and the cell dimensions become very close to‘metrically cubic’. This note first summarizes thegeometrical relations among the face-centred cubic,
primitive rhombohedral, and triple-hexagonal cells
and then provides a practical procedure to choose the‘true’ 3-fold axis, and accordingly the ‘true’ rhombohedral cell, from the face-centred pseudo cubic
cell.
The motivation of writing this note came in thecourse of very difficult single-crystal structure analyses of SrTiO3-LaAlO3 solid solution in collaboration with Prof. H. Ohsato and his laboratory members. Most serious problem was that the crystals
containing 0.5 and 20 mole% SrTiO3 seem rhombohedral but their unit cells are close to metrically
cubic. They have appreciable intensities for h-hl
reflections with l=odd on the basis of the triple
hexagonal obverse, suggesting that they do not belong to the ͵ത group, but does to either ͵ത or͵ത. However we suffered from the difficulty inchoosing the correct cell from various candidates.
This was partly due to the lack in accuracy of theinitial cell obtained, for example, by ‘DETERMINEUNITCELL’ procedure of the Smart Apex II software.
To overcome various difficulties in addition to this,we realized the necessities of clarifying the underlying geometrical relations and of establishing asystematic approach to identify the rhombohedralcell having almost metrically cubic dimensions. We
hope this note serves to people who suffer fromsimilar problems in different pseudo-cubic crystals.

(a) Intensities of reflections for the obverse and reverse settings satisfy the conditions below.
–h+k+l = 3n for hkl in obverse
h-k+l=3n for hkl in reverse.
(b) To check whether the current cell is recorded
either obverse or reverse, we need to look at intensities of the h-hil (i=-h-k=2h) reflection parity
groups.
Group A: Ͳʹʹതͳǡ ʹʹതͲͳǡ ʹതͲʹͳǡ Ͳʹതʹͳതǡ ʹതʹͲͳതǡ ʹͲʹതͳത
They all satisfy the reflection condition, -h+k+l=3n(obverse).
Group B: Ͳʹʹതͳതǡ ʹʹതͲͳതǡ ʹതͲʹͳതǡ Ͳʹതʹͳǡ ʹതʹͲͳǡ ʹͲʹതͳ
They do not satisfy the reflection condition,–h+k+l=3n, whereas do the condition, h-k+l=3n(reverse).
(c) If all reflections of the parity group A have thesame appreciable intensities whereas those ofgroup B have none, then the cell setting is the obverse. Vice versa is the reverse.
(d) To change from reverse to obverse, reverse thedirections of a1 and a2 unit cell vectors in hexagonal representation:‫܉‬െͳ Ͳ Ͳ ‫܉‬ቆ‫܊‬ቇൌ ൭ Ͳ െͳ Ͳ൱ ቆ‫܊‬ቇ‫ ܋‬୭ୠ୴ͲͲ ͳ ‫ ܋‬୰ୣ୴

Eq. 2.1

2. The ‘obverse’ and ‘reverse’ settings for the
triple-hexagonal cell
The geometrical relation of the triple-hexagonalcells in obverse and reverse settings with respect tothe rhombohedral cell is shown in Fig. 2.1. The figure is essentially the same as given in Fig. 5.1.3.6 in
the International Tables for Crystallography Vol. A,
p84 [1]. The ‘triple’ means that there are three lattice
centering points in the hexagonal cell. In this sense,
the rhombohedral cell are sometimes called ‘primitive-rhombohedral’. The triple-hexagonal obverse
has centering points at 0,0,0; 2/3,1/3,1/3; 1/3,2/3,2/3,
whereas the triple-hexagonal reverse at 0,0,0;
1/3,2/3,1/3; 2/3,1/3,2/3. The obverse setting is recommended to use in modern crystallography. Several notes regarding the two settings are given below.

Fig. 2.1. Geometrical relation of the triple-hexagonal cells
in obverse and reverse settings with respect to the rhombohedral cell
― 36 ―

Nobuo Ishizawa and Yumi Inagaki(e) The reflection parity groups A and B are used to
distinguish the ͵ത (͵ത, etc) and ͵ത groups.

Note that if l=2n for hhl in P cell, then the converted
hkl in F cell become all even. If l=2n+1 for hhl in P
cell, then the converted hkl in F cell become all odd.

͵ത etc: h+l=3n and l=2n for h-hl͵ത, ͵ത etc: h+l=3n for h-hl

3.2. Four possible ‘rhombohedral’ orientations

Since –h+k+l=3n in triple-hexagonal obverse, thereflection condition for h-hl are transformed into-h+(-h)+l=-2h+l=3{-h+(h+l)/3}=3n. This meansthat h+l in small parentheses should be a multipleof 3. In other words, the condition h+l=3n for h-hl
is not a special one, but automatically derived fromthe general reflection condition, –h+k+l=3n. Important is that the ͵ത group has an additional
special condition, l=2n for h-hl, whereas the ͵തgroup (incl. ͵ത) has no special conditions otherthan the general one, –h+k+l=3n.

If the true symmetry is rhombohedral, then weshould identify the true 3-fold axis from four suspected 3-fold axes in pseudo FCC. In other words,
we should examine four rhombohedral cells, eachcontaining one of the four suspected 3-fold axes in
FCC. The geometrical relation among four possible
rhombohedral cells and the pseudo FCC cell isshown in Fig. 3.1. The filled and open circles standfor the lattice points at z=0 and ½, respectively. The
transformation from the pseudo FCC (pF) into corresponding four rhombohedral ones (R1, R2, R3 andR4) can be expressed as follows. (The first one is thesame as given in Section 2.)

Note that, when the parity group A (h-hl, l=odd) has
appreciable intensities, the space group is limited tothe ͵ത group (incl. ͵ത), and cannot be the ͵തgroup.

ͲͳȀʹ ͳȀʹ ‫܉‬‫܉‬ԢͲͳȀʹ൱ ቆ‫܊‬ቇ൭‫܊‬Ԣ൱ ൌ ൭ͳȀʹ‫ ܋‬୮୊
ͳȀʹ ͳȀʹͲ‫܋‬Ԣ ୖଵെͳȀʹͲͳȀʹ ‫܉‬‫܉‬ԢͳȀʹ ͳȀʹ൱ ቆ‫܊‬ቇ൭‫܊‬Ԣ൱ ൌ ൭ Ͳ‫ ܋‬୮୊
െͳȀʹ ͳȀʹͲ‫܋‬Ԣ ୖଶ

3. Transformation between the face centred celland the primitive cell3.1. Transformation matrixͲെͳȀʹ ͳȀʹ ‫܉‬‫܉‬ԢͲͳȀʹ൱ ቆ‫܊‬ቇ൭‫܊‬Ԣ൱ ൌ ൭െͳȀʹ‫ ܋‬୮୊
െͳȀʹ െͳȀʹͲ‫܋‬Ԣ ୖଷ

Our concern lies in the transformation between
the face-centred pseudo cubic (F) and the primitive
rhombohedral (P) cells, but the transformation matrix between F and P is applicable to general case andnot limited to the case of our concern.

ͳȀʹͲͳȀʹ ‫܉‬‫܉‬ԢെͳȀʹ ͳȀʹ൱ ቆ‫܊‬ቇ൭‫܊‬Ԣ൱ ൌ ൭ Ͳ‫ ܋‬୮୊
ͳȀʹ െͳȀʹͲ‫܋‬Ԣ ୖସ

‫܉‬
‫܉‬
െͳ ͳ
ͳ
ቆ‫܊‬ቇ ൌ ൭ ͳ െͳ ͳ ൱ ቆ‫܊‬ቇ‫ ܋‬୊ͳͳ െͳ ‫୔ ܋‬

Eqs. 3.3

െͳ ͳ
ͳ
݄
݄
൭݇൱ ൌ ൭ ͳ െͳ ͳ ൱ ൭݇൱ͳͳ െͳ ݈ ୔݈ ୊

Eqs. 3.1

Or alternatively,ͲͳȀʹ ͳȀʹ ݄݄ͲͳȀʹ൱ ൭݇൱൭݇൱ ൌ ൭ͳȀʹͳȀʹ ͳȀʹͲ݈ ୔݈ ୊

Eq. 3.2
Note that the transformation matrix for the real cell
vectors is the same as that for reflection indices hkl.
Ex. 3-1. Conversion of several hkl’s.
݄൭݄ ൱ ֜ ൭  ൱݈ ୔ʹ݄ െ ݈ ୊ͳͳ൭ͳ൱ ֜ ൭ͳ൱
ͳ ୔
ͳ ୊

ͳ
ʹ
൭ʹ൱ ֜ ൭ͳ൱ 
͵ ୊
ͳ ୔

Ͳ
ʹ
൭Ͳ൱ ֜ ൭ʹ൱ 
ʹ ୔
ʹത ୊

ͳ
ͳ
൭ͳ൱ ֜ ൭ͳ൱ͳ ୔ͳ ୊Fig. 3.1. Four possible rhombohedral cell orientations, Ri(ai, bi, ci) with i=1~4, with respect to the pseudo FCC cell
(apF, bpF, cpF). Lattice points at z=0 and 1/2 of the pseudo
FCC cell are denoted as solid and open circles, respectively.

ͳͳത൭ͳത൱ ֜ ൭ͳ൱
͵ത ୊
ͳ ୔

The orientation relationship among the fourrhombohedral cells, R1, R2, R3 and R4, is summarized in Table 3.1 using Eqs. 3.4.
― 37 ―

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell‫܉‬‫܉‬ቆ‫܊‬ቇ ൌ‫ ܀‬௜՜௝ ቆ‫܊‬ቇ ǡ‫ୖ ܋‬೔‫ୖ ܋‬ೕ

This equation is convenient in analyzing the four
twin components in a crystal in rhombohedral form.
See more details given in Section 7.2.
Note that the rotation matrices {(1 0 -1), (1 0 0),(1 -1 0)} (palely hatched in Table 7.1) and {(0 1 0),(0 1 -1), (-1 1 0)} (no hatched in Table 7.1) are allowed for permuting generation of the four unit cells,
while {(0 1 -1 }, (1 0 -1), (0 0 -1)} (darkly hatched inTable 7.1) is NOT. The last matrix can only convert
two unit cells, for example, cells 1 and 3, or cells 2
and 4.

‫ ܀‬௜՜௝ ൌ ‫܀‬௝՜௜ ିଵ

Eqs. 3.4
See Table 7.1 to find similar representation for
triple-hexagonal cells. Table 3.1 indicates that thefour rhombohedral cells can be expressed by a single
3x3 matrix, for example, as:‫܉‬ͳ Ͳ െͳ ‫܉‬ቆ‫ ܊‬ቇൌ ൭ͳ ͲͲ ൱ ቆ‫܊‬ቇ‫ୖ ܋‬೔శభ‫ୖ ܋‬೔ͳ െͳ Ͳ

Eq. 3.5

Table 3.1. Orientation relationship (Riĺj) among the four possible rhombohedral cells. The subscripts i(after transformation) and j (before transformation) run from 1 through 4.

i =1

j=1

j=2

j=3

ԟ

Ͳ ͳ Ͳ
൭ Ͳ ͳ െͳ൱െͳ ͳ Ͳ

Ͳ
൭ͳ
Ͳ

ͳ െͳ
Ͳ െͳ൱
Ͳ െͳ

ͳ Ͳ െͳ൭ͳ ͲͲ൱ͳ െͳ Ͳ

Ͳ
൭Ͳ
െͳ

ͳ Ͳ
ͳ െͳ൱
ͳ Ͳ

Ͳ
൭ͳ
Ͳ

ͳ െͳ
Ͳ െͳ൱
Ͳ െͳ

Ͳ
൭Ͳ
െͳ

ͳ Ͳ
ͳ െͳ൱
ͳ Ͳ

i =2

ͳ Ͳ െͳ൭ͳ ͲͲ൱ͳ െͳ Ͳԟi =3

Ͳ ͳ െͳ൭ͳ Ͳ െͳ൱Ͳ Ͳ െͳͳ Ͳ െͳ൭ͳ ͲͲ൱ͳ െͳ Ͳ

ԟ

i =4

Ͳ ͳ Ͳ
൭ Ͳ ͳ െͳ൱െͳ ͳ ͲͲ ͳ െͳ൭ͳ Ͳ െͳ൱Ͳ Ͳ െͳͳ Ͳ െͳ൭ͳ ͲͲ൱ͳ െͳ Ͳ

j=4

ԟ

Ex. 4.1. Conversion of several hkl’s.
Ͳͳ൭ʹ൱֜൭ ͳ ൱ͳ ୌ୭ୠ୴
െͳ ୖ

4. Transformation between the triple-hexagonalcell in obverse setting (H) and the rhombohedral primitive cell (R)

െʹ
െͳ
൭Ͳ൱
֜൭ ͳ ൱ͳ ୌ୭ୠ୴ͳ ୖͷȀ͵ʹcf. ൭Ͳ൱֜ ൭െͳȀ͵൱െͳȀ͵ ୖͳ ୌ୰ୣ୴

4.1. Cell vectors and indicesʹȀ͵ͳȀ͵ ͳȀ͵ ‫܉‬‫܉‬ቆ‫܊‬ቇ ൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵൱ ቆ‫܊‬ቇ‫ୖ ܋‬െͳȀ͵ െʹȀ͵ ͳȀ͵ ‫ ܋‬ୌThe above examples tell that all the reflections allowed for the triple-hexagonal in obverse setting canbe converted to the rhombohedral primitive cell withinteger values, whereas part of reflections allowed
for the triple-hexagonal in reverse setting result in
non-integer indices. This happens because the above
201 reflection does lie on the primitive hexagonallattice in reverse setting (Hrev), but never lie on the
triple-hexagonal lattice in obverse setting (violation
to the rule: -h+k+l=3n).

ʹȀ͵ͳȀ͵ ͳȀ͵ ݄݄൭݇൱ ൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵൱ ൭݇൱െͳȀ͵ െʹȀ͵ ͳȀ͵ ݈ ୌ݈ ୖ

Eqs. 4.1
Or alternatively,ͳ െͳ Ͳ݄݄൭݇൱ ൌ ൭Ͳ ͳ െͳ൱ ൭݇൱
ͳ ͳ
ͳ
݈ ୌ
݈ ୖ

4.2. Conversion between the rhombohedral andthe triple-hexagonal cell parameters

Eq. 4.2

The equations for conversion are given below. In
rhombohedral, a1=a2=a3(=aR), Į=ȕ=Ȗ(=Į). In hex― 38 ―

Nobuo Ishizawa and Yumi Inagaki

agonal, a1=a2(=aH ), c=cH, Į=ȕ=90°, Ȗ=120°. For
details, refer to Eqs. 10.40-10.43 in ‘Crystal Structures: A Working Approach’ by H. D. Megow [2].

ͳ
Ͳ
Ͳ
‫܉‬
ͳ
Ͳ ൱ ȉ ቆ‫܊‬ቇ‫ ܀‬ଷ ൌ ‫۾‬ଷ ȉ ۶ ൌ ൭ ͳʹȀ͵ ͳȀ͵ ͳȀ͵
‫ ܋‬ୌ

ଶ ଵȀଶ

Ͳ
ͳ
Ͳ
‫܉‬
Ͳ
Ͳ ൱ ȉ ቆ‫܊‬ቇ‫ ܀‬ସ ൌ ‫۾‬ସ ȉ ۶ ൌ ൭ െͳെͳȀ͵ ͳȀ͵ ͳȀ͵
‫ ܋‬ୌ

ͳ
ୌ
ୖ ൌ ୌ ቈ͵ ൅ ൬ ൰ ቉ୌ͵Ƚ ൌ ʹ ିଵ

Eqs. 4.4

ଷ
ౙ
౗ౄ

మ భȀమ

ଶቈଷା൬ ౄ ൰ ቉ୌ ൌ ʹୖ ȽʹͶ ଶ Ƚ ଵȀଶ͵ቂͳെୌ͵  ʹቃ
ൌ
Ƚ
ୌ
ʹ 
ʹ

Eqs. 4.3
See Appendix 1 for the relation between the rhombohedral angle Į and the hexagonal c/a ratio whenc/a is close to ¥6 (=2.44949) corresponding to Į=60°(metrically FCC).
4.3. Four possible rhombohedral orientations
from the experimentally-obtained ‘pseudo’
hexagonal cellFig. 4.1. Three pseudo primitive-‘rhombohedral’ cells, R2,
R3 and R4, on the triple-hexagonal (obverse) lattice. The[001]H direction (cH) is perpendicular to this paper. Lattice
points of filled black circles are located at z=0, filled red
ones at z=1/3, open red ones at 2/3 (-1/3). The R1 cell (a0,b0, c0) with its 3-fold axis parallel to [001]H is omitted to
avoid complexity.

The experimentally obtained ‘triple-hexagonal’
cell could be a wrong choice (due to poor accuracy),
if the ratio of the cell dimensions, c/a, is very close to¥6. Figure 4.1 shows three possible alternative
choices, R2, R3 and R4, of the ‘rhombohedral’ cell in
addition to R1 which can be obtained by the normaltransformation procedure given in Section 2, fromexperimentally-obtained triple-hexagonal cell tocoincide the 3-fold axis, i.e., [111]R parallel to[001]H.
Among these four candidates, the only one cell is‘TRUE’ and the other three cells are ‘false’, if thecrystal is really rhombohedral. The transformation
matrices from the experimentally obtained
triple-hexagonal (obverse) cell, H = (aH, bH, cH) T, tothe four possible ‘rhombohedral’ cells , R1 = (a1, b1,c1) T, R2 = (a2, b2, c2)T, R3 = (a3, b3, c3)T, and R4 = (a4,b4, c4) T are as follows:ഥ
4.4. Equivalent reflections for the Laue group ૜
The reflections equivalent to triple-hexagonal andrhombohedral settings for the Laue group ͵ത aregiven in Tables 4.1 and 4.2.
Table 4.1. Equivalent reflections for theLaue group ͵ത in hexagonal and rhombohedral settings (a).
hexagonal hkl
rhombohedral hkl
h-k
k-l
h+k+l
h
k
l
-h-l h-k
h+k+l
l
h
k
k-l
-h-l
h+k+l
k
l
h
-h+k -k+l-h-k-l-h-k-lh+l -h+k -h-k-l-l-h-k-k+l h+k-h-k-l
-k
-l
-h

ʹȀ͵
ͳȀ͵ ͳȀ͵‫܉‬‫܀‬ଵ ൌ ‫۾‬ଵ ȉ ۶ ൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵൱ ȉ ቆ‫܊‬ቇ‫ ܋‬ୌെͳȀ͵ െʹȀ͵ ͳȀ͵

െͳ
െͳ
Ͳ
‫܉‬
െͳ
Ͳ ൱ ȉ ቆ‫܊‬ቇ‫ ܀‬ଶ ൌ ‫۾‬ଶ ȉ ۶ ൌ ൭ ͲെͳȀ͵ െʹȀ͵ ͳȀ͵‫ ܋‬ୌTable 4.2. Equivalent reflections for the Laue group ͵ത in hexagonal and rhombohedral settings (b).
hexagonal hkl
h
k
l
-h-k
h
l
k
-h-k
l
-h
-k
-l
h+l
-h
-l
-k
h+k
-l

2h/3+k/3+l/3-h/3-2k/3+l/3-h/3+k/3+l/3-2h/3-k/3-l/3
h/3+2k/3-l/3h/3-k/3-l/3

rhombohedral hkl-h/3+k/3+l/3
2h/3+k/3+l/3-h/3-2k/3+l/3
h/3-k/3-l/3
2h/3-k/3-l/3
h/3+2k/3-l/3― 39 ―-h/3-2k/3+l/3-h/3+k/3+l/3
2h/3+k/3+l/3
h/3+2k/3-l/3h/3-k/3-l/3-2h/3-k/3-l/3

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell

5. Transformation between the triple-hexagonalcell in obverse setting and the face-centred
pseudo cubic cellʹȀ͵ͳȀ͵ ͳȀ͵ ‫܉‬
‫܉‬
െͳ ͳ
ͳ
‫܊‬
െͳȀ͵
ͳȀ͵ ͳȀ͵൱ ቆ‫܊‬ቇ ቆ ቇ ൌ ൭ ͳ െͳ ͳ ൱ ൭‫ ܋‬୊ͳͳ െͳ െͳȀ͵ െʹȀ͵ ͳȀ͵ ‫ ܋‬ୌെͶȀ͵ െʹȀ͵ ͳȀ͵ ‫܉‬ൌ ൭ ʹȀ͵ െʹȀ͵ ͳȀ͵൱ ቆ‫܊‬ቇʹȀ͵ͶȀ͵ ͳȀ͵ ‫ ܋‬ୌThe conditions, (ii), (iii), (iv), are derived from thegeneral FCC condition, (i). In other words, ͵ത
has no special reflection conditions other than thegeneral FCC one, (i).
The relation between the triple-hexagonal obverse cell and the FCC cell has been derived in Eqs.
5.1. Let us check here the conditions for h-hl reflections (k=-h).
െͶȀ͵ െʹȀ͵ ͳȀ͵݄݄൭݇ ൱ൌ ൭ ʹȀ͵ െʹȀ͵ ͳȀ͵൱ ൭െ݄൱ʹȀ͵ͶȀ͵ ͳȀ͵݈ ୊େେ ୌ୭ୠ୴

െͶȀ͵ െʹȀ͵ ͳȀ͵ ݄݄൭݇൱ ൌ ൭ ʹȀ͵ െʹȀ͵ ͳȀ͵൱ ൭݇൱ʹȀ͵ͶȀ͵ ͳȀ͵ ݈ ୌ݈ ୊ʹͳെ ݄൅ ݈
͵
͵ ‫ۊ‬
‫ۇ‬
ͳ
Ͷ
ൌ ‫ۈ‬൅ ݄ ൅ ݈ ‫ۋ‬
͵ ‫ۋ‬
‫͵ ۈ‬
ͳ
ʹ
‫ۉ‬െ ͵ ݄ ൅ ͵ ݈ ‫ی‬୊େେ

Eqs. 5.1
Or alternatively,ͲͳȀʹ ͳȀʹ ݄ͳ െͳ Ͳ݄ͲͳȀʹ൱ ൭݇ ൱൭݇൱ ൌ ൭Ͳ ͳ െͳ൱ ൭ͳȀʹͳȀʹ ͳȀʹ
Ͳ
ͳ ͳ
ͳ
݈ ୊
݈ ୌ

Eq. 6.2
Substituting l with 3n-h using the equation, h+l=3n,െͳȀʹ ͳȀʹͲ݄ൌ൭ ͲെͳȀʹ ͳȀʹ൱ ൭݇ ൱
݈ ୊
ͳ
ͳ
ͳ

ʹ
ͳ
ʹ
ͳ
െ ݄൅ ݈െ ݄ ൅ ሺ͵ െ ݄ሻ
͵ ‫ۊ‬
͵
‫͵ ۇ‬
‫͵ ۇ‬
‫ۊ‬
ͳ
ͳ
Ͷ
Ͷ
‫݄ ۈ‬൅ ݈ ‫ۋ‬‫ۈ‬ൌ ൅ ݄ ൅ ሺ͵ െ ݄ሻ‫ۋ‬
͵ ‫ۋ‬
͵
‫͵ ۈ‬
‫͵ ۈ‬
‫ۋ‬
ͳ
ͳ
ʹ
ʹ
‫ۉ‬െ ͵ ݄ ൅ ͵ ݈ ‫ی‬୊େେ ‫ۉ‬െ ͵ ݄ ൅ ͵ ሺ͵ െ ݄ሻ‫ی‬୊େେ

Eq. 5.2
Ex. 5.1. Conversion of several hkl’s.

െͷȀ͵
െͳ
Ͳ
Ͳ
൭ʹ൱
֜ ൭െͳ൱  Ǥ ൭ʹ൱֜ ൭െͷȀ͵൱െ͹Ȁ͵ ୊͵ ୊ͳ ୌ୭ୠ୴ͳത ୌ୰ୣ୴െ݄ ൅ ൌ ൭൅݄ ൅ ൱െ݄ ൅  ୊େେ

The above examples show again that all the reflections allowed for the triple-hexagonal in obverse
setting can be converted to the face-centred indiceswith integer values, whereas a part of reflections
allowed for the triple-hexagonal in reverse setting
result in non-integer. This occurs by the same reason
as given in Section 4.1. The conversion of selectedreflection indices are tabulated in Appendix 2.

First we consider the case, l = even.
݈ ൌ ͵ െ ݄ ‫ʹ ؠ‬

Eq. 6.3݄ ൌ ͵ െ ʹെ͵ ൅ ʹ ൅ െ݄ ൅ ൌ ൭൅͵ െ ʹ ൅ ൱൭൅݄ ൅ ൱െ͵ ൅ ʹ ൅  ୊େେെ݄ ൅  ୊େେെʹ ൅ ʹൌ ൭൅Ͷ െ ʹ൱െʹ ൅ ʹ ୊େେ

6. Reflection conditions in the triple-hexagonal
cell


െ ൅ ൌ ቆቇ
=2൭൅ʹ െ ൱ ୊େେെ ൅  ୊େେ

6.1. Reflection conditions
In the triple-hexagonal obverse cell, the spacegroup ͵തhas a reflection condition of h+l=3n for
h-hl, while ͵ത has that of h+l=3n and l=2n for h-hl.
How these conditions are modified in the transformed face-centred pseudo cubic cell, for example,
represented by ͵ത?
Before proceed, note that ͵ത has reflection
conditions:(i) h+l=2n, k+l=2n, l+h=2n for general hkl.
(ii) k=2n, l=2n for 0kl,(iii) h+l=2n for hhl,(iv) l=2n for 00l.

Eq. 6.4
This indicates that the h-hl (h+l=3n and l=2m) reflections indexed on the triple-hexagonal cell aretransformed into pseudo FCC with indices being alleven values.
Similarly, we find that the h-hl (h+l=3n and
l=2m+1) reflections indexed on the triple-hexagonalcell are transformed into pseudo FCC with indicesbeing all odd values.
Ex. 6.1. Conversion of several hkl’s.

― 40 ―

Nobuo Ishizawa and Yumi Inagaki

The h-hl reflections with h=0 in triple-hexagonal
obverse:െ݄ ൅ 
ͳ൭ ݄൅ ൱֜  ൭ͳ൱
ǡെ݄ ൅  ୊େେ
ͳ ୬ୀଵ

ʹ
൭ʹ൱
ǡ
ʹ ୬ୀଶ

axes in the pseudo cubic cell (see Section 4.3). A
mere conjecture of the ‘true’ 3-fold axis results inmetrically cubic c/a or Į value (see Sections
7.3-7.4).

͵
൭͵൱ 
͵ ୬ୀଷ

7. General approach to discriminate ‘TRUE’
rhombohedral cell

The h-hl reflections with h=1 in triple-hexagonal
obverse:7.1. Transformation from the pseudo cubic (pF)cell into four triple-hexagonal cells (Hobv1 ~Hobv4)

െͳ ൅ Ͳെͳ൭ ͳ൅ ൱֜൭ ͳ ൱ǡ ൭ʹ൱ǡെͳ ൅  ୊େେͲ ୬ୀଵെͳ ୬ୀ଴ͳʹ൭͵൱ǡ൭Ͷ൱ 
ͳ ୬ୀଶ
ʹ ୬ୀଷ

The matrices can be obtained from Eqs. 3.3 andEq. 4.2.
ͲͳȀʹ ͳȀʹ ‫܉‬‫܉‬ͳ െͳ Ͳ
ͲͳȀʹ൱ ቆ‫܊‬ቇቆ‫܊‬ቇൌ ൭Ͳ ͳ െͳ൱ ൭ͳȀʹ‫ ܋‬୮୊‫ ܋‬ୌ୭ୠ୴ଵ
ͳȀʹ ͳȀʹͲͳ ͳͳ

The h-hl reflections with h=2 in triple-hexagonal
obverse:െʹ ൅ െ͵െͶ൭ ʹ൅ ൱֜൭ Ͳ ൱ǡ ൭ ͳ ൱ǡെʹ ൅  ୊େେെ͵ ୬ୀିଵെͶ ୬ୀିଶെʹǡ൭ ʹ ൱െʹ ୬ୀ଴
Ͳ
ǡ
൭Ͷ൱
Ͳ ୬ୀଶ

‫܉‬
െͳȀʹ ͳȀʹͲൌ൭ ͲെͳȀʹ ͳȀʹ൱ ቆ‫܊‬ቇ‫ ܋‬୮୊
ͳ
ͳ
ͳ

െͳ
൭͵൱
ǡ
െͳ ୬ୀଵെͳȀʹͲͳȀʹ ‫܉‬‫܉‬ͳ െͳ Ͳ
ͳȀʹ ͳȀʹ൱ ቆ‫܊‬ቇቆ‫܊‬ቇൌ ൭Ͳ ͳ െͳ൱ ൭ Ͳ‫ ܋‬୮୊‫ ܋‬ୌ୭ୠ୴ଶ
െͳȀʹ ͳȀʹ
Ͳ
ͳ ͳ
ͳ

ͳ
൭ͷ൱ 
ͳ ୬ୀଷ

‫܉‬
െͳȀʹ െͳȀʹͲൌ ൭ ͳȀʹͲͳȀʹ൱ ቆ‫܊‬ቇ‫ ܋‬୮୊
െͳͳͳNote that the h-hl (h+l=3n and l=2m+1) reflections indexed on the triple-hexagonal cell are transformed into pseudo FCC with indices being all odd
values. This never means that any hkl reflection withall odd values on the pseudo FCC cell is forbidden in
the ͵ത group (see Section 6.2).

ͲെͳȀʹ ͳȀʹ ‫܉‬‫܉‬ͳ െͳ Ͳ
ͲͳȀʹ൱ ቆ‫܊‬ቇቆ‫܊‬ቇൌ ൭Ͳ ͳ െͳ൱ ൭െͳȀʹ‫ ܋‬୮୊‫ ܋‬ୌ୭ୠ୴ଷ
െͳȀʹ െͳȀʹͲͳ ͳͳͳȀʹ െͳȀʹ
ൌ൭ Ͳ
ͳȀʹ
െͳ
െͳ

ഥ and ‫܀‬૜ഥ‫܋‬
6.2. Procedure to discriminate ‫܀‬૜groups‫܉‬ͲͳȀʹ൱ ቆ‫܊‬ቇ‫ ܋‬୮୊ͳͳȀʹͲͳȀʹ ‫܉‬‫܉‬ͳ െͳ Ͳ
െͳȀʹ ͳȀʹ൱ ቆ‫܊‬ቇቆ‫܊‬ቇൌ ൭Ͳ ͳ െͳ൱ ൭ Ͳ‫ ܋‬୮୊‫ ܋‬ୌ୭ୠ୴ସ
ͳȀʹ െͳȀʹͲͳ ͳͳ(a) Check hkh reflections and their permutations like
hhk, khh indexed on the face-centred pseudo cubic.
(b) If you observe only hkh (and hhk, khh) reflections
with both h and k even and no significant intensities for reflections with both h and k odd, thestructure then may not be cubic and probably belongs to the ͵ത group. The systematic absence
like this is ‘abnormal’, i.e., over-extinct for the͵ത group.
(c) On the other hand, if you observe significant intensities for hkh (and hhk, khh) reflections with all
odd values, then the crystal can be the ͵ത group.
Note that no systematic absence for hkh and its
permutations in FCC never exclude the possibilityof ͵ത.
(d) The discrimination between the ͵ത and ͵തgroups based only on the systematic reflection
condition is impossible. Check distortion from the
cubicity, for example, deviation from 60° in theprimitive rhombohedral cell, or the deviation from
¥6 (=61/2) for c/a in the triple-hexagonal cell.
(e) When converting the pseudo cubic cell torhombohedral or triple-hexagonal cells, pay special attention in choosing the ‘true’ 3-fold axis
because there are the other three ‘pseudo’ 3-foldͳȀʹ ͳȀʹൌ ൭െͳȀʹ
Ͳ
ͳ
െͳ

‫܉‬
Ͳ
ͳȀʹ൱ ቆ‫܊‬ቇ‫ ܋‬୮୊
ͳ

Eqs. 7.1
These four transformations are important in the discrimination of the true rhombohedral cell from thepseudo FCC when the c/a ratio is close to ¥6 intriple-hexagonal, or Į is close to 60° in primitive
rhombohedral. To convert a candidate of triple
hexagonal cell, for example, Hobv1 to the othercandidates, i.e., Hobv2, Hobv3 and Hobv4, see Section 7.2.
7.2. Geometrical relationship among the four
triple-hexagonal cells
The orientation relationship among the four
triple-hexagonal cells, Hobv1, Hobv2, Hobv3 andHobv4, is summarized in Table 7.1, using Eqs. 7.2.
‫܉‬
‫܉‬
ቆ‫܊‬ቇ
ൌ‫ ܀‬୧՜୨ ቆ‫܊‬ቇǡ‫ ܋‬ୌ୭ୠ୴ሺ୧ሻ‫ ܋‬ୌ୭ୠ୴ሺ୨ሻ
― 41 ―‫ ܀‬୧՜୨ ൌ ‫ ܀‬୨՜୧ ିଵ

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell

Eqs. 7.2

Note that the TWIN line may truncate if you use too
many digits for the values. The rotation matrix [1/3
2/3 -1/3 -1/3 1/3 1/3 8/3 4/3 1/3] is then applied
3(=4-1) times to generate the indices of twin components from the input reflection indices which apply to the first component. The scale factors km forthe twin components are calculated from :Table 7.1 indicates that the four twin components
can be represented iteratively as follows:‫܉‬ͳȀ͵ ʹȀ͵ െͳȀ͵ ‫܉‬ቆ‫܊‬ቇൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵ ൱ ቆ‫܊‬ቇ‫ ܋‬ୌ୭ୠ୴ሺ୧ାଵሻ‫ ܋‬ୌ୭ୠ୴ሺ୧ሻͺȀ͵ ͶȀ͵ ͳȀ͵

୬
ଶ
ଶ
ଶ
ୡ ൌ ሺሻ ෍  ௠ ୡ೘௠ୀଵEq. 7.3

Eq. 7.5

Or using the ‘i’-times repeated multiplication of thematrix,In the above equation, ‘osf’ is the overall scale factor,
and n=4. Initial scale factors, k2 , k3, and k4 for thetwin components for variants 2, 3 and 4 are necessary as inputs (0km1). The scale factor k1 for thetwin component 1 is calculated from the equation:‫܉‬ͳȀ͵ ʹȀ͵ െͳȀ͵ ୧ ‫܉‬ቆ ‫܊‬ቇൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵ ൱ ቆ‫܊‬ቇ‫ ܋‬ୌ୭ୠ୴ሺ୧ሻ‫ ܋‬ୌ୭ୠ୴ሺଵሻͺȀ͵ ͶȀ͵ ͳȀ͵

Eq. 7.4୬ଵ ൌ ͳ െ ෍  ௠

These expressions are useful for twin analysis because the four twin components can be represented
by iterating a single transformation matrix up to fourtimes. If you are the SHELX user and want to determine the amounts of these four possible twin
components in a crystal, you may add the following
lines, for example, in the instruction file of .ins as:௠ୀଶEq. 7.6

TWIN   0.3333 0.6667 Ͳ0.3333  Ͳ0.3333 0.3333
0.33332.66671.33330.33334
BASFk2k3k4

Note that the rotation matrices {(1/3 2/3 -1/3), (-1/3
1/3 1/3), (8/3 4/3 1/3)} and {(-1/3 -2/3 1/3), (1 1 0),(-4/3 4/3 1/3)} in Table 7.1 are allowed for permutation in generating four twin components, while{(-1 0 0 }, (2/3 1/3 1/3), (4/3 8/3 -1/3)} is NOT. The
last matrix can convert between only two twin
components, for example, components between 1
and 3, or those between 2 and 4.

Table 7.1. Orientation relationship (Riĺj) among the four possible triple-hexagonal cells. The subscripts i (after transformation) and j (before transformation) run from 1 through 4.

i =1(Hobv1)i =2(Hobv2)i =3(Hobv3)i =4(Hobv4)

j =1 (Hobv1)

j =2 (Hobv2)

j =3 (Hobv3)

j =4 (Hobv4)ԟെͳȀ͵ െʹȀ͵ ͳȀ͵
ͳ
Ͳ ൱
൭ ͳ
െͶȀ͵ ͶȀ͵ ͳȀ͵െͳͲͲ൭ʹȀ͵ ͳȀ͵ ͳȀ͵ ൱ͶȀ͵ ͺȀ͵ െͳȀ͵ͳȀ͵ ʹȀ͵ െͳȀ͵൭െͳȀ͵ ͳȀ͵ ͳȀ͵ ൱ͺȀ͵ ͶȀ͵ ͳȀ͵ͳȀ͵ ʹȀ͵ െͳȀ͵൭െͳȀ͵ ͳȀ͵ ͳȀ͵ ൱ͺȀ͵ ͶȀ͵ ͳȀ͵ԟെͳȀ͵ െʹȀ͵ ͳȀ͵
ͳ
Ͳ ൱
൭ ͳ
െͶȀ͵ ͶȀ͵ ͳȀ͵െͳͲͲ൭ʹȀ͵ ͳȀ͵ ͳȀ͵ ൱ͶȀ͵ ͺȀ͵ െͳȀ͵െͳͲͲ൭ʹȀ͵ ͳȀ͵ ͳȀ͵ ൱ͶȀ͵ ͺȀ͵ െͳȀ͵ͳȀ͵ ʹȀ͵ െͳȀ͵൭െͳȀ͵ ͳȀ͵ ͳȀ͵ ൱ͺȀ͵ ͶȀ͵ ͳȀ͵ԟെͳȀ͵ െʹȀ͵ ͳȀ͵
ͳ
Ͳ ൱
൭ ͳ
െͶȀ͵ ͶȀ͵ ͳȀ͵െͳȀ͵ െʹȀ͵ ͳȀ͵
ͳ
Ͳ ൱
൭ ͳ
െͶȀ͵ ͶȀ͵ ͳȀ͵െͳͲͲ൭ʹȀ͵ ͳȀ͵ ͳȀ͵ ൱ͶȀ͵ ͺȀ͵ െͳȀ͵ͳȀ͵ ʹȀ͵ െͳȀ͵൭െͳȀ͵ ͳȀ͵ ͳȀ͵ ൱ͺȀ͵ ͶȀ͵ ͳȀ͵ԟ― 42 ―

Nobuo Ishizawa and Yumi Inagaki

7.3. Practices -Case 1-

The primitive rhombohedral cell (R) can beconverted to a ‘TRUE’ triple-hexagonal cell (Hobv1)
and a ‘TRUE’ pseudo FCC cell (pF) using transformations given in Eqs. 4.1 and Eqs. 5.1, respectively. Using Eqs. 7.2 and 7.3, the triple-hexagonalcell (Hobv1) can be transformed into Hobv2, Hobv3
and Hobv4. Several interaxial angles of these‘FALSE’ hexagonal cells have small deviations of
0.01-0.02° from ideal values. In addition, there is asmall difference of less than 0.002 Å in a and b
lengths.

It would be a good idea to get the knack of successful identification through practices. Let us startfrom a ‘TRUE’ rhombohedral cell with a=5.400 Å,
Į=60.02°. This is a cell likely to occur in the
SrTiO3-LaAlO3 solid solution system, especially inthe crystal containing 20 mole% SrTiO3. Table 7.2
shows how the cell parameters vary depending on
the cells under assumption.

Table 7.2. Various cell parameters obtained for the Case 1. See text for the cell abbreviation.
cell

a

b

c

Į

ȕ

Ȗ

comment

R5.40005.40005.400060.02060.02060.020TRUE

pF

7.63797.63797.637990.01790.01790.017

TRUE

Hobv1

5.40165.401613.2252

90.00090.000120.000

TRUE

Hobv2

5.3996

5.399813.2305

90.01389.984119.979

FALSE

Hobv3

5.4016

5.399913.2303

89.98490.000120.010

FALSE

Hobv4

5.39985.401613.2303

90.00090.013120.008

FALSE

Table 7.3. Various cell parameters obtained for the Case 2. See text for the cell abbreviation.
cell

a

b

c

Į

ȕ

Ȗ

comment

Hobv1
R1
pF1

5.39985.40007.63775.40165.4015
7.637813.23035.4016
7.638090.00060.00190.01790.01359.99189.985120.008
59.98989.981

FALSE
FALSE
FALSE

Hobv2
R2
pF2

5.40125.4001
7.63735.40135.39987.637413.22515.39977.638189.99760.01990.01389.99860.02090.018119.997
60.01590.016

TRUE
TRUE
TRUE

Hobv3
R3
pF3

5.39985.4016
7.6376

5.39995.40007.637613.23055.4014
7.638090.01159.99089.98289.98660.00090.016119.980
59.98889.983

FALSE
FALSE
FALSE

Hobv4
R4
pF4

5.40135.4015
7.63755.40005.4015
7.637513.2300
5.39977.638289.98359.99389.98490.00059.99389.984120.008
59.99890.017

FALSE
FALSE
FALSE

7.4. Practices -Case 2-

you can find some characteristic tendencies. Thisappears most typically in the primitive rhombohedralcell. The answer is that the ‘TRUE’ cell is Hobv2.
Supporting evidences are given below.

Let us consider another case (Case 2). Suppose wehave obtained experimentally the data of Hobv1 inTable 7.3. According to the procedure given in 7.2,
we can obtain the other three candidates of Hobv2,
Hobv3 and Hobv4 as listed in Table 7.3, with theircorresponding primitive rhombohedral (Rn) andpseudo cubic (pFn) cell dimensions, n=1~4. Which isthe ‘TRUE’ one?
For the first time you feel a difficulty in answeringto this question because none of the four candidates
have the exact geometrical relations required for theideal cells. However, by looking into data closely,(a) The ‘TRUE’ primitive rhombohedral cell has thelargest (and mostly equal) deviations from 60° forthe three interaxial angles. FALSE cells haveusually plus-and-minus mixed deviations from 60°
due to its distorted cell nature. In addition, thedeviations from 60° in ‘FALSE’ cells are usuallysmaller by an order of magnitude and come closer― 43 ―

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell

to a cubic cell in average (recall ĮR=60° for the
metrically FCC cell).
(b) The cell lengths of the ‘TRUE’ primitive rhombohedral have the smallest deviation among them.
In Case 2, the deviation for R2 is less than 0.0004 Å,
while R1, R3 and R4 are 0.0016, 0.0016, 0.0018 Å,
respectively. The deviation in R2 is one figure
small compared with the ‘FALSE’ cells.
If you look into the cell dimensions in the pseudo
cubic setting, you will find more information to tell
about the discrimination of the ‘TRUE’ cell. An
important thing is that even a ‘TRUE’ cell does nothave the ‘exact’ geometrical relations in experiments.
This, of course, never means that it is useless to improve accuracy. On the contrary, we should try toenhance the quality of experiment to improve accuracy as much as possible. After that, find most distorted cell from the candidates. The distortion is ameasure of departure from cubicity. The FALSE
cells tend to assume cubicity in a sort of the averaged
way. If the distortions occur most systematically andprominently in only one candidate cell, then the cell
is most plausibly the right answer.
Finally we would like to reveal that the numerical
data used in Case 2 are essentially made from thosein Case 1. Actually we just picked up the Hobv4 dataof Case 1, assumed that they are obtained experimentally, and put them as Hobv1 in Case 2. The recalculation from this new Hobv1 to Hobv2, Hobv3
and Hobv4 brought slightly different values fromthose listed in Case 1, probably due to the termination errors during numerical operations, for example,
an input of 0.3333 instead of 1/3. As a result, the‘TRUE’ answer lost its exactness slightly, which
served, in turn, for usage as a ‘practical’ example.
8. A convenient technique of discrimination using
the Smart Apex II software(a) Collect data assuming any temporary cell foundin the process ‘DETERMINE UNIT CELL’. Assumption of a primitive cell with the same cell volume
as that of the ‘true’ cell could be recommended forrunning ‘STRATEGY’. Collect as many frames as
possible up to the highest diffraction angle available for the time you are allowed to use the machine.
For inorganic crystals the Ȧ and ĳ scan widths lessthan 0.3º are recommended when the crystal-to-detector distance is approximately 60 mm. If
necessary, increase the total scan range.
(b) After data collection, run again ‘DETERMINEUNIT CELL’, using much more frames than the default values if the starting cell is supposed to contain large errors. A series of successive runs (a setof frames) with lower diffraction angles is recommended. The number of successive runs can be
3 to 5, and the number of frames in a single run canbe 100-200, for example. This will take a slightly
longer time but would give much better initial cellparameters than the previous results obtained in a).
(c) If you get a plausible cell like triple-hexagonal,
primitive rhombohedral or face-centred pseudo― 44 ―

cubic cells, check carefully the deviation of cellparameters from their ‘ideal’ values, for example,
deviation of c/a ratio from ¥6 and angular deviations from 90° and 120 ° for the triple-hexagonalcell. Check three Į angles of the primitive rhombohedral cell using ‘BRAVAIS’. If necessary, use
Eqs. 4.3 in calculating Į. They should be deviatedfrom 60° with the equal amount, for example,
60.03°, 60.03°, 60.03°. If the Į values are scatteredin various ways, like, 59.96, 60.03, 60.02, the cellis probably FALSE. But never mind and proceed.
(d) Go to the procedure ‘INTEGRATE’ and click‘REFINEMENT OPTION’. Enter 200 (number offrames) , for example, instead of 50 (default) in the‘FREQUENCY (Images)’ in the ’Periodic RefineͲment’ panel. The periodic refinement of various
system parameters at longer intervals (i.e., using
more frames) is important if the crystal contains asmall number of reflections per frame. The groupsof frames (RUNs) with low numbers are usually
collected at relatively high angles (see STRATEGY).
These frames are less crowded with weak diffraction spots. These peaks are highly possible to go
missing in the ‘INTEGRATE’ procedure if theorientation matrix (UB matrix) is inaccurate.
Therefore, the periodic refinement using a small
number of frames like default value is dangerous
for many inorganic crystals and may introduce serious errors in the system parameters to be refined.
Do not forget to set the d (Å) limit in top rightcorner of the window panel to an appropriate value.
It is recommended to choose ‘TRICLINIC’ for the
option ‘CONSTRAINT METRIC SYSTEM’ in both
‘PERIODIC
REFINEMENT’and‘GLOBAL
REFINEMENT’ panels. Be careful in using the default crystal system for ‘CONSTRAINT METRICSYSTEM’, because the default is the one previously
determined in the ‘DETERMINE UNIT CELL’ procedure.
(e) After ‘INTEGRATE’, check carefully the createdfile ‘XXX_0m._ls’. You find three sets of the refined cell parameters in the very last part of the file.
(f) Go to ‘EXAMINE DATA’ and run ‘XPREP’(IMPORTANT)(g) The first thing to do on entering ‘EXAMINE
DATA’, is to specify the p4p file and raw data file
you are going to use. They are usually‘ XXX_0m.p4p and ‘XXX_0m.raw’ in the working directory. You will find on the screen the initialcheck results about the ‘Lattice Exceptions’. Lookcarefully into the O (obverse) and R (reverse) exceptions when you use the triple-hexagonal cell. If
you find many exceptions in both columns, it neverbe triple-hexagonal (though the possibility oftwinning of triple-hexagonal cells still remains).
Whichever is the cell obverse or reverse, just keep
it in mind and accept the recommendation for thistime. There is no need to transform the cell fromreverse to obverse (see step (i) for the reason).
Then ‘XPREP’ shows various options to proceed.
(h) Choose ‘UNIT CELL TRANSFORMATION (U)’ and‘FIND NIGGLI REDUCED CELLS (N)’. This routine
finds reduced cells from multi-centred cells, forNobuo Ishizawa and Yumi Inagaki

example, triple-hexagonal or face-centred pseudo
cubic by the Niggli method [3]. If your original cellis triple-hexagonal, then you will find a reduced
primitive rhombohedral Niggli cell, which indicates the direction of the ‘true’ 3-fold axis in highprobability. Check three Į angles of the primitive
rhombohedral cell shown in the results carefully.
They may be deviated from 60° with approximately the equal amount, for example, 60.03,
60.03, 60.04° (Congratulation!). Now your current
cell is primitive rhombohedral.
(i) If you prefer to transform once again from theprimitive rhombohedral to triple-hexagonal in
obverse setting, type ‘U’ to run ‘UNIT CELL
TRANSFORMATION (U)’. Enter transformation
matrix like ‘ 1 -1 0 0 1 -1 1 1 1 ‘, as given in
Section 4.1, Eq. 4.2. This will convert the current
primitive rhombohedral cell to triple-hexagonalcell in obverse. Even though the original cell was
reverse, your latest current cell in the top panel is
now obverse after the procedures of (N) and (U) as
above. The procedure described in the steps (h) and(i) is the key of this technical note.
(j) Go back to the ‘XPREP’ main menu and do necessary things to prepare .ins and .hkl files using
the options:‘ENTERSPACEGROUP(S)’ to enter ͵ത etc‘DEFINE UNIT CELL CONTENTS (C)’ to enter Z
value‘ABSORPTION CORRECTOOPN etc. (A)’ for absorption correction‘SETUP SHELX FILES (S)’ to create XXXX.hkl(any file name with a suffix of .hkl).
(k) Finally, if you want to make consistent various
files created during this process, especially in orderto keep the consistency of reflection indices, precession images, crystal faces for absorption, I
recommend to start over the whole processes
maybe from INTEGRATE and PRECESSION, importing the latest .p4p to the system.
The steps from (g) through (k) will solve mostdifficult and intrinsic problems associated with theidentification of the real symmetry of theface-centred pseudo cubic cells.

‫܋܉‬
‫܋܊‬൱
‫܋܋‬

  Ȗ ૛ൌ ቌ  Ȗ૛  ȕ   ȕ

 ȕ
 Įቍ
૛

Eq. 9.2
Suppose that cell vectors are transformed by the 3x3
matrix P as:‫܉‬‫܉‬Ԣ൭‫܊‬Ԣ൱ ൌ ‫ ۾‬ȉ ቆ‫܊‬ቇ
‫܋‬
‫܋‬Ԣ

Eq. 9.3
Then the relation between the G and G’ matrices ofthe old and new cell vectors, respectively, can be
written as:
୘

۵ൌ‫۾‬

ିଵ

‫܉‬Ԣ
‫܉‬Ԣ
൭‫܊‬Ԣ൱ ൝‫ି ۾‬ଵ ൭‫܊‬Ԣ൱ൡ
‫܋‬Ԣ
‫܋‬Ԣ

‫܉‬Ԣ
ൌ ‫ି ۾‬ଵ ൭‫܊‬Ԣ൱ ሺ‫܉‬Ԣ‫܋‬Ԣ‫܊‬Ԣ ‫܋‬Ԣሻ ȉ ሺ‫ି ۾‬ଵ ሻ୘ൌ ‫ି ۾‬ଵ ۵Ԣሺࡼିଵ ሻ୘

Eq. 9.4
Applying P from left and PT from right to the bothsides of Eq. 9.4:
‫۾‬۵‫ ۾‬୘ ൌ ‫ି ۾۾‬ଵ ۵Ԣሺ‫ି ۾‬ଵ ሻ୘ ‫ ۾‬୘ ൌ ‫ି ۾۾‬ଵ ۵ᇱ ሺ‫ି ۾۾‬ଵ ሻ୘ൌ ۳۵ᇱ ሺ۳ሻ୘The experimentally obtained ‘triple-hexagonal’
cell could have been obtained based on a wrong
choice (due to poor accuracy), if the ratio of the celldimensions, c/a, is very close to ¥6. As mentioned in
Section 4.3, there are four possible ‘rhombohedral’cells. Among these four candidates, the only one cellis ‘TRUE’ and the other three cells are ‘FALSE’, if
the crystal is really rhombohedral. The transformation matrices from the experimentally obtained
triple-hexagonal (obverse) cell, H =(aH, bH, cH) T, tothese four possible rhombohedral cells , R1 =(a1, b1,c1) T, R2=(a2, b2, c2)T, R3 =(a3, b3, c3)T, and R4 =(a4,b4, c4) T are as follows:

‫כ‬

െ

‫܊܉‬
‫܊܊‬
‫܊܋‬

9.2. Four possible rhombohedral cells from theexperimentally-obtained triple-hexagonal cell

The orientation of the crystal on the diffractometer is expressed by the matrix UB where the matrix B is defined as:
‫כ‬

‫܉܉‬
ൌ ൭‫܉܊‬‫܉܋‬

Eqs. 9.5

9.1. UB and G matrices‫  כ‬ɀ‫כ‬‫  כ‬ɀ‫כ‬Ͳ۵ ൌ ሼሺ‫܃‬۰ሻ୘ ȉ ‫܃‬۰ሽିଵ ൌ ሺ۰ ୘ ‫ ܃‬୘ ‫܃‬۰ሻିଵ ൌ ሺ۰ ୘ ۰ሻିଵ‫ ׵‬۵Ԣ ൌ ‫۾‬۵‫ ۾‬୘

9. A practical guide to finding ‘TRUE’ rhombohedral cell from the UB matrix‫כ‬۰ ൌ ቌͲ
Ͳ

Lattice vectors are obtained from of the G matrix
which is the inverse matrix of a product of UB and its
transposition, UBT : Ⱦ‫כ‬ Ⱦ‫  כ‬ȽቍͳȀ

Eq. 9.1― 45 ―

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell

Therefore, we applied the conversion matrix fromreverse to obverse using the transformation matrix U
as given in Eq. 2.1:ʹȀ͵ͳȀ͵ ͳȀ͵ ‫܉‬‫܀‬ଵ ൌ ‫۾‬ଵ ۶ ൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵൱ ቆ‫܊‬ቇെͳȀ͵ െʹȀ͵ ͳȀ͵ ‫ ܋‬ୌ
െͳ
െͳ
Ͳ
‫܉‬
െͳ
Ͳ ൱ ቆ‫ ܊‬ቇ‫ ܀‬ଶ ൌ ‫۾‬ଶ ۶ ൌ ൭ ͲെͳȀ͵ െʹȀ͵ ͳȀ͵ ‫ ܋‬ୌെͳ Ͳ Ͳ‫ ܃‬ൌ ൭ Ͳ െͳ Ͳ൱
Ͳ
Ͳ ͳ

ͳ
Ͳ
Ͳ
‫܉‬
ͳ
Ͳ ൱ ቆ‫ ܊‬ቇ‫ ܀‬ଷ ൌ ‫۾‬ଷ ۶ ൌ ൭ ͳʹȀ͵ ͳȀ͵ ͳȀ͵ ‫ ܋‬ୌ۵ᇱ ൌ ࢁ۵ࢁ୘
ʹͻǤͲͳ͹͹ െͳͶǤͷͳʹ͵۵ᇱ ൌ ۵ ൌ ൭െͳͶǤͷͳʹ͵ ʹͺǤͻͶͶͶെͲǤͲͲ͹ͻ
െͲǤͲ͹Ͳ͹

Ͳ
ͳ
Ͳ
‫܉‬
Ͳ
Ͳ ൱ ቆ‫ ܊‬ቇ‫ ܀‬ସ ൌ ‫۾‬ସ ۶ ൌ ൭ െͳെͳȀ͵ ͳȀ͵ ͳȀ͵ ‫ ܋‬ୌ

Eqs. 9.9
Further transformations were applied to the four
possible rhombohedral cell models, Rn (n=1, 2, 3,
and 4), to obtain Gn:

Eqs. 9.6
The converted cell dimensions can be obtained by
the equation, ۵Ԣ ൌ ‫۾‬۵‫ ۾‬୘ , where P is substituted by
P1, P2, P3 or P4. These cell dimensions should be
carefully examined in the light of metric constraints
for rhombohedron, a=b=c, Į=ȕ=Ȗ. For this evaluation, you have to use an accurate UB matrix, and thusG, because the small metrical offsets from the ideal
values have of vital importance. On Smart Apex II
software, it is better to use ‘UNCONSTRAINED’ setting parameter (UB matrix). You can find it in the
output file, XXX._ls, which is created after the
INTEGRATE procedure.

۵୬ ൌ ‫۾‬୬ ۵୬ ‫۾‬୬ ୘

Eq. 9.10ʹȀ͵ͳȀ͵ ͳȀ͵If ‫۾‬ଵ ൌ ൭െͳȀ͵ ͳȀ͵ ͳȀ͵൱, thenെͳȀ͵ െʹȀ͵ ͳȀ͵ʹͺǤͻ͹ͶͲ۵ଵ ൌ ൭ͳͶǤͶ͸ͻͲͳͶǤͷͳͻʹʹͺǤͻ͵͹͸۵ଶ ൌ ൭ͳͶǤͶ͵ʹͳͳͶǤͶ͵ͲͶ9.3.1. Direct conversion from the pseudo hexagonal to rhombohedral candidatesʹͻǤͲͳ͹͹۵ଷ ൌ ൭ͳͶǤͷͲͷͶͳͶǤͷͳͲ͵ͳͶǤͶ͵ʹͳʹͺǤͻͶͶͶ
ͳͶǤͶ͵ͷ͵

ͳͶǤͶ͵ͲͶͳͶǤͶ͵ͷ͵൱ʹͺǤͻ͵ͷ͸

ͳͶǤͷͲͷͶʹͺǤͻ͵͹͸
ͳͶǤͷͲ͹ʹ

ͳͶǤͷͳͲ͵ͳͶǤͷͲ͹ʹ൱ʹͻǤͲͳʹͶͲͳͲͲͲ ൱, thenIf ‫۾‬ସ ൌ ൭ െͳെͳȀ͵ ͳȀ͵ ͳȀ͵ͲǤͲʹͲͲ͸ͻͺെͲǤͲʹͶͶ͵͸ͳ൱ͲǤͲ͸ͺͻͲ͸ͷʹͺǤͻͶͶͶ۵ସ ൌ ൭ͳͶǤͷͳʹ͵ͳͶǤͷͲͻͳEq. 9.7ͳͶǤͷͳʹ͵
ʹͻǤͲͳ͹͹
ͳͶǤͷͲ͹͵

ͳͶǤͷͲͻͳͳͶǤͷͲ͹͵൱ʹͻǤͲͲͻͷ

Eqs. 9.11

The matrix G then became:ʹͻǤͲͳ͹͹ െͳͶǤͷͳʹ͵۵ ൌ ൭െͳͶǤͷͳʹ͵ ʹͺǤͻͶͶͶെͲǤͲͲ͹ͻ
െͲǤͲ͹Ͳ͹ͳͶǤͷͳͻʹ
ͳͶǤͷͳͻ͸൱ʹͻǤͲͲʹͲͳͲͲͳͲ ൱, thenIf ‫۾‬ଷ ൌ ൭ ͳʹȀ͵ ͳȀ͵ ͳȀ͵Below is an example obtained from the single
crystal synchrotron experiment at BL14A using a
0.5mol%SrTiO3-99.5mol%LaAlO3 mixed crystal(ST05_2) in June 5, 2008. The wavelength of X-rays
is 0.671710Å. The crystal was ground into sphericalshape with 70-80 µm in diameter. The orientation
matrix UB of the initial hexagonal cell was refined as
below:
െͲǤͲ͹ͷʹ͸ͻ͹ͲǤͳͺͳͷʹͲ͵ͲǤͲͺ͸ͷͺʹͳͳͶǤͶ͸ͻͲʹͺǤͻͺͳ͸ͳͶǤͷͳͻ͸െͳെͳͲെͳͲ ൱, thenIf ‫۾‬ଶ ൌ ൭ ͲെͳȀ͵ െʹȀ͵ ͳȀ͵

9.3. ExampleͲǤͳʹͻͳͲͻ͹‫܃‬۰ ൌ ൭ͲǤͳ͸ͻ͹ʹͻͷ
ͲǤͲʹʹ͹ͷʹʹ

െͲǤͲͲ͹ͻ
െͲǤͲ͹Ͳ͹ ൱ͳ͹͵Ǥͻ͹͵ͲThe cell parameters obtained from the Gn matrices are given in Table 9.1. The cell angles of thefour candidates suggest that the TRUE cell is R2 withits 3-fold axis running parallel to [-3,-6, 1] directionof the ‘FALSE’ triple-hexagonal (obverse) cell in Fig.
4.1.

െͲǤͲͲ͹ͻ
െͲǤͲ͹Ͳ͹ ൱ͳ͹͵Ǥͻ͹͵ͲEq. 9.8
The G matrix gave the cell dimensions; a=5.3868,
b=5.3800, c=13.1899 Å, Į=90.057, ȕ=90.006,
Ȗ=120.050º. This cell looked apparently hexagonal,
but actually it was not the TRUE triple hexagonal,i.e., the cH is not the ‘TRUE’ three-fold axis. First ofall, we found that this setting was ‘reverse’ bylooking into the systematic conditions of reflections.

9.3.2. Indirect conversion from the pseudo hexagonal to rhombohedral candidates via
pseudo FCC cell― 46 ―

Nobuo Ishizawa and Yumi Inagaki

We can of course find a true cell through the routeof pseudo FCC cell as described in Section 7. In thisroute we first transform the experimentally-obtained
triple-hexagonal cell in reverse setting into obverse,
and then transform the reverse into pseudo FCC. The
four ‘rhombohedral’ cells with different orientations
are then generated and given as R’1, R’2, R’3, R’4 inTable 9.2. Since we found a satisfactory result for theR’3, the cell was further transformed back into thetriple-hexagonal obverse.
Comparing the rhombohedral models obtained

from the direct conversion from obverse to rhombohedral (Table 9.1) and from the indirect conversion via pseudo FCC (Table 9.2), we find a good
agreement between R1 ļ R’1, R2 ļ R’3, R3 ļ R’2,
and R4 ļ R’4, respectively. No significant difference
can be seen between these two routes except for thesequence of a, b, c, and some small truncation errors.
This is because they are essentially the same procedures from mathematical point of view.

Table. 9.1. Cell dimensions of the four ‘rhombohedral’ cells (R2 in hatched line is the ‘TRUE’ cell).
cell

a

b

c

Į

ȕ

Ȗ

comments

initial Hrev

5.38685.380013.1899

90.05790.006120.050= R1

R15.38635.38605.379259.96859.95860.008FALSE

R25.37945.38005.379260.07960.08760.088TRUE

R3

5.38685.37945.386359.95559.99459.962FALSE

R45.38005.38685.386

59.99959.95359.950

FALSE

final Hobv

5.38685.386013.1686

89.99290.010120.001

from R2

Table 9.2. Cell dimensions of the four ‘rhombohedral’ cells using an intermediate conversion into pseudo
FCC cell. R’3 in the hatched line is the TRUE rhombohedral cell.
initial Hrev
FCC
R’1
R’2
R’3
R’4
final Hobv3

a

b

c

Į

ȕ

Ȗ

comments

5.38687.61215.38635.37945.38005.38605.38685.38
7.61345.38605.38635.37945.38005.386313.18997.61255.37925.38685.37925.386813.168690.05789.93359.96859.99460.08759.95
90.00190.00689.92959.95859.96260.07959.99989.991120.0590.08160.00859.95560.08859.953120.006

=R’1
=R’1
FALSE
FALSE
TRUE
FALSE
FALSE

10. Closing remarks

ciety of Promotion of Science. Part of the authors(YI) appreciates the Research Fellowship of the Japan Society for the Promotion of Science for Young
Scientists. Synchrotron X-ray experiments werecarried out at KEK-PF under the programs, No.
2007G027 and 2007G028.

Discrimination between the rhombohedral andface-centred pseudo cubic unit cells is not difficult inprinciple, if we have fully accurate data about thecell dimensions. The accuracy of data is improving
year by year through the various developments inexperimentation, which favors the analysis of thiskind. The repeated trials of this sort of discriminationon the Smart Apex II single-crystal diffractometer
have suggested that we could distinguish the truerhombohedral cell orientation if the rhombohedral
distortion angle of ɲ departs more than 0.004° from
60°.

References[1] International Tables for Crystallography Vol. A,
p84 (2002).
[2] H. D. Megow, ‘Crystal Structures: A Working
Approach’, W.B. Saunders Company, Philadelphia(1973).
[3] P. Niggli, ‘Handbuch der Experimentalphysik,
Vol.7, Part 1’, Leipzig: Akademische Verlagsgesellschaft, (1928).

Acknowledgements
This work is supported by Grants-in-Aid for
Scientific Research No. 18206071 from Japan So― 47 ―

A Guide to Discriminating the Rhombohedral Cell from the Face-Centred Pseudo Cubic Cell

Appendix 1. Relation between the c/a ratio for the triple-hexagonal cell and the interaxial angle
Į in degree for the primitive rhombohedral cell. Data line at Į=60º and c/a=¥6 is hatched.
ɲ

c/a

ɲ

c/a

ɲ

c/a

ɲ

c/a

ɲ

c/a

59.70059.702
59.704
59.706
59.708
59.710
59.712
59.714
59.716
59.718
59.72059.722
59.724
59.726
59.728
59.73059.732
59.734
59.736
59.738
59.74059.742
59.744
59.746
59.748
59.75059.752
59.754
59.756
59.758
59.76059.762
59.764
59.766
59.768
59.77059.772
59.774
59.776
59.778
59.78059.782
59.784
59.786
59.788
59.790
59.792
59.794
59.796
59.798
59.80059.802
59.804
59.806
59.808
59.81059.812
59.814
59.816
59.818

2.46622.46612.46602.46582.46572.46562.46552.46542.46532.46522.46512.46492.46482.46472.46462.46452.46442.46432.46422.46402.46392.46382.46372.46362.46352.46342.46332.46322.46302.46292.46282.46272.46262.46252.46242.46232.46212.46202.46192.46182.46172.46162.46152.46142.46132.46112.46102.46092.46082.46072.46062.46052.46042.46022.46012.46002.45992.45982.45972.459659.82059.822
59.824
59.826
59.828
59.830
59.832
59.834
59.836
59.838
59.840
59.842
59.844
59.846
59.848
59.85059.852
59.854
59.856
59.858
59.86059.862
59.864
59.866
59.868
59.87059.872
59.874
59.876
59.878
59.880
59.882
59.884
59.886
59.888
59.89059.892
59.894
59.896
59.898
59.90059.902
59.904
59.906
59.908
59.910
59.912
59.914
59.916
59.918
59.920
59.922
59.924
59.926
59.928
59.93059.932
59.934
59.936
59.938

2.45952.45942.45922.45912.45902.45892.45882.45872.45862.45852.45842.45822.45812.45802.45792.45782.45772.45762.45752.45732.45722.45712.45702.45692.45682.45672.45662.45652.45632.45622.45612.45602.45592.45582.45572.45562.45552.45532.45522.45512.45502.45492.45482.45472.45462.45442.45432.45422.45412.45402.45392.45382.45372.45362.45342.45332.45322.45312.45302.452959.94059.94259.94459.94659.94859.95059.95259.95459.95659.95859.96059.96259.96459.96659.96859.97059.97259.97459.97659.97859.98059.98259.98459.98659.98859.99059.99259.99459.99659.99860.00060.00260.00460.00660.00860.01060.01260.01460.01660.01860.02060.02260.02460.02660.02860.03060.03260.03460.03660.03860.04060.04260.04460.04660.04860.05060.05260.05460.05660.0582.45282.45272.45262.45242.45232.45222.45212.45202.45192.45182.45172.45162.45142.45132.45122.45112.45102.45092.45082.45072.45062.45042.45032.45022.45012.45002.44992.44982.44972.44962.44942.44932.44922.44912.44902.44892.44882.44872.44862.44842.44832.44822.44812.44802.44792.44782.44772.44762.44742.44732.44722.44712.44702.44692.44682.44672.44662.44642.44632.446260.06060.06260.06460.06660.06860.07060.07260.07460.07660.07860.08060.08260.08460.08660.08860.09060.09260.09460.09660.09860.10060.10260.10460.10660.10860.11060.11260.11460.11660.11860.12060.12260.12460.12660.12860.13060.13260.13460.13660.13860.14060.14260.14460.14660.14860.15060.15260.15460.15660.15860.16060.16260.16460.16660.16860.17060.17260.17460.17660.1782.44612.44602.44592.44582.44572.44562.44552.44532.44522.44512.44502.44492.44482.44472.44462.44452.44432.44422.44412.44402.44392.44382.44372.44362.44352.44332.44322.44312.44302.44292.44282.44272.44262.44252.44232.44222.44212.44202.44192.44182.44172.44162.44152.44142.44122.44112.44102.44092.44082.44072.44062.44052.44042.44022.44012.44002.43992.43982.43972.439660.18060.182
60.184
60.186
60.188
60.19060.192
60.194
60.196
60.198
60.20060.202
60.204
60.206
60.208
60.21060.212
60.214
60.216
60.218
60.22060.222
60.224
60.226
60.228
60.23060.232
60.234
60.236
60.238
60.24060.242
60.244
60.246
60.248
60.25060.252
60.254
60.256
60.258
60.26060.262
60.264
60.266
60.268
60.27060.272
60.274
60.276
60.278
60.28060.282
60.284
60.286
60.288
60.29060.292
60.294
60.296
60.298

2.43952.43942.43932.43912.43902.43892.43882.43872.43862.43852.43842.43832.43812.43802.43792.43782.43772.43762.43752.43742.43732.43722.43702.43692.43682.43672.43662.43652.43642.43632.43622.43602.43592.43582.43572.43562.43552.43542.43532.43522.43512.43492.43482.43472.43462.43452.43442.43432.43422.43412.43402.43382.43372.43362.43352.43342.43332.43322.43312.4330― 48 ―

Nobuo Ishizawa and Yumi Inagaki

Appendix 2. Selected reflections indexed by the pseudo FCC, rhombohedral, and triple-hexagonal obverse. N=h2+k2+l2,
i=-h-k. The equivalent reflection groups are hatched with the same tone. The reflection conditions for ͵ത groups are indicated (‘x’ stands for extinct). All reflection appears in ͵ത.
pseudo
FCC
h k l

h

k

l

h

k

i

l

3

1
-1
1
1

1
1
-1
1

1
1
1
-1

1
1
0
0

1
0
1
0

1
0
0
1

0
1
-1
0

0
0
1
-1

0
-1
0
1

4

2
0
0

0
2
0

0
0
2

0
1
1

1
0
1

8

0
2
2
0
2
-2

2
0
2
-2
0
2

2
2
0
2
-2
0

2
1
1
0
-1
1

11 3
1
1
3
-1
-1
3
-1
1
3
1
-1

1
3
1
-1
3
-1
1
3
-1
-1
3
1

1
1
3
-1
-1
3
-1
1
3
1
-1
3

12 2
-2
2
2

2
2
-2
2

2
2
2
-2

N

pseudo
FCC
h k l

͵ത

N

h

k

l

h

k

i

l

3
1
1
1

x
x
x
x

1
1
0

-1 0 1 2
1 -1 0 2
0 1 -1 2

ż
ż
ż

1
2
1
1
0
-1

1
1
2
-1
1
0

1
-1
0
-1
-1
2

0
1
-1
2
-1
-1

-1
0
1
-1
2
-1

4
4
4
0
0
0

ż
ż
ż
ż
ż
ż

19 1
3
-3
1
-3
3
1
3
3
1
-3
-3

-3
1
3
3
1
-3
3
1
3
-3
1
-3

3
-3
1
-3
3
1
3
3
1
-3
-3
1

0
-1
2
0
2
-1
3
2
2
-3
-1
-1

2
0
0
-1
0
2
2
3
2
-1
-3
-1

-1
2
-1
2
-1
0
2
2
3
-1
-1
-3

-2
-1
3
1
2
-3
1
-1
0
-2
2
0

3
-2
-1
-3
1
2
0
1
-1
0
-2
2

-1
3
-2
2
-3
1
-1
0
1
2
0
-2

1
1
1
1
1
1
7
7
7
-5
-5
-5

ż
ż
ż
ż
ż
ż
x
x
x
x
x
x

1
2
2
-1
1
1
0
2
1
0
1
2

2
1
2
1
-1
1
1
0
2
2
0
1

2
2
1
1
1
-1
2
1
0
1
2
0

-1
1
0
-2
2
0
-1
2
-1
-2
1
1

0
-1
1
0
-2
2
-1
-1
2
1
-2
1

1
0
-1
2
0
-2
2
-1
-1
1
1
-2

5
5
5
1
1
1
3
3
3
3
3
3

x
x
x
x
x
x
ż
ż
ż
ż
ż
ż

2
2
0
0

2
0
2
0

2
0
0
2

0
2
-2
0

0
0
2
-2

0
-2
0
2

6
2
2
2

ż
ż
ż
ż

35 5
1
3
5
3
1
5
-1
3
5
3
-1
5
-1
-3
5
-3
-1
5
1

3
5
1
1
5
3
3
5
-1
-1
5
3
-3
5
-1
-1
5
-3
-3
5

1
3
5
3
1
5
-1
3
5
3
-1
5
-1
-3
5
-3
-1
5
1
-3

2
4
3
2
3
4
1
4
2
1
2
4
-2
1
2
-2
2
1
-1
1

3
2
4
4
2
3
2
1
4
4
1
2
2
-2
1
1
-2
2
3
-1

4
3
2
3
4
2
4
2
1
2
4
1
1
2
-2
2
1
-2
1
3

-1
2
-1
-2
1
1
-1
3
-2
-3
1
2
-4
3
1
-3
4
-1
-4
2

-1
-1
2
1
-2
1
-2
-1
3
2
-3
1
1
-4
3
-1
-3
4
2
-4

2
-1
-1
1
1
-2
3
-2
-1
1
2
-3
3
1
-4
4
-1
-3
2
2

9
9
9
9
9
9
7
7
7
7
7
7
1
1
1
1
1
1
3
3

ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż
ż

-3
-5
-1
3

1
3
-5
-1

5
-1
3
-5

3
1
-1
-3

1
-3
1
-1

-1
-1
-3
1

2
4
-2
-2

2
-2
4
-2

-4
-2
-2
4

3
-3
-3
-3

ż
ż
ż
ż

R

Hobv

― 49 ―

R͵ത

Hobv