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Tables of oriented site symmetries in space groups
CHEMICALPHYSICS6 (1974) l-18. Q NORTH-HOLLANDPUBLISHINGCOMPANY
TABLESOFORlENTEDSlTESYMMETRlF3INSPACEGROUPS J.D.H. DONNAY Dejbrtement
de G.!ologie. Universithde
Monfrtkl,
Mont&~/,P.Q., Caruda
and
c. TURRELL Deprtement de C’hitnie, Univetsit.! de Montr&l. and Department of Chemistry, McGill University, Montreal,
P.Q., cclnadp
Received 1 April 1974 Revisedmanuscriptreceived29 July 1974 Double+ntry tables are presented in which the columns correspondto oriented site symmetriesand the rows to the space groupswhose factor groupsare isomorphic with a given crystallographicpoint group. The site SYtUmettkS are representedby oriented Hermann-Mauguinpoint-groupsymbols. Ekh entry shows the multiplicity of the pointpositions that have the required site symmetq and, far each point-position, the Wyckoff letter and th:: fracticnal aordinates of one site. An additional table gives the supergroupsof every crystallographicpoint gmup, thus indicating the tables that are to be consulted Spectroscopicapplications of the tables are illustratedby two examples. In the ikt, the tables are employed to facilitate factor-groupanalyses when complete crystallographic data are available. In the second example, structuralinformation is obtained from vibrationalspectraby systematic use of the tables.
I. Introduction The following tabulations are offered as companion tables to those listingcoprime multiplicities of spacegroup point-positions [ 11. Like their predecessors,the present tables have been compiled with the needs of the spectroscopist in mind. They provide the answer to the following question: “In which space groups does a point-position of known site symmetry occur?” At the same time they also give the multiplicity of the point-position, its Wyckoff letter and the fractional coordinates of one of its sites. In single-crystalspectroscopy some crystallographic data usually are availableto the spectroscopist. They are the point group of the crystal, its Laue classor its crystal system. The tables are based on these fundamental subdivisions.The cubic system requires three tables: the first for 0, - 4lm 2 2/m. the second for T,,-a3m and O-432, and the third one for the two point groups of Laue class Th - 2/m 3. The hexagonal system fits into two tables: one for the seven point groups in which the lattice must be hexagonal
(table IV) and the other for the five trigonal point groups where the lattice may be hexagonal or rhombohedral (table V); each of these two tables covers two Laue classes. In the tetragonal system, the high-symmetry point groups are treated individually(tables v1 to IX) while the three point groups of Laue classC4h-4/m are presented in table X. In the low-symmetry systems, the orthorhombic has three tables (Xl, XII and XDD; the monoclinic has one (table XIV) and so has the triclinic (table XV). This arrangementhas no special significance;it was dictated by space considerations and reasons of convenience.
2. The tabular presentations In table VI, which is entitled “Tetragonal D4h - 4/m 2/m 2/m”, the rows correspond to the
twenty space groups whose factor groups are isomorphic with point group D4h-4/m 2/m 2/m; the space groups are listed along the left margin.The number
2
J.D.H. hmayand
G. Tunell. Tablesof oriknred sire symmemks in space groups
under Ddh is the superscript in the Schoeties
symbol; it is followed by the Hermann-Mauguin symbol. ‘Ihe column headingsrefer ,to the site symmetries available in the point-positions of the various space groups. lhese point-group symmetries are also given in both notations, and the oriented Hermann-Mauguin short symbols are used here. Consider the column headed Ca. The site symmetry C, - 2/m may be oriented in three different ways, according as the 2-fold axis is parallel to the primary direction Z (2/m..), to one of the secondary directions X or Y (.2/m.) or to one of the tertiary directions (..2/m). In the oriented symbols the dummy dots serve to situate the 2/m direction within the tetragonal symmetry of the crystal: they stand for the “unused” directions. Site symmetry D2-222 can likewise be oriented in two ways, and the Hermann-Mauguin symbol can be “written tetragonally” either 222. or 2.22, according as two of the three a-fold axes lie along the secondary or the tertiary directions. The first 2 in either symbol refers to the primary direction, Z, of the tetragonal system. At the intersection of a given column with one of the appropriate rows, the entry gives the following information on one or several point-positions that have the required site symmetry: Multiplicity of the pointposition(s), Wyckoff letter(s), coordinates of one site from each point-position. Thz choice of origi.i is thus immediately apparent, as the coordinate triplet 000 will be found under the site symmetry of the origin. The Wyckoff letters are given to facilitate crossreference to other tables, such as Infernafional Tables for X-my Crystallography 121 and Space Croupsnnd Lotrice C’omp&es 131. The entries in the tables ale read as follows. In table VI, for example, consider the TOW that corresponds to space group 0:: - P4*/mcm. Some of the entries give only one point-position, like 4e (042) under D2- 222., 8n (~0) under C, - m.. and 16~ (x_~z)under Ct - 1. Dividing a point-position multiplicity into the order (16) of point group 4/m 2/m 2/m gives the index of the site-symmetry subgroup. For the above site symmetries the subgroup indices are as follows: for D2, 1614 = 4; for C,, 16/S = 2; forC1, 16/16=I.Theentry4~underC2,-m.2m gives two point-positions, 4i and 43, for which the listed site coordinates are xx0 and xx4, respectively.
‘Ilte fust entry in f&, -P4/mmm is lad; it is followed by four triplets of coordinates, for positions la, 1b, lc and Id, respectively. This compact presentation can be used whenever there is no gap in the alphabetical sequence. If there is, the alphabetical order is reversed, as in 2db and 2cu in space group D$ - P42/mcm. Occasionally three point positions have Wyckoff letters that cannot be written in a gapless alphabetical sequence. In-such cases the letter precedes the site coorciinates in the entry. In space group D$, -P&2 (table IV), under site symmetry a.., the entry lists three point-positions, b (003], d (483) and f (843), under their common multiplicity 2. The site coordinates are fractions of unit-cell* edges, expressed as numbers of halves, fourths, eighths or twelfths of these edges.The legendsto the tables give thz necessary information in each case. They also explain a few abbreviations that had to be used for the sake of compactness: Q = 10. /3= 11, X = 4 fx, and the like.
3. Subgroups and supergroups of crystal point groups Since a given site symmetry can only occur among the space groups whose factor group is a supergroup of that site symmetry, it is necessary to find which point groups are supergroups of a given point group. A special table is provided for this purpose (table XVI)**. An illuminating example is provided by site symmetry Da -2/m 2/m 2/m. Locate this symmetry l
Althoughmodem crystrdlograpbicwge tends to substitute “cell” for “‘onitcell”, the term “unit cell” will be retained
here on purpose, to mean the parallelepipedalcell (whether primitiveor multiple) used by crystallographersand to distinguishit from the spectroscopists’primitive cell, or the Wlgnn-Seitz “cell” (41. The latter refersto the smallestpolyhedron containingone lattice point and possessingfull lattice symmetry at thal point. The content of the Wignn-Seitz “cell”, crystallographicallyspeaking, is the “translationrepeat” (French: morif. German: B&). ** Simibu tables atreadyeti in the tilerature.Rogers’ table [S] appearedin 1926. C. Hermann’sdiagramcan be found in the fnst edition oflnferMfionuI Tables (1935) and has been reproducedin subsequenteditions [ 21. More recently (1971) Boyle [6] publisheda table for spect~ scophts. The presenttable YM aims at giving more :klformation, in a more easily afcwibls form.
3.D. H. Donno~ and G. lheU.
Tables of oriented sire symmetries in spcrcegmups
symbol among the 32 point-group symbols listed along the diagonal(table XVI). The row to the left of the symbol 2/m 2/m2/mcontains four entries, which refer to the desired supergroupsabove and at the same time indicate how many times Da occurs in symmetrically nonequivalent orientations in the supergroup: twicein D4h-41m 21m21m,once in D6h-6/m 21m 2/m, once in Th - 2/m 5 and twice in Oh - 4/m 3 2/m. The group 2/m 2/m2/mis, of course, a trivial supergroup of itself and must also be considered; this fact is recalledby the I X in front of, and the I behind 2/m 2/m2/m.Thetables to be consulted are, therefore, tables I, 111,IV, VI and XI.
Note how effectively the Hermann-Mauguin notation showsthe various nonequivalentorientations that the site-symmetry point group can assumewith respect to the coordinate axes (taken along the edges of the smallestconventional crystallographicunit cell). In this respect the Hermann-Mauguinsymbols, especially in their oriented form, are obviously superior to the Schoenfliessymbols. Table XVI, of course, can be used to read off the subgroupsof any point group; they appear in the column below the givenpoint-group symbol. For the above example, 2/m2/m 2/m,each of the entries in the column refers to a subgroup(to the right) and givesthe number of different orientations in which it can occur.
4. The scaffold of symmetry elements of a space WuP At a pinch, if drawingsof the symmetry elements of the space groups are not available,the present tables can be used as a help to reconstruct the space-group diagrams,though it must be admitted such a task will not alwaysbe straightforward.In all cases,however, the tables unambiguouslyindicate the choice of the origin.
5. Spectroscopic applications The application of these tables to the vibrational spectroscopy of crystals can be made in variousways dependingon the amount of crystallographicdata availablefor the system being studied. If an X-ray
3
analysishas been made which includes the detennination of the space group of the crystal and the positions of all atoms in the cell (that is, a complete crystal strutture), infrared and Raman selection rule-s for the vibrational fundamentalscan be determined by the factorgroup method in any of its several forms [4). The traditional analysisdue to Bhagavantarnand Venkatarayudu [7] involvesthe determination Gf the effect of each symmetry operation of the factor group on the atoms contained in the primitive cell of the crystal. This geometricalapproach, which requires the visualizationof the appropriate symmetry operations, is greatly facilitated by reference to the diagramsin the fIItemntio?ru~Tables [2]. More recently, however, two methods have been developed which allow the factor-groupanalysisto be carried out without direct use of the geometry of the primitive cell. Both the correlation method [8] and the tables * developedby Adamsand Newton [9, 101 employ the Wyckoff letters to identify the various point-positions(sets of equivalent sites) in the crystal space group. The point-position and its multiplicity must be known for every set of symmetry-related
atoms (and for the center of gravity of a molecule or complex ion) before these methods can be applied. This information, which is givenin detail in the fnrernutiond Tables [2] ,is presented here (tables I-XV) in a more convenient and compact form. Ihe tabular form allowscomparison to’be made of the various availablepoint-positionswithin a givencrystal class,a feature that is especiallyimporiant when the space group is unknown. As illustrationsof the use of the tables, consider the molecular compounds urea 0C(NH2)2 and thiourea SC(NI-LJ2.I”neircrystals belong to space groups DL-P42,m, with 2 molecules per unit ceil,
* Thefollowingstatement is made in ref. 1101 @. 1): “FadorgroupanalysisaIways dealswith the smallestunit ccIi, Le.. the rhombohedralcell for the hexagonal system. the primitive cell for other systems.” Note that the rhombohedrai cell is a primitivecell.Notealsothat only a crystai that has a rhombohedrallattice can be desxiid by a primitive rhombohedralunit cell. Any ttigonal or hexagonalspace group whose Hermann-Mauguin symbol begins with a P has a primitiveunit cell that is not a rhombohedmnoc B rhombohcdraiunit celt that is not primitk Cp. foatnote on p. 2.
2e 000
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Fd3m
Fdlc
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7
8
9
10
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Film
1
2
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4
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161 xxx
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. . . LBf x02
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1921 xyz
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4YZ . . . . . .
.. .
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. . .
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128 xlm
. . .
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lab 000 444
Ii
‘h
Table 111. Th -2/m
X,
.
. . .
.
. . .
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.. .
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24d 022
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x:
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c2
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24~ x00
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6ef . . . x00 x44 . . . . . . 6b 044
...
. . . . . . 32P. . . . . . .,. . . . . . . . . .
. . .
.. .
. . mlcd... 044 400
4
‘3
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. . .
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ta
000
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...
... . .. ... ... ... ...
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CoI,rdlnatee
. . .
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Table II. T,j - 53m % 0 - 432
:
. . .
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Fm3c
6
Legend
Lab 8c . . . . . . 000 222 444 . . . . . . Be Bb 222 000
Fmlrn
5
000
2,,
.. .
pn,m
0
. . .
. . .
. . .
2a 000
. . .
.. .
Pm3n
3
.. .
. . .
Pn3n
2
.. .
lab 000 444 ..;
Pm3m
1
TabIa I. 4 -4/m 3 2/m
I I
I
I
P
J.D.H. Donnny and G. Smell,
Tables of oriented site symmcrnes m spncegmups
J.D.H. Donnay and G. IIureli. Tablesof oriented site symmetis
:
n
.
*
in qwe groups
I. D.H. Don~y
and G. lbrrell, Tables of oriented site Eymmetries in spx
gmups
I
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..
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6d
XYZ
. . . . . . . . . 6c
. .. . . .
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9b .. . .. . . . . .. . 1Bc xxdlr XYZ
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. . . .. .. . . ,, ... ...
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001
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002
. . . .., ,.. 2ac
2b IBZ
Warning: ‘i%a unit cell of a rhombohedral space group is here the tdple cell referred to hexagonal coordinates. To use the rirombohedral primitive cell, divide all the sitemultlpkitler by their common factor 3.
. . .
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be
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4
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kYZ
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. . . 18fR.n. x00 X06
6f x03
6Rh X00 X06
Id XPZ
.. . ... ... . ..
. . . 1Bh XRz
XYZ
. . . . . . . . . . . . 1Be . . . 18d 36f X03 600 xyz
9de 606 600
_ cl
. . . 121
c1
. . . 6h 68 121 xx3 600 xyr
. . . 6lj 110 xX6
cz
. . . . . . ,. . . . . . . . . ..
P3cl
lac . . . 001 4BZ 841 la ooz
... ...
... ...
3
1
4cd 002 482
,.. . . . . . . . . . 6e . . . bb 12~ 000 OOZ 003
3sb .. . be 000 002 006
.. .
f&
6
6k X02
. . . . .. . . . . . .
. . . 2b 000
. . . . . . .. . . . .
. ..
. ..
.
.., . . . . . . . . . 2e 003
lab .. . Zcd .. . 000 001 DLl6 4BZ
P3lm
Rb
5
_
c.
2 Zb Lef . . . . . . . . . . . . a003 oao 00; CA83 682 d843 lot 61 ... 600 rPr 60b
2
P&l
I4
'Zh _I--
. . . . . . . . . . . . ._.
1
Pi.1
3
cl1 c3
. . . Zcd .a. &h . . . 3fR 600 LB0 caz 606 486
-
D3
5”
PIlc
2
-
,.. lab . . . 2e 000 002 006
-
c3"
301 3ln P3al
PJlm
1
flflalBoHeoIuL
'3d
Table V. Hexagonal & rhombohedml
P3221
I4
2,3
c3 1
2
1
R3
P31,2
3 P3
R3
P3
51 _~ J
R32
6 7
PI212
PI121
x,12
PJZl
P312
5
4
3
2
I
D3d -
SY -
T
'2h q
'.
‘!
cl
cl
.y.
x00 X06
. . . . . . 6~
. . . . . . . . . . . . 3-f
XYZ
. . . 6L
l-i F! #..d r*
c2
. . . . . . a.. . . . . . . 3Jk XI0 x26
*
-I_-
08
P3121
except
for
. . . . . . . . . . . . 9de x00 x06
3ob(xOB)(x02). ..a .., 6c 002
XYZ
. .. . . . 1Bf
.. .
. ..
. ..
3ab 6c 000 002 006
. . . .,. . . . . . . . . . . . . 9de 1Bf 606 xyr 600
XYL
LegendI
Coi3rdInetes
In tuelfths
of
cell
EdBOB.
o-10,
6.11.
.. . . . . . . . . . . . . . . . . .,. 35 . . . . . . . . . . . . . . . .. . ..a 9b 002 *Yr
. . . . . . . . . . . . . ..
. . .
1.x . . . . .. . . . . . . . . . . . . ,.a Id 001 XYZ 4BZ B4Z . . . . . . . . . . . . . . . . . . . .. . . . . . . .., . . . . . . . . . . . . ,.. $3
.. .
.. . . . .
.. .
. . . . . . . . . . . . . . . . . . lab Zcd . . . .., ,.. . . . . . . . . . 3af bg 000 oar 600 xyr 006 48~ bob
.. . . . . . . . . . . 3ab 000 006
Same
3ell .*. 6C xal XYZ XPO . .. . . . . . . .a. . . . . . . . . . . . . . . . . . . . . . . . . 3ab . .. . .. 6c X04 =YZ X00 Same aa P3112 except for lab(riB)(xR2).
. . . . . . ,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zcd 001 482
2gl O& LB2 842
c3i c3
. . . . . . .a. . . . . . . lef . . . 000 006 080 486 840 846 . .. . . . .._ . . . lab . . . . . . 000 006
lulc4aIlHeDnAL -
HU(ACOHAl.
Table VI. Dqh - 4/m 2/m 2/m
TETIUCONAL 122
%h 1
2
K Q&/wan
Q4/mcc
. ..
. . . .. . . . . 2ef . . . ‘La Ljk Ci x00 xx0 OL?! OLL 010 x0.4 XXI XL0 XLL . .. . . . 2ca 2db . . . Lgh . . . . . . . . . . . . . . . . . . af LL2 LLO 002 au ‘4.z 002 000
lad 000 004 440 LL4 . . .
I
PLlnbm
. . .
4
QLlnnc
. . . . . _.. . .
5
P4lmbm
. . .
6
7
QLlmnc
QLlnmm
2cd OS0 014
.. .
. . .
2gh . . . 002 LLZ
. . .
. ..
. ..
. . . 2ab .._ 000 OOL
. . . Lg 002
2ab .._ ‘d 000 LO2 OOL . . . . . . Zab .,. 000 LLO
Le 002 Lef
001 OLZ
.. . . . . . .. . . . . . . .. .
. . . 2eb 000 004
2c OLZ
. . .
. . .
2sb . . . 6e 000 002 QOL . . . . . . . .. .. .
Ic OLZ
.. .
. ..
. . .
. . .
. . . . . . Oh OLZ
.._
. . .
_..
. . .
.. .
. ..
. . .
. . .
.. .
. ..
. . .
. ..
PS/ncc
. . .
9
QL2/mnc
IO
QL2/mcm
. . . 2ef . . . . . . . . . . . . . . . 2ad . . . DO2 000 442 LLO 000 OLC . . . 2db . . . . . . . . . . . . . . . . . . .., 2ce LL? LLO 002 000
.. .
. . . Lb 000
.. .
11
QL2labc
. . . . . . . . . . . . . . . ._.
12
QL21nnm
.. .
2ab 000 001
. ..
. . .
. . .
. ..
13
QL21mbc
.. .
.. .
. . .
. . .
. . .
. . . 4b
QL2/unm
15
PL2/nme
lb
QO2Inmc
17
14lraKu
1s
s.lmcm
. . .
.. .
._.
. . . . . . 2& 000 004 .. . .. . . . .
. . .
. ..
. ..
. . .
_..
._.
.. . . . . ‘e LOO
.._
. . .
. . .
. . .
. ..
._.
. . .
. . . 4d Lc O&2 040
.. .
..*
8x7
. . .
._.
. . . ‘dc 220 22L
.__
. ..
. . .
. . .
.._
._.
. . .
. . .
Lgi . . . 002 LLZ CJLZ
.. .
. . .
. . .
.. .
. . . Bq rye
. . . L$, 002 LLr
Oe 042
. . . Lf 010
.. .
...
. ..
. . .
. . .
.-.
. . .
.. .
. . .
. . . 4s 4c OOZ 040
._.
. ..
_..
. . .
_..
. ..
. . .
. . .
4fg
. . . Gab &c OL2 040 002 xOb
. . .
. . . .. .
20
lL,lecd
.. . . . . . . . . . . .. . . . .
4ab . . . OGU 004
.-.
...
. . . Oe
CotJrdinares
8iven
In
eighths
160 XYZ
16n =yz
._.
. . .
Ei Oxr
8, xxx
. . . 8gh
. . . 2bk
.._
. . . Be oar
. . . BE n2
Bop ox2 LXL
. . .
. ..
. . . lbr . . . Bn XXL ryr
. . . . . . 8n
8k 04.
8&, x02 x42
. . . 80
xx1
rY0
...
. . . 8gf 002 0:~
Ld oc2
. . .
. . . Lef 222 666
. . .
. . . E,, Bh XXL 04.
Ld 042
‘c. . . . OLO 000 ...
. . . Bh
...
. .. . .. . ..
.. .
. . . ae . .. OO.?
.. .
8a
.. .
. . .
. . .
. . .
. . .
. . . Bb 002
. . .
K rtandr
for
4
+
. ..
x.
...
Ed 220
L6g qr
. . . 16~
Bhl Bj x02 xx0 rob
9e 222
Ibk XVL
8ij XW rOL
. . .
16~1 =y=
8kE xx2 xX6
. . . 8ef . . . Bg 002 xx2 OCZ
. ..
Bj
Bh oh
WC
=Yr
_..
WO
. . . Bi
.. .
IbL xv
na x%4
.. .
...
. . .
. ..
. ..
...
edges.
. . . 86 xx.7
.. .
.. .
cell
.. .
.. .
. ..
of
. . . 8ke ei, xoono x04 xx6
. ..
. . . . . . Oc
000
Lewd:
. ..
xx0 007. 010 xx2 WO XEO .. . 4cd _.. . . . .. . . . . . . . . . . . .. 8, .. . oxz Oar Oi; . . . . . . . . . Le . . . La . . . . . . Led . .. . . . 81 OLr 002 222 xx2 22b
. . . . . . . . . . Ic . . . 81J
ILllamd
. . .
. . .
. ..
. . .
.._
‘a 002
. . ah ag . .. Zab . . . Ld le 042 oar 000 040 x00 a0 O‘Z OOL XL0 .. . Lb . . . _.. La Lc . . . Bf .._ Id _.. bh . .. 8s 042 002 000 001 060 xx0 OLZ
19
. . . Ei Bkl 81 OLZ x02 xx2 XL2 BP xx2
lbu
rlrr
. . . Bf 042
WJ
_..
.._
. . .
. . .
XYO
.._
.. .
. . . Lb 000
. . 80
. . . Lef 220 22L
id 000
. . . 4d OL2
.. .
.._
. .. . Gf 00s
Ljm . . . x00 XLL XOL XL0 . . . 4ij xx0 %x4
. . . 2ab . . . 000 006 . . . .. . . . . . . . . . . . . . . . . .. .
. . .
.. .
Le . . 040
.. .
002
14
nyo x02 xx* XyL XCZ
. . . 8g si, 8b Bf L6k &Or x00 xx0 222 EYL X04 .. . . . . . . . .. . IbL . . . 2cd . . . Lgh . . . . . . .. . .. . . . . . . . . . . BiJ . . . Ek xx0 xX7. Eyr WJ XXL *YL
8
.__
. . . . . . . . . . 8pqBar 8r __. .._ __. .._
.._
.. .
. . . 16L xyr
.. .
lbk ryr
. . . . . . 8f no
EC 222
L6h q-2
8f “L
. . . 1bJ YiYr
. .. 8gh xx2 nb
16L lbn lba .. . . .. 16k . . . 320 . . . Bf 222 rye oxz m v. . be
. . . 8cd 021 “7% . . . :::
lbk . . . 16L .._ 16j 161 .__ ,in, x02
sxz
xx2
222
xyll
.. .
. . .
lbh 0x2
.. .
. ..
.. .
. ..
...
.-.
16d lbc 161 oar 2x1 xx2
lbf ‘68 rZ1no
ryr
. . .
37.5 xya
16~
32% ryr
021
J.D.H. mntq
10 Table VTL Du
and G. Well,
Tables of oriented site symmetnSes in space groups Table VlIL cd,, -
- 42~1 ii am2
DZd
‘4
‘2v
D2
5
%v=: c,
5
c2
4mm
C
c2
-YA
ElRAGoNN.
:Lv 1 lad
...
...
. . . 23b
2ef . . . . . . 4n 002 4w ax2 64Z 404
OQO 444 004 040 2
PZ2.z
. . .
. . .
. ..
. . .
. .. . . . . . .
6
7
8
9
P&2
Phb2
PTn2
l&2
10
IZc2
11
IS2m
12
Ir2d
. . .
. . .
. . .
lrd
2cf 000 440
2ab . . . WO 004 . . . 2eg 002 442 042
. . .
. . .
. ..
Zab a00 004
. . .
. . . 2ab 000 oo4
. ..
. . . 2ad
...
. . .
. . .
. . . hef 002 042
. ..
. ..
. ..
iii a42 a46 . . . . . . 4bc Ml0 042 2ab . . . 4d OOO 042 oob
. . . .. .
. . . 2ad 002 402 442
042 2ab . . . 2e . . . 000 042 004
%I 444 000 . . . Zcd 000 440
..,
. . .
Cab
. ..
. ..
. . .
. . .
. . .
...
r42nm
. ..
. ..
5
Phcc
.,.
2ab . . . OOL 44L
*Y*
6
P:nc
. . . 2a 002
3i xyr
7
Ph2mc
._.
a..
2cd 040 044
.. .
2cd 042 046
. . .
. . . 43i oar 442 042
. . . 4hl xx0 n4
. . . 4ef 3J a2 xyz xx6
. . . 81 . . . XOZ
.. .
. . . he oar
he 040
.. .
. . . 81 -
. . .
. ..
..,
8
9
. ..
31 vz
4f3
Yi
Xx6 xx2
xyz
. . . 83h
16J
no xx2
xyyr
Bfg . . . Ehc 002 xx0 a42 xx2
161 lyyr
8h 8fg 04X x00 xc4
v=
04. oaz
.. .
xyz
. . . 4ef . . . 4gh oar rxa our xx4 . . . 4he
10
11
12 . ..
. . . Bc oar
8d x21
2ab 002 442
. . . Cd xx*
4c 8e 042 xyr
. . . 2a aor
. . . 4c IXL
Lb 042
8d ryr
. . . 8n xyz
. . .
. . .
cob
Warning: Space gxoups Dz2
4
. . . 8e
. . .
. . .
. . . . .. . . .
,..
2ab 002 442
. ..4ef4d . . . 88 xor xxz XYZ x42 . . . ?b . . . 4c . . . 8d 052 Xx7. ryr
047.
P42cm
. . . 8f
. . . 4jk XOZ x42
Iab...?c 002 44r . . . 2a oar
3
X06 . . . 4c 4d . . . xx2 002
. . .
P&xln
Wbm
. ..
. . . Led OOr 042 .. . . . .
43J xat x46 x42
Rtjp:i-
hurl
2
. . .
.. .
. . . Lkm 002 442 042
4if . . . 60 x00 XYZ x44 x04 X60
. . .
. . . 4da 040 002
COO
4m 042
and $“I2 refer to a double ~4.
16j
16~ ayr
cl
P42bc
14csn
Ilcm
IClmd
Ibled
. ..
.. .
. . .
2ec aar kdr 042 . . . . . .
2a oar
. . . 4b 042
. ..
La 001
. ..
. ..
.,.
.. .
. . . 4c 042
Bd xyr
. ..
...
. . . Lb 042
6c xyz
__.
4de x0: xc2
. . .
.. .
..
.
. . . 4b 042
. .
. ..
. . ._ 4ab
8d xoz
8c XXL
. . . 8c
002 04. . ..
. . .
xx2
16e
16-j XYZ
. . . 8b 0x2
. ..
.. .
...
. . . 8a oar
...
8c xyz
XYZ
. . . La 001 .. .
8f XYL
. ..
1bc xy7. 16b xyr
J.D.H. DOIIM~ and G. 7Uncrell.Tables of orier~ed sire symmetries in space group Table X. c& -4/m. S4 - 1% C4 - 4
Table UC.04 - 422
mRAcMuL C
P.422
lad Zsh 000 aor OOL ‘4L
Zef LOO 501.
. . .
&i osz
&lo L.!k x00 xx0 x84 UL x0: X&O
160 u:
/
2c wt
. . .
2ab 000 005
Ld Oor
. . .
Lef 8R xx0 xyr XXL
~4122
. . .
_..
. . .
. . .
. ..4ab4c 0x0 6x0
a3
Bd xyz
ba no
8b xyz
b
Pl12,2
i
Pi222
. . .
. . .
. . .
040 O&L . . . . . . Zob 000 006
5
PL2Z12
7
P&,22
Same
B
P43212
Same
9
lb22
?ab 000 004
0
2
. . .
14 12?
. . .
. ..
. . .
._.
. . .
2ad 000 L40
Zef 002 b&2
Lgl 002 SLr 042
ojm Ino x00 a2 al!.0 xx6 x0:.
as
P&,22
55
Pbl212.
Le 002
. . .
X40 . . .
81 Bhi OLZ x00 x&i
Lab 8c 000 002 OOL
bef xx0 xx4
8Je xX2 xx0
16k xyr
8f
8de
16~
x21
xx0 XX0
xyr
of
cell
edges.
‘s
Cl
CL
2
I
1
. . .
2gi-1 2ef
Ojk
li
002 142
040 0.44
xyo Iy4
WL
. . .
2ad OM 460 040 040 . . .
Oj ry0
Ogi OOr 442 war.
. . . 0k
. . .
If oar
Ide 220 224
8g xyz
. . .
. . .
. . .
4ef a42 Ooz
4cd 222 226
8g xyz
4e elk
4c 040
Bh xya
86 042
8f 222
16t XYZ
.__
_._
. . .
8e 002
8cd 021 025
16E qz
. . .
. . .
. . .
2e,3 OOZ
. . .
140 644 .._
2ef 002 462
. . .
Zab 2c 000 042 00.5
4
P02/Il
. . .
2ab 000 00.4
5
I4/=.
6
IlllIe
2ab 4d 000 042 004 . . . lab 000 004
. . .
BE =Y=
rgr
G Pz
. . .
lad 000 004 440 460
2
C2
lad
PO/n
=4 1
‘2h
12lem
3
C4 1 Legend: Coordinarcs La clghths X stsnda for 4 + II_
P021rn
8p xyz
‘3~ xyz
‘4
T
for bc(xx5
except
Lc Od O&o 062
. .. ...
4cd 002 OLZ
‘4
000 004
8P xyr
P&Z12
.. .
Oh oh 1 PPll
‘&h 4/m
IS
4 Pb
. . .
2ad a00 000 042 W6
4h =v
442 042
. . .
. ..
. . .
bef
..,
OQ 042
8g =Yz
. . .
. ..I&
. . .
. . .
2c a42
._.
. . .
. . .
. . .
. . . 4.9
2x oar L&r 042 . . .
. . . 46 xYr
2
Pdl
,..
002 Chr . . . . . .
3
P42
. . .
. . .
. . .
. . .
. . .
4
PL3
. . .
. ..
_..
. . .
. . .
4d ryr
ryz
. . . 4. w=
5
Wmlng: Space
groups
04 9S’o,
@,
IL
S$ and c’4 refer to a double cell.
. . .
. . .
2a 002
. . .
. . .
Cb o&r
. . . EC xyr
. ..
. ,. . . . . ., ,.. . . .
. ..
. . .
_..
~hna
Rna
Pee.
Pbam
pccn
pbca
6
7
a
9
10
11
~~
'2h
cm
.. .
044 ... ...
.. .
bd x22
.. .
. . . 4c oy2
46 212
4dc 202 242
Lob 000 040
Ef xyr
4ab 8c 000 xyr 004 . . . . . . ai XYZ
. . . 4c 2nr
.._
. . .
. ..,. 1..*.P. . . ,.
. . .
.. .
.. .
._.
. 413
. . .
2nd . . . .,. ... _. . ..,. . ,___ .~
.. .
Ahm2
.. .
...
,_A ^
. . . 4nb 80 oon ry* LO'J Icf . . . 8h
Abo2 17 --TV‘-Pilrl’l
AmD2
Amm2 14
ccc2
Cmc21 12
13
-2
Pnn2
Prl.aZl
Pba2
11
10
9
8
-21
hlcz
b
7
PcaZ1
Pm&?
Pcc2
5
16
. . . bgh . . . ,.. 4e.f . . . 81 002 wr XYD 042 XY4
.. .
XYr
8m
. . . Er XYZ
OIj 4kL 4ef oyo oar 220 oy4 04z 224
4mp 002 44r 04r 402
81 . . . OYO oy4 . . . . . . XYZ 4sh
x04
XOa
4Rh
LkC oy2 4y2
. . .
2nd . . . OOD 004 040 044 . . . .. .
...
4lj
x02 x42
. . . bcf x00 x40
.a,
. ..
..-
48 rye
4c x20
. . .
...
. . . 111 . . . OYZ
.. .
xor x42 4ij
. . .
. ..
. . . . . . . .. . .._ . . . ._. . . . . . . . . . . . . 4d xy2
. . .
. . .
...
2yr . . . 4k
.. .
.,.
15
. . .
.
. ..
2nd . . . 000 400 ‘LO 060 . ... .. .
...
004 040 2nd . . . 000
...
2ad 000 440 060 400
4
3
Pmczl
1
2
-2 R2
Z2”
. . . 4cd 4ab 80 222 000 xyr 262 owl
. . .
. ,, .. .
,..
. .. 2ef 242 004 . . . 202
.. .
.,.
ORllM-
RHOMBIC
. ..
. . .
000
2nd
400 404
. . .
4k: 4ef Em 007. 22.7 ryr O&r 6b6
XYZ
eu
tii L1
. . . . . . -.. . . .
=2 ”
C2
C. .y-;
2eb 002 202 .. .
2ab 002 02r ...
..
2ab ooz 022
be Bf Oyr XYL
. . . Lc XYL
... ...
Be xyr . . . Bb . . rl%
. . . 4b 1yr .. . . . . ba -... Hn ..T
. . . 4e 00s
Ed XYZ
8f xyr
Bd XYZ
Bb Oyr rys
. . . . 40
Id ror
...
4a XYZ
..,
. ..
4b xyr . . . 4e =Yr
21 oyz
. . . 4c XYL
. . . 40 XYZ
bd xyr
. . .
. ..
. ..
. ..
. . . 2c 1yr
4e XYZ
4c xyr
4i x7=
-
Cl
411~ 001 022 112 2ab . . . 4c 4dc xoz oyz 002 202 2YZ . . . 4ab4c .. . 002 XlZ 201
...
2eb 4c Ooa 112 02r . . . ..
.
Pab 002 022 . . . .,.
.. .
._.
. . .
..,
.. .
lad . . . 2ef Z6h aor oyr 001 x2; 2yr u2r ZOZ 22r I.. . . . ..; 2ab oyr 2YZ .._ 2nd . . . . . . 002 021 201 222
“d
c2v
Table XII. C2, - mm2
.. .
. .. 12 mnm ..x*-LsL,u> .CI-L .
I
I
...
. ..
. ..
. ..
...
Peb
002 402 042 462
Pm
5
. ..
...
. . .
Rlan
4
. . .
OOL 040
. . .
Pccm
3
---
. . . . . . ..a . . . 4U" OWL 4yr oyr nor XYO 4yr x4r XY4
%
L44 . . . . . . . . . . . . 2.9~ . . . . . . . .. . . . . . . . . . 6g.b 411 x00 oyo x04 4~0 400
040
2mp 2qr
no0 oyo ooz x04 oy4 04; x40 fbyo 402 x44 4y4 442
lmh Zil
000 400 004 404 040 040
c2v
. . .
Pnnn
2
222 iii R
--
.._
II
I
D2h 1
'2h
Table Xl. Dm - 2/m 2/m 2/m
6
5
4
3
2
1
c 2h
c2/c
P~,/c
P2/C
c2/m
pZIIm
2’a P2lrn
MWLINIC
2c xl2
1
‘1
Zcf 0yl 2y1
. . . be Oyl
4~ IYL
Lad Bf onn XY7. 020 110 112
2ad 40 nor) xy, 200 002 202
2nd 000 220 n20 200
4gh 4cf BJ oY0 iin XYZ OY2 112
. . . 2nd 41 000 XYI 200 002 202
... ... ...
. ..
i
‘1
2mn Zfi . . . 40 x07. 3Yo XYZ x2x 2yo OY2 2Y2
2ed 4‘ 000 xoz 020 002 022 ... . ..
lah 000 020 no2 200 220 022 202 222 ...
'2h ',I ‘2 21~ m 2
Table XlV. Monoclinic
P
&
. . . ... ...
. . .
uanm
cccn,
as&4
Ccc,,
F_
Fddd
m
lbam
2bcm
m
19
M
21
22
23
24
25
26
27
2a
wunlng:
L
. . . ,. . . . . . . . . . .
f%,ca
oar
4kL
Bh oyo
...
81 oar
8ab 000 004
8f 222
. ..
. ..
. ..
. . . 40 001
.. .
.
.
. . .
. . .
...
...
...
...
. ..
ac 022
. ..
,.
L:
..*.
.. .
. ..
. ..
. . . 4c x2r
.. .
. . .
2YL x22
._..
. . . 8f Oyr
...
. ..
.. .
.
&‘a” .
.
.. . ...
...
. ..
.
.
..,
Ed, oyz
. .. 81 i2r
.,.
...
.
.
.
_ .
ah1 x00 x04
88 x02
.. .
. . . a, x00
.._
BL rvt
Bpq xyo xv4
an
. . .
.. .
8~ lY2
. . . Bf x00
.a.
8d 8a 2YO 02x
16k xyr
160 XYZ
Bob 16f 000 XYL 222 . . . LbJ RYL
88 8111 Bc Lly2 002 222 OlZ
222
32h ryr
. . . 32p XYZ
. . . . . . . . . ak
Bf 102
160 xyz
L6m XYZ
XYL
Ihr
16h XYZ
ad lbl: 220 xyz
ad XYL
ac 000 r/7. Cl”4
16gl6cd 001 111 555
. . . Be x02
Ej xyo
XYO
16,~ 16f x00 oyo
p
BRII Bed 161 OOZ 202 xyx 22r 022
16k 16J 2Y2 222
Bf oyo
_.
4nb ad on0 XY?. 040
alk . . . 002 067. 221
Bjk at 2yo 202 2Y4
Bh oy2
.
E
. . . ac 210
. . . 8m 221
ac 2Y2
.
ODD XYZ 004
. . . 4ob 000
moo
Ed x00
.
. . . 4Db
.
xy2
. ..
. ..
.
l3e . . . . . .
... .. . 8m noi
.
ag
. . .
161~ 16n 160 I6t cyr xoz XYO x22
. ..
. . . aI oyz
...
ac 220
...
. . . 4cd . . . OW 400
...
,..
8d 202
. ..
Bn x2z
.. .
4ef Bn 80 220 OYZ xor 224
. . . 4ef . . . 000 040 220 260 4ef . . . am 220 oyz 224
_..
OYZ
.
. . . 4c ..* OY2
.
. . .
.
. ..
. ..
. ..
.
-IT.
. . . . . . Bt . . . . . .
. . . 4ab 4cd 000 222 Eoz 226
. ..
.
Lab . . . 000 OLD
. ..
. . .
.,
. . .
.
J
22
Imoz
Ibn2
lImn2
. . . be 002
2ab . . 001 02z . . . 4nb oar 022
9
a
7
1
%
222
12,22Z1
1222
P222
2nd 000 200 ml2 020 .,.
lad 000 002 111 113
2ad 000 020 202 002
c222
4a no1
4cf x00 102
ajc xl1 x00
4cf x00 x02
. . . 4a x00
c2221
. ..
4c XYZ
4c Olr
48h 4Lj OYO 00; 2Yfl o2r
Bd ry=
8k xyt
16k ryr
at xyr
. . . BE rys
lab Lc OO?! xyz 02r . . . 411 aY=
. ..
tlif Q,h 1y1 00s OYO Ilr
4b ly0
c1
8c xyr
241 4u oar xyr 2Or 02r 222
N :_
48h 4lk OYO 00. oy2 022 1lZ
4b fly1
_. .
.. .
P212121
,
2lC 2mp x00 oyo x02 oy2 x20 2yo ~22 2Y2
P21212
lab 000 200 020 002 220 202 022 222 ...
:” N
--
c2
8c xyr
XYZ
. . . BC XYC
4d oyz
. . . 4b lyr
.. .
4c xor
2ab 2cd x00 oy1 x20 2y1 ... . .. . ..
P222,
P222
A N
D2
Table XIII. D7 - 222
L
21
20
oar
Unit cella (I at am A, B, Cor I centered are double cells, those that are F centered arc quadruple cells.
. . .
2nd 4aC 4gh 411 . . . 000 x00 oyo 001 044 x40 oy4 402 440 404 . . . . . . . . . . . . 4ab 002 402
. ..
.
lab 4cd 200 Km 204 004
. . . 4eb 000 004
46 021
.
.. .
.
Lab 000 ‘00 . . . . ..
. . .
. . . 4ab 002 042
. . . .a. . . .
4ab 8s 000 x00 004 . ..
. . .
,..
-
oyo
. . .
ov4 04.
4fJ
400 x04 404 004 . . . ..,
x00
2ad 4Rh
090
w2
LC
. . .
.. .
18
. . .
.. .
. . .
. . .
...
...
(Lacln
~.
17
. . .
.
Fnma
.. .
.
.. .
.
16
._.
. ..
222 261
LIlY
F&x
. ..
.
,,.,,
. ..
.
15
.. .
. ..
. .
Pbcn
.
14
.
.
.:.
. . . .. . . . . . . . . . . .. . . . .
CZ
P21
..
I
1 PI
000 001 010 I00 210 io1 011 111
Ish
la XYl
21 ryr
5I 51
. ...
Lo8end: CoUrdinatcm In halvoa of cell cdgoa.
Cl 1
=i
i 1 pi
TRlCLlHlC
. ..
. ..
ba XYZ
. . . 4b w=
2eb . OYO OY2
. 4c xyr
OY2 2YO 2Y2 . . . . . . . . . 20 XYZ
.. .
. . .
. . . lad . . PC OYO XYZ
. ..
..
.. .
. . . 2a X02
2e XYZ
... ... ... ,..
Inb . . . XOZ x2z
2c XYZ
. ...
..
_.._ _..-_-
Table XV. Tdclinic
3
2
2 PZ
1
CC
Cm
PC
~m
m
C_ L
*
3
2
i
9
c-
I.D.H. Donnn~ and G. lhrell, Tables of oriented sire symmetries in spucegroups
14
Table XVI. Subgroups and supergroups of crystal point groups
lx.
.
.
..Lx..lx.
.
.
.
.
.
.
.
Ix .
.
.
.
.
.
.
.
.
.
.
.
The ale of mutual exclusion applies Lothe fust point group in each Laue class (see the fm Schoenflies symbol below each horizontal dashed line), ptis 0 ~432. l * In these tables rbombohedml space groups are described in hexagonal coordinates (triple unit cell); the laltice type is then called “hexagonal R”. B desigr&ion that is equivaint to “rhombohedral P’. In the monoclinic system iho symmetry direction is chosen as the b axis. l
J.D.H.
Donnay and G. Thell,
Tables of oriemed sire symmetries
in space gmupo
15
Notes ro table XVI (1) Synonymy of Schoenflies symbols: Ci= S2= /I, Clh =sl q 12,D2 = V= p, Du = Vh, Du= vd, CJi’&j =fj.s4 =f4, C3h = S3 = 16, in which I stands for a rotatory inversionaxis (German:inversion)and S for a rotatory reJ7ecrionaxis (German: Spiegelung).(In = ii in Hermann-Mauguinnotation). (2) Dashedlines separate the Laue classes. (3) This table presents only the 32 crystallographic point groups (also known as “ctystrd classes” or, morepncirety,“geametic crystalclasses”)so that no provisionis made for the variouspossible settings in which a given crystal may be oriented. Speciftcally. Ch may have to be written mm2.2mm or m2m, if the uniteelI edgs ate chosen according to the metric convention c < a C 6. LikewiseDw has two possible orientationsif the smallest tetragonalunit cell (P or I) is used; they are written&m and int2. SimilarlyD3h can be.written 52m and 6m2. Furthenore, three trigonal point groupsalso need alternate_orientatiatts, though only in the case of a hexagonal lattice; they are: 32 (321 or 312). 3m (3ml or 3lm). ? 2/m (3 2/m 1 or 3 1 2/m). (4) Occasionallysome subgroupof a factor g~oup does not occur as site symmetry in any of the isomorphic space groupr_Such subgroupsshould be disregardedand have not been entered in the table. Thus omitted were:3 in 6/m 2/m 2/m. 2/m in 4/m 3 2/m and-i in 4/m !i Z/m. 2/m 2 and 6/m 2/m Z/m.
and D$ - Prima*,, with 4 molecules per unit cell, respectively [ll] . The letter Pin the HermannMauguin symbol immediately indicates that the crystallographic unit cell is primitive and, therefore, can he used without subdivision in the factor-group analysis. This explicit slatement of the lattice type is another advantage of the Hermann-Mauguin notation over that of Schoenflies. The latter, however, provides a direct relation between the space group and the crystal class(and hence the factor group and its isomorphic point group); simply dropping the superscript of the Schoenflies symbol yields the symbol of the point group to be used in applications of the factor-group method. Thus, Du -&II and 3% -2/m 2/m 2/m are, respectively, the point groups that describe the crystal classes of urea and thiourea. The Herrnann-Mauguin symbols do not provide this information quite as readily**. It is known that both urea and thiourea have symmetry Ch- mm2 in the free state. It follows that either molecule must occupy a site of symmetry
Two orientations haw been used in the literatureto descni thiourea:Pnma and Pbnm. The latter is brought onto the former by transfotmationmatrix 010(001/1M). Pnma is the conventionalorientation of D$, in which the unitceUedgesarelabelledsoartoobeyc
I in thecrystalline state, as indicated in the column below the Cb-nvn2 entry in table XVI. In the case of urea the crystal point group is Dzd- 32m. Scanning the row to the right of the entry Du- d2m, shows that only sites of symmetries S4 - 4, C, - mm2 and D2 -222 have multiplicity 2. Hence,_C, is the site group of urea. Since the Du- 42m row points to tabibleVII, one can turn to that table and locate, in &, the site orientation (2.mm) for a point-position of multiplicity 2 with Wyckoffs letter c and site coordinates (042). The table thus provides the information necessary to complete the factor-group calculation, for example. hy the tabular method of Adams and Newton [9, IO]. Similar arguments for thiourea follow from table XVI, where possible sites of multiplicity 4 in D, are C,, C2 and Ci. The first two of these point-groups are subgroups of C, and. therefore, are acceptable. However, reference to table Xl (indicated by the Dz row in table XVI) reveals that the space group DE has no sites of symmetry C2. Therefore, the correct site group is C,, its orientation is .m. and the pointposition is 4c with coordinates (x2z). From these examples it should be apparent that the tables presented here are to be used as an adjunct to the tables of Adams and Newton [IO] or the correlation method [S] .
Cb-mm2,2,-m,C2-2orC,-
6. Spectroscopic crystallogrephy In genera!, vibrational spectroscopy cannot Claim
16
1. D.H. Donnay and C. nrrell, Tables of oriented site symmerries in spacegroups
to be a direct method of determining crystal structure. Nevertheless, it is often possible to obtain a certain amount of structural information from the infrared and Raman spectra of crystalline systems. Such information may be e-specially useN where few or no diffraction data are available. The use of spectroscopy in assigning the molecules to their proper sites in the correct space group depends on the following arguments: 1. Thesite symmetry of a given molecule (or complex ion) is a subgroup of the point group of the free molecule (as in the vapor phase). 2. Certain vibrational fundarner&&, which are spectroscopically forbidden (infrared or Raman) for a free molecule, may become active for the molecule in a crystal. This observation indicates that the symmetry of the site is lower than that of the free molecule. 3. Splitting of a degenerate molecular vibration in the crystm shows that a~ least one axis, of order 3 or higher, of the free molecule has been removed by the (lower) symmetry of the site. 4. If a nondegenerate fundamental vibration becomes split in the spectrum of the crystal, the primitive cell must contain more than one molecule. The number of components of such a split fimdamental is a lower bound to the number of molecules per primitive cell. 5. In the case of dilute isotopic species in a crystal any splitting arises from the site effect, while in a neat crystal splittings can be due to site effect, to correlation effect, or to both. 6. The absence of coincidence of any infraredand Rarnan-active fundamental suggests that the factor group contains a center of symmetry*. However, the definition of coincidence depends on the magnitudes of experimental frequency errors; hence, this argument must be used with caution. 7. Infrared or Rarnan Polarization studies can often be used to determine the crystal system, as well as the molecular orientation within a monocrystalline sample. As an example of the application of the above principles, consider cyclopropane, C,H,. The mole* Note that infraredand Raman coincidenceis absent in cwtals of classO-432 even though theseayhls center of symmetry.
lack a
cule is of symmetry L& - k?nr in the vapor phase, hence, the irreducible representations of the 21 vibrational fundamentals have the structure C,& =_3_A_T?tA;+_~I+A'it2At+_3_E_'I.
Thespecies of the six infrared-active fundamentals are underlined, while those of the 10Raman-active fundamentals are underdashed. The four vibrations of species E'arecoincident in the two spectra. Polarized infrared studies of a single crystal show conclusively that it belongs to the orthorhombic system [ 121. Data on one of the El’-species fundamentals (ut4) in the 750 cm -I region show that it exhibits separate polarization properties along three mutually perpendicular axes for each component of the absorption band. Reference to table XVI indicates that the only possible crystal classes in the orthorhombic system are Da-2/m 2/m2/m,t&,-mm2 and D2 - 222. These, then are the point groups that are isomorphic with the possible factor groups of the crystal. Furthermore, as at least one nondegenerate f_mdamental (~2, for example) is split into a doublet, the primitive cell contains at least two molecules. The determination of the site group of cyclopropane was initially made by Brecher et al. [ 121, also by infrared polarization measurements. The conclusion that the site group is C,-m was confirmed by recent spectroscopic studies [ 131 of solid solutions of CJH~ in CsD6. Furthermore, as site splitting of the E’ and E” modes was observed, the plane of symmetry of the site corresponds to one of the vertical planes of the D3,, - g2rn symmetry group of the free molecule. (Compare the two correlations L+h * C, of fig. 1.) Once the site group has been determined as C,- m, the factor group D, - 222 can be eliminated, as it contains no planes of symmetry. (Table XVI shows that there are no C,-m sites in a crystal of class D2 - 222.) Of the two remaining possible classes, D* -2/m 2/m 2/mcontains a center of symmetry, while C, - mm2 does not. Comparison of the infrared data on crystalline cyclopropane with the recent Raman data [14] shows coincidence of the vibrational fundamentals to within + 1 cm-r. Thus, although accidental coincidence cannot be completely ruled out, this result and the rule of mutual exclusion [4] provide strong support for a noncentrosymmetric structure. The ensemble of spectroscopic evidence
J.D.H. Donnay and G. Turrell. Tables of oriented sire symmerries in spncegmups
II3h
c, $,~O)
17
C2v(m.-1
Fig. 1. Correlation of irreducible representations of the molecular point group (Djh - hn) of cyclopropane with possible C3 - 3 sites and the C2,, - mm2 factor group. Species of infrared active fundamentals are underlined; &man-active sgcies are underdashed. suggests a factor group that is isomorphic
with the
point group c,, even though the group Ds cannot be rigorously eliminated [ 131. Table XVI indicates that the class Cti contains two C, sites per primitive cell and that table XU should be consulted to find their orientations and the possible space grou s. There, it is seen that CL, $ ~4 ~7 c” ~1 P ~‘4 ~15 c’6 ~‘8 c20and 2uJ 2lP 2ul 2ul 2w 2lJr 2w 2U* 2w 2u G$ are the only space groups compatible with the
spyctroscopic results for cyclopropane. Thus, 218 space groups can be virtually eliminated on spectroscopic grounds alone, so that twelve possibilities remain. It has been found by Brecher et al. [12] that solid cyclopropane is denser than the liquid. Packing considerations based on reasonable molecular dimensions have been used to estimate the densities of structures belonging to the various possible space groups, and thus to eliminate structures whose densities are less than that of the liquid. All the ahove space groups other than C$,-fW2 and C&-Pmn21 are thus eliminated. Of the remaining two, CL-Anna* gives a structure denser by approximately 5%. If the space group of cyclopropane is indeed CL, reference-to table XII indicates that the pointposition is ~~(OJU)and that the molecular orientation m.. is such that the plane of symmetry of the site group is parallel to the b - c plane of the crystal and
hence parallel to the plane designated a(yz) in the CL point group. (Note, however, that this molecular orientation postulates the space group CL of the crystal to be oriented Penn?, .) The resulting correlatitn of the symmetry species is shown in fig. 1. The CL structure, which was predicted primarily from the spectroscopic data, has been discussed in detail by Brecher et al. [ 121: Recent X-ray work [ 131 on polycrystalline C& consists of a IS-line powder pattern which has been indexed on an orthorhombic cell (a = 10.0, b = 6.8, c = 5 .l A, at 85 K) so that the calculated density D, equals 0.40 or 0.81 g/cm3 for 2 or 4 molecules, respectively, in the unit cell. Choice of the higher density, which demands 4 molecules in the known unit-cell volume, and the spectroscopic requirement of site symmetry C2 -M led to the factor group Dzh - 2/m 2/m 21~. As can eassilybe seen from table XVI, this result is correct provided the unit cell is primitive. Indeed it is possible to place 4 molecules in the primitive cell for C, sites in factor group Du,. In Ch- mm?, however,the fourmoleculeswould haveto occupy a general point.position, with site symmetry Ct - 1, contraryto spectroscopic evidence*. Table XVI provides the possible space groups in &, namely those with Schoenflies superscripts 1,3,5,7, l
Footnote: see next page.
18
9,11,12,13,16.
etal. [13] who
J.D.H. Dannay ond G. ntne$
Tables of oriented site symmetries in qpncegmups
They are the ones given by Bates Rnmm “by distance
calculations”. Neither the space group nor even the lattice type could be determined from the powder pattern with certainty [ 131. It follows that the requirement of 4 moleculesper unit ceil can also be satisfied in factor group C,- mm2,for site symmetry C,- m,if the unit cell is double, either one.face
l
Acknowkdgement
eliminate r>$, -
The infrared polarization study j121 of singlecrystal C,H, provides very strong support for C,- m site sym-
metry. However, the possibility that the molecule occupies a general position (C, - 1) cannot be rigorously excluded, as pseudosymmetry could accountfor the spectxoscopic results;
Wewish to thank one of the referees for his helpful suggestions,which have been incorporated in the statement of the argumentsnumbered 3 to 5 in section 6. References [ 11J.D.H. Donnay and G. Donnay, Can. J. Spectr. 17 ..(1972) 45. Note the foIlowing errata, communicated by Professor Yosio Sakamoto (Hiroshima University). Tables 2.3 and 2.4; restore omitted heading (32) of the last column. Table 2.5, &P%l:the2incoLlshould be in col. 2. Table 2.6, first column: instead of C2 - 6, read C6 - 6; instead of C3h - 6, read C3h - d. (21 N.F.M. Henry and K. Lonsdale, eds.. International tablesfor X-ray crystallography. Vol. I: Symmetry groups, 3rd Ed.(pubIished for the international Union of Crystallography, Kynoch Press. Birmingham, England. 1969). 131 W. Fiiher, H. Burzlaff, E. HelIner and J.D.H. Donnay. Space groups and lattice complexes, Nat. Bur. Stand. (U.S.A.) Monogr. 134 (1973). 141 C. Turrell, Infrared and Raman spectra of crystals (Academic Press, London. 1972). 151 A.F. Rogers, Proc. Am. Acad. Arts&i. 61 (1926) 161. [6] L.L. Boyle, Acta Cryst. A27 (1971) 76. 171 S. Bhagavantam and T. Venkatarayudu, Theory of groups and its application to physical problems, 3rd Ed. (Andhra University, Waltair, 1962). [ 81 W.G. Fateley, F.R. Dollish, N.T. McDevitt and F.F. Bentley, Infrared and Raman selection NkS for molecular and lattice ~&rations: ?he correlation method (Wiley-Interscience. New York, 1972). 191 D.M. Adams and DC. Newron. J. Chem. Sot. (A) 17 (1970) 2822. [ 101 D.M. Adamsad D.C. Newton, Tables for factor-group and point-group analysis (Beckman-RHC Ltd., Croydon, England, 1970). [ 111 J.D.H. Donnay and H.M. Ondik,gen. eds., Crystal data, determinative tables. Vol. 1: Organic compounds, 3rd Ed. [NSRDS-JCpDS. Nat. Bur. Stand. (USA), 19721. [ 12) C Brecher. E. Krikorian. J. Blanc and R.S. Halford. I. Chem. Phys. 35 (1961) 109. [ 131 J.B. Bates, DE. San& and W.H. Smith, J. Chem. Phys. 51 (1969) 105. [ 141 J.B. Bates, J. Chcm. Phys. 58 (1973) 4236.
TABLESOFORlENTEDSlTESYMMETRlF3INSPACEGROUPS J.D.H. DONNAY Dejbrtement
de G.!ologie. Universithde
Monfrtkl,
Mont&~/,P.Q., Caruda
and
c. TURRELL Deprtement de C’hitnie, Univetsit.! de Montr&l. and Department of Chemistry, McGill University, Montreal,
P.Q., cclnadp
Received 1 April 1974 Revisedmanuscriptreceived29 July 1974 Double+ntry tables are presented in which the columns correspondto oriented site symmetriesand the rows to the space groupswhose factor groupsare isomorphic with a given crystallographicpoint group. The site SYtUmettkS are representedby oriented Hermann-Mauguinpoint-groupsymbols. Ekh entry shows the multiplicity of the pointpositions that have the required site symmetq and, far each point-position, the Wyckoff letter and th:: fracticnal aordinates of one site. An additional table gives the supergroupsof every crystallographicpoint gmup, thus indicating the tables that are to be consulted Spectroscopicapplications of the tables are illustratedby two examples. In the ikt, the tables are employed to facilitate factor-groupanalyses when complete crystallographic data are available. In the second example, structuralinformation is obtained from vibrationalspectraby systematic use of the tables.
I. Introduction The following tabulations are offered as companion tables to those listingcoprime multiplicities of spacegroup point-positions [ 11. Like their predecessors,the present tables have been compiled with the needs of the spectroscopist in mind. They provide the answer to the following question: “In which space groups does a point-position of known site symmetry occur?” At the same time they also give the multiplicity of the point-position, its Wyckoff letter and the fractional coordinates of one of its sites. In single-crystalspectroscopy some crystallographic data usually are availableto the spectroscopist. They are the point group of the crystal, its Laue classor its crystal system. The tables are based on these fundamental subdivisions.The cubic system requires three tables: the first for 0, - 4lm 2 2/m. the second for T,,-a3m and O-432, and the third one for the two point groups of Laue class Th - 2/m 3. The hexagonal system fits into two tables: one for the seven point groups in which the lattice must be hexagonal
(table IV) and the other for the five trigonal point groups where the lattice may be hexagonal or rhombohedral (table V); each of these two tables covers two Laue classes. In the tetragonal system, the high-symmetry point groups are treated individually(tables v1 to IX) while the three point groups of Laue classC4h-4/m are presented in table X. In the low-symmetry systems, the orthorhombic has three tables (Xl, XII and XDD; the monoclinic has one (table XIV) and so has the triclinic (table XV). This arrangementhas no special significance;it was dictated by space considerations and reasons of convenience.
2. The tabular presentations In table VI, which is entitled “Tetragonal D4h - 4/m 2/m 2/m”, the rows correspond to the
twenty space groups whose factor groups are isomorphic with point group D4h-4/m 2/m 2/m; the space groups are listed along the left margin.The number
2
J.D.H. hmayand
G. Tunell. Tablesof oriknred sire symmemks in space groups
under Ddh is the superscript in the Schoeties
symbol; it is followed by the Hermann-Mauguin symbol. ‘Ihe column headingsrefer ,to the site symmetries available in the point-positions of the various space groups. lhese point-group symmetries are also given in both notations, and the oriented Hermann-Mauguin short symbols are used here. Consider the column headed Ca. The site symmetry C, - 2/m may be oriented in three different ways, according as the 2-fold axis is parallel to the primary direction Z (2/m..), to one of the secondary directions X or Y (.2/m.) or to one of the tertiary directions (..2/m). In the oriented symbols the dummy dots serve to situate the 2/m direction within the tetragonal symmetry of the crystal: they stand for the “unused” directions. Site symmetry D2-222 can likewise be oriented in two ways, and the Hermann-Mauguin symbol can be “written tetragonally” either 222. or 2.22, according as two of the three a-fold axes lie along the secondary or the tertiary directions. The first 2 in either symbol refers to the primary direction, Z, of the tetragonal system. At the intersection of a given column with one of the appropriate rows, the entry gives the following information on one or several point-positions that have the required site symmetry: Multiplicity of the pointposition(s), Wyckoff letter(s), coordinates of one site from each point-position. Thz choice of origi.i is thus immediately apparent, as the coordinate triplet 000 will be found under the site symmetry of the origin. The Wyckoff letters are given to facilitate crossreference to other tables, such as Infernafional Tables for X-my Crystallography 121 and Space Croupsnnd Lotrice C’omp&es 131. The entries in the tables ale read as follows. In table VI, for example, consider the TOW that corresponds to space group 0:: - P4*/mcm. Some of the entries give only one point-position, like 4e (042) under D2- 222., 8n (~0) under C, - m.. and 16~ (x_~z)under Ct - 1. Dividing a point-position multiplicity into the order (16) of point group 4/m 2/m 2/m gives the index of the site-symmetry subgroup. For the above site symmetries the subgroup indices are as follows: for D2, 1614 = 4; for C,, 16/S = 2; forC1, 16/16=I.Theentry4~underC2,-m.2m gives two point-positions, 4i and 43, for which the listed site coordinates are xx0 and xx4, respectively.
‘Ilte fust entry in f&, -P4/mmm is lad; it is followed by four triplets of coordinates, for positions la, 1b, lc and Id, respectively. This compact presentation can be used whenever there is no gap in the alphabetical sequence. If there is, the alphabetical order is reversed, as in 2db and 2cu in space group D$ - P42/mcm. Occasionally three point positions have Wyckoff letters that cannot be written in a gapless alphabetical sequence. In-such cases the letter precedes the site coorciinates in the entry. In space group D$, -P&2 (table IV), under site symmetry a.., the entry lists three point-positions, b (003], d (483) and f (843), under their common multiplicity 2. The site coordinates are fractions of unit-cell* edges, expressed as numbers of halves, fourths, eighths or twelfths of these edges.The legendsto the tables give thz necessary information in each case. They also explain a few abbreviations that had to be used for the sake of compactness: Q = 10. /3= 11, X = 4 fx, and the like.
3. Subgroups and supergroups of crystal point groups Since a given site symmetry can only occur among the space groups whose factor group is a supergroup of that site symmetry, it is necessary to find which point groups are supergroups of a given point group. A special table is provided for this purpose (table XVI)**. An illuminating example is provided by site symmetry Da -2/m 2/m 2/m. Locate this symmetry l
Althoughmodem crystrdlograpbicwge tends to substitute “cell” for “‘onitcell”, the term “unit cell” will be retained
here on purpose, to mean the parallelepipedalcell (whether primitiveor multiple) used by crystallographersand to distinguishit from the spectroscopists’primitive cell, or the Wlgnn-Seitz “cell” (41. The latter refersto the smallestpolyhedron containingone lattice point and possessingfull lattice symmetry at thal point. The content of the Wignn-Seitz “cell”, crystallographicallyspeaking, is the “translationrepeat” (French: morif. German: B&). ** Simibu tables atreadyeti in the tilerature.Rogers’ table [S] appearedin 1926. C. Hermann’sdiagramcan be found in the fnst edition oflnferMfionuI Tables (1935) and has been reproducedin subsequenteditions [ 21. More recently (1971) Boyle [6] publisheda table for spect~ scophts. The presenttable YM aims at giving more :klformation, in a more easily afcwibls form.
3.D. H. Donno~ and G. lheU.
Tables of oriented sire symmetries in spcrcegmups
symbol among the 32 point-group symbols listed along the diagonal(table XVI). The row to the left of the symbol 2/m 2/m2/mcontains four entries, which refer to the desired supergroupsabove and at the same time indicate how many times Da occurs in symmetrically nonequivalent orientations in the supergroup: twicein D4h-41m 21m21m,once in D6h-6/m 21m 2/m, once in Th - 2/m 5 and twice in Oh - 4/m 3 2/m. The group 2/m 2/m2/mis, of course, a trivial supergroup of itself and must also be considered; this fact is recalledby the I X in front of, and the I behind 2/m 2/m2/m.Thetables to be consulted are, therefore, tables I, 111,IV, VI and XI.
Note how effectively the Hermann-Mauguin notation showsthe various nonequivalentorientations that the site-symmetry point group can assumewith respect to the coordinate axes (taken along the edges of the smallestconventional crystallographicunit cell). In this respect the Hermann-Mauguinsymbols, especially in their oriented form, are obviously superior to the Schoenfliessymbols. Table XVI, of course, can be used to read off the subgroupsof any point group; they appear in the column below the givenpoint-group symbol. For the above example, 2/m2/m 2/m,each of the entries in the column refers to a subgroup(to the right) and givesthe number of different orientations in which it can occur.
4. The scaffold of symmetry elements of a space WuP At a pinch, if drawingsof the symmetry elements of the space groups are not available,the present tables can be used as a help to reconstruct the space-group diagrams,though it must be admitted such a task will not alwaysbe straightforward.In all cases,however, the tables unambiguouslyindicate the choice of the origin.
5. Spectroscopic applications The application of these tables to the vibrational spectroscopy of crystals can be made in variousways dependingon the amount of crystallographicdata availablefor the system being studied. If an X-ray
3
analysishas been made which includes the detennination of the space group of the crystal and the positions of all atoms in the cell (that is, a complete crystal strutture), infrared and Raman selection rule-s for the vibrational fundamentalscan be determined by the factorgroup method in any of its several forms [4). The traditional analysisdue to Bhagavantarnand Venkatarayudu [7] involvesthe determination Gf the effect of each symmetry operation of the factor group on the atoms contained in the primitive cell of the crystal. This geometricalapproach, which requires the visualizationof the appropriate symmetry operations, is greatly facilitated by reference to the diagramsin the fIItemntio?ru~Tables [2]. More recently, however, two methods have been developed which allow the factor-groupanalysisto be carried out without direct use of the geometry of the primitive cell. Both the correlation method [8] and the tables * developedby Adamsand Newton [9, 101 employ the Wyckoff letters to identify the various point-positions(sets of equivalent sites) in the crystal space group. The point-position and its multiplicity must be known for every set of symmetry-related
atoms (and for the center of gravity of a molecule or complex ion) before these methods can be applied. This information, which is givenin detail in the fnrernutiond Tables [2] ,is presented here (tables I-XV) in a more convenient and compact form. Ihe tabular form allowscomparison to’be made of the various availablepoint-positionswithin a givencrystal class,a feature that is especiallyimporiant when the space group is unknown. As illustrationsof the use of the tables, consider the molecular compounds urea 0C(NH2)2 and thiourea SC(NI-LJ2.I”neircrystals belong to space groups DL-P42,m, with 2 molecules per unit ceil,
* Thefollowingstatement is made in ref. 1101 @. 1): “FadorgroupanalysisaIways dealswith the smallestunit ccIi, Le.. the rhombohedralcell for the hexagonal system. the primitive cell for other systems.” Note that the rhombohedrai cell is a primitivecell.Notealsothat only a crystai that has a rhombohedrallattice can be desxiid by a primitive rhombohedralunit cell. Any ttigonal or hexagonalspace group whose Hermann-Mauguin symbol begins with a P has a primitiveunit cell that is not a rhombohedmnoc B rhombohcdraiunit celt that is not primitk Cp. foatnote on p. 2.
2e 000
, , .
Fd3m
Fdlc
lmlm
Ield
7
8
9
10
hm Pbm
Film
1
2
‘d
CUBIC
.,.
. ..
4ad OW 444 222 666
5
’
lab OW 444
,;
‘d
+
‘3,
. ..
16~ xxx
. . . . 4e x)oL
4
’
. . .
. . ,
. . .
;
222 666
4bc
.. .
. . .
.. .
xxx
130
. . .
. . .
. . .
161 XX=
. . .
,..
lie 222
161 xxx
. . . Bc 222
,. ,
,:.
. ..
., .
86 xxx . . .
. ..
. . .
6d b44
. . . 6cd 420
led 044 400 ,..
. ..
. . .
. . .
,z
rJ’
a..
I*
:
8.N
. ..
: .a
‘4
cell
2.418 x00 x44
6f6 x00 x44
$ N
c2v
. . .
. . .
D2
X
...
. . .
.. .
12.l 420
. . . 4Eh xx1
121 XII
c.
4 +X,
. . .
. . .
.. .
,..
..,
,..
24~ 200
..a
.. .
N
n
. . .
bb 044
;-;t--; 2 N
32~ xxx
. . .
6411 . . . xxx
. . .
6LX xxx
edpc.
16~ 000
. . .
of
. . .
16f . . . XXX 16b 111
32~ 333
. . . 32b 111
‘4
eighths
. . .
Bc 222
. . .
. . .
32~ xxx
.. . ... ...
D2d D4
In
. . .
. . .
. . . 16cd 111 555 lbn... 000
. . .
xxx
. ..
12h x40
?. -
..,
. . .
. . .
. . .
. . .
..a
. . .
..a
. . .
_..
. . .
XYZ
XYZ
12~ rOil
.. .
. . .
.. .
2
l
.,.
IZh X40
. .. .
.
. .. .
24X x04
..,.
. . . .
2 +X.
. . .
.
. ..
.. .
. . . 48~
a..
12fh x00 x04 X40 . . . _...
6b 044
. . .
. . .
..-
74h oxx
. . .
. . .
48ht oxx 4X% . . .
. . .
. . .
121j OX% 4xX . . .
.._~
Th 1
PE3
5
CUBlC
. . .
i
-
26k XXL
OYZ
961
. . .
4bc 222 666
. . .
,”
‘3L
24h 0x1
. ..
481 ryz
4811 XYZ
.
.
:
%h
. .._
‘2”
6d OL4
.. .
D2
...
.. . -
‘2h
. ..
12tg x00 x40
24h XYL _
XYZ
‘1
. . . 24L
‘2
OYZ 4YI
‘II
12Jk
96h xyr
LBg IXY
. . . LBf x02
961
192h xyz
1921 xyz
192) XYZ
MY2
. . .
966 1xY
96) 1rY
...
IYZ
1921
bl3L ryz
ry?.
60! 2xX
48k xx=
. . . 961 xoil
.
96h 2u
2011 24ij x04 2xX ZXX . . . . . .
24%
. . . . 24) Let
240 x04
. . .
968 . . XX.?
.,
96k xxz
. .
..
,..
24m IXL
8$ 3cd 6eh xxx044rOO 400 x04 X40 x44 Be . . xxx
”
‘3
23
OYZ
. . . 48J
. . .
... .. .
3 R T-
T
24k OXY
. . . 96j oyr
12f 420
. . .
. ..ZLc... 210
. . .
24kL Oyr
4YZ . . . . . .
.. .
. ...
. . .
48f x00
. .
4Es X22
128 xlm
. . .
._.
. . .
. . . 2n 000
lab 000 444
Ii
‘h
Table 111. Th -2/m
X,
.
. . .
.
. . .
. . .
.s.
.. .
. ..
24d 022
. ..
. . .
. . .
2 -
24d 230
. . .
4Bd 200
. . .
. . . 48f roe
. . .
.
,.,
. . .
. . . . . ,,,
. . .
Y -
24J
_
cl
x:
. ..
.,.
. ..
. ..
246 022
. . .
. . .
. . .
. . . 1Zd 420
. . .
. . . 961
. ..
4 -
c2
l?c x00
. . .
. . .
. . .
24~ x00
. . .
. . .
6ef . . . x00 x44 . . . . . . 6b 044
...
. . . . . . 32P. . . . . . .,. . . . . . . . . .
. . .
.. .
. . mlcd... 044 400
4
‘3
. . .
. . .
. . .
ta
000
,*.
...
... . .. ... ... ... ...
. . .
‘3
CoI,rdlnatee
. . .
. . .
Table II. T,j - 53m % 0 - 432
:
. . .
. . . Bab . . . OM) 444 . . . .._......
Fm3c
6
Legend
Lab 8c . . . . . . 000 222 444 . . . . . . Be Bb 222 000
Fmlrn
5
000
2,,
.. .
pn,m
0
. . .
. . .
. . .
2a 000
. . .
.. .
Pm3n
3
.. .
. . .
Pn3n
2
.. .
lab 000 444 ..;
Pm3m
1
TabIa I. 4 -4/m 3 2/m
I I
I
I
P
J.D.H. Donnny and G. Smell,
Tables of oriented site symmcrnes m spncegmups
J.D.H. Donnay and G. IIureli. Tablesof oriented site symmetis
:
n
.
*
in qwe groups
I. D.H. Don~y
and G. lbrrell, Tables of oriented site Eymmetries in spx
gmups
I
z:p : :. .. . gs
85 s5 ; . : . .. .. . . ;
P””
G6 . . .. . :
sJt.20’ -
. : . .. .
. . . ...
. : : :. . : .. . .
.
:
:. .. . : .. . . u
.
PYY
d.59
2%
:
*
,
: .
. .
. . . .
. . . .
:
.
.
z!H :. : . . : .. . . .
.
: . : .
: .
.
:
i .. . . . . .. . * .
. :
r
2 ;c
:
: : , -
.
: ;
. .
: :
. . .
; :
.
-
*
.
I
I : I ; : :
Y) . ‘g
g-
3 -
; :
10
4 _
.-t
P-
.-c
IlexAmAL
.. .
. . . . . . ,.. . . .
R3m
R3C
5
6
. . .
. .,
38 002
. ..
2eb
.a. . . . . . . 60
It XOZ ,.I
.
..
XYZ
NY2
6d
XYZ
. . . . . . . . . 6c
. .. . . .
. . . . . . *.. . .. . . . . . . ,.. 18b
9b .. . .. . . . . .. . 1Bc xxdlr XYZ
. . . .. . . . . . . .
. . . .. .. . . ,, ... ...
1BZ
001
IBZ BLZ
002
. . . .., ,.. 2ac
2b IBZ
Warning: ‘i%a unit cell of a rhombohedral space group is here the tdple cell referred to hexagonal coordinates. To use the rirombohedral primitive cell, divide all the sitemultlpkitler by their common factor 3.
. . .
... ...
. ..
. . . . . . . . . . . . .,a .., . . .
be
XYZ
,.. . . . . . . . . . . . . 6d
.I.
... ...
. .. . . .
. ..
..a
. . . . ..
P3lc
161 NY2
.., . . . . . .
4
. ..
.. . 60 128 600 xyz
kYZ
.. . . . . 12j
. . . 18fR.n. x00 X06
6f x03
6Rh X00 X06
Id XPZ
.. . ... ... . ..
. . . 1Bh XRz
XYZ
. . . . . . . . . . . . 1Be . . . 18d 36f X03 600 xyz
9de 606 600
_ cl
. . . 121
c1
. . . 6h 68 121 xx3 600 xyr
. . . 6lj 110 xX6
cz
. . . . . . ,. . . . . . . . . ..
P3cl
lac . . . 001 4BZ 841 la ooz
... ...
... ...
3
1
4cd 002 482
,.. . . . . . . . . . 6e . . . bb 12~ 000 OOZ 003
3sb .. . be 000 002 006
.. .
f&
6
6k X02
. . . . .. . . . . . .
. . . 2b 000
. . . . . . .. . . . .
. ..
. ..
.
.., . . . . . . . . . 2e 003
lab .. . Zcd .. . 000 001 DLl6 4BZ
P3lm
Rb
5
_
c.
2 Zb Lef . . . . . . . . . . . . a003 oao 00; CA83 682 d843 lot 61 ... 600 rPr 60b
2
P&l
I4
'Zh _I--
. . . . . . . . . . . . ._.
1
Pi.1
3
cl1 c3
. . . Zcd .a. &h . . . 3fR 600 LB0 caz 606 486
-
D3
5”
PIlc
2
-
,.. lab . . . 2e 000 002 006
-
c3"
301 3ln P3al
PJlm
1
flflalBoHeoIuL
'3d
Table V. Hexagonal & rhombohedml
P3221
I4
2,3
c3 1
2
1
R3
P31,2
3 P3
R3
P3
51 _~ J
R32
6 7
PI212
PI121
x,12
PJZl
P312
5
4
3
2
I
D3d -
SY -
T
'2h q
'.
‘!
cl
cl
.y.
x00 X06
. . . . . . 6~
. . . . . . . . . . . . 3-f
XYZ
. . . 6L
l-i F! #..d r*
c2
. . . . . . a.. . . . . . . 3Jk XI0 x26
*
-I_-
08
P3121
except
for
. . . . . . . . . . . . 9de x00 x06
3ob(xOB)(x02). ..a .., 6c 002
XYZ
. .. . . . 1Bf
.. .
. ..
. ..
3ab 6c 000 002 006
. . . .,. . . . . . . . . . . . . 9de 1Bf 606 xyr 600
XYL
LegendI
Coi3rdInetes
In tuelfths
of
cell
EdBOB.
o-10,
6.11.
.. . . . . . . . . . . . . . . . . .,. 35 . . . . . . . . . . . . . . . .. . ..a 9b 002 *Yr
. . . . . . . . . . . . . ..
. . .
1.x . . . . .. . . . . . . . . . . . . ,.a Id 001 XYZ 4BZ B4Z . . . . . . . . . . . . . . . . . . . .. . . . . . . .., . . . . . . . . . . . . ,.. $3
.. .
.. . . . .
.. .
. . . . . . . . . . . . . . . . . . lab Zcd . . . .., ,.. . . . . . . . . . 3af bg 000 oar 600 xyr 006 48~ bob
.. . . . . . . . . . . 3ab 000 006
Same
3ell .*. 6C xal XYZ XPO . .. . . . . . . .a. . . . . . . . . . . . . . . . . . . . . . . . . 3ab . .. . .. 6c X04 =YZ X00 Same aa P3112 except for lab(riB)(xR2).
. . . . . . ,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Zcd 001 482
2gl O& LB2 842
c3i c3
. . . . . . .a. . . . . . . lef . . . 000 006 080 486 840 846 . .. . . . .._ . . . lab . . . . . . 000 006
lulc4aIlHeDnAL -
HU(ACOHAl.
Table VI. Dqh - 4/m 2/m 2/m
TETIUCONAL 122
%h 1
2
K Q&/wan
Q4/mcc
. ..
. . . .. . . . . 2ef . . . ‘La Ljk Ci x00 xx0 OL?! OLL 010 x0.4 XXI XL0 XLL . .. . . . 2ca 2db . . . Lgh . . . . . . . . . . . . . . . . . . af LL2 LLO 002 au ‘4.z 002 000
lad 000 004 440 LL4 . . .
I
PLlnbm
. . .
4
QLlnnc
. . . . . _.. . .
5
P4lmbm
. . .
6
7
QLlmnc
QLlnmm
2cd OS0 014
.. .
. . .
2gh . . . 002 LLZ
. . .
. ..
. ..
. . . 2ab .._ 000 OOL
. . . Lg 002
2ab .._ ‘d 000 LO2 OOL . . . . . . Zab .,. 000 LLO
Le 002 Lef
001 OLZ
.. . . . . . .. . . . . . . .. .
. . . 2eb 000 004
2c OLZ
. . .
. . .
2sb . . . 6e 000 002 QOL . . . . . . . .. .. .
Ic OLZ
.. .
. ..
. . .
. . .
. . . . . . Oh OLZ
.._
. . .
_..
. . .
.. .
. ..
. . .
. . .
.. .
. ..
. . .
. ..
PS/ncc
. . .
9
QL2/mnc
IO
QL2/mcm
. . . 2ef . . . . . . . . . . . . . . . 2ad . . . DO2 000 442 LLO 000 OLC . . . 2db . . . . . . . . . . . . . . . . . . .., 2ce LL? LLO 002 000
.. .
. . . Lb 000
.. .
11
QL2labc
. . . . . . . . . . . . . . . ._.
12
QL21nnm
.. .
2ab 000 001
. ..
. . .
. . .
. ..
13
QL21mbc
.. .
.. .
. . .
. . .
. . .
. . . 4b
QL2/unm
15
PL2/nme
lb
QO2Inmc
17
14lraKu
1s
s.lmcm
. . .
.. .
._.
. . . . . . 2& 000 004 .. . .. . . . .
. . .
. ..
. ..
. . .
_..
._.
.. . . . . ‘e LOO
.._
. . .
. . .
. . .
. ..
._.
. . .
. . . 4d Lc O&2 040
.. .
..*
8x7
. . .
._.
. . . ‘dc 220 22L
.__
. ..
. . .
. . .
.._
._.
. . .
. . .
Lgi . . . 002 LLZ CJLZ
.. .
. . .
. . .
.. .
. . . Bq rye
. . . L$, 002 LLr
Oe 042
. . . Lf 010
.. .
...
. ..
. . .
. . .
.-.
. . .
.. .
. . .
. . . 4s 4c OOZ 040
._.
. ..
_..
. . .
_..
. ..
. . .
. . .
4fg
. . . Gab &c OL2 040 002 xOb
. . .
. . . .. .
20
lL,lecd
.. . . . . . . . . . . .. . . . .
4ab . . . OGU 004
.-.
...
. . . Oe
CotJrdinares
8iven
In
eighths
160 XYZ
16n =yz
._.
. . .
Ei Oxr
8, xxx
. . . 8gh
. . . 2bk
.._
. . . Be oar
. . . BE n2
Bop ox2 LXL
. . .
. ..
. . . lbr . . . Bn XXL ryr
. . . . . . 8n
8k 04.
8&, x02 x42
. . . 80
xx1
rY0
...
. . . 8gf 002 0:~
Ld oc2
. . .
. . . Lef 222 666
. . .
. . . E,, Bh XXL 04.
Ld 042
‘c. . . . OLO 000 ...
. . . Bh
...
. .. . .. . ..
.. .
. . . ae . .. OO.?
.. .
8a
.. .
. . .
. . .
. . .
. . .
. . . Bb 002
. . .
K rtandr
for
4
+
. ..
x.
...
Ed 220
L6g qr
. . . 16~
Bhl Bj x02 xx0 rob
9e 222
Ibk XVL
8ij XW rOL
. . .
16~1 =y=
8kE xx2 xX6
. . . 8ef . . . Bg 002 xx2 OCZ
. ..
Bj
Bh oh
WC
=Yr
_..
WO
. . . Bi
.. .
IbL xv
na x%4
.. .
...
. . .
. ..
. ..
...
edges.
. . . 86 xx.7
.. .
.. .
cell
.. .
.. .
. ..
of
. . . 8ke ei, xoono x04 xx6
. ..
. . . . . . Oc
000
Lewd:
. ..
xx0 007. 010 xx2 WO XEO .. . 4cd _.. . . . .. . . . . . . . . . . . .. 8, .. . oxz Oar Oi; . . . . . . . . . Le . . . La . . . . . . Led . .. . . . 81 OLr 002 222 xx2 22b
. . . . . . . . . . Ic . . . 81J
ILllamd
. . .
. . .
. ..
. . .
.._
‘a 002
. . ah ag . .. Zab . . . Ld le 042 oar 000 040 x00 a0 O‘Z OOL XL0 .. . Lb . . . _.. La Lc . . . Bf .._ Id _.. bh . .. 8s 042 002 000 001 060 xx0 OLZ
19
. . . Ei Bkl 81 OLZ x02 xx2 XL2 BP xx2
lbu
rlrr
. . . Bf 042
WJ
_..
.._
. . .
. . .
XYO
.._
.. .
. . . Lb 000
. . 80
. . . Lef 220 22L
id 000
. . . 4d OL2
.. .
.._
. .. . Gf 00s
Ljm . . . x00 XLL XOL XL0 . . . 4ij xx0 %x4
. . . 2ab . . . 000 006 . . . .. . . . . . . . . . . . . . . . . .. .
. . .
.. .
Le . . 040
.. .
002
14
nyo x02 xx* XyL XCZ
. . . 8g si, 8b Bf L6k &Or x00 xx0 222 EYL X04 .. . . . . . . . .. . IbL . . . 2cd . . . Lgh . . . . . . .. . .. . . . . . . . . . . BiJ . . . Ek xx0 xX7. Eyr WJ XXL *YL
8
.__
. . . . . . . . . . 8pqBar 8r __. .._ __. .._
.._
.. .
. . . 16L xyr
.. .
lbk ryr
. . . . . . 8f no
EC 222
L6h q-2
8f “L
. . . 1bJ YiYr
. .. 8gh xx2 nb
16L lbn lba .. . . .. 16k . . . 320 . . . Bf 222 rye oxz m v. . be
. . . 8cd 021 “7% . . . :::
lbk . . . 16L .._ 16j 161 .__ ,in, x02
sxz
xx2
222
xyll
.. .
. . .
lbh 0x2
.. .
. ..
.. .
. ..
...
.-.
16d lbc 161 oar 2x1 xx2
lbf ‘68 rZ1no
ryr
. . .
37.5 xya
16~
32% ryr
021
J.D.H. mntq
10 Table VTL Du
and G. Well,
Tables of oriented site symmetnSes in space groups Table VlIL cd,, -
- 42~1 ii am2
DZd
‘4
‘2v
D2
5
%v=: c,
5
c2
4mm
C
c2
-YA
ElRAGoNN.
:Lv 1 lad
...
...
. . . 23b
2ef . . . . . . 4n 002 4w ax2 64Z 404
OQO 444 004 040 2
PZ2.z
. . .
. . .
. ..
. . .
. .. . . . . . .
6
7
8
9
P&2
Phb2
PTn2
l&2
10
IZc2
11
IS2m
12
Ir2d
. . .
. . .
. . .
lrd
2cf 000 440
2ab . . . WO 004 . . . 2eg 002 442 042
. . .
. . .
. ..
Zab a00 004
. . .
. . . 2ab 000 oo4
. ..
. . . 2ad
...
. . .
. . .
. . . hef 002 042
. ..
. ..
. ..
iii a42 a46 . . . . . . 4bc Ml0 042 2ab . . . 4d OOO 042 oob
. . . .. .
. . . 2ad 002 402 442
042 2ab . . . 2e . . . 000 042 004
%I 444 000 . . . Zcd 000 440
..,
. . .
Cab
. ..
. ..
. . .
. . .
. . .
...
r42nm
. ..
. ..
5
Phcc
.,.
2ab . . . OOL 44L
*Y*
6
P:nc
. . . 2a 002
3i xyr
7
Ph2mc
._.
a..
2cd 040 044
.. .
2cd 042 046
. . .
. . . 43i oar 442 042
. . . 4hl xx0 n4
. . . 4ef 3J a2 xyz xx6
. . . 81 . . . XOZ
.. .
. . . he oar
he 040
.. .
. . . 81 -
. . .
. ..
..,
8
9
. ..
31 vz
4f3
Yi
Xx6 xx2
xyz
. . . 83h
16J
no xx2
xyyr
Bfg . . . Ehc 002 xx0 a42 xx2
161 lyyr
8h 8fg 04X x00 xc4
v=
04. oaz
.. .
xyz
. . . 4ef . . . 4gh oar rxa our xx4 . . . 4he
10
11
12 . ..
. . . Bc oar
8d x21
2ab 002 442
. . . Cd xx*
4c 8e 042 xyr
. . . 2a aor
. . . 4c IXL
Lb 042
8d ryr
. . . 8n xyz
. . .
. . .
cob
Warning: Space gxoups Dz2
4
. . . 8e
. . .
. . .
. . . . .. . . .
,..
2ab 002 442
. ..4ef4d . . . 88 xor xxz XYZ x42 . . . ?b . . . 4c . . . 8d 052 Xx7. ryr
047.
P42cm
. . . 8f
. . . 4jk XOZ x42
Iab...?c 002 44r . . . 2a oar
3
X06 . . . 4c 4d . . . xx2 002
. . .
P&xln
Wbm
. ..
. . . Led OOr 042 .. . . . .
43J xat x46 x42
Rtjp:i-
hurl
2
. . .
.. .
. . . Lkm 002 442 042
4if . . . 60 x00 XYZ x44 x04 X60
. . .
. . . 4da 040 002
COO
4m 042
and $“I2 refer to a double ~4.
16j
16~ ayr
cl
P42bc
14csn
Ilcm
IClmd
Ibled
. ..
.. .
. . .
2ec aar kdr 042 . . . . . .
2a oar
. . . 4b 042
. ..
La 001
. ..
. ..
.,.
.. .
. . . 4c 042
Bd xyr
. ..
...
. . . Lb 042
6c xyz
__.
4de x0: xc2
. . .
.. .
..
.
. . . 4b 042
. .
. ..
. . ._ 4ab
8d xoz
8c XXL
. . . 8c
002 04. . ..
. . .
xx2
16e
16-j XYZ
. . . 8b 0x2
. ..
.. .
...
. . . 8a oar
...
8c xyz
XYZ
. . . La 001 .. .
8f XYL
. ..
1bc xy7. 16b xyr
J.D.H. DOIIM~ and G. 7Uncrell.Tables of orier~ed sire symmetries in space group Table X. c& -4/m. S4 - 1% C4 - 4
Table UC.04 - 422
mRAcMuL C
P.422
lad Zsh 000 aor OOL ‘4L
Zef LOO 501.
. . .
&i osz
&lo L.!k x00 xx0 x84 UL x0: X&O
160 u:
/
2c wt
. . .
2ab 000 005
Ld Oor
. . .
Lef 8R xx0 xyr XXL
~4122
. . .
_..
. . .
. . .
. ..4ab4c 0x0 6x0
a3
Bd xyz
ba no
8b xyz
b
Pl12,2
i
Pi222
. . .
. . .
. . .
040 O&L . . . . . . Zob 000 006
5
PL2Z12
7
P&,22
Same
B
P43212
Same
9
lb22
?ab 000 004
0
2
. . .
14 12?
. . .
. ..
. . .
._.
. . .
2ad 000 L40
Zef 002 b&2
Lgl 002 SLr 042
ojm Ino x00 a2 al!.0 xx6 x0:.
as
P&,22
55
Pbl212.
Le 002
. . .
X40 . . .
81 Bhi OLZ x00 x&i
Lab 8c 000 002 OOL
bef xx0 xx4
8Je xX2 xx0
16k xyr
8f
8de
16~
x21
xx0 XX0
xyr
of
cell
edges.
‘s
Cl
CL
2
I
1
. . .
2gi-1 2ef
Ojk
li
002 142
040 0.44
xyo Iy4
WL
. . .
2ad OM 460 040 040 . . .
Oj ry0
Ogi OOr 442 war.
. . . 0k
. . .
If oar
Ide 220 224
8g xyz
. . .
. . .
. . .
4ef a42 Ooz
4cd 222 226
8g xyz
4e elk
4c 040
Bh xya
86 042
8f 222
16t XYZ
.__
_._
. . .
8e 002
8cd 021 025
16E qz
. . .
. . .
. . .
2e,3 OOZ
. . .
140 644 .._
2ef 002 462
. . .
Zab 2c 000 042 00.5
4
P02/Il
. . .
2ab 000 00.4
5
I4/=.
6
IlllIe
2ab 4d 000 042 004 . . . lab 000 004
. . .
BE =Y=
rgr
G Pz
. . .
lad 000 004 440 460
2
C2
lad
PO/n
=4 1
‘2h
12lem
3
C4 1 Legend: Coordinarcs La clghths X stsnda for 4 + II_
P021rn
8p xyz
‘3~ xyz
‘4
T
for bc(xx5
except
Lc Od O&o 062
. .. ...
4cd 002 OLZ
‘4
000 004
8P xyr
P&Z12
.. .
Oh oh 1 PPll
‘&h 4/m
IS
4 Pb
. . .
2ad a00 000 042 W6
4h =v
442 042
. . .
. ..
. . .
bef
..,
OQ 042
8g =Yz
. . .
. ..I&
. . .
. . .
2c a42
._.
. . .
. . .
. . .
. . . 4.9
2x oar L&r 042 . . .
. . . 46 xYr
2
Pdl
,..
002 Chr . . . . . .
3
P42
. . .
. . .
. . .
. . .
. . .
4
PL3
. . .
. ..
_..
. . .
. . .
4d ryr
ryz
. . . 4. w=
5
Wmlng: Space
groups
04 9S’o,
@,
IL
S$ and c’4 refer to a double cell.
. . .
. . .
2a 002
. . .
. . .
Cb o&r
. . . EC xyr
. ..
. ,. . . . . ., ,.. . . .
. ..
. . .
_..
~hna
Rna
Pee.
Pbam
pccn
pbca
6
7
a
9
10
11
~~
'2h
cm
.. .
044 ... ...
.. .
bd x22
.. .
. . . 4c oy2
46 212
4dc 202 242
Lob 000 040
Ef xyr
4ab 8c 000 xyr 004 . . . . . . ai XYZ
. . . 4c 2nr
.._
. . .
. ..,. 1..*.P. . . ,.
. . .
.. .
.. .
._.
. 413
. . .
2nd . . . .,. ... _. . ..,. . ,___ .~
.. .
Ahm2
.. .
...
,_A ^
. . . 4nb 80 oon ry* LO'J Icf . . . 8h
Abo2 17 --TV‘-Pilrl’l
AmD2
Amm2 14
ccc2
Cmc21 12
13
-2
Pnn2
Prl.aZl
Pba2
11
10
9
8
-21
hlcz
b
7
PcaZ1
Pm&?
Pcc2
5
16
. . . bgh . . . ,.. 4e.f . . . 81 002 wr XYD 042 XY4
.. .
XYr
8m
. . . Er XYZ
OIj 4kL 4ef oyo oar 220 oy4 04z 224
4mp 002 44r 04r 402
81 . . . OYO oy4 . . . . . . XYZ 4sh
x04
XOa
4Rh
LkC oy2 4y2
. . .
2nd . . . OOD 004 040 044 . . . .. .
...
4lj
x02 x42
. . . bcf x00 x40
.a,
. ..
..-
48 rye
4c x20
. . .
...
. . . 111 . . . OYZ
.. .
xor x42 4ij
. . .
. ..
. . . . . . . .. . .._ . . . ._. . . . . . . . . . . . . 4d xy2
. . .
. . .
...
2yr . . . 4k
.. .
.,.
15
. . .
.
. ..
2nd . . . 000 400 ‘LO 060 . ... .. .
...
004 040 2nd . . . 000
...
2ad 000 440 060 400
4
3
Pmczl
1
2
-2 R2
Z2”
. . . 4cd 4ab 80 222 000 xyr 262 owl
. . .
. ,, .. .
,..
. .. 2ef 242 004 . . . 202
.. .
.,.
ORllM-
RHOMBIC
. ..
. . .
000
2nd
400 404
. . .
4k: 4ef Em 007. 22.7 ryr O&r 6b6
XYZ
eu
tii L1
. . . . . . -.. . . .
=2 ”
C2
C. .y-;
2eb 002 202 .. .
2ab 002 02r ...
..
2ab ooz 022
be Bf Oyr XYL
. . . Lc XYL
... ...
Be xyr . . . Bb . . rl%
. . . 4b 1yr .. . . . . ba -... Hn ..T
. . . 4e 00s
Ed XYZ
8f xyr
Bd XYZ
Bb Oyr rys
. . . . 40
Id ror
...
4a XYZ
..,
. ..
4b xyr . . . 4e =Yr
21 oyz
. . . 4c XYL
. . . 40 XYZ
bd xyr
. . .
. ..
. ..
. ..
. . . 2c 1yr
4e XYZ
4c xyr
4i x7=
-
Cl
411~ 001 022 112 2ab . . . 4c 4dc xoz oyz 002 202 2YZ . . . 4ab4c .. . 002 XlZ 201
...
2eb 4c Ooa 112 02r . . . ..
.
Pab 002 022 . . . .,.
.. .
._.
. . .
..,
.. .
lad . . . 2ef Z6h aor oyr 001 x2; 2yr u2r ZOZ 22r I.. . . . ..; 2ab oyr 2YZ .._ 2nd . . . . . . 002 021 201 222
“d
c2v
Table XII. C2, - mm2
.. .
. .. 12 mnm ..x*-LsL,u> .CI-L .
I
I
...
. ..
. ..
. ..
...
Peb
002 402 042 462
Pm
5
. ..
...
. . .
Rlan
4
. . .
OOL 040
. . .
Pccm
3
---
. . . . . . ..a . . . 4U" OWL 4yr oyr nor XYO 4yr x4r XY4
%
L44 . . . . . . . . . . . . 2.9~ . . . . . . . .. . . . . . . . . . 6g.b 411 x00 oyo x04 4~0 400
040
2mp 2qr
no0 oyo ooz x04 oy4 04; x40 fbyo 402 x44 4y4 442
lmh Zil
000 400 004 404 040 040
c2v
. . .
Pnnn
2
222 iii R
--
.._
II
I
D2h 1
'2h
Table Xl. Dm - 2/m 2/m 2/m
6
5
4
3
2
1
c 2h
c2/c
P~,/c
P2/C
c2/m
pZIIm
2’a P2lrn
MWLINIC
2c xl2
1
‘1
Zcf 0yl 2y1
. . . be Oyl
4~ IYL
Lad Bf onn XY7. 020 110 112
2ad 40 nor) xy, 200 002 202
2nd 000 220 n20 200
4gh 4cf BJ oY0 iin XYZ OY2 112
. . . 2nd 41 000 XYI 200 002 202
... ... ...
. ..
i
‘1
2mn Zfi . . . 40 x07. 3Yo XYZ x2x 2yo OY2 2Y2
2ed 4‘ 000 xoz 020 002 022 ... . ..
lah 000 020 no2 200 220 022 202 222 ...
'2h ',I ‘2 21~ m 2
Table XlV. Monoclinic
P
&
. . . ... ...
. . .
uanm
cccn,
as&4
Ccc,,
F_
Fddd
m
lbam
2bcm
m
19
M
21
22
23
24
25
26
27
2a
wunlng:
L
. . . ,. . . . . . . . . . .
f%,ca
oar
4kL
Bh oyo
...
81 oar
8ab 000 004
8f 222
. ..
. ..
. ..
. . . 40 001
.. .
.
.
. . .
. . .
...
...
...
...
. ..
ac 022
. ..
,.
L:
..*.
.. .
. ..
. ..
. . . 4c x2r
.. .
. . .
2YL x22
._..
. . . 8f Oyr
...
. ..
.. .
.
&‘a” .
.
.. . ...
...
. ..
.
.
..,
Ed, oyz
. .. 81 i2r
.,.
...
.
.
.
_ .
ah1 x00 x04
88 x02
.. .
. . . a, x00
.._
BL rvt
Bpq xyo xv4
an
. . .
.. .
8~ lY2
. . . Bf x00
.a.
8d 8a 2YO 02x
16k xyr
160 XYZ
Bob 16f 000 XYL 222 . . . LbJ RYL
88 8111 Bc Lly2 002 222 OlZ
222
32h ryr
. . . 32p XYZ
. . . . . . . . . ak
Bf 102
160 xyz
L6m XYZ
XYL
Ihr
16h XYZ
ad lbl: 220 xyz
ad XYL
ac 000 r/7. Cl”4
16gl6cd 001 111 555
. . . Be x02
Ej xyo
XYO
16,~ 16f x00 oyo
p
BRII Bed 161 OOZ 202 xyx 22r 022
16k 16J 2Y2 222
Bf oyo
_.
4nb ad on0 XY?. 040
alk . . . 002 067. 221
Bjk at 2yo 202 2Y4
Bh oy2
.
E
. . . ac 210
. . . 8m 221
ac 2Y2
.
ODD XYZ 004
. . . 4ob 000
moo
Ed x00
.
. . . 4Db
.
xy2
. ..
. ..
.
l3e . . . . . .
... .. . 8m noi
.
ag
. . .
161~ 16n 160 I6t cyr xoz XYO x22
. ..
. . . aI oyz
...
ac 220
...
. . . 4cd . . . OW 400
...
,..
8d 202
. ..
Bn x2z
.. .
4ef Bn 80 220 OYZ xor 224
. . . 4ef . . . 000 040 220 260 4ef . . . am 220 oyz 224
_..
OYZ
.
. . . 4c ..* OY2
.
. . .
.
. ..
. ..
. ..
.
-IT.
. . . . . . Bt . . . . . .
. . . 4ab 4cd 000 222 Eoz 226
. ..
.
Lab . . . 000 OLD
. ..
. . .
.,
. . .
.
J
22
Imoz
Ibn2
lImn2
. . . be 002
2ab . . 001 02z . . . 4nb oar 022
9
a
7
1
%
222
12,22Z1
1222
P222
2nd 000 200 ml2 020 .,.
lad 000 002 111 113
2ad 000 020 202 002
c222
4a no1
4cf x00 102
ajc xl1 x00
4cf x00 x02
. . . 4a x00
c2221
. ..
4c XYZ
4c Olr
48h 4Lj OYO 00; 2Yfl o2r
Bd ry=
8k xyt
16k ryr
at xyr
. . . BE rys
lab Lc OO?! xyz 02r . . . 411 aY=
. ..
tlif Q,h 1y1 00s OYO Ilr
4b ly0
c1
8c xyr
241 4u oar xyr 2Or 02r 222
N :_
48h 4lk OYO 00. oy2 022 1lZ
4b fly1
_. .
.. .
P212121
,
2lC 2mp x00 oyo x02 oy2 x20 2yo ~22 2Y2
P21212
lab 000 200 020 002 220 202 022 222 ...
:” N
--
c2
8c xyr
XYZ
. . . BC XYC
4d oyz
. . . 4b lyr
.. .
4c xor
2ab 2cd x00 oy1 x20 2y1 ... . .. . ..
P222,
P222
A N
D2
Table XIII. D7 - 222
L
21
20
oar
Unit cella (I at am A, B, Cor I centered are double cells, those that are F centered arc quadruple cells.
. . .
2nd 4aC 4gh 411 . . . 000 x00 oyo 001 044 x40 oy4 402 440 404 . . . . . . . . . . . . 4ab 002 402
. ..
.
lab 4cd 200 Km 204 004
. . . 4eb 000 004
46 021
.
.. .
.
Lab 000 ‘00 . . . . ..
. . .
. . . 4ab 002 042
. . . .a. . . .
4ab 8s 000 x00 004 . ..
. . .
,..
-
oyo
. . .
ov4 04.
4fJ
400 x04 404 004 . . . ..,
x00
2ad 4Rh
090
w2
LC
. . .
.. .
18
. . .
.. .
. . .
. . .
...
...
(Lacln
~.
17
. . .
.
Fnma
.. .
.
.. .
.
16
._.
. ..
222 261
LIlY
F&x
. ..
.
,,.,,
. ..
.
15
.. .
. ..
. .
Pbcn
.
14
.
.
.:.
. . . .. . . . . . . . . . . .. . . . .
CZ
P21
..
I
1 PI
000 001 010 I00 210 io1 011 111
Ish
la XYl
21 ryr
5I 51
. ...
Lo8end: CoUrdinatcm In halvoa of cell cdgoa.
Cl 1
=i
i 1 pi
TRlCLlHlC
. ..
. ..
ba XYZ
. . . 4b w=
2eb . OYO OY2
. 4c xyr
OY2 2YO 2Y2 . . . . . . . . . 20 XYZ
.. .
. . .
. . . lad . . PC OYO XYZ
. ..
..
.. .
. . . 2a X02
2e XYZ
... ... ... ,..
Inb . . . XOZ x2z
2c XYZ
. ...
..
_.._ _..-_-
Table XV. Tdclinic
3
2
2 PZ
1
CC
Cm
PC
~m
m
C_ L
*
3
2
i
9
c-
I.D.H. Donnn~ and G. lhrell, Tables of oriented sire symmetries in spucegroups
14
Table XVI. Subgroups and supergroups of crystal point groups
lx.
.
.
..Lx..lx.
.
.
.
.
.
.
.
Ix .
.
.
.
.
.
.
.
.
.
.
.
The ale of mutual exclusion applies Lothe fust point group in each Laue class (see the fm Schoenflies symbol below each horizontal dashed line), ptis 0 ~432. l * In these tables rbombohedml space groups are described in hexagonal coordinates (triple unit cell); the laltice type is then called “hexagonal R”. B desigr&ion that is equivaint to “rhombohedral P’. In the monoclinic system iho symmetry direction is chosen as the b axis. l
J.D.H.
Donnay and G. Thell,
Tables of oriemed sire symmetries
in space gmupo
15
Notes ro table XVI (1) Synonymy of Schoenflies symbols: Ci= S2= /I, Clh =sl q 12,D2 = V= p, Du = Vh, Du= vd, CJi’&j =fj.s4 =f4, C3h = S3 = 16, in which I stands for a rotatory inversionaxis (German:inversion)and S for a rotatory reJ7ecrionaxis (German: Spiegelung).(In = ii in Hermann-Mauguinnotation). (2) Dashedlines separate the Laue classes. (3) This table presents only the 32 crystallographic point groups (also known as “ctystrd classes” or, morepncirety,“geametic crystalclasses”)so that no provisionis made for the variouspossible settings in which a given crystal may be oriented. Speciftcally. Ch may have to be written mm2.2mm or m2m, if the uniteelI edgs ate chosen according to the metric convention c < a C 6. LikewiseDw has two possible orientationsif the smallest tetragonalunit cell (P or I) is used; they are written&m and int2. SimilarlyD3h can be.written 52m and 6m2. Furthenore, three trigonal point groupsalso need alternate_orientatiatts, though only in the case of a hexagonal lattice; they are: 32 (321 or 312). 3m (3ml or 3lm). ? 2/m (3 2/m 1 or 3 1 2/m). (4) Occasionallysome subgroupof a factor g~oup does not occur as site symmetry in any of the isomorphic space groupr_Such subgroupsshould be disregardedand have not been entered in the table. Thus omitted were:3 in 6/m 2/m 2/m. 2/m in 4/m 3 2/m and-i in 4/m !i Z/m. 2/m 2 and 6/m 2/m Z/m.
and D$ - Prima*,, with 4 molecules per unit cell, respectively [ll] . The letter Pin the HermannMauguin symbol immediately indicates that the crystallographic unit cell is primitive and, therefore, can he used without subdivision in the factor-group analysis. This explicit slatement of the lattice type is another advantage of the Hermann-Mauguin notation over that of Schoenflies. The latter, however, provides a direct relation between the space group and the crystal class(and hence the factor group and its isomorphic point group); simply dropping the superscript of the Schoenflies symbol yields the symbol of the point group to be used in applications of the factor-group method. Thus, Du -&II and 3% -2/m 2/m 2/m are, respectively, the point groups that describe the crystal classes of urea and thiourea. The Herrnann-Mauguin symbols do not provide this information quite as readily**. It is known that both urea and thiourea have symmetry Ch- mm2 in the free state. It follows that either molecule must occupy a site of symmetry
Two orientations haw been used in the literatureto descni thiourea:Pnma and Pbnm. The latter is brought onto the former by transfotmationmatrix 010(001/1M). Pnma is the conventionalorientation of D$, in which the unitceUedgesarelabelledsoartoobeyc
I in thecrystalline state, as indicated in the column below the Cb-nvn2 entry in table XVI. In the case of urea the crystal point group is Dzd- 32m. Scanning the row to the right of the entry Du- d2m, shows that only sites of symmetries S4 - 4, C, - mm2 and D2 -222 have multiplicity 2. Hence,_C, is the site group of urea. Since the Du- 42m row points to tabibleVII, one can turn to that table and locate, in &, the site orientation (2.mm) for a point-position of multiplicity 2 with Wyckoffs letter c and site coordinates (042). The table thus provides the information necessary to complete the factor-group calculation, for example. hy the tabular method of Adams and Newton [9, IO]. Similar arguments for thiourea follow from table XVI, where possible sites of multiplicity 4 in D, are C,, C2 and Ci. The first two of these point-groups are subgroups of C, and. therefore, are acceptable. However, reference to table Xl (indicated by the Dz row in table XVI) reveals that the space group DE has no sites of symmetry C2. Therefore, the correct site group is C,, its orientation is .m. and the pointposition is 4c with coordinates (x2z). From these examples it should be apparent that the tables presented here are to be used as an adjunct to the tables of Adams and Newton [IO] or the correlation method [S] .
Cb-mm2,2,-m,C2-2orC,-
6. Spectroscopic crystallogrephy In genera!, vibrational spectroscopy cannot Claim
16
1. D.H. Donnay and C. nrrell, Tables of oriented site symmerries in spacegroups
to be a direct method of determining crystal structure. Nevertheless, it is often possible to obtain a certain amount of structural information from the infrared and Raman spectra of crystalline systems. Such information may be e-specially useN where few or no diffraction data are available. The use of spectroscopy in assigning the molecules to their proper sites in the correct space group depends on the following arguments: 1. Thesite symmetry of a given molecule (or complex ion) is a subgroup of the point group of the free molecule (as in the vapor phase). 2. Certain vibrational fundarner&&, which are spectroscopically forbidden (infrared or Raman) for a free molecule, may become active for the molecule in a crystal. This observation indicates that the symmetry of the site is lower than that of the free molecule. 3. Splitting of a degenerate molecular vibration in the crystm shows that a~ least one axis, of order 3 or higher, of the free molecule has been removed by the (lower) symmetry of the site. 4. If a nondegenerate fundamental vibration becomes split in the spectrum of the crystal, the primitive cell must contain more than one molecule. The number of components of such a split fimdamental is a lower bound to the number of molecules per primitive cell. 5. In the case of dilute isotopic species in a crystal any splitting arises from the site effect, while in a neat crystal splittings can be due to site effect, to correlation effect, or to both. 6. The absence of coincidence of any infraredand Rarnan-active fundamental suggests that the factor group contains a center of symmetry*. However, the definition of coincidence depends on the magnitudes of experimental frequency errors; hence, this argument must be used with caution. 7. Infrared or Rarnan Polarization studies can often be used to determine the crystal system, as well as the molecular orientation within a monocrystalline sample. As an example of the application of the above principles, consider cyclopropane, C,H,. The mole* Note that infraredand Raman coincidenceis absent in cwtals of classO-432 even though theseayhls center of symmetry.
lack a
cule is of symmetry L& - k?nr in the vapor phase, hence, the irreducible representations of the 21 vibrational fundamentals have the structure C,& =_3_A_T?tA;+_~I+A'it2At+_3_E_'I.
Thespecies of the six infrared-active fundamentals are underlined, while those of the 10Raman-active fundamentals are underdashed. The four vibrations of species E'arecoincident in the two spectra. Polarized infrared studies of a single crystal show conclusively that it belongs to the orthorhombic system [ 121. Data on one of the El’-species fundamentals (ut4) in the 750 cm -I region show that it exhibits separate polarization properties along three mutually perpendicular axes for each component of the absorption band. Reference to table XVI indicates that the only possible crystal classes in the orthorhombic system are Da-2/m 2/m2/m,t&,-mm2 and D2 - 222. These, then are the point groups that are isomorphic with the possible factor groups of the crystal. Furthermore, as at least one nondegenerate f_mdamental (~2, for example) is split into a doublet, the primitive cell contains at least two molecules. The determination of the site group of cyclopropane was initially made by Brecher et al. [ 121, also by infrared polarization measurements. The conclusion that the site group is C,-m was confirmed by recent spectroscopic studies [ 131 of solid solutions of CJH~ in CsD6. Furthermore, as site splitting of the E’ and E” modes was observed, the plane of symmetry of the site corresponds to one of the vertical planes of the D3,, - g2rn symmetry group of the free molecule. (Compare the two correlations L+h * C, of fig. 1.) Once the site group has been determined as C,- m, the factor group D, - 222 can be eliminated, as it contains no planes of symmetry. (Table XVI shows that there are no C,-m sites in a crystal of class D2 - 222.) Of the two remaining possible classes, D* -2/m 2/m 2/mcontains a center of symmetry, while C, - mm2 does not. Comparison of the infrared data on crystalline cyclopropane with the recent Raman data [14] shows coincidence of the vibrational fundamentals to within + 1 cm-r. Thus, although accidental coincidence cannot be completely ruled out, this result and the rule of mutual exclusion [4] provide strong support for a noncentrosymmetric structure. The ensemble of spectroscopic evidence
J.D.H. Donnay and G. Turrell. Tables of oriented sire symmerries in spncegmups
II3h
c, $,~O)
17
C2v(m.-1
Fig. 1. Correlation of irreducible representations of the molecular point group (Djh - hn) of cyclopropane with possible C3 - 3 sites and the C2,, - mm2 factor group. Species of infrared active fundamentals are underlined; &man-active sgcies are underdashed. suggests a factor group that is isomorphic
with the
point group c,, even though the group Ds cannot be rigorously eliminated [ 131. Table XVI indicates that the class Cti contains two C, sites per primitive cell and that table XU should be consulted to find their orientations and the possible space grou s. There, it is seen that CL, $ ~4 ~7 c” ~1 P ~‘4 ~15 c’6 ~‘8 c20and 2uJ 2lP 2ul 2ul 2w 2lJr 2w 2U* 2w 2u G$ are the only space groups compatible with the
spyctroscopic results for cyclopropane. Thus, 218 space groups can be virtually eliminated on spectroscopic grounds alone, so that twelve possibilities remain. It has been found by Brecher et al. [12] that solid cyclopropane is denser than the liquid. Packing considerations based on reasonable molecular dimensions have been used to estimate the densities of structures belonging to the various possible space groups, and thus to eliminate structures whose densities are less than that of the liquid. All the ahove space groups other than C$,-fW2 and C&-Pmn21 are thus eliminated. Of the remaining two, CL-Anna* gives a structure denser by approximately 5%. If the space group of cyclopropane is indeed CL, reference-to table XII indicates that the pointposition is ~~(OJU)and that the molecular orientation m.. is such that the plane of symmetry of the site group is parallel to the b - c plane of the crystal and
hence parallel to the plane designated a(yz) in the CL point group. (Note, however, that this molecular orientation postulates the space group CL of the crystal to be oriented Penn?, .) The resulting correlatitn of the symmetry species is shown in fig. 1. The CL structure, which was predicted primarily from the spectroscopic data, has been discussed in detail by Brecher et al. [ 121: Recent X-ray work [ 131 on polycrystalline C& consists of a IS-line powder pattern which has been indexed on an orthorhombic cell (a = 10.0, b = 6.8, c = 5 .l A, at 85 K) so that the calculated density D, equals 0.40 or 0.81 g/cm3 for 2 or 4 molecules, respectively, in the unit cell. Choice of the higher density, which demands 4 molecules in the known unit-cell volume, and the spectroscopic requirement of site symmetry C2 -M led to the factor group Dzh - 2/m 2/m 21~. As can eassilybe seen from table XVI, this result is correct provided the unit cell is primitive. Indeed it is possible to place 4 molecules in the primitive cell for C, sites in factor group Du,. In Ch- mm?, however,the fourmoleculeswould haveto occupy a general point.position, with site symmetry Ct - 1, contraryto spectroscopic evidence*. Table XVI provides the possible space groups in &, namely those with Schoenflies superscripts 1,3,5,7, l
Footnote: see next page.
18
9,11,12,13,16.
etal. [13] who
J.D.H. Dannay ond G. ntne$
Tables of oriented site symmetries in qpncegmups
They are the ones given by Bates Rnmm “by distance
calculations”. Neither the space group nor even the lattice type could be determined from the powder pattern with certainty [ 131. It follows that the requirement of 4 moleculesper unit ceil can also be satisfied in factor group C,- mm2,for site symmetry C,- m,if the unit cell is double, either one.face
l
Acknowkdgement
eliminate r>$, -
The infrared polarization study j121 of singlecrystal C,H, provides very strong support for C,- m site sym-
metry. However, the possibility that the molecule occupies a general position (C, - 1) cannot be rigorously excluded, as pseudosymmetry could accountfor the spectxoscopic results;
Wewish to thank one of the referees for his helpful suggestions,which have been incorporated in the statement of the argumentsnumbered 3 to 5 in section 6. References [ 11J.D.H. Donnay and G. Donnay, Can. J. Spectr. 17 ..(1972) 45. Note the foIlowing errata, communicated by Professor Yosio Sakamoto (Hiroshima University). Tables 2.3 and 2.4; restore omitted heading (32) of the last column. Table 2.5, &P%l:the2incoLlshould be in col. 2. Table 2.6, first column: instead of C2 - 6, read C6 - 6; instead of C3h - 6, read C3h - d. (21 N.F.M. Henry and K. Lonsdale, eds.. International tablesfor X-ray crystallography. Vol. I: Symmetry groups, 3rd Ed.(pubIished for the international Union of Crystallography, Kynoch Press. Birmingham, England. 1969). 131 W. Fiiher, H. Burzlaff, E. HelIner and J.D.H. Donnay. Space groups and lattice complexes, Nat. Bur. Stand. (U.S.A.) Monogr. 134 (1973). 141 C. Turrell, Infrared and Raman spectra of crystals (Academic Press, London. 1972). 151 A.F. Rogers, Proc. Am. Acad. Arts&i. 61 (1926) 161. [6] L.L. Boyle, Acta Cryst. A27 (1971) 76. 171 S. Bhagavantam and T. Venkatarayudu, Theory of groups and its application to physical problems, 3rd Ed. (Andhra University, Waltair, 1962). [ 81 W.G. Fateley, F.R. Dollish, N.T. McDevitt and F.F. Bentley, Infrared and Raman selection NkS for molecular and lattice ~&rations: ?he correlation method (Wiley-Interscience. New York, 1972). 191 D.M. Adams and DC. Newron. J. Chem. Sot. (A) 17 (1970) 2822. [ 101 D.M. Adamsad D.C. Newton, Tables for factor-group and point-group analysis (Beckman-RHC Ltd., Croydon, England, 1970). [ 111 J.D.H. Donnay and H.M. Ondik,gen. eds., Crystal data, determinative tables. Vol. 1: Organic compounds, 3rd Ed. [NSRDS-JCpDS. Nat. Bur. Stand. (USA), 19721. [ 12) C Brecher. E. Krikorian. J. Blanc and R.S. Halford. I. Chem. Phys. 35 (1961) 109. [ 131 J.B. Bates, DE. San& and W.H. Smith, J. Chem. Phys. 51 (1969) 105. [ 141 J.B. Bates, J. Chcm. Phys. 58 (1973) 4236.