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Resetting a Triclinic Unit-Cell in the Conventional Orientation

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Resetting a Triclinic Unit-Cell in the Conventional Orientation RESETTING A TRICLINIC UNIT-CELL IN THE CONVENTIONAL ORIENTATION J.D.H.DoNNav, Ex.periment Station, H ercules P owderC omp any, Wilmington, Delaware. INrnooucrroN The rules for the conventional orientation of a triclinic crystal (Don- nay and M6ion, 1933;Donnay, Tunell, and Barth, 1934;Donnay,1943) are the following. The axes must be: (1) chosenalong the shortest three translations; (2) named so as to satisfy the condition c 1a{b; (3) oriented so that the axial cross is right-handed, with a and 0 both obtuse. The first of these rules has been followed generally in the past, so that the problem of reorienting a triclinic crystal usually reducesto resetting the unit-cell, that is to say, renaming and reorienting the axes.Although this problem admits of a straightforward solution, it is encountered so often in crystallographic work that its treatment at this place should be of practical value. RnsBrrrxc A TRrcuNrc CELL The original axial elements are usually given in the form (right- handed axial cross): ao:7.99 bo:7.27 co:7.03 a:89'59', A:95"16,, t:103"25', Copy them, without the letters, as follows: 7.88, ia7 7.O3 89'59" 95"16" r03"25',. Reiabel the axesc, a, b, in increasing order (so as to satisfy the condition c1a1b). The new labelsof the interaxial anglesare thereby determined (respectivelyi .y, dt B). Thus: l,:7.88 a:7.27, c:7.03, B:89"59', a:95o16', t:1O3"25'. Two casesmay occur: The arrangement of letters, read from Ieft to right, either is or (as in the above example) is not a cyclic permutation ofabc. I. If the arrangement is one of the three cyclic permutations (a b c, b c a, c o b), the axial cross has remained a right-handed one. If a re- orientation of the axes is found to be necessaryin order to comply with the convention "a and 0 both obtuse," two axes must change their signs in order that the axial crossremain right-handed. 508 J, D. H, DONNAY Any one of four possibilities may arise: (1) a and 0 both obtuse; (2) c obtuse, but B acute; (3) B obtuse,but a acute; (4) a and 0 both acute. (1) a and 0 both obtuse.-No further modification of the setting is needed. (2) a obtuse, B acute.-A reorientation of axes is necessaryto make the angle p obtuse. As two axes must be reversed, only one of the three interaxial angles can retain its character (obtuse or acute), namely, the angle comprised between the two axes that change their signs. Since a must remain obtuse, the axes to be reversedare b and c. After this trans- formation, B has changed from acute to obtuse (as desired), and 7 has also changed its character (which is immaterial). (3) B obtuse, o acute.-This caseis similar to the one just considered. Reverse the signs of a and c; take the supplements of a and 7. (4) a and 0 both acute.-The only angle that should retain its char- acter (obtuse or acute) is 7. Reverse the signs of a and. D,' take the supplements of a and p. II. If the arrangement of letters, obtained after renaming the axes, is not a cyclic permutation of o b c, but is one of the other three permu- tations (a c b, c b a, b o c), the axial crosshas become left-handed. A re- orientation of the axes is imperative in any case in order to restore the right-handed character of the axial cross; it may also be neededto bring the new setting into agreement with the rule "a and 0 both obtuse." The left-handed axial cross can be made right-handed in two ways only: either by changing the sign of a single axis, or by reversing all three axes. The four possibilities to be examined are the same as before: (1) a and F both obtuse.-Reverse all three axes. The axial cross again becomesright-handed. Ail interaxial angles keep the same values. (2) a obtuse, B acute.-Reversing one axis only changesthe character (obtuse or acute) of the two angles which this axis makes with the other two axes. The angle a cannot be changed; B must, and 7 may, become obtuse. Reverse the sign of o; take the supplements of B and 7. (3) p obtuse, d acute.-For the foregoing reasons, reverse the sign of b; take the supplements of a and 7. (4) a and B both acute.-Change the sign of c; take the supplements of a and 6. The above discussionis summarized in a double-entry table (Table 1), which also gives the old-to-new transformation matrices for the 24 possible right-handed settings. This table should prove worthwhile as a labor-saving device, whenever a large number of triclinic cell reorienta- tions are to be performed. The procedure is simple: rename the axes so that c1a1b; if, after renaming the axes, a and B are both acute, take RESETTING A TRICLINIC UNIT-CELL 509 their supplements; if only one of them is acute, take its supplement and also that of 7; the old-to-new transformation matrix is then found in the table. Exeupr,us The triclinic speciesgiven in Wyckofi (1931, 1935) have been worked out as examples. In each caseit has been assumedthat the unit lengths were the shortest three translations. The examples are listed in groups, each under the appropriate transformation, indicated by its letter (cp' Table 1). These examples probably constitute a representative sampling of the various types of setting used for triclinic crystals. They are instructive in providing a statistical survey and they also bring out certain special caseswhich require additional conventions. If one of the two interaxial angles u and B is 90o (as in pectolite and wollastonite), two settings are found, as 90o is consideredobtuse or acute in turn (transformations W and X, respectively). That setting which leads to 7 obtuse is obviously the more desirable. If both o and p are 90o(as in 3KPbCI3'H2O),two setsof axial elements are likewise obtained, with supp'Iementaryvalues for 7 (transformations O or R, and P or Q). Again the setting that makes 7 obtuse should be preferred. A casewhere o:D has been encountered(H3BO3). Two settings are obtained (transformations A and W), which differ from each other by the interchange of a and B. To cope with such cases (which seldom occur), an additional convention may be devised (a smaller than p, for instance), although refinement of the measurements will usually reveal a difference between the unit lengths. RnlrntNcss DoNNAY, J. D. H., aNo M6r-oN, J., Haiiy-Bravais lattice and other crystallographic data for sodium molybdotellurate: Arn. Mineral., f8' 225 (1933). TuNELL, G., nrn B,rxrn, T. F. W., Various modes of attack in crystallographic investigation: Am. Mineral., 19, 437 (1934). Rules for the conventional orientation of crystals: Arn. Minerol', 28,313 (1943). Wvcxorl, R,u.pn W. G., The Structure of Crystals, 2nd. Ed., I - Am. Chern.Soc., Mono- graph No. f9, (1931). --t Supplement for 1930-1934 to the second edition: Jour- Am- Chem. Soc',Mono' graph No. l9A (1935). 510 J. D. H. DONNAY Tlsr,B 1. TnaxslonuerroN MATRTCES (Old setting to new setting) In order to satisfy the condition c1a{b, the original After renaming aobocobecome: the axes: abc bca cab acb cba bac E K o S w a and B both obtuse: 100 010 001 100 00I 010 Leave aII angles un- 010 001 100 001 0T0 100 unchanged 00 | 100 010 0T0 T00 007 B F L P X a obtuse, B acute: 100 010 001 I00 00I 010 Take supplements of 0T0 001 T00 001 010 100 BandT 00I T00 0T0 010 100 001 C G M a U Y p obtuse, a acute: 100 010 001 100 00 | 010 Take supplements of 010 001 100 001 0T0 100 a andT 001 100 0r0 010 100 001 D H N R V Z a and B both acute: 100 010 001 100 001 010 Take supplements of 010 001 T00 001 010 100 aandB 001 100 010 0T0 T00 00I Lrsr ol Exelrprns Transfornation and Substance Old Setting New Setting (A) Albite, NaAlSi,Oa 8 t4 12.86 7 17 880 9', Calcium sulfate urea, CaSOa.CO(NHr)r 14.74 14.95 6 47 91"26', 90022' 86042' 4 Nitro-2-Methylaminotoluene, (red /.o 6.5 /.J form) 1130 98" 109' Aenigmatite 18.3 18.3 10 6 96030', 96030' 113'30' Kyanite, AlzSiOa 7.o9 7.72 5.s6 90o 5/ 1010 2' 105"44, Ammonium hydrogen lumarate, CrHzNOr 7.00 7.44 6.56 1070 1' 117058', 69"16', Fumaric acid 7.60 15.11 6.61 90'40' 1110 Boric acid, HsBOr 7.04 7.04 6 56 920 30' 10to to' 120" Celsianr 8.63 1.3.10 7.29 ca.90o 1160 ca.90o Andesine 8.14 72.86 7.17 93023' 116028' 89'59', Tungstic trioxidez WOr 7 -28 7.48 3.82 ca.90' ca.90" ca.90" RESETTING A TRICLINIC UNIT-CELL 511 Lrsr ol Exaupr,ns-Continued Transformation and Substance Old Settirs New Setting (B) Stilbene*2 mol. 1,3,5 Trinitrobenzene, 12.7 15.4 7.7 12.1 15.4 7.7 coII5CH:CHC,H,.2[C,H3(NO',] 102"16' 85030' 87035' 102" 16', 94'30' 92"25'. (C) Copper sulfate pentahydrate, 6.07 10 78 5.89 6 07 10.78 5.89 CuSOn'sHrO 82o S' 1070 8' 102041' 97"55', 107'8', 7?o19', Rhodonite,MnrCa(SiO)s 7.77 t2.45 6.74 7.77 12.45 6.74 85'10', 940 4' 111"29' 94050', 940 4' 68331' Racemicacid (anhydrous), 7.18 9.71 4.98 7.18 9.71 4.98 cooH(cHoH),cooH 82"20' 1180 0' 72"58', 97o40' 1I8o o' 1o1" 2' Racemic acid monohydrate, C+HrO6'HrO 8 09 10.03 4 81 8.09 10.03 4.81 160 2', 96057' 120" 8' 103058' 96"51', 59'52', (D) Thallium mesotartrate, ThCrHnOu 16.12 7.63 8.26 16,12 7 63 lr.;?i, 86037' 82014' 1040 6' 93"23' 82014' Cholesterylsalicylate, CuH;oOr 9.68 12 52 6.31 9 68 12.52 6.31 85'53', 77"4t', 84'1', 94" 1', lD20r9' 840 1', (E) No example found (F) Hexamethylbenzene,Co(CHr)o 9.01 8.926 5.34r 8.926 9.01 5.34r 44"27', 116"43' r19"34', 116043', 135033' 60"26' (G) Bisethylene diamino platinous chloride, 8.37 4.95 6 86 6.86 8 37 4.95 Pt (C,HsN,),CI, 100046' t11"40' 8t'56', 980 4', 100"46', 68"20', Racemic methyl ephedrine hydroiodide, 11.2 7.67 7.68 7.68 1r.2 7.67 CuH'0N'HI 108'55' 95036', 84"15', 95"45' 108045' 84"24', (H) Cyclododecane,CrrH* 7.84 5.4+ 7.82 7.82 7.8+ 5-44 98" 18', 64" 810 990 98"18', 64" (K) Anorthite, CaALSirOe 8.2t 12.95 14.16 12 95 14 16 8.21 93" 13' 115056' 91012' 1i5056', 91"12' 9301.3' (L) Labradorite 8.21 12.95 14.16 12.95 14 16 8.21 93"2r', 11603' 89"55', 1160 3' 90' 5' 86429' Babingtonite 6-73 7.54 12.43 7.54 12.43 6.73 112022', 93048', 860 9' 93"48', 93051' 67"ii' (M) (N) No example found.
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