THE CRYSTAL STRUCTURE of CLAUDETITE (MONOCLINIC Aszoa)

THE CRYSTAL STRUCTURE OF CLAUDETITE (MONOCLINIC AszOa)

Arrnnr J. Fnunn, Jn., Uniaersity of Chicago,Chi,cago, Illinois Assrnecr From a complete difiraction record of a single crystal of claudetite taken by equi-incli- nation Weissenburg and Precession techniques, the unit cell dimensions were conf.rmed as: a:5.25 it, b:12 87 L, e:4.54 A, 9:93'49'. The cell contains four AsrOs.Due to the appar- ent faint appearance of an odd order (0&0) refiection, doubt was cast upon the existence of a twofold screw axis (2r). The presenceof the 21,and hence of the space group P21/n,was confirmed by the comparison of the Harker-Patterson sections P (x0z) and P (tclz). From these sections and from Patterson projections on (001) and (100) the eight arsenic atoms were found to be in two general fourfold positions, one set at r:.258, y:.102, z:.040; the other set at r:.363, y: .352, z:.C07; and their symmetry equivalents. From electron density projections and intensity checks, the twelve oxygen atoms were found to be in thtee generalfourfold positions at t:.45, y:.22, z:.03; x:.62, y:'41, z:.18; and r:.95, y: .16, z: .13; and their symmetry equivalents. The structure is one of linked AsOsgroups sharing all oxygens to fcrm an irregular sheet parallel to the perfect (010) cleavage.

INrnonucrtoN Claudetite, the monoclinic form of AszOa,is found in nature as a sec- ondary mineral, formed by the oxidation of realgar, arsenopyrite, or other arsenic minerals (1). The habit of its crystals is tabular, forming thin plates on [010] ; both penetration and contact twins on { 100} are very common. The crystals show perfect micaceouscleavage parallel to { 010} . According to Schulman and Schumb (2), claudetite, the high-tempera- ture form, is the stable form of the dimorphous pair, claudetite-arseno- lite, at temperaturesas low as 20 C. By calculationsbased upon the measured temperature dependenceof the solubilities, these authors con- clude that an enantiotropic relation exists between the two forms, the transition temperature being approximately -13" C. The unit cell and spacegroup of claudetite was investigated by Buerger (3) and the follow- ing cell dimensions were determined: o: 5.25A b:12.87A c: 4.54A B:93"49', The cell contains four formula weights of AszOa.The difiraction symbol is 2/mP2r/n, which uniquely fixes the space group as P2'/n(Czn6).

ExpBnruBNrar, Pnocpouno The major portion of the data for the present work was obtained from a naturally occurring crystal of claudetite from Jerome, Lrizona. Because of the tabular habit, the extreme easewith which these crystals will de- 833 834 ALFRED J. FRUEH, JR. form plastically, and the scarcity of untwinned crystals, it was impossible to obtain a crystal of optimum size and shape or to peel a crystal down to a desiredshape. Consequently, the crystal usedhad a 6 axis many times shorter than the other axes.No attempt was made to correctfor relative absorption.A completerecord of the zero, first, secondand third levels of the c axis was taken by the equi-inclinationWeissenburg method using filtered copper radiation. Zero ievel a and b axis records were obtained by the Buergerprecession method using filtered molybdenum radiation. Zero level c axis data were also obtained by the Weissenburgmethod from a minute tabular crystal grown from a KOH solution of AsrOa.fntensity data were gathered from the entire diffraction record in a manner similar to that first suggestedby Dawton (4) and adapted to Eastman No- ScreenX-Ray film by workers in the MassachusettsInstitute of Tech- nology crystallographiclaboratory (5). The intensitieswere made pro- portional to Fnr"t2by correcting for Lorentz and polarization factors as describedby Buerger and Klein (6, 7).

CoNlrnlrlrroN ol. Sylrunrny Close examination of the zero levei c axis Weissenburgphotographs showed increasedblackening along the generalradiation streak in the vicinity of (030), which cast doubt on the existenceof a true 2r axis.'Io ascertain the existence of this symmetry element and thereby confirm the space group as P21fn, two Harker-Pattersonsections (8), one on P(r0z) and the other on P(rlz), were computed by meansof Patterson- 'Iunell strips (9) and are shownin Figs. la and 10.It is apparentfrom the

Frc. 1. Harker-Patterson sections of claudetite. (a) p (*02). (b) p (x*z). larger peaks on the section P(xlz), representing a 6 axis translation com- ponent of f, that this translation component must exist in the crystal; hence the twofold screw axis and therefore the space group P21fn were CRYSTALSTRUCTURE OF CLAUDETITE 835 confirmed. The high peak about the origin of the P(c0z) would be present regardlessof symmetry since it is the zero vector representing each atom to itself. Loc.qrrox ol rHE AnsBNrc Arolrs The arsenicatoms were located from Patterson projections on the (001) and (100) planes illustrated in Figs. 2 and 3 and from the Harker- Patterson section P(rlz) mentioned above. In the latter section one elongatedpeak appeared,due to the presenceof two interactions slightly

Frc. 2. Patterson projection of claudetite on (001). displacedalong the axis of elongation.In order to obtain the center of density of eachof theseinteractions and hencethe end of the interatomic distance vectors, two graphical sectionswere made across one of the other peaksknown to be a singleinteraction, and comparedat intervals until it wasseen that their sum approximatedthe elongatedpeak (Fig' a). The eight arsenicatoms were found to lie in two generalfourfold posi- tions, and the following coordinateswere determined: As1, x: .27, y: .10, z:.03; As1v,r:.35, y:.35, z:.00. The four equivalentpoints for the

Frc. 3. Pattersonprojection of ciaudetite on (100). 836 ALFRED J. FRUEH. JR.

-x, -!, -z) generalposition are:n, !, zl fr++, +-y, zl|; and |-r, y*L, t-2. Not only are all the interactions between arsenic atoms in these Iocations satisfactorily found on the three vector maps, but also all the sizablepeaks on both the Pattersonprojections can be attributed to interactions between the arsenic atoms. Consequently,it must be as- sumed that the oxygen-oxygeninteractions and the arsenic-oxygeninter- actions are either masked by arsenic-arsenicinteractions or are too com- paratively weak to give rise to substantialpeaks.

- singlepeok -- sumof singlepeoks .---doublepeok

\/ \,/

Frc. 4. Graphical sectionsthrough singleand double Harker-Pattersoninteraction peaks to determineseparation in doublepeak.

DnrBnlrrNarroN or, THE OxyGEN Panauprons Electron density projections on (001) and (100) were computed, again by Patterson-Tunell strips, using only those terms whose signs could be unequivocably determined by the arsenic atoms in the locations as found above, the sign determinations being made from the structure factor formula:

Fw,t:16!7^.o,z,(n, +, *'tl#) .o,zo(oo -'*1*').

The Thomas-Fermi scattering factor for heavy atoms and the Hartree scattering factor for light atoms (10) were used in these calculations. Only about 70/6 of the available reflections were used in constructing these first electron density projections. These projections yielded one -- possiblegeneral fourfold oxygen location, at x: .45, ! .22, z: .03, and its symmetry equivalent. This location permitted the determination of the signs of.additional terms which were in turn employed in making ad- ditional electrondensity projections(Figs. 5o and 5b). Considerablefalse detail was presentabout the arsenicpeaks in the projectionson the (001) and (100) planes. Frc. 5. Electron density projections of claudetite. (a) p (001) I cell' (6) p (100) | cell. (c) p (001) from calculated arsenic values only. I cell. (d) p (001) from observed values modi- fied by Cesaro Means. Entire cell. 838 ALFRED J. FRUEH,JR.

An attempt was made to distinguish between the real oxygen peaks and those peaks due to the false detail about the arsenic atoms. An electrondensity projection on (001) (Fig. 5c) was made using only those terms used in making the same projection as in Fig. 5a with F values calculated from the Thomas-Fermi scattering factors of arsenic in the locations determined above. This method was used rather than directlv making a differencemap (11, 12) expressedby the function:

D,a: po- t": +lf r,1not1- Fc(h0r.)sztt@a+t'v1 for in the employment ,nir"fr,r.. method it would be necessaryto de- "f termine an atomic form factor, usually expressedin terms of a tempera- ture coefficient,to make the observed values fall off at the same rate as the calculatedvalues. It would also be necessaryto scale the Fo (ob- served) values with the F" (calculated) values. By comparing Fig. 5a directly with Fig. 5c it will be observed that, with the exception of the oxygen peak noted above, most of the peaks on the map made from the observedvalues are present on the map made from the calculated arsenic values. Therefore, it can be concluded that either the projection of the oxygen peaks coincideswith the peaks of arsenic false detail, or the pro- jections of the oxygen peaks are so close to those of the arsenic that they are masked by the arsenic peaks themselves or by the depressionssur- rounding the peaks. It is also evident that since both maps produced similar amounts of false detail, this false detail can be attributed for the most part to the abrupt termination of the seriesdue to experimental limitations determined by the small cell dimensions in two directions. The application of the temperature factor or absorption corrections would be insufficient to make the Fourier seriesclose. The seriescould be made to close falsely by applying the method of the Cesdromeans to the coefficientsof the series(13). Thus the two- dimensionalprojection becomes: -+)('- ,.":+T T . ( +) e2.i,b+hu, where 11 and K are one greater than the maximum indices Z and ft used in the series.The resultsof this seriesfor the (001) projection, as solved by X-RAC, are shownin Fig. 5d, and it will be noted that the falsedetail has been removed. unfortunately, this seriesexhibits greatly diminished resolving power over the normal series,and any oxygen peaks lying close to arsenicsare not resolved. However, the lack of more than one oxygen peak permits the assumption that the other two oxygens lie close to the arsenicin this proiection. CRYSTAL STRUCTURE OF CLAUDETITE 839

b prc. O'nt"ltron densityprojections of claudetite.(o) p (101)i cell. (D)p(101)lcell.

Electron density projections were then made on the (101) and (101) planes(Figs. 6o and 6b),and from thesethe remainingoxygen parameters weredetermined to be approximately:r:.65, y: .40,z:.14; and r:.94, y: .14,z: .I0. As the contribution of any one oxygenatom has a negligi- ble effect on the signs of the terms of the Fourier seriesin the electron density projections, the Fourier technique of parameter refining could not be used. Refining was carried out by a system of comparing observedwith calculated intensities. The final parameters are listed in Table 1 and in Table 2. The intensitiescalculated from theseparameters are compared with those observedand corrected.

T'rsla 1

Asr .258 .102 .040 Asrr .363 .352 .007 Or t< .22 .03 orr .62 .41 .18 Orrr .95 .16 .13

Becausethe scattering power of arsenic is four times that of oxygen at Iorv angles,increasing to over seventimes that of oxygen at higher angles, the oxygenparameters can only be determinedwith limited accuracy. 840 ALFRED ]. FRUEH, JR.

Te.eLE 2a. Znno Lrvnl c Axts. Ossanvnn Dera r.RoM Eeur-rNcLrNATroN WETssENBURG Rrcom Tlrnn wnu Coppnn Ka R,l,orarrox AND CoRREcTEDloR LoRENTZ AND Polenrzerrox Fncrons

Index Sin I l.&'l Calculated F Observed

0 2 0 .120 A + 0 40 .240 AA 17 0 60 .360 I ,

0 80 .480 8 IJ 0100 .600 0 0 012 0 710 n 6 0140 .839 0 0 0160 .959 11 18

1 10 .159 9 8 120 .190 4 6 130 .232 10 8 140 .28r 13 11 150 .339 3 160 .389 1 2 170 .M5 9 12 180 .502 19 2l 190 .561 6 1100 .ol/ I 1 | lt 0 .o/o z 1120 - /J.) 12 ZJ 1130 .794 5 8 1140 .852 I 0 1150 .9lr 0 0 1160 .971 9 12

2 00 .294 10 6 210 .300 12 t2 2 2 0 . Jl / 6 7 2 3 0 .345 I z

2 40 .379 10 IL 2 5 0 .420 _tl l2 2 60 .464 7 9 2 70 .513 3 A 2 80 .563 n 6 2 90 .612 8 l1 2100 .668 7 11 211 0 .722 6 10 2120 I 6 2r30 .833 + o 2140 .889 6 11 2150 .946 6 11 CRVSTAL STRUCTL|RE OF CLAUDETITE

TrsrE 2a (Continued)

Index Sin d lFl Calculated P Observed

310 ,M5 4 n 3 2 0 .457 IJ lo 3 30 .476 8 11 3 40 .502 3 3 50 . JJJ 0 0 3 60 .569 10 13 3 70 .609 6 9 3 80 .651 z 4 3 90 .695 3 6 3100 .7M 0 0 311 0 .794 6 3120 .844 7 3130 .895 6 10 3140 .948 , 7

4 00 .588 0 0 4 | 0 .591 0 0 4 2 0 .600 4 7 4 30 .616 3 4 4 40 .635 I 0 4 50 .660 z 4 60 .689 tl 15 4 7 0 .722 1 0 4 80 .759 0 0 4 90 .796 1 0 4100 .840 IJ 18 4tl 0 .884 0 4120 .929 0 0 4130 .976 I 1

510 .738 A 5 20 .744 13 IO 5JU .756 4 z 5 40 - l tJ 0 I 5 50 .794 3 n 5 60 .818 8 9 5 70 .846 z 0

6 00 .882 A 2 610 .884 1 0 6 2 0 .890 1 6 30 .900 0 0 6 40 .914 0 6 50 .932 6 6 '7 6 60 .952 10 ( 6 70 .976 4 842 ALFRED J. FRUEE, JR.

Ter:l2b,znxoLrwraAxrs. oasrwro Dere lnou pnrcnssrowRrcoro T.q.rnNwnH Mor,vsnrwuM Ka RanrerroN axo connrcrno lon Lon-owrzenn polhnrzarrox Fecrons

Index Sin d lFl Calculated F Observed

0 20 .055 l 0 40 .111 18 0 60 .166 1 2 0 80 .221 7 7 0100 .27? 0 0 012 0 .332 4 0 0140 .387 0 0 0160 .442 11 6

011 .083 l

0 21 .096 2 J 0 31 .rl+ 5 o 41 .IJJ z 0 0 51 . lJv 7 7 0 61 .184 0 0 0 71 .211 13 10 0 81 .235 0 0 0 91 .260 13 1l 010 1 .287 0 0 011 1 .Jl4 10 6 012 1 .341 1 0 013 1 .368 9 014 1. .393 1 0 015 1 .422 9

0 02 . t.)/ 23 18 012 .159 8 8

o 22 . too J J o 32 3 6 0 42 .192 15 12 0 52 .209 2 0 o 62 .228 1 0 0 72 .249 1 o 82 .27| 7 I 0 92 .293 1 0 0102 .319 1 0 011 2 .342 1 z 0122 .sot 2 I 0132 .393 4 2 o142 .418 1 0 0152 .444 1 0 0162 .469 8 2 CRYSTAL STRUCTURE OF CLAUDETITE 84il

T;rnt-x.2b(Continueil)

Index Sin 0 I Fl Calculated F Observed

013 .236 0 0 0 2 3 .241 6 6 0 33 .249 6 I 0 43 .260 J 0 53 .273 7 6 0 63 .288 J + 0 73 .304 12 8 0 83 .323 6 0 93 . J+l IL 7 0103 .363 0 0 011 3 .384 9 6 012 3 .407 2 0133 .429 8 q 0143 .453 2 0 0153 .477 5 6

0 04 . JTJ 12 8 014 .314 J o 24 .318 t 0 0 34 .324 4 0 44 .3t2 I 0 54 .342 1 0 .354 1 2 0 64 ( 0 74 .368 0 84 .384 n 0 0 94 .400 4 0 0104 .418 0 i 011 4 .+J I I ot24 .456 0 2 0134 ifi 2

015 .393 T) 0 0 25 .395 5 J 0 3s .400 6 0 45 .407 4 0 5s .415 3 0 0 65 .426 4 0 0 7 5 .437 1 0 85 .450 I 6 0 95 .463 5 8M ALFRED J. FRAEH, JR.

Tr ]-n 2c. srcoNo Lnwr, c Axrs. onsrnwo Dere r.nou Equr-rNcr,rNerror wnrssENBURc Rncon-oTaxrN wnr Copprn Ka RaorerroN eNl Connrcmn lon LonnNrz eNn polanrzlrroN Facrons

Index Sin I j.R'l Calcutated .F Observed

0 02 .340 23 012 .345 8 o 22 .360 3 z

0 32 .384 J 7 0 42 .416 _tJ 15 0 52 .453 2 a o 62 .495 I 1 0 72 .540 5 8 0 82 .588 7 8 0 92 .636 z 0 0r02 .689 2 0

011 2 .742 1 J 0122 .796

I 0132 .852 J 0742 .905 0 0152 .961 2 0

I t2 .379 2 2 | 22 lo? 4 3 tJz .415 ( 3 1 42 .M5 7 11 | 52 .480 5 6 1 62 .519 z 172 .563 10 12 182 .608 11 1.9

192 . o.).) 8 IJ 1102 .707 0 0 tlt2 .758 7 8 1122 .8t2 11 18 1132 .864 6 7 1142 .919 1 0 r152 .974 5

T 12 .370 J 10 I22 . J6.) 3 z T 32 111 6 7 aln .+J I 9 10 I52 .473 6 ( I62 .513 1 2 T 72 .557 q A 182 .603 15 22 T 92 .&9 0 0 Tr02 .702 2 0 I11 2 .754 0 0 CRVSTAL STRUCTURE OF CLAUDETITE 845

Tearr 2c (Continued)

Index Sin d lFl Calculated F Observed

I122 .807 12 20 1132 .861 7 1 1142 .915 I 0 T152 .971 n

2 02 .457 8 10 2 t2 .460 t2 t2 2 22 .472 o A 2 32 .491 I 8 2 42 .516 9 t2 2 52 .546 8 10 2 62 .581 6 6 2 72 .620 4 \) 2 82 .663 3 J 2 92 . /UJ 4 J 2102 . t54 7 7 211 2 .802 1 I 2122 .852 I J 2732 ,9M 0 z 2142 .955 6 7

2 02 M' II r7 21,2 .446 10 12 2 22 .458 2 o 2 32 .477 0 0 qAa .503 10 12 11 2 52 . JJ+ 9 2 62 .570 2 J 2 72 .609 5 6 2 82 .652 i 4 2 92 .695 8 12 2lQ2 . /+J A A 2172 .794 8 10 2122 .8M 0 2132 .896 5 7 2142 .949 I J t 1< 7 .999 8 I5

312 .518 0 0 3 22 .588 12 13 3 32 .593 4 3 42 .614 3 3 3.52 .640 6 8 3 62 .670 9 8 3 72 .7M 6 7 846 ALFRED J. FRaEH, rR.

T.rsrr 2c (Continued)

Index Sin d lFl Calculated F Observed

3 82 .l+L A 3 92 .780 10 3102 .824 I 0 311 2 .868 8 10 3122 .915 J 4 3132 .963 15

312 .551 7 10 3 22 .565 I4 1i 3 32 -5/O 10 IJ 3 42 .598 2 a 3 52 .624 J 3 62 .655 9 9 3 72 .690 8 1l 3 82 .728 2 2 3 92 .lot 2 2 3102 .8r2 0 0 311 2 .857 4 3 3122 .904 1 4 3132 .953 6 3142 .999 I 5

4 02 .689 I I 4 t2 .69r 4 22 600 + ^ 4 32 .712 7 4 42 .729 I 0 4 52 .7sr 3 4 62 .777 6 6 t11 .807 J A 4 82 .83S 1 0 4 92 .873 0 0 4102 .913 12 2l

4 02 .669 0 z 4 t2 .672 0 I 4 22 .680 5 6 4 32 .693 I 0 4 42 .71r 0 0 4 52 . /JJ 3 4 4 62 .754 13 15 4 72 .790 5 J 4 82 .823 0 0 4 92 .858 3 4r02 .898 l.) 2l 4rr2 .939 3 4 CRYSTAL STRUCTURE OF CLAUDETITE 847

Tenr,r 2c (Continued,)

Index Sin d l,r'l Calculated F Observed

512 .822 4 4 5 22 .828 12 20 5 32 .839 4 2 5 42 .854 1 0 5 52 .872 I 4 5 62 .895 8 5 72 .921 0 5 82 .950 0 0 5 92 .980 3 3

512 .801 0 0 5 22 .808 15 20 5 32 .819 t 0 5 42 .834 0 0 551 .854 n 0 5 62 .876 9 10 5 72 .903 J 5 82 .932 2 0 5 92 .964 4 z

6 02 .955 0 0 612 -951 1 0 6 22 .963 o 3 6 32 .972 1 z 6 42 .985 0 0

6 02 .934 J 612 .936 6 6 22 .942 2 I 6 32 .952 5 2 6 42 .965 J 0 6 52 .981 6 5 6 62 .999 8 15

DrscussroN oF THE Srnucrunr The structure of claudetite can be looked upon as consisting of infinite zigzag chains of As ,/\ ooo ./\,/ As As 848 ALFRED J. FRIIEH, JR.

extending in the direction of the c axis and being linked together into irregular sheetsperpendicular to the 6 axis by additional As-O-As bonds. Thus each arsenic atom is surrounded by three oxygen atoms, and each oxygen has two close arsenic neighbors. The structure is illustrated in Fig. 7.

Fro. 7. Crystal structure of claudetite.

Table 3 contains the important interatomic distances. The As-O distance found in claudetite is the same as the As-O distance of 1.80*.02 A found in the AsaOomolecule in the gaseousstate and the distanceof 1.80*.05 A found in arsenolite(14). Becauseof the difficulty in obtaining accuracy in the determination of the oxygen parameters, it would be misleading to quote any greater accuracy in the O-O parameters than that given in the table. However, it can be seen that the greater density of claudetite over arsenolite can be accounted for by the closer O-O distances.The closestO-O distancein arsenoliteis 2.73+.07 A. rt can be seenthat the prominent cleavageand tabular habit of claude- tite can be explained by the above structure. The polymorphic relation- ship between claudetite and arsenolite will be the subiect of a future paper. CRYSTAL STRUCTURE OF CLAUDETITE 849

Tasln 3. Soun lupont.lxr INrrn-lrolrrc DrsraNcns rN Cr-eulerrrc

Atom Neighbor Distance

Asr Asr 3.79 Asrt Asu 4.07 Asr Asrr 3.02 Asrr 3.27 Asrr 3.30 Asr Or 1.82 Orr r.79 Orrr 1.82 Asrr Or 1.77 orr r.74 Orrr 1.77 Or Ot 3.0 orr 2.8 Onr 2.8 Orrr 2.4 Orr Orr 2.9 Orrr 2.8 Onr 2.4

ACKNOWLEDGMENTS The author is indebted to ProfessorMartin J. Buerger for his counsel especially during the initial part of the work which was performed in his laboratory in partial fulfillment of the Master's degree at the Massa- chusetts Institute of Technology. Thanks are also due to Professor R. Pepinsky for suggestingthe method of the Cesiro means and for the use of X-RAC. The technicalassistance of Anne T. Frueh and W. L. Peterson is gratefully acknowledged.

R-olmemcns 1, P.lr-lcrre, C., Bnnuen, H., nxn Fnonou, C., Dano's SystemoJ Mineralogy, Seventh Edition, l, 195-197(lg44). 2. ScnuluaN, J., evo Scnuun, W., The polymorphismof arsenicoide: Jow. Am- Chem. Soc.,65, 878-883(1943). 3. Bunncnn, M. J., The unit cell and spacegroup of claudetite, As2Oa:Am- Mineral.,27, 216 (1942). 4. Drwrox, R., The integration of large numbersof r-ray crystal reflections:Proe . Phys' Soc.Lond.on, 50, 919-925(1938). 5. Kr,rrN, G. 8., The crystal structureof nepheline:Master's thesis,Department of Geol- ogy, MassachusettsInstitute of Technology(1947). 6. Butncen, M. J., aNn KluN, G., Correctionof r-ray difiraction intensitiesfor Lorenu and polarizationfactors: Jou'r. App. Phys.,16,408-418(1945). 7. Bunncnn, 1\r[.J., axo Kr,nrr, G., Correctionof diffraction amplitudesfor Lorentz and polarizationfactors : J ota. A pp. P hys., 17, 285-306(1946). 850 ALFRED J, FRUEH. JR.

8. Hanxnn, D., The application of the three-dimensionalPatterson method and the crystal structureof proustiteand pyrargyrite:Jour. Chem.Phys.,4, 381-390(1936). 9. P.l,rrrnsoN, A. L., em TuNnr.r-,G., A method for the summation of Fourier series usedinther-rayanalysisof crystalstructures:Ant..Mineral.,27r655-679(1942). 10. Inter. Tabellenz. Best. v. Kristohlstrukluren,Band.Il,571-572 (1935). 11. FrNnar, C., aNn Nonuax, N., Subtractionof atom imagesfrom a Fourier synthesisof a c,rystallattice: Acta Ckem.Scand.,2, 813-827(1948). 12. Cocnn,x, W., The structure of pyrimidinesand purines,Part V: Aeta Cryst.,418l-92 (19s1). 13. Pnerxsrv, R., The applicationof Cesiro meansand Parseval'srelations in the Fourier analysisof crystals: LbsI. Am. Cryst.Assn., Winter meeting,Feb., 1951. 14. Alurn, K. E., axo Wnsrcnnx, A., The iattice parametersof the cubic AszOaand SbzOz:Ark'it for Kemi, Mineralogi.eoch Geologie,l5B, No. 22, l-6 (1942). Manuscriptreceioed May 28, 1951.