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The Crystal structure of Anhydrous sodium Chromate

Approved:

Approved:

Dean graduate school.

The Crystal structure of Anhydrous Sodium Chromate

Thesis

presented to the Faculty of the Graduate school of The University of rexas in rartial rulfill-

ment of the requirements

for the Degree of

Doctor of rhilosophy

i3y

John Jaimerson Miler, 8.A., M.A. 706 nest 35th st., Austin, Texas.

Austin, Texas June, 1936

Preface

Almost all solid substances are crystalline, and in most cases it is possible to obtain single crystals that are large enough to make physical measurements upon. The structure of crystals is, then, of great importance in the study of the structure of matter.

Although they were limited to the measurement of the interfacial angles that appeared on the crystals, early experimental crystallographers obtained a vast amount of valuable crystallographic data. As early as 1908 P. Groth published several large volumes of such data. This was before Laue’s discovery of x-ray diffraction by crystals. These data have been very helpful to the modern x-ray crys tallographer. 'rhe early crystallographers were able to classify all crystals into six systems which they further divided into thirty-two classes.

The mathematical crystallographers, working with the problem of possible arrangements of points in space, have shown that there are fourteen distinct space lattices upon which point groups may be arranged in a continuous manner. By applying every possible element of symmetry that each point group could possess when placed upon the various lattices, the mathematical crystallographers have shown that there are two hundred thirty different space groups possible. It remained for the x-ray crystallographers to identify these points as atoms and to determine their positions in the unit cells. T-,U

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After Laue's discovery in 1912 great progress began in the experimental determination of the structure of crystals. Under the leadership of such investigators as the Braggs in England, Deßroglie in France, Debye, Scherer, schiebold and fwald in Germany, and Wyckoff, null, rauling and zsachariasen in this country a vast amount of information has been gathered in this field.

It is the desire of the writer that the crystallographic data obtained during this investigation may prove to be of value in future crystallographic or chemical research.

The writer wishes to express his sincere gratitude to Dr. M. Y. Colby, under whose direction this work has been done, for his helpful guidance throughout the investigation, and to thank Dr. L. J. B. LaCoste for many helpful suggestions.

Contents

Page

Introduction — 6

Preparation of the Crystals 8

Apparatus and Methods Used 9

Determination of the Unit Cell ———.—-— 11

Determination of the Space Group 21

Determination of the Parameters —— 52

Discussion of the Structure 69

Summary of the Structure 70

Bibliography — 71

Introduction

The Crystal Structure of Anhydrous Sodium Chromate

While the complete structures of a very great number of crystals have been determined, from which much valuable data in the fields of crystallography, atomic physics and physical chemistry have been derived, very little information of this type concerning the chromate group has been available. In fact, potassium chromate is the only crys tai in this group whose complete structure has been published.

Empirically derived relations concerning atomic structures are statistical. It is desirable, then, to hive a large supply of data. In the selection of a problem in crystal structure it was thought advisable to determine the complete structure of a member of a group for which there was a very limited amount of crystallographic data. A member of the chromate group, anhydrous sodium chromate, was chosen.

There has been a difference of opinion among chemists concerning sodium chromate. says it is completely isomorphous with sodium o sulphate. Retgers says the two are not isomorphous. Furthermore, some of the crystallographic data that have been obtained seem to be in error.

Ilie purpose of this investigation is to determine the complete crystal structure of anhydrous sodium chromate, thus contributing to the ever growing collection of crystallographic data definite information concerning a member of a group that has been studied very little, correcting certain errors in the recorded data, and settling the question as to whether or not sodium chromate and sodium sulphate are isomorphous.

From collected crystallographic data students of the structure of matter may derive valuable information about such topics as atomic radii, minimum separation of ions in a compound, physical properties of matter, arrangements of atomic groups and explanations of chemical reactions. These in turn are useful to the crystallographer in his search for additional data.

X-rays, of previously determined wave-lengths, are essential in the study of the structure of crystals whose atomic plane spacings are not known. Likewise, crystals of known structure are useful in the study of x-rays. X-ray spectroscopy is thus dependent upon the results of the study of the structure of crystals. Until quite recently crystals were the only diffraction gratings of use in x-ray spectroscopy and they are still the most dependable and widely used x-ray diffraction gratings. Crystallographic data are of service, then, not only in the study of the structure and properties of matter but in the study of radiant energy as well.

1) Flach, S.: Kaliumchroma t und llatriumchromat, ihre Fahigkeit zur mischkristall-und Doppelsalzbildung and ihre Beziehungen zu den entsprechenden Sulfaten. Leipzig, 1918.

Retgers, J. W.: Beit. Phys. Chern. 8, pp. 47-51. 1891.

Preparation of the Crystals

The crystals used, in this investigation were grown from an aqueous solution that was kept at a constant temperature of 76 C. This was achieved, by lowering the vessel containing the solution into a thermo- 3 statically controlled, oil bath, according to the data listed by Mellor, sodium chromate is anhydrous if it comes out of an aqueous solution at o a temperature above 68 C, Samples of the crystals were powdered and heated to test them for water of crystallization. There was no appreciable loss of weight. It appeared quite obvious that they were anhydrous .

A large number of interfacial angles were measured by means of a contact goniometer, but these measurements failed to reveal the identity of any of the faces. This was due to errors in the crystallographic data. After many crystals had been broken into bits a well defined

cleavage face, the b-face, was found. X-ray reflection photographs from this face were made with the axis of rotation in various positions until a symmetrical pattern was obtained on the photographic plate. In this way one of the other principal axes of the crystal was found. A face was then ground so that it was perpendicular to both the b-face and the axis about which the crystal was rotated when a symmetrical pat. tern was obtained. The third principal face was then ground so that it was perpendicular to the other two.

Sodium chromate crystals left exposed to the atmosphere soon take up water into which they dissolve. For this reason it was found necessary to keep the crystals covered with vaseline while they were not in use.

3) Mellor, J. W.: Comprehensive Treatise on Inorganic and 'Theoretical Chemistry, 11, p. 344. London, 1931

Apparatus and Methods Used

The data used in the determination of the crystal structure of anhydrous sodium chromate were taken from reflection photographs of the oscillation type. Reflection photographs from each principal face were taken as the crystal oscillated at constant angular velocity between zero and thirty degrees about each principal axis parallel to that face. Comparison reflection photographs, in which the reflections from each principal face were allowed to fall upon the same photographic plate as those from the principal face of calcite, were taken in which the oscillations were between zero and twenty degrees.

The equipment used in making these photographs is shown in figure 1. The source of the monochromatic x-rays is a water cooled Coolidge type x-ray tube equipped with a molybdenum target. It is inside the housing shown at A. It operates at 30000 volts, R. M. S., and about 20 milliamperes. This equipment is by General Electric. The spectrograph, which was constructed in this laboratory, is shown at B. The motor driven cam supports a lever rigidly attached to the crystal holder, thus supplying to the crystal the desired type of motion. The plate

holder is some five centimeters behind the crystal. This distance was increased to about ten centimeters for the comparison photographs.

Reflecting planes were identified by means of the reciprocal meth-4 od introduced by Ewald. To facilitate the work of projecting the lines from the photographic plate onto the reciprocal lattice, use was made 5 of the convenient ruler and chart devised by LaCoste. The chart was constructed on graph paper, thus eliminating the necessity of using the T-square.



Figure 1. X-ray Tube and Spectrograph

Determination of the Unit Cell

6 According to the crystallographic data recorded by Groth, the anhydrous form of sodium chromate is rhombic bipyramidal, has a specific gravity of 2.710-2.736, well defined cleavage along the b-face, and an axial ratio, a:b:c, of .4643:1:.7991.

An experimental check upon the specific gravity was made. By weigh ing some of the crystals in air and in toluene and calculating the specific gravity a value of 2.73 was obtained, which is in good agreement with the value given by Groth.

From the comparison photographs shown in figures 2,3, and 4 data were obtained for the determination of a possible unit cell. These data,

Figure 2. Comparison Photograph. Reflection from a-faca of Sodium Chromate and a-face of Calcite.

Figure 3. Comparison Photograph. Reflection from b-face of Sodium Chromate and a-face of Calcite.

Figure 4. Comparison Photograph. Reflection from c-face of Sodium Chromate and a-face of Calcite.

together with certain calculated results are shown in tables 1,2, 3, 4, 5 and 6.

The number of molecules per unit cell is given by the relation:

a. t 0 c 0 /° N m - M

where m is the number of molecules per unit cell, M is the molecular weight, N is Avagadro’s number, the density in grams per cubic centimeter, and a O; b 0) c* are the dimensions of the unit cell in centimeters .

If the first order on the photographic plate is taken as the first order reflection of the principal plane in each case the result is

m x 4,614 x 3.b94 x 2.73 x 6.06 x 10 _ ,5005. 162

nils is not a possible unit cell. At least one of these reflections must be of the second or higher order. If one of these reflections is of the second order and the other two of the first order the result is m = 1. This is a possible unit cell. Likewise, if two are of the second order and the third of the first order or if all are second order reflections m will be an integer and the condition for a possible unit cell will be satisfied. There are many other possible unit cells. It will be seen later from consideration of the reciprocal lattice projections that each first order on the photographic plate is a second order reflection. This gives for the dimensions of the unit cell of anhydrous

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Table 1. Determination of r from Comparison rhotograph o (Figure 2) Using Calcite (d = 3.0268 a)

Maximum Deviation from Average =

Maximum Deviation from Average = ,35%

Maximum Deviation from Average = .23%

Maximum Deviation, from Average =

Maximum Deviation from Average = .24%

Maximum Deviation from Average = .39%

sodium chromate

a = 5.912 A, b = 9.228 1, c = 7.189 A, O 7 o 7 c 7

and for the number of molecules per unit cell, m = 4. The density calculated from the above equation is 2.727 grams per cubic centimeter.

The axial ratio, a:b:c, is .641:1:.779. This does not agree with the crystallographic data. Furthermore, the two ratios will not agree when each dimension is multiplied by a small integer. This explains why planes could not be identified by the measurement of the interfacial angles. There are several possible explanations for this lack of agreement. The crystals measured by the early crystallographers may not have been anhydrous. Again, it my be possible that this crystal, like calcium carbonate, has more than one modification. But most likely, in view of the agreement in density, the crystallographic data may be in error. It is entirely conceivable that other sets of planes almost orthogonal might have been mistaken for the principal planes. After the correct axial ratio had been determined, some interfacial angles were calculated and some planes identified by the angles they made with the b-face.

4) Ewald, P. P.: Zeit. f. Krist. 56, p. 129. 1921.

5) LaCoste, L. J. B.: R. S. I. 3, p. 356. 1932.

6J Groth, P.: Chemiache Krystallographie, 2, p. 333. Leipzig, 1908.







Mo. Line Order on Plate 2s sinO 9 tan29 r 1 4.300 .1023 5° 52 20" .2079 10.342 % 1 4.380 .1042 o 1 " 5 59 00 6° 42 30" .2119 10.335 K 0<i 1 4.940 .1168 .2385 10.356 0 / // 6 45 10 V 1 4.980 .1175 .2402 10.362 11 48' 20* K y 2 9.010 .2046 .4371 10.307 2 9.205 .2084 12° 1 30" .4463 10.312 K 2 10.530 .2337 13° 31 Oo" .5103 10.318 K. 2 10.625 .2351 13° 36 00" .5139 10.338 Average 10.334

Mo. Line Order on Plate 2s 0 sinO k _ a 0 n &O n 1 4.475 0 / // 6 6 30 .1064 2.9643 5.928 2 V 1 5.040 a / H 6 51 00 .1193 2.9665 5.933 2 1 5.080 ° ! II 6 54 00 .1201 2.9646 5.929 2 2 9.305 6 / II 12 7 00 .2099 1.4762 5.905 4 2 9.510 12* 21 00" .2139 1.4752 5.901 4 K 2 10.890 13° 53' 30" .2400 1.4743 5.897 4 K 2 10.980 13* 59' 30" .2417 1.4728 5.891 4 Average 5.912

Table Z, Determination of a o from Comparison Photograph (Figure 2) (r = 10.334 cm.)

Mo. Line Order on Plate 2s sinO 0 tan20 r K > 1 4.300 .1023 o / It 5 52 20 .2079 10.342 1 4.380 .1042 0 / ,l 5 59 00 .2119 10.335 1 4.940 .1168 g i U 6 42 30 .2385 10.356 1 4.975 .1175 a 1 ~ // 6 45 10 .2402 10.356 K. 2 9.205 .2084 Q f H 12 1 30 .4463 10.312 a / H 2 10.525 .2337 13 31 00 .5103 10.313 2 10.600 .2351 13 36 00 .5139 10.313 Average 10.332

Table 3. Determination of r from Comparison Photograph (Figure 3) Using Calcite (d = 3.0288 A)

Mo. Line Order on Plate 2s tan29 0 sinO /L = 2sin8 n b. n 1 2.855 .1382 3° 56 .0686 4.600 9.200 2 V 1 3.225 .1560 4° 26 .0773 4.606 9.212 2 2 5.700 .2761 7° 43 Z .1343 2.307 9.228 4 2 5.820 .2817 7° 52' .1369 2.305 9.210 4 k 2 6.570 .3179 8° 49' .1533 2.308 9 .232 4 K 2 6.620 .3204 8° 53 Z .1544 2.306 9.224 4 3 8.855 .4286 11° 36' .2011 1.541 9.246 6 3 9.050 .4383 11° 50 Z .2051 1.538 9.228 6 K 3 10.335 .5000 13° 17' .2298 1.540 9.240 6 K 3 10.425 .5044 13° 26 Z .2315 1.538 9.228 6 4 12.750 .6168 15° 50* .2728 1.157 9.256 8 Average 9.228

Table 4. Determination of b o from Comparison Photograph (Figure 3) (r = 10.332 cm.)

Mo. Line Order on Plate 2s sinO 0 tan2O r 1 4.285 .1023 6 5 / 52 .2079 10.305 1 4.360 .1042 o 5 59 .2119 10.288 K 1 4.925 .1168 _ o 6 / 42 .2385 10.325 1 4.965 .1175 a 6 45' .2402 10.335 K r 2 9.010 .2046 0 11 48' .4371 10.306 2 9.200 .2084 0 12 / 2 .4463 10.307 K 2 10.515 .2337 Q 13 31 .5103 10.303 K 2 10.600 .2351 13° / 36 .5139 10.313 Average 10.310

Table 5, Determination of r from Comparison Photograph (Figure 4) Using Calcite (d = 3.0D88 A)

Mo. Line Order on Plate 2s tan29 6 sinO A, Co 2sin0 ~ n n 1 3.650 .1770 a 5 i 1 .0874 3.608 7.217 2 K 1 4.110 .1993 0 5 38' .0983 3.602 7.204 2 V 1 4.130 .2003 a 5 40' .0987 3.609 7.218 2 K r 2 7.450 .3613 0 9 56' .1725 1.796 7.184 4 2 7.605 *3688 9 10 / 7 .1758 1.795 7.180 4 K X, 2 8.650 .4195 o 11 23' .1976 1.794 7.176 4 2 8.720 .4229 0 11 0 28' 19* .1987 1.792 7.168 4 3 12.210 .5921 15 .2642 1.195 7.170 6 Average 7.189

Table 6. Determination of c o from Comparison Photograph (Figure 4) (r = 10.310 cm.)

Determination of the Space Group

The space group to which a crystal belongs depends upon both the space lattice and the arrangement of points upon the lattice. It has been shown by mathematical crystallographers that there are two hundred thirty different extended crystallographically symmetrical arrangements of equivalent points. Each such arrangement, placed upon one of the fourteen space lattices, constitutes a space group. It is necessary to eliminate from consideration two hundred twenty-nine of them.

The crystallographic data tell us that anhydrous sodium chromate is rhombic bipyramidal and the symmetry exhibited by the reflection photographs verifies this statement. This eliminates from consideration all but the twenty-eight space groups of the orthorhombic system which exhibit the holohedral type of symmetry.

Due to destructive interference, not every atomic plane that is in position to reflect x-rays will give rise to lines on the photographic plate. Each space group will exhibit its characteristic types of absences. Space group criteria for all the orthorhombic space groups were 7 published by Wyckoff in 1925. The characteristic absences for each of the orthorhombic space groups are listed in these criteria.

Certain space groups may be eliminated from further consideration if reflections required by them to be absent are found to be present. Space groups that cannot be eliminated by reflections present may in some cases be eliminated because they fail to exhibit a suffient

number of characteristic absences. In other cases it is necessary to compare observed and calculated intensities before elimination of space groups is complete.

In assigning indices to the reflections present which are to be used in connection with space group criteria it must be kept in mind that there are six possible permutations, namely: x = a, x = b, x = c, right-handed indices; and x = a, x = b, x = c, left-handed indices. For a space group to be eliminated by reflections present, then, it must be eliminated for all the six permutations.

The Miller indices used to specify reflecting planes are the numerators of the reciprocals of the intercepts of the planes on the three principal axes when these reciprocals are reduced to the lowest common denominator.

The six reflection photographs are shown in figures 5,6, 7,8, 9, and 10. The reciprocal lattice projections of the lines in these photographs are shown in figures 11, 12, 13, 14, 15, and 16. In table 7 are listed the reflections identified by means of the reciprocal lattice projections with every possible permutation of indices.

It will be observed from a study of the reciprocal lattice projections that in the case of the principal spectra, all the even orders and none of the odd orders occur. This shows that the correct values of n were used in calculating the dimensions of the unit cell.

The criteria for the four body-centered orthorhombic space groups, V^ 7 , and V^ 8 , require that hkl reflections be absent in odd orders if h+k+l is odd. Some reflections present that eliminate these space groups are: 1-9-1, 1-9-5, 5-1-1, 5-1-3, and 5-1-5. The criteria

Figure .11 • Reciprocal Lattice Projection ( a-face about b-axis )

Figure 12. Reciprocal Lattice Projection ( a-face about c—axis )

Figure 14. Reciprocal Lattice Projection ( b-face about c-axis )

Figure 15. Reciprocal Lattice Projection ( c-face about a-axis )

Table 7. Planes Identified from Reciprocal Lattice Projections

fable 7. (Continued) Planes Identified

for the two face-centered orthorhombic space groups, and require that all reflections be absent in odd orders for which h+k, k+l, or h+l is odd. Some reflections that eliminate these space groups, regardless of permutation of indices, are: 1-1-2, 1-1-4, 1-1-6, 3-1-2, 5-1-2, 0-2-1, 0-2-3, and 0-2-5. One of the base-centered space groups, 22 \ , can be eliminated by planes present. Its criteria require that all reflections Oki, hOl, and hkO be absent in odd orders. Reflections that eliminate this space group, regardless of permutation of indices, are: 1-1-0, 1-3-0, 1-9-0, 0-2-1, 0-2-3, and 0-2-5. Space group requires no absences and therefore cannot be eliminated by reflections present•

In tables 8,9, 10, 11, 12, and 13, the remaining twenty space groups are listed together with reflections that will eliminate them for each permutation of indices. From these tables it may be observed that the space groups V?, V?, vj s ylB y2O an s y2l are elimih’ h* h ’ h * h h h nated for all permutations of indices. There remain fourteen space groups that cannot be eliminated by reflections present.

The characteristic absences for anhydrous sodium chromate, as may be observed from a careful study of table 7, are as follows: hkO if h+k is odd, Oki if k is odd, and hOl if h, 1, or h+l is odd. It is due to the very large number of absences that so few space groups can be eliminated by reflections present.

Eleven of the remaining fourteen space groups require two or less characteristic absences each. They may be eliminated on the basis that they exhibit too few types of absences. The three remaining space groups,

V 4 6 T l 4 and , each exhibit three types of characteristic absences. Each of these three space groups must be tested to determine whether or not the atoms can be placed into the equivalent positions in such a way that the calculated and observed intensities agree and the interatomic distances are not too small.

The calculated intensity of reflection from a reflecting plane, hkl, is proportional to

rr > ~ / p Q 2iri(hXj+kyj+lz .) 1 + cos — a J sin 30 La J J L J

where F is the scattering power of one of the a kinds of atoms, x., y., a J J z. are the coordinates of the jth atom, and 9 is the Bragg angle. F and J a the last factor, an attenuation factor, both vary with 9.

4 The eight equivalent positions for the space group V , as given by 8 Wyckoff, are: x, y, z; x, -y, -z; -x, y, -z; -x, -y, z; 1/2-x, i/2-y, -z; 1/2-x, y+l/2, z; x+l/2, 1/2-y, z; x+l/2, y+l/2, -z. By shifting the origin to a center of symmetry one may reduce these to: ±(x, y, z) ; ±(x, 1/2-y, -z); i(l/2-x, y, -z); i(l/2-x, 1/2-y, z).

When the exponential term above is expressed in terms of sines and cosines and the equivalent positions are put into the expression, the summation becomes 2cos 2ir(hx+ky+lz) + 2cos 2ir(hx-ky-lz+k/2) + 2cos 2nT~hx+ky-lz+h/2) + 2cos 2m(-hx-ky+lz + h+k/2). By expanding each of these terms and collecting similar functions of x, y, and z, one obtains the following expression:

2cos2mhx cos2irky cos2/rlz £1 + cos2?rh/2 + cos2tfk/2 + cos2ii (h+k)/£j + 2sin2nhx sin2irky cos2nlz£.l + cos2mh/2 + cos2irk/2 - cos2ir(h+k)/2] + 2sin2irhx cos2irky sin2mlz £.l - cos2mh/2 + cos2irk/2 + cos2ir(h+k)/2] + 2cos2irhx sin2irky sin2irlz £1 + cos2mh/2 - cos2/rk/2 + cos2m(h+k)/2J .

For planes with indices (eee) or (eeo) the last three brackets become zero and the sum becomes Bcos2-jrhx cos2nky cos2'irlz. Similarly, for planes with indices (ooe) or (ooo) the sum is -bsin2 4 7hx sin2irky cos2mlz; for planes with indices (eoe) or (eoo) it is -Bsin2irhx cos2nky sin2irlz; and for planes with indices (oeo) or (oee) the sum is -Bcos2irhx sin2irky sin2irlz.

For large values of sinO, the scattering power of chromium is far greater than the combined scattering power of sodium and oxygen. An indication of the positions of the chromium atoms may then be obtained by considering some of the more intense reflections corresponding to large values of sin 9. Also, the very intense lines on the photographic plate may be considered as due to reflections from every kind of atom.

A study of the reflection photographs reveals that the more intense lines correspond to reflections 2-0-0, 4-0-0, 1-3-0, 3-3-0, 0-0-4, 0-2-1, and 0-2-3. For large values of sinO the stronger reflections are: 0-8-1, 1-9-1, 1-9-5, 0-2-9, 0-10-0, 0-10-2, and 5-1-2. By applying the structure factor worked out above, and expressing the coordinates in degrees, one finds that the following values should be large: cos2x, cos4x, sinx, sin3x, sinsx, siny, sin3y, cos2y, cosBy, coslOy, sin9y, cosz, cos2z, cos3z, cos4z, and cosbz. parameters that will make these values large are: x = 90° or 270°, y = 90° or 270°, and z = 0° or 180°. These parameters

give large calculated intensities to the reflections 0-0-5, 0-0-7, and 0-0-9, which are entirely missing. These reflections correspond to values of sinO for which the scattering power of chromium is very large in comparison with that of sodium or oxygen. Interference due to reflections from sodium and oxygen atoms could not account for these absences* Consequently the space group is not the correct one.

The eight equivalent positions for the space group V^ 4 , when the h origin is shifted to the center of symmetry, are: i(x, y, z); ±(x4l/2, IA-y, 1/2-z); (x, 1/2-y, z+l/2); (1/2-x, -y, z) . When these points are placed into the summation above, the result is as follows: Bcos2jrhx cos2irky cos2irlz for reflections with indices (eee) or (eoo), -Bsin2nhx sin2irky cosSulz for reflections with indices (eeo) or (eoe), -Bsin2jrhx cos2irky sin2/rlz for reflections with indices (ooo) or (oee), and -Bcos2irhx sin2ffky sin2fflz for reflections with indices (ooe) or (oeo).

The reflections used above in seeking possible chromium parameters, 14 when the indices are changed to fit the permutation for which V could h not be eliminated, are: 0-2-0, 0-4-0, 4-0-0, 1-0-8, 9-0-2, 0-0-10, 0-1-3, 0-3-3, 1-0-2, 3-0-2, 1-1-9, 5-1-9, and 2-1-5. In order for calculated and observed intensities to agree, the following values must be large: sinx, sin3x, sin9x, cos2x, cos4x, cosy, cos2y, cos3y, cos4y, sin2z, sinBz, sin9z, cos3z, cosSz, and coslOz, where the coordinates are expressed in degrees. Parameters that will make these values large are: x = 90 or 270*, y= 0° or 180, and z = 55.

Since there are only four chromium atoms they must be in special positions. There is only one set of special positions for space group

that vdll fit these parameters. When the origin is shifted to the center of symmetry the four equivalent points of this special set are; i-(l/4, 0, u); rt (-1/4, 1/2, 1/2-u)• The chromium atoms can be placed in this set of special positions with the parameter u = 551 The sixteen oxygen atoms can be placed in two sets of general positions in such a way that they will surround the chromium atoms tetrahedrally. The eight sodium atoms cannot be placed in general positions because such an arrangement would place some of the sodium atoms too close to the other atoms and leave a large unoccupied space in the unit cell. One set of four sodium atoms may be placed in the special positions given above with the parameter u = 235. No other set of special positions for V^ 4 will distribute the remaining sodium atoms at a sufficient distance from the chromium atoms. Since this space group does not offer a symmetrical distribution of sodium and chromium atoms when parameters for the chromium atoms are so chosen as to make calculated and observed intensities check, it must not be the correct space group. This leaves only one space group,

7) Wyckoff, R. W. G.: Amer. J. Sci., pp. 151-164. 1925.

8) Wyckoff, R. W. G.: The analytical Expression of the Results of the Theory of Space Groups, p. 60. Washington, 1922.



Figure 5. Reflection from a-face Rotated abouv b-axis.



Figure 6. Reflection from a-face Rotated about c-axis.



Figure 7. Reflection from b-face Rotated about a-axis.



Figure 8, Reflection from b-face Rotated about



Figure 9, Reflection from c-face Rotated about a-axis.



Figure 10. Reflection from c-face Rotated about b-axis.







Figure 13. Reciprocal Lattice Projection ( b-face about a-axis )







Figure 16. Reciprocal Lattice Projection ( c-face about b-axis )

Right-handed Indices Left-handed Indices x = a x = b x = c x = a x = b X = c 0-0-2 0-2-0 2-0-0 0-2-0 . 0-0-2 2-0-0 0-0-4 0-4-0 4-0-0 0-4-0 0-0-4 4-0-0 0-0-6 0-6-0 6-0-0 0-6-0 0-0-6 6-0-0 0-0-8 0-8-0 8-0-0 0-8-0 0-0-8 8-0-0 0-0-10 0-10-0 10-0-0 0-10-0 0-0-10 10-0-0 0-2-0 2-0-0 0-0-2 0-0-2 2-0-0 0-2-0 0-2-1 2-1-0 1-0-2 0-1-2 2-0-1 1-2-0 0-2-3 . 2-3-0 3-0-2 0-3-2 2-0-3 3-2-0 0-2-4 2-4-0 4-0-2 . 0-4-2 2-0-4 4-2-0 0-2-5 2-5-0 5-0-2 0-5-2 2-0-5 5-2-0 0-2-6 2-6-0 6-0-2 0-6-2 2-0—6 6-2-0 0-2-7 2-7-0 s 7-0-2 0-7-2 2-0-7 7-2-0 0-2-8 2-8-0 8-0-2 0-8-2 2-0-8 8-2-0 0-2-9 2-9-0 9-0-2 0-9-2 2-0-9 9-2-0 0-2-10 . 2-10-0 10-0-2 0-10-2 2-0-10 10-2-0 0-4-0 4-0-0 0-0-4 0-0-4 4-0-0 0-4-0 0-4-1 4-1-0 1-0-4 0-1-4 4-0-1 1-4-0 0-4-2 4-2-0 2-0-4 0-2-4 4-0-2 2-4-0 0-4-3 4-3-0 3-0-4 0-3-4 4-0-3 3-4-0 0-4-4 4-4-0 4-0-4 0-4-4 4-0-4 4-4-0

Ri gilt-handed Indices Left-handed Indices x = a 0-4-6 x = b 4-6-0 x = c 6-0-4 x = a 0-6-4 x = b 4-0-6 X = c 6-4-0 0-4-8 4-8-0 8-0-4 0-8-4 4-0-8 8-4-0 0-4-10 4-10-0 10-0-4 0-10-4 4-0-10 10-4-0 0-6-0 6-0-0 0-0-6 0—0—6 6-0-0 0-6-0 0-6-1 6-1-0 1-0-6 0-1-6 6-0-1 1-6-0 0-6-2 6-2-0 2-0-6 0-2-6 6-0-2 2-6-0 0-6-3 6-3-0 3-0-6 0-3-6 6-0-3 3-6-0 0-6-4 6-4-0 4-0-6 0-4-6 6-0-4 4-6-0 0-6-8 6-8-0 8-0-6 0-8-6 6-0-8 8-6-0 0-8-0 8-0-0 0-0-8 0-0-8 8-0-0 0-8-0 0-8-1 8-1-0 1-0-8 0-1-8 8-0-1 1-8-0 0-8-2 8-2-0 2-0-8 0-2-8 8-0-2 2-8-0 0-8-3 8-3-0 3-0-8 0-3-8 8-0-3 3-8-0 0-10-0 10-0-0 0-0-10 0-0-10 10-0-0 0-10-0 0-10-1 10-1-0 1-0-10 0-1-10 10-0-1 1-10-0 0-10-2 10-2-0 2-0-10 0-2-10 10-0-2 2-10-0 0-10-3 10-3-0 3-0-10 0-3-10 10-0-3 3-10-0 0-12-0 12-0-0 0-0-12 0-0-12 12-0-0 0-12-0 1-1-0 1-0-1 0-1-1 1-0-1 1-1-0 0-1-1 1-1-2 1-2-1 2-1-1 1-2-1 1-1-2 2-1-1

Right-handed Indices Left-handed Indices x = a 1-1-4 x = b 1-4-1 x = c 4-1-1 x = a 1-4-1 x = b 1-1-4 x = c 4-1-1 1-1-5 1-5-1 5-1-1 1-5-1 1-1-5 5-1-1 1-1-6 1-0-1 6-1-1 1-6-1 1-1-6 6-1-1 1-1-8 1-8-1 8-1-1 1-8-1 1-1-8 8-1-1 1-1-10 1-10-1 10-1-1 1-10-1 1-1-10 10-1-1 1-3-0 3-0-1 0-1-3 1-0-3 3-1-0 0-3-1 1-3-6 3-6-1 6-1-3 1-6-3 3-1-6 6-3-1 1-3-8 3-8-1 8-1-3 1-8-3 3-1-8 8-3-1 1-5-3 5-3-1 3-1-5 1-3-5 5-1-3 3-5-1 1-5-6 5-6-1 6-1-5 1-6-5 5-1-6 6-5-1 1-5-7 5-7-1 7-1-5 1-7-5 5-1-7 7-5-1 1-7-2 7-2-1 2-1-7 1-2-7 7-1-2 2-7-1 1-7-4 7-4-1 4-1-7 1-4-7 7-1-4 4-7-1 1-7-6 7-6-1 6-1-7 1-6-7 7-1-6 6-7-1 1-9-0 9-0-1 0-9-1 1-0-9 9-1-0 0-9-1 1-9-1 9-1-1 1-1-9 1-1-9 9-1-1 1-9-1 1-9-3 9-3-1 3-1-9 1-3-9 9-1-3 3-9-1 1-9-4 9-4-1 4-1-9 1-4-9 9-1-4 4-9-1 1-9-5 9-5-1 5-1-9 1-5-9 9-1-5 5-9-1 1-11-0 11-0-1 0-1-11 1-0-11 11-1-0 0-11-1

Table 7. Planes Identified

Right- •handed Indices Left- ■handed Indices x = a x = b x = c x = a x = b x = c 2-0-0 0-0-2 0-2-0 2-0-0 0-2-0 0-0-2 2-0-4 0-4-2 4-2-0 2-4-0 0-2-4 4-0-2 2-0-6 0—6—2 6-2-0 2-6-0 0-2-6 6-0-2 2-0-8 0-8-2 8-2-0 2-8-0 0-2-8 8-0-2 2-2-0 2-0-2 0-2-2 2—0—2 2-2-0 0-2-2 2-2-6 2-6-2 6-2-2 2-6-2 2-2-6 6-2-2 2—6—0 6-0-2 0-2-6 2-0-6 6-2-0 0-6-2 2-6-8 6-8-2 8-2-6 2-8-6 6-2-8 8-6-2 2-8-0 8-0-2 0-2-8 2-0-8 8-2-0 0-8-2 2-12-0 12-0-2 0-2-12 2-0-12 12-2-0 0-12-. 3-1-1 1-1-3 1-3-1 3-1-1 1-3-1 1-1-3 3-1-2 1-2-3 2-3-1 3-2-1 1-3-2 2-1-3 3-1-5 1-5-3 5-3-1 3-5-1 1-3-5 5-1-3 3-1-6 1-6-3 6-3-1 3-6-1 1-3-6 6-1-3 3-3-0 3-0-3 0-3-3 3-0-3 3-3-0 0-3-3 3-3-8 3-8-3 8-3-3 3-8-3 3-3-8 8-3-3 3-5-2 5-2-3 2-3-5 3-2-5 5-3-2 2-5-3 3-7-2 7-2-3 2-3-7 3-2-7 7-3-2 2-7-3 3-9-0 9-0-3 0-3-9 3-0-9 9-3-0 0-9-3 3-11-0 11-0-3 0-3-11 3-0-11 11-3-0 0-11-:

Table 7. (Continued) rlanes Identified

Right- ■handed Indices Left- ■handed Indices x = a x = b x = c x = a x = b x = c 4-0-0 0-0-4 0-4-0 4-0-0 0-4-0 0-0-4 4-0-2 0-2-4 2-4-0 4-2-0 0-4-2 2-0-4 4-0-4 0-4-4 4-4-0 4-4-0 0-4-4 4-0-4 4-2-0 2-0-4 0-4-2 4-0-2 2-4-0 0-2-4 4-2-3 2-3-4 3-4-2 4-3-2 2-4-3 3-2-4 4-2-5 2-5-4 5-4-2 4-5-2 2-4-5 5-2-4 4-2-6 2—6—4 6-4-2 4-6-2 2-4-6 6-2-4 4-6-0 6-0-4 0-4-6 4-0-6 6-4-0 0-6-4 4-8-1 8-1-4 1-4-8 4-1-8 8-4-1 1-8-4 5-1-0 1-0-5 0-5-1 5-0-1 1-5-0 0-1-5 5-1-1 1-1-5 1-5-1 5-1-1 1-5-1 1-1-5 5-1-2 1-2-5 2-5-1 5-2-1 1-5-2 2-1-5 5-1-3 1-3-5 3-5-1 5-3-1 1-5-3 3-1-5 5-1-4 1-4-5 4-5-1 5-4-1 1-5-4 4-1—5 5-1-5 1-5-5 5-5-1 5-5-1 1-5-5 5-1-5 5-1-6 1-6-5 6-5-1 5-6-1 1-5-6 6-1-5 5-3-0 3-0-5 0-5-3 5-0-3 3-5-0 0-3-5 5-3-3 3-3-5 3-5-3 5-3-3 3-5-3 3-3-5 5-7-2 7-2-5 2-5-7 5-2-7 7-5-2 2-7-5 6-0-0 0-0-6 0-6-0 6-0-0 0-6-0 0-0-6

Table 7. (Continued) Planes Identified

Right-handed Indices Left-handed Indices x = a x = b x = c x = a x = b X = c 6-0-4 0-4-6 4-6-0 6-4-0 0-6-4 4-0-6 6-0-6 0-6-6 6-6-0 6-6-0 0-6-6 6-0-6 6-2-0 2-0-6 0-6-2 6-0-2 2-6-0 0-2-6 6-2-1 2-1-6 1-6-2 6-1-2 2-6-1 1-2—6 6-2-2 2-2-6 2—6—2 6-2-2 2—6—2 2-2-6 6-2-3 2-3-6 3-6-2 6-3-2 2-6-3 3—2—6 6-2-6 2-6-6 6-6-2 6-6-2 2-6-6 6-2-6 6-4-0 4-0-6 0-6-4 6-0-4 4-6-0 0-4-6 6-6-0 6-0-6 0-6-6 6-0-6 6-6-0 0-6-6 7-1-0 1-0-7 0-7-1 7-0-1 1-7-0 0-1-7 7-3-0 3-0-7 0-7-3 7-0-3 3-7-0 0-3-7 7-5-1 5-1-7 1-7-5 7-1-5 5-7-1 1-5-7 7-7-2 7-2-7 2-7-7 7-2-7 7-7-2 2-7-7 6-0-0 0-0-8 0-8-0 8-0-0 0-8-0 0-0-8

fable 7. (Continued) Planes Identified

Space Group Eliminating Reflections 0-2-1, 0-2-3, 0-2-5, 0-4-1, 0-4-3, 0-6-1 0-2-1, 0-2-3, 0-2-5, 0-4-1, 0-4-3, 0-6-1 Cannot be eliminated by reflections present. Cannot be eliminated by reflections present. Cannot be eliminated by reflections present. 1-1-0, 1-3-0, 3-3-0, 5-1-0, 5-3-0, 7-3-0 1-1-0, 1-3-0, 3-3-0, 5-1-0, 5-3-0, 7-3-0 Cannot be eliminated by reflections present. 0-2-1, 0-2-3, 0-2-5, 0-4-1, 0-4-3, 0-6-1 1-1-0, 1-3-0, 3-3-0, 5-1-0, 5-3-0, 7-3-0 0-2-1, 0-2-3, 0-2-5, 0-4-1, 0-4-3, 0-6-1 Cannot be eliminated by reflections present. 1-1-0, 1-3-0, 3-3-0, 0-2-1, 0-4-3, 0-6-1 y 15 ii 1-1-0, 1-3-0, 3-3-0, 3-9-0, 5-1-0, 5-3-0 1-1-0, 1-3-0, 3-3-0, 3-9-0, 5-1-0, 5-3-0 I 7 Cannot be eliminated by reflections present. ylB v h 1-1-0, 1-3-0, 3-3-0, 3-9-0, 5-1-0, 5-3-0 V 19 h Cannot be eliminated by reflections present. 0-2-1, 0-2-3, 0-2-5, 0-4-1, 0-4-3, 0-6-1 1-1-0, 1-3-0, 3-3-0, 3-9-0, 5-1-0, 5-3-0

Table 8. Elimination of Space Groups by Reflections Present Right-handed Indices, x = a.

Space Group Eliminating Reflections Cannot be eliminated by reflections present. 1-0-1, 1-0-5, 1-0-7, 3-0-3, 3-0-5, 3-0-7 1-0-5, 1-0-7, 3-0-.3, 2-1-0, 2-3-0, 2-5-0 1-0-1, 1-0-5, 1-0-7, 3-0-3, 3-0-5, 3-0-7 2-1-0, 2-3-0, 2-5-0, 4-3-0, 6-1-0, 6-3-0 $ Cannot be eliminated by reflections present. 1-0-1, 1-0-5, 1-0-7, 2-1-0, 2-3-0, 2-5-0 1-0-1, 1-0-5, 1-0-7, 3-0-3, 3-0-5, 3-0-7 vj° 1-0-1, 1-0-5, 1-0-7, 2-1-0, 2-3-0, 2-5-0 1-0-1, 1-0-5, 1-0-7, 2-1-0, 2-3-0, 2-5-0 Cannot be eliminated by reflections present. 2-1-0, 2-3-0, 2-5-0, 4-1-0, 4-3-0, 6-1-0 1-0-1, 1-0-5, 1-0-7, 3-0-3, 3-0-5, 3-0-7 yl5 v h 1-0-1, 1-0-5, 1-0-7, 3-0-3, 3-0-5, 3-0-7 yl6 h 2-1-0, 2-3-0, 2-5-0, 4-1-0, 4-3-0, 6-1-0 h 1-2-3, 1-2-5, 1-4-5, 1-6-5, 1-4-1, 1-6-1 v 18 h 1-2-3, 1-2-5, 1-4-5, 1-6-5, 1-4-1, 1-6-1 1-2-3, 1-2-5, 1-4-5, 1-6-5, 1-4-1, 1-6-1 1-2-3, 1-2-5, 1-4-5, 1-6-5, 1-4-1, 1-6-1 1-2-3, 1-2-5, 1-4-5, 1-6-5, 1-4-1, 1-6-1

Table 9. Alimination of Space Groups by Reflections Present. Right-handed. Indices, x =b.

Space Group Eliminating Reflections G 1-0-2, 1-0-4, 1-0-6, 3-0-2, 3-0-4, 3-0-6 0-1-1, 0-1-3, 0-1-9, 0-3-3, 0-3-9, 0-5-1 0-1-1, 0-1-3, 0-1-9, 1-0-2, 1-0-4, 1-0-6 Cannot be eliminated by reflections present. 1-0-2, 1-0-4, 1-0-6, 0-1-1, 0-1-3, 0-1-9 1-0-2, 1-0-4, 1-0-6, 3-0-2, 3-0-4, 3-0-6 0-1-1, 0-1-3, 0-1-9, 0-3-3, 0-3-9, 0-5-1 v h 1-0-2, 1-0-4, 1-0-6, 0-1-1, 0-1-3, 0-1-9 V 1O h 0-1-1, 0-1-3, 0-1-9, 0-3-3, 0-3-9, 0-5-1 yll h 1-0-2, 1-0-4, 1-0-6, 3-0-2, 3-0-4, 3-0-6 yl2 h 1-0-2, 1-0-4, 1-0-6, 3-0-2, 3-0-4, 3-0-6 Cannot be eliminated by reflections present. 7 h 4 Cannot be eliminated by reflections present. yl5 h 0-1-1, 0-1-3, 0-1-9, 0-3-3, 0-3-9, 0-5-1 V 16 n 1-0-2, 1-0-4, 1-0-6, 3-0-2, 3-0-4, 3-0-6 vi 7 n 2-1-1, 2-3-1, 2-5-1, 4-1-1, 6-1-1, 8-1-1 2-1-1, 2-3-1, 2-5-1, 4-1-1, 6-1-1, 8-1-1 2-1-1, 2-3-1, 2-5-1, 4-1-1, 6-1-1, 8-1-1 2-1-1, 2-3-1, 2-5-1, 4-1-1, 6-1-1, 8-1-1 2-1-1, 2-3-1, 2-5-1, 4-1-1, 6-1-1, 8-1-1

Table 10. Elimination of Space Groups by Reflections rresent. Right-handed Indices, x= c.

Space Group Eliminating Reflections 0-1-2, 0-1-4, 0-1-6, 0-3-2, 0-3-4, 0-5-2 £ 1-0-1, 1-0-3, 1-0-9, 3-0-3, 3-0-9, 5-0-1 1-0-1, 1-0-3, 1-0-9, 0-1-2, 0-1-4, 0-1-6 1-0-1, 1-0-3, 1-0-9, 3-0-3, 3-0-9, 5-0-1 0-1-2, 0-1-4, 0-1-6, 0-3-2, 0-3-4, 0-5-2 Cannot be eliminated by reflections present. $ 1-0-1, 1-0-3, 1-0-9, 0-1-2, 0-1-4, 0-1-6 1-0-1, 1-0-3, 1-0-9, 0-1-2, 0-1-4, 0-1-6 Vjo 1-0-1, 1-0-3, 1-0-9, 3-0-3, 3-0-9, 5-0-1 1 1-0-1, 1-0-3, 1-0-9, 3-0-3, 3-0-9, 5-0-1 0-1-2, 0-1-4, 0-1-6, 0-3-2, 0-3-4, 0-5-2 $ Cannot be eliminated by reflections present. 1-0-1, 1-0-3, 1-0-9, 0-1-2, 0-1-4, 0-1-6 v h 5 n 1-0-1, 1-0-3, 1-0-9, 0-1-2, 0-1-4, 0-1-6 Cannot be eliminated by reflections present. 1-4-1, 1-4-7, 1-4-9, 1-6-1, 1-6-3, 1-8-3 1-4-1, 1-4-7, 1-4-9, 1-6-1, 1-6-3, 1-8-3 vi 9 1-4-1, 1-4-7, 1-4-9, 1-6-1, 1-6-3, 1-8-3 y20 h 1-4-1, 1-4-7, 1-4-9, 1-6-1, 1-6-3, 1-8-3 1-4-1, 1-4-7, 1-4-9, 1-6-1, 1-6-3, 1-8-3

Table 11. Elimination of Space Groups by Reflections Present. Left-handed Indices, x =a.

Space Group Eliminating Reflections 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 Cannot be eliminated by reflections present. 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 1-1-0, 3-1-0, 3-3-0, 2-0-1, 2-0-3, 2-0-5 1-1-0, 3-1-0, 3-3-0, 2-0-1, 2-0-3, 2-0-5 Cannot be eliminated by reflections present. V 1O 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 1-1-0, 1-5-0, 3-1-0, 3-3-0, 3-7-0, 9-1-0 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 n Cannot be eliminated by reflections present. vF v h 1-1-0, 1-5-0, 3-1-0, 2-0-1, 2-0-3, 2-0-5 V F n. 1-1-0, 1-5-0, 3-1-0, 2-0-1, 2-0-3, 2-0-5 1-1-0, 1-5-0, 3-1-0, 2-0-1, 2-0-3, 2-0-5 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 V l 8 1-1-0, 1-5-0, 3-1-0, 2-0-1, 2-0-3, 2-0-5 Cannot be eliminated by reflections present. h 2-0-1, 2-0-3, 2-0-5, 4-0-1, 4-0-3, 6-0-1 h 1-1-0, 1-5-0, 3-1-0, 3-3-0, 3-7-0, 9-1-0

Table 12. Elimination of Space Groups by Reflections rresent. Left-handed Indices, x= b.

Space Group Eliminating Reflections Cannot be eliminated by reflections present. 0-1-1, 0-1-5, 0-3-1, 0-3-3, 0-3-5, 0-3-7 0-1-1, 0-1-5, 0-3-1, 1-2-0, 3-2-0, 5-2-0 T h Cannot be eliminated by reflections present. 0-1-1, 0-1-5, 0-3-1, 1-2-0, 3-2-0, 5-2-0 1-2-0, 1-4-0, 1-6-0, 3-2-0, 3-4-0, 5-2-0 0-1-1, 0-1-5, 0-3-1, 0-3-3, 0-3-5, 0-3-7 v h 0-1-1, 0-1-5, 0-3-1, 0-3-3, 0-3-5, 0-3-7 V 1O h 1-2-0, 1-4-0, 1-6-0, 0-1-1, 0-1-5, 0-3-1 yll h Cannot be eliminated by reflections present. Cannot be eliminated by reflections present. 1-2-0, 1-4-0, 1-6-0, 3-2-0, 3-4-0, 5-2-0 vi 4 n 1-2-0, 1-4-0, 1-6-0, 3-2-0, 3-4-0, 5-2-0 v 15 1—2—0, 1-4-0, 1-6-0, 0-1-1, 0-1-5, 0-3-1 Cannot be eliminated by reflections present. 2-1-1, 2-7-1, 4-1-1, 4-7-1, 6-1-1, 6-1-1 v“ n 2-1-1, 2-7-1, 4-1-1, 4-7-1, 6-1-1, 8-1-1 yl9 h 2-1-1, 2-7-1, 4-1-1, 4-7-1, 6-1-1, 8-1-1 TT 20 V 2-1-1, 2-7-1, 4-1-1, 4-7-1, 6-1-1, 8-1-1 n 2-1-1, 2-7-1, 4-1-1, 4-7-1, 6-1-1, 8-1-1

Table 13. Elimination of Space Groups by Reflections Present. Left-handed Indices, x= c.

Determination of the Parameters

9 The eight equivalent positions given by Wyckoff for the space group are: (xyz); (x, -y, -z}; (-x, y, 1/2-z); (-x, -y, z+l/2); (1/2-x, 1/2-y, -z); (1/2-x, y+l/2, z); (x+l/2, 1/2-y, z+l/2); and (x+l/2, y+l/2, 1/2-z). If 1/4 is added to each x and y parameter, which is equivalent to shifting the origin to the center of symmetry, these eight positions reduce to ± (x+l/4, y+l/4, z); ±(x+l/4, 1/4-y, -z); ±(1/4-x, y+l/4, 1/2-z); ± (1/4-x, 1/4-y, z+l/2). If we let x+l/4 = u, y+l/4 =v, and z =w, these positions become _± (uvwj; ± (u, 1/2-v, -w) ; J:(l/2-u, v, 1/2-w); ± (1/2-u, 1/2-v, w+l/2).

The calculated intensity of a reflection, hkl, is given by the equation

p j = C F 1 + costae ■ ■ a J J J sin. 26 / J .—I L J

where ois a constant, is the scattering power of atom a, Vp and w. are the coordinates of the jth atom, and 6 is the Bragg angle. J

When the equivalent positions given above are substituted into the equation the value of the exponential term is

e 2iri(huj+kVj+lw^) _ 2cos2ir(hu+kv+lw) + 2cos2ir(hu-kv-lw+k/2) J . + 2cos2u(-hu+kv-lv?+h+l/2) +2cos2tf(-hu-kv+lw+h+k+l/2)-

2cos2irlhu+kv+lw) + 2cos2irihu-kv-lw) cos2irk/2 sin2irk/2 + 2cos2irl-hu+kv-lw) cos2ir(h+l)/2 - 2sin2nl-hu+kv-lw) sin2irlh+l)/2 + 2cos2irl-hu-kv+lw) cos2ir(h+k+l)/2 - 2sin2jrl-hu-kv+lwj sin2irlh+k+l)/2 =

2cos2n(hu+kv) cos2irlw - 2sin2niliu+kv) sin2irlw + 2cos2ir(hu-kv) cos2irlw cosirk + 2sin2iHhu-kvj sin2irlw cosirk + 2cos2ir(-hu+kv) cos2irlw cosir(h+l) + 2sin2jri-iiu+kv) sin2irlw cosir(k+l) + 2cos2ir(-hu-kv) cos2irlw cosn(h+k+l) - 2sin2it(-ku-kv) sin2nlw cosir(h+k+l) =

cos2jrkv cos2irlw - 2sin2irhu sin2irkv cos2irlw - 2sin2irhu cos2irkv sin2irlw - 2cos2irh.u sin2irkv sin2nlw + 2cos2irhu cos2irkv cos2irlw cosirk + 2sin2irhu sin2irkv cos2irlw cosirk + 2sin2irhu cos2irkv sin2irlw cosirk - 2cos2nhu sin2irkv sin2irlw cosirk + 2cos2irh.u cos2irkv cos2irlw cosir(h+l) + 2sin2irhu sin2irkv cos2irlw cosn(h+lj - 2sin2irhu cos2irkv sin2nlw cosirih+l) + 2cos2irhu sin2irkv sin2irlw cosir(h+lj + 2cos2irhu cos2irkv cos2irlw cosirdi+k+l)-2sin2irhu sin2;rkv cos2irlw cosir(h+k+l) + 2sin2irhu cos2irkv sin2irlw cosir(h+k+l)+2cos2irhu sin2irkv sin2irlw cosrHh+k+l) =

2cos27rhu cos2irkv cos2irlw Qi + cosirk + cosiHh+l) + cos ir (h+k+l/J +■ 2sin2irh.u sin2irkv £-1 + cosirk + cosiH h+l) - cosir th+k+lj + 2sin2irhu cos2irkv £-1 + cosirk - cosiHh+l) + cosir(h+k+l)J + 2cos2jrhu sin2 4 rkv sin2irlw f-1 - cosirk + cosn(h+l) + cosir(k+k+l) j.

This summation is equal to Bcos2irhu cos2irkv cos2irlw for reflections with indices (eee) or loeo); -Bsin2irhu sin2irkv cos2irlw for reflections with

indices (eoo) or (ooe); -Bsin2ffhu cos2irkv sin2aTw for reflections with indices (eoe) or (ooo); and -Bcos2irhu sinZirkv sin2irlw for reflections with indices (eeo) or (oee). This structure factor is worked out for the eight general positions. For special positions, in which there are only four atoms, the coefficient becomes 4 instead of 8.

The F-curves, in which the scattering power of the atoms is plotted against sin 6, are shown in figure 17. The attenuation curve is shown in figure 18. Values of F and (I+cos 2Q] /sinßQ, used in calculating intensities, were taken from these curves. The F-curve data for chromium were calculated by the Thomas method as given by Bragg and Westls These data, together with the F-curve data for sodium (taken from Wyckoff ) and ox-12> ygen (taken from nest , are shown in table 14. attenuation curve data are shown in table 15.

In determining the chromium parameters, consideration was given to two types of reflections; namely, the very intense reflections, in which all kinds of atoms must have contributed, and the stronger reflections corresponding to large values of sinO in which the scattering power of chromium is much greater than the combined scattering power of sodium and oxygen. Of the first group some outstanding reflections were; 2-0-0, 4-0-0, 0-0-4, 1-3-0, 3-3-0,0-2-1, and 0-2-3. Of the second group some strong reflections were: 0-8-1, 0-2-7, 0-2-9, 0-10-0, 1-9-1, and 1-9-5. In order to make the calculated intensities large for these reflections the following values should be large: cos2u, cos4u, cos4w, sinu, sin3u, sin3v, sin2v, sinw, sin3w, sinBv, sin7w, sin9w, coslOv, cos9v, and sinsw.

j ior Chromium,

Figure 18. Attenuation Curve.

o o O « Parameters that make these values large are: u = 90 or 270, v = 33 - 36, and w = 90 or 270.

The four chromium atoms must be placed in special positions. The one set of special positions for the space group 7$ that can be made to correspond to these parameters is given by Wyckoff^ s as 0, u, 1/4; 0,-u, 3/4; 1/2, 1/2-u, 3/4; 1/2, u+l/2, 1/4. In order to shift the origin to the center of symmetry it is necessary to add 1/4 to each of the first two parameters. This gives i (1/4, u+l/4, 1/4); ±(1/4, 1/4-u, 3/4). If we now let u+l/4 = v, we get ±(1/4, v, 1/4); ±ll/4, 1/2-v, 3/4). The chromium atoms may then be placed into this set of special positions with the parameter ir o 0 about 33 - 36. The exact value of v has to be determined by a process of successive trials. Intensities are calculated for different values of the unknown parameters until the best agreement results. This process cannot be undertaken however until the other atoms are tentatively located, since the intensities depend upon reflections from all the atoms.

In locating the sodium atoms two Shings must be kept in mind: the calculated intensities must agree with observed intensities; an atom must not be too close to another atom. An empirical set of atomic radii in which the minimum separation of ions in crystals is given has been published by 14 Zachariasen. Placing the sodium atoms into any set of general positions either puts sodium atoms too close to the chromium atoms or else too close to other sodium atoms. It might be expected that the sodium atoms should be distributed in the larger spaces that are left vacant after the chromium atoms have been placed in the unit cell. The larger vacant spaces are:

from u » 180 to 360, v =o° to 180°, w =o° to 360; from u «0° to 180°, v = 180 to 360°, w» 0 to 360; from u =o° to 180°, v =o* to 90°, w = 180° to 360; from u = 0 to 180, v = 90 to 180, w» 0 to 180; from u = 180 to 360, v = 270 to 360, w» 0° to 180; and from u » 180 to 360, v * 180 to O o o 270, w = 180 to 270. This may be seen from figure 19 in which the chromium parameters are shown.

Figure 19. Projection on the C-face Showing Chromium Parameters

Comparison of the intensities of the 0-0-2 and 0-0-4 reflections indicates that the w parameter of at least one sodium group is of such value that the reflections from sodium and chromium aid each other in the fourth order 001 reflection but not in the second order. This suggests a value of o o 0 or 180 for the w parameter of one sodium group. An examination of figure 19 may suggest the u and v parameters of this sodium group. For u » 270 and

v = 90, the sodium would be widely separated from the chromium atoms. A set of special positions for Vs, given by that can be made to fit these parameters is as follows: x, 0,0; -x, 0, 1/2; 1/2-x, 1/2, 0; x+l/2, 1/2, 1/2 When 1/4 is added to the first two coordinates of each point these positions reduce to ±(x+l/4, 1/4, 0) ±(1/4-x, 1/4, 1/2), or ±(u, 1/4, 0) ±(1/2-u, 1/4, 1/2) where u = x+l/4. One set of four sodiums may be placed in this set of special positions, where u = 270.

The intensities of the 0-0-2 and 0-0-6 reflections indicate that the o o other sodium group must have a w parameter near 90 or 270. Placing the atoms of this group near the centers of the remaining open spaces puts them into the special positions given above for the chromium atoms with the parameter O O v between 90 and 180, For the best distribution of the atoms this v para- o meter is near 160. Figure 20 shows the approximate distribution of both the chromium and sodium atoms.

figure 20. Projection on C-face Showing Chromium and Sodium Parameters

In the determination of the oxygen parameters three things were kept in mind: first, four oxygen atoms surround each chromium atom tetrahedrally; second, the oxygen atoms must not be too close to each other, to the sodium atoms, or to the chromium atoms; third, calculated and observed intensities must agree. According to the results worked out by the formula given by Zach-16 ariasen , the minimum Cr-0 distance for coordination number four is 1.55 A, the minimum Na-0 distance for coordination number six is 2.4 A, and the minimum 0-0 distance for oxygen atoms of different tetrahedra is 3.2 A.

The very large intensities of the hOO and the 001 reflections suggest that the oxygen atoms lie in the same planes as the chromium atoms along a and c; that is, the u and w parameters are the same as the corresponding parameters for chromium. One group of oxygen atoms may be placed in general <7 positions with w = 90. This places the oxygen atoms in the same plane as the chromium atoms along the c direction. Another group of oxygen atoms may be o placed in general positions with u » 90. This places the other group of oxygen atoms in the same plane as the chromium atoms along the a direction.

In order to separate the oxygen atoms of one tetrahedron as much as possible from those of another it is necessary to assign a value of v to the first group that is greater than the v parameter of chromium and a value of v to the second group that is less than the v parameter of chromium.

The final values of the unknown parameters are determined by successive trials. Intensities are calculated for all observed reflections with different values of the unknown parameters until the best agreement between calculated and observed intensities results, Por anhydrous sodium chromate, the best agreement was obtained for parameters as shown in table 16.

The process of calculating an intensity may be illustrated as follows. Let us calculate the intensity of the reflection 5-1-2, using the formula for intensity given above and the parameters given in table 16. The value of sinO may be determined by means of Bragg’s law,

nA. « 2dsin9,

where d for a set of planes hkl in an orthorhombic crystal is given by the relation,

a hkl ” Tt::"."-,. >

whence

sin 6 « , 2

where n is the order of the reflection, hkl the indices of the reflecting plane, the wave length of the x-rays and a o b o and c o the dimensions of the unit cell. For the reflection 5-1-2, then,

sine . » .318. 2

By referring to the F-curves one finds that, corresponding to this value of sinQ, F Cr » H. 4, Fjj a = 3.3, and Fq = 1.8. From the attenuation curve the value of (l+cos~28)/sin20 is found to be 2.7. For planes with indices (ooe), as seen from the structure factor worked out above for Vs, the value of the exponential term is -Bsinhu sinkv coslw, when u, v, and w are expressed in degrees. For the reflection 5-1-2 this becomes -Bsinsu sinv cos2w. For atoms in special positions, where there are only four atoms, the coefficient becomes -4 instead of -8. The calculated intensity for the 5-1-2 reflection is, then,

F » C £-4(11.4) (sin4so sin3s cos!80) - 4(3.3)(5in-450 sin9o cosO) - 4(3.3)(5in450 sinl62 cosl80) -8(1.8)(sin50 sin 72 cosl80) - 8(1.8)(sin450 sinO cosso)J [2.7] = 7875 C.

Calculated and observed intensities are shown in tables 17, 18, 19, and 20. The value of the constant C was taken as .01 for convenience. In figure 21 is shown the projection upon the b-face of all the atomic positions.

Figure 21. Projection upon the b-face. The numbers represent the positions along b ( b = 9.228 A ) in degrees. TxU

9) Wyckoff, R. W. G.: Loe. cit. p. 61.

10) Bragg, W. L. and West, J.: Zeit. f. Krist. 69, pp. 134-137. 1929.

11) Wyckoff, R. W. G.: The Structure of Crystals, p. 100. N. Y., 1931.

12) West, J.: Zeit. f. Krist. 74, p. 306. 1930.

13) Wyckoff, R. h. G. Log. cit. p. 61.

14) Zachariasen, W. H.: Zeit. f. Krist. 80, pp. 144 - 153. 1931.

15) Wyckoff, R. W, G.; Loc. cit. p. 61.

16) Zachariasen, W. H.: Loc. cit. pp. 144-153.











sin© F„ Or sin© F Na sin© ? 0 .0539 22.30 .1420 7.88 .0000 8.00 .1078 19.14 .1770 6.65 .1000 6.50 .1618 16.41 .2130 5.60 .2000 3.30 .2156 14.15 .2840 3.90 .3000 1.80 .3235 11.26 .3550 2.60 .4000 1.20 .4310 9.08 .4260 1.70 .5000 .80 .5390 7.42 .4970 1.05 .6000 .55

Table 14. F-Curve Data for Chromium, Sodium, and Oxygen. Table 15. Attenuation Curve Data.

sin© 20 cos 26 sin 26 1 + sin 26 .08 9 10 Z .987 .1600 12.33 .10 11 30 Z .980 .2000 9.80 .20 23° 6 .920 .3920 4.70 .30 35° OO' .820 .5750 2.91 .40 00' .683 .7320 2.00 .50 60 0Q Z .500 .8660 1.44

u V w •90 90 0 Na„ 90 162 90 Ct 90 36 90 °1 10 72 90 °2 90 0 25

Table 16. Parameters in Degrees

Indices sinO Observed Intensity F Indices sin© Observed Intensity F 1-1-0 .072 m 124 4-6-0 .332 w 36 0-2-0 .077 m 81 1-9-0 .352 w 37 2-0-0 .120 vvs 1730 6-0-0 .360 m 113 1-3-0 .130 vvs 1220 6-2-0 .368 VW 16 2-2-0 .144 vs 527 0-10-0 .386 w 39 0-4-0 .154 m 84 3-9-0 .388 VW 18 3-3-0 .215 vs 456 6-4-0 .390 vvw 3 0-6-0 .231 m 78 7-1-0 .422 vvw 2 4-0-0 .240 vs 673 6-6-0 .425 VW 20 4-2-0 .251 w 17 1-11-0 .429 VW 9 2-6-0 .260 m 106 7-3-0 ,434 w 46 3-5-0 .260 nil .5 3-11-0 .456 vvw 4 5-1-0 .302 w 24 0-12-0 .463 VW 14 0-8-0 .308 VW 6 2-12-0 .477 VW 14 5-3-0 .320 m 100 8-0-0 .480 W+ 43 2-8-0 .328 w+ 46

Table 17. Calculated Intensities for (hkO) Zone

Indices si nd Observed Intensity F Indices sinO Observed Intensity F 0-0-2 .098 vs 930 6-0-0 .360 ID 113 2-0-0 .120 vvs 1730 6—0—2 .374 VW 16 0-0-4 .196 vvs 980 0-0-8 .392 w+ 64 2-0-4 .230 S— 180 2-0-8 .410 w 25 4-0-0 .240 vs 673 6-0-4 .410 w 40 4-0-2 .260 m 95 6-0-6 .465 w 23 0-0-6 .294 s— 183 8-0-0 .480 43 4-0-4 .310 s— 188 0-0-10 .490 V/ 26 2-0-6 .318 w+ 52

Table 18. Calculated Intensities for (hOl) Zone

Indices sin8 Observed Intensity F Indices sinO Observed Intensity F 0-2-0 .077 m 81 0-6-4 .307 m 89 0-2-1 .092 vs 725 0-8-0 .308 vw 6 0-0-2 .098 vs 930 0-8-1 .312 m 104 0-2-2 .125 nil .3 0-8-2 .324 w 29 0-4-0 .154 m 84 0-4-6 .330 w 16 0-4-1 .162 w 28 0—8—3 .350 m 73 0-2-3 .165 s 253 0-2-7 .351 w+ 40 0-4-2 .183 s 250 0-10-0 .386 w 39 0-0-4 .196 vvs 980 0-10-1 .389 ww 1 0-2-4 .210 vw 5 0-0-8 .392 w+ 64 0-4-3 .210 VW 6 0-10-2 .398 w 35 0-6-0 .231 hl 78 0-2-8 .399 vvw 1 0-6-1 .237 w 31 0-10-3 .413 vvw 1 0-4-4 .246 vw 5 0-4-8 .420 vw 11 0-6-2 .252 m 55 0-2-9 .448 w 20 0-2-5 .256 m 89 0-6-8 .452 w 31 0-6-3 .275 w 24 0-12-0 .463 vw 14 0-4-5 .292 nil .3 0-0-10 .490 w 26 0-0-6 .294 s— 183 0-2-10 .496 ww 5 0-2-6 .303 w 51 0-4-10 .513 vvw 8

Table 19. Calculated Intensities for (Oki) Zone

Indices sinO Observed Intensity F Indices sin6 Observed Intensity F 1-1-2 .121 vs 745 5-1-4 .360 VW 31 3-1-1 .191 m- 59 6-2-1 .372 VW 18 3-1-2 .208 m 114 1-9-3 .381 VW 14 1-1-4 .209 VW 22 6-2-2 .381 w 19 1-5-3 .250 m 169 4-2-6 .387 VW 23 1-1-5 .256 66 5-1-5 .390 VW 22 3-5-2 .282 w 41 4-8-1 .394 w+ 44 4-2-3 .290 w+ 60 6-2-3 .396 VW 17 1-7-2 .293 w+ 61 1-1-8 .398 VW 7 1-1-6 .302 m- 67 1-5-7 .398 VW 20 3-1-5 .307 w 35 1-9-4 .402 w 29 5-1-1 .307 w 35 1-7-6 .403 VW 24 5-1-2 .318 m 79 5-7-2 .414 VW 22 2-2-6 .326 m 98 5-1-6 .422 VW 25 1-7-4 .338 vvw 7 1-9-5 .428 w+ 29 5-1-3 .338 VW 2 6-2-5 .442 vvw 13 3-7-2 .338 w+ 53 3-3-8 .447 VW 25 3-1-6 .347 VW 26 7-5-1 .462 VW 23 4-2-5 .352 w 36 2-6-8 .471 w 28 1-9-1 .354 w+ 50 6-2-6 .471 vvw 14 5-3-3 .354 VW 32 1-1-10 .496 ww 4 1-5-6 .357 VW 15 7-7-2 .508 VW 16

Table 20. Calculated Intensities for (hkl) Reflections

Discussion of the Structure

Each chromium atom is surrounded by four oxygen atoms ( two at 1.58 A and two at 1.62 A) which form a nearly regular tetrahedron with the chromium atom at the center. The average Or-O distance in the o ° same tetrahedron is 1.6 A, while the 0-0 distance is 2.61 A. This is in excellent agreement with the corresponding distances for potassium chro-17 mate as determined by Zachariasen.

Each atom is surrounded by six oxygen atoms (four at 2.44 A 0 . a and two at 2.36 A) at an average distance of 2.41 A. Each atom is 0 * surrounded by six oxygen atoms (two 0, at 2.82 A, two 0. at 2.55 A, and X two 0 at 2.37 A) at an average distance of 2.58 A. The Na-C distances are consistently less than the K-0 distances in potassium chromate. In fact, the minimum Na-0 distance is slightly less than the minimum given 18 by Zachariasen*s empirical formulae, which may be expected, since the unit cell is small and the crystal is not very stable. The coordination number, six, is equal to the maximum coordination number for sodium and oxygen.

0 The minimum Na-Ma distance is 3.3 A, the minimum Cr-Ka distance is o o 3.25 A, the minimum Cr-Cr distance is 4.56 A, and the minimum C-0 distance for oxygen atoms of different tetrahedra is 3.2 A. These are in agreement with the empirical formulae.

Since anhydrous sodium chromate belongs to one of the simple ortho- rhombic space groups, it cannot be isomorphous with sodium sulphate 19 which, as shown by Colby, belongs to the face-centered orthorhombic space group

17) Zachariasen, W. H.: Zeit. f. Krist. 80, pp. 164-173. 1931.

18) Ibid : pp. 137-153.

Summary of the Structure

Anhydrous sodium chromate is built upon the holohedral type of the simple orthorhombic lattice. It belongs to the space group The unit h cell, containing four molecules of , has the dimensions:

a = 5.912 A, o

b o = 9.228 A,

c = 7.189 A.

The four chromium atoms are in the special positions; ±(1/4, v, 1/4); it (1/4, 1/2-v, 3/4); with v = 35°. The eight sodium atoms are in two sets of special positions, Four sodium atoms are in the special positions: ±(u, 1/4, 0); ±(l/2-u, 1/4, 1/2); with u = -90 (or 270). The other four sodium atoms are in the special positions: ±(1/4, v, 1/4); ±(1/4, 1/2-v, Q 3/4); with v = 162. The sixteen oxygen atoms, which surround the chromium o atoms tetrahedrally, at an average distance of 1.6 A, occupy wo sets of general positions: ±(u, v, w); ±(u, 1/2-v, -w); ±(l/2-u, v, 1/2-w); o 0 ±(l/2-u, 1/2-v, w+l/2); with the following parameters: = 10, = 72, wt = 90, U-, = 90, v =O, and w o = 25°.

The density of anhydrous sodium chromate at room temperature (24 C) is 2.727 grams per cubic centimeter.

19) Colby, M. Y.: Zeit. f. Krist. 77, pp. 49-60. 1931

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