THE AMERICAN MINERALOGIST, VOL 53, NOVEMBER_DECEMBER, I968

A SYSTEMATIC APPROACH TO INDEXING POWDERPATTERNS OF LOWER SYMMETRY USING

DE WOLFF'S PRINCIPLES

K. Vrswexa rutN, I ns titwt fcir M,iner al.o gie, (J niv er sitrit M ii.nster,Miinsler, W est Germany.

Anstne.cr

Using De Woltr's (1957) equations two centrosymmetrical reciprocal lattice planes arefound which intersect along a common axis. Assuming the Q-value of a particular latticepoint to be R, planes parallel to the centrosymmetrical planes are constructed. The valueof R is found by considering the differences in the Q-values of a particular row of tlleparallei planes. The nature of distribution of equal Q-values on the parallel planes exhibitthe symmetry of the crysta, .

INrnooucrroN

Runge (.1917) and Ito (1949, 1950) presented methods for indexingpowder photographs regardless of symmetry, which were based on thereiationship between the reciprocal lattice vectors and sin2 I values.Based on this approach De Woltr (1957) suggested a more elegantmethod to construct directly the reciprocal lattice of a crystal fromQ-values (0:sin'9X109. De Wolff has given the following four im-portant equations, which give the relationship between different recip-rocal lattice vectors.

Q&,, -l hz) t Q(h - h.,) : 2Q(h) + 2Q(h,)

n'zQ(mh) : m'Q(nh\

Q(h + nh') - Q(h - xh') : n. lQ(h + h') - Q(h - h ') l

Q(h + h, * h) -t Q(h') - Q@, * h,t) - Q@z * ha): Q@'. * hz) - Q@) - Q&,)

De Wolff has also given three examples to show how these equationscan be used for different purposes. But his approach gives the impres-sion that it is necessary to find three reciprocal Iattice planes withEq. (1), if a powder pattern of a monoclinic or tricl inic crystal is to beindexed. It is found that in some cases it is almost impossible to getrnore than two planes. Therefore this method has been suggested whichrequires only two intersecting centrosymmetrical planes to start with.Further an attempt has been made to evolve a systematic procedure toindex powder patterns of unknown symmetry.

First the p-values are tested for cubic, tetragonal and hexagonalsymmetries according to the methods described by Azaroff and Buerger(1958). After eliminating these three possibil i t ies, the procedure of Dc

(1 )( ) \

(3)

(4)

2047

K. VISWANATIIAN

Wolff is followed to find two intersecting planes using Eq. (1). In asystematic procedure it is not advisable to use Eq. (3) to find a parallelplane as done by De Wolff for his example 1 because of the followingreasons: (1) If the first plane found by Eq. (1) is an orthogonal array,there wil l be success only if the crystal is orthorhombic and only whenthe orthogonal plane contains two of the three principal axes o*, Dxand c*. (2) If the first plane is not an orthogonal array, it is possible tofind a parallel plane with Eq. (3) only if the crystal is monoclinic andonly when the first plane contains the axes o* and c*.

It may be noted that if four Q-values satisfy Eq. (3) and if they givea periodicity equal to one of the p-values, say 0A, of the first planefound by Eq. (1), the same data can be used to construct a second planeintersecting the first one along the vector giving the value Qa.

After f inding four p-values with Eq. (1) and forming a plane, theother Q-values of it can be found with Eqs. (2) and (3) as suggested byDe Wolff. Then the two planes must be tested for the following:(1) Whether any one of them show pairs of equal p-values and if so,whether it can be reduced to an orthogonal array. (2) Assuming thatthe two planes intersect along the hz-a.xis and one has Zr and ,2 as trans-lation directions and the other hs and /22, it nrust be checked whether

Q(ht), Q(hr) and Q(fu) are really the "unit"l p-values in the respectivedirections and do not correspond to higher orders (probably secondorder). De Wolff formed the first plane of his example III by consideringthe Q-values 936.0 and 166.1. Later it was found that 234.0 and not936.0 is the fundamental Q-value in that direction. Such cases mayoccur especially if the Q-values selected for Eq. (1) occur among thehigher values.

MBluou

The method of constructing the reciprocal lattice after finding twointersecting planes is described by considering the powder data (Table1) given by De Wolfi for tricl inic KBOa.H2O2 and orthorhombicprogesterone.

Eromple I. KBQs.HtOz. De Wolff f irst found the following two planes.

1044.41313.2 620.4

Plane,4

824 | 936.0 1380 1 1az)261.1 234.0 539.1 1176.4 (o')

0 0 166.1 664.4 (oo)

r r'Unit" Q-value has been used in the sense that it will be the first Q-value in that

particular direction considered from the origin and all other Q-vahres in that direction willbe hieher orders of this value.

INDEXI NG POWDER PATTERNS

Plane B

702 .0 838 .1175 .5 326 .6h i T

0 . 0 - + 1 6 6 . 1h2

1306.4809.9

664.4 1494.9

An exar.rination of the planes A and B reveals that they intersectalong the row 0-166.1-664.4. Now the three axes 234.0, 166.1 and175.5 are assumed to be the a'xlshs,h2,andhandthe plane,4 isimaginedto stand on plane B along the common axis hz (i.e. the rows 4s and bocoincide) at some angle. Then a plane ,41 is constructed along the firstrow (b1) of plane B parallel to the plane ,4 using Eq. (a)

e(h + h, t h,) + ee) - e(h, * ha) - e(hz I hi: Q(h, t hz) - Q(h) - Q(h,)

The positive direction of he is above the plane of paper. The positivedirections of hr and hz are indicated in the table. The value Q(h1lhs)in this equation is assumed to be some value R and is the Q-value ofthe point lying above 175.5. Q@r+hz*h') l ies above the point 326.6.Therefore using Eq. (4) a relationship between R and Q(fulhtlh)can be established.

Q(h r * hz * hs ) +234 .0 - R - 539 .1 : s26 .6 - 175 .5 - 166 .1

Q ( h ' * h , l h t ) : R + 2 9 0 . 1

The point Q@r-h) which l ies below 175.5 (i.e. below the plane ofthe paper) is equal to 819.0-R according to the equation

Q(h, - h,) I R: 2(r7s.s + 2s4.0)

Similarly the point Q(ht+hr-h) below 326.6 gets the value 831.1-R.The p-values above and below the point 809.9 can be determined by asimilar procedure. Thus the following values of the plane ,4r have beendetermined.

R R+290.r R+9r2.4175.5 326.6 809.9819.0-R 831. 1-R t t75.4-R

Now with the help of the Eq. (3) the other values in the plane ,41are calculated. To find the value of R it is enough if Q-values are cal-culated for two or three rows parallel to the row R-175.5-(819.0-R)and for two rows parallel to the row (b1). Thus the plane,41 is con-structed.

(bz)(b')

(br)

K. VISWANATIIAN

Plane,4r

430.9+2R r95 6+2R 292.5+2R 721.6+2R r482.9+2R (cz)( t r47.7) (100s.s) (1436.s)

R +416.4 R +42.1 R R +290.1 R +912.4 \q)(77s.7) (3s6.0) (643.e)

869.9 356 6 175.5 326.6 809.9 (b, )

1791 .4 -R 1139 .1 -R 819 .0 -R 831 .1 -R L r75 4 -R \q )(143s.1) (463.e) (474.0)

2389 .6-2R 1930. 5 - 2R 1803 .6-2R 2008 .9 -2R (cz)(r2r7.e) (r0e2 4)

(Values in parentheses are the corresponding observed Q-values).If the value of R is found, the other values can be calculated. The best

method of obtaining the value of rR is to proceed systematically asfollows:

The lowest p-values on the plane ,41 must be lying in the neighbor-hood of the central point 175.5. ft can therefore be expected that manyof the low unidentified Q-values in Table 1 can be from the first rowswhich run parallel to the middle row (01).Ilence two tables of differencesare prepared, one between the Q-values of the first upper row (c1) andthe other between those of the lower row (c1) :

Difierences in Q-values (cy)

R+42.r 870.3 374.3 248 0R+29O.r 622 s 126 3R+416.4 496.0R+912.4

Differences in Q-values (c1)

819.0-R 972.4 356.4 320 | 12.1831 1-R 960.3 3M.3 308 01139 1-R 652 3 36 .31 1 7 5 . 4 - R 6 1 6 . 0179t .4 - R

The lowest f ive unidenti f ied p-values are tabulated and the dif fer-ences I isted.

Differences in lowest five unidentified Q-values

463 9 541.6 309.8 180 0 10 1474.0 531.5 299.7 t69 9&3.9 361 .6 t29 .8773.7 231 .8

1005.5

INDEXING POWDER PATTL,RNS 2051

On comparison of these three tabulations it is found that there arethree possible combinations.

l .129.8 (near to 126.3) re lates 643.9(R1290.1) and 773.7(R+416.4)giving an average R-value 355.6.

2. 309.8 (near to 308.0) re lates 463.9(831.1-R) and 773.7(1t39.1-R)giving an average R-value 366.3.

3. 10.1 (near to 12.1) re lates 463.9(819.0- R) and 474.0(331.1- R)giving an average R-value 355.7.

It is clear that observations (1) and (3) agree with each other and(2) is not compatible with either (1) or (3). If the orher p-values arecalculated with R:355.7, many of them are found to agree with theobserved values which are given in parentheses on plane ,41. Thereforethe advantage of this method as compared with that of De Wolff isthat there is no necessity of finding the third plane and later orientingi t .

If a plane ,4r has been drawn through the first row of the plane -Bparallel to plane,4, it is comparatively easy to draw a plane -81 throughthe first row (a1) of ,4 parallel to B. There is no necessity to constructthe plane 81 if the value of R could be obtained from the first rows (cr)or (c1) of plane -4r and if the symmetry of the crystal could be un-equivocally ascertained from the same. For the example under con-sideration plane B1 is constructed to check whether it shows any sym-metry because the plane ,4r does not give any useful information re-garding the symmetry.

ft must be expected that, as the two parallel planes -4r and 81 aredrawn through the first rows of B and ,4 respectively, they will intersectalong the first row above the respective central lines (rows D1 and a1)and their first rows below the respective central lines will be centro-svmmetrical. This will become clear if the planes .4r and -B1 &r€ conr-pared.

2R+563.4(1277 .0)

Plane 81

2R+174.1 2R+1r7 .0 2R+392 .1 2R+999.4 (d.)(1 103 .3)

R+4r6 .4 R+42 . r R R+2e0.r R+9t2.4 (d)

620.1 2 6 t . 1 234.O 539. I 1176.4 (a)

1175 .4 -R 831 1 -R 819 .0 -R 1139 .1 -R 1791 .4 -R (d . )

2081 4-2R 1752.r-2R 1755.0-2R 2090.r-2R (d.z)(1367 .r)

(Values in parentheses are corresponding observed p-values).

2052 K VISWANATHAA

It is seen that row (d1) is identical to (c1) because i t is the row along

which the two paral lel planes intersect each other and the row (dr) is

centrosynrmetrical to the row (c1). p-values of (dz) and (d) of plane -B1

can be calculated within minutes with Eq. (3) and we get three more

Q-values which agree with those given in Table 1.

It may- also be noted that the plane Br also gives a row (dr) of lattice

points with Q-values in terms oI 2R*. . . Therefore i f the value of R

could not be found from the f i .rst rows (c1 or c1), then the second rows

(cz) and (d2) can be considered, which give ten p-values ranging between

2R+lt7 and 2R11482.9. Omitt ing the two high p-values, the re-

Tarlr,n 1. OesrRvno Q-V,tr-t'ns (Dn Wor,rl, 1957)

12

456789

t 01 1t 2I J

I4l . )

16

T

KBOg.HrOz

1 6 5 . 9176.1261.2356 .0463.9474 05 3 9 . 0620 6643.9665.5701 .9

823.+835 .4869. 5899.2

II.Progesterone

6 9 . 38 7 . 39 3 . 5

12+.3150 1162.0r 7 9 . 5205.92 1 7 . 22 2 t . 7237 .42 5 2 . 22 7 + . 5290.43 2 9 . 8346.2

Line No.

t 71819202 l22Z J

2425262 7282930. ) l

I .KBOr.HzOz

II .Progesterone

9 3 5 . 81005 51045 21092.+1103 3rr47 .71178 2r 2 l 7 . 9t 2 7 7 . 0l 3 l 2 . l1 3 6 7 . 11380.6r+36.81479.91497 .O

3 7 3 . 1384 8393 2402.6430.4486. 1+96.5500.35 1 8 . 5530. 2539 .0552 9567 .05 9 1 . 4

maining eight have accounted for f ive of the observed Q-values. Con-

versely, if the differences between these eight Q-values had been com-

pared, the five observed Q-values could have been identified, thereby

obtaining the value of R.A consideration of all p-values of plane 81 does not indicate the

presence of a binary axis or a symmetry plane. (A discussion of sym-

metries that can be expected in such reciprocal lattice planes is given

later.) Therefore the crystal is tricl inic. A consideration of the p-values

of planes At and Br indicates that the three assumed translations 234.0,

166.1 and 175.5 are the shortest and hence the conventional axes. After

finding the true symmetry of the reciprocal lattice and ascertaining

whether the assumed axes are the true axes, indexing can be attempted'

INDEXING POWDER PATTERNS 2053

I f we assume the t ranslat ions 234.0,166.1 and 175.5 to be the cx, b*and ox axes, the planes A, IJ, Ar and Br give all the reflections withindices (lkl), (hk}), (lhl) and (i2fr1) respectively. For KBOa.HzOz, allthe observed p-values given in Table 1 could be indexed only with thesefour planes. But if there are sti l l higher Q-values, they can be indexedby constructing planes parallel either to 41 or 81. While constructingthe second plane ,42 parallel to ,4y, the p-values of the rows (d2) and (dz)of .B1 can be used.

Example I I. Progesterone.

889.0

Plane ,4538 5 346.0 290 5 372.0 (a.)

484.5 2r7 .o 86 5 93 .0 236.5 517.0 (a , )0 .0 68 .5 274.0 616.5 (oo)

86 5 2r7 .0 484.5 889.0

Plane B536.5 496.0 590 5 (bz)

372.0 r79 5 r24 0 205.5 424.0 779.5 (br)0 68 .5 274.0 616.5 (b , )

De Wolff noted systematic difierences in the p-values of the tworows (o2) and (b1) and suggested an elegant method to determine thereciprocal lattice. The same powder pattern is considered here to showthat if such differences had not been observed for some reason or otherthis method could be used as an alternative. Furthermore this exampleshows how the derived parallel planes (At, Bt) exhibit the symmetry ofthe crystal.

The first parallel plane ,41 was constructed through the first row (fu)of the plane B and is parallel to plane,4. Four of the first seven un-identif ied Q-values were found to be from the first rows, three beingfrom the lower row (cr) and one from the upper one (c1). The .R-valuewas found to be 161.5,

868 0 (551.s)533 .5 279 .0372 0 179.5(383.s) (2s3.0)(s68.0) (4ee.s)

533 5484. 5

Plane ,4r(372.0)(161. s)124.O259 .5

(s68.0)

(32e s) (424.0)1 8 1 0 3 3 7 . 5205.5 424 0

(403 .0)/ / J . J

(cz)

(",)(b')

\cr)(cz)

Plane Brs89.0 (484.s) (517 0) (d")279.0 (161.5) 181 0 337.s (d . )2 1 7 . 0 8 6 5 9 3 . 0 2 3 6 . 5 5 1 7 . 0 ( o r )(403.0) 2s9.s (253.0) (383.5) (d.)

(Q-values in parentheses are observed values.)

2054 K. VISWANATHAN

Frc. 1. Portion of the reconstructed reciprocal iattice of progesterone.q1, q2, q3, Qq Qs a:nd. qrl represent the lattice points with Q-values 86.5, 93.0, 253.0, 259.5,

68.5 and 124.0 respectively.q6g7q8qe represents a plane centrosymmetrical to plane Bt. A and B are the midpoints

of the sides qr(la and QzQs and C is the midpoint of ,48.qrO represents a lattice point along the lattice row OC above the plane ,B1.

A consideration of the plane ,41 shows the emergence of three binaryaxes, namely the lattice points having the p-values 279.0, 337.5 and499.5. AII three points show pairs of equal Q-values on their sides, one(337.5) along the vertical row, the second (279.0) along the diagonalrow and the third (499.5) along the horizontal row (c2).

The plane .B1 seems to show apparently only two axes with values279.0 and 337.5. But a consideration of the differences between thecorresponding Q-values of the rows (o1) and (dr) indicates the presenceof the third axis as i l lustrated in Fisure 1. In accordance with theobservation

(86.s - e3.0) : (2s3.0 - 2se.s)

the lengths of the lines Oqt, Oqt, Oqa and Ogl in this figure must be soassumed that

(oq,)' - (oqr)' : (oqD2 - (o$)2

2055I N DDXI NG POW DER PATTERNS

Rearranging,

Theref ore

(oqn),:z\oq,), + (oqa)rl - (q,q,), - (AB),

But by construction

AB : Oq_, and Qilz : OQn

Theref ore

(oqdz : 2\oq,), a (oqa),) - (oq,)2 - (oq),

Substituting the values

(Oqr)' : 93, (O$)2 : 253.0, (Oqu)' : 6S.Sand (Oqi' : 124,0

(oq,), + (oqr), : (oq,), * (oq),

Considering the triangle Oqrq4, we get

(Oqr) , + (Oqo), : 2 [ (OA), I (Aq, . ) r l

Similarly

(oqr) '+ (oq,) ' : 2l(oB)' + (Bq,) ' l

Therefore

(oA), + (Aq,) ' : (OB)2 + (Bq,),

But,4q1:Bgz; and hence

O A : O B

Therefore OAB is an isosceles triangle and OC is perpendicular to AB.Hence OC is perpendicular to Oq-" of plane B (by construction). There-fore OC represents the direction of the third axis. As the lines qeqr,

QrQz, QaQa and qzq+ are parallel to OC, there should be a lattice point {roalong the axis OC at a distance Oqto (equal to qaq).Butqaq1:2OC.

This means that the plane .B1 cuts the third axis OC exactly betweensome two lattice points. Now

(Oqrc)2 : 4(oC)' : 4[(OB)' - (BC)']

l - (ABvf:41 (oB) ' - - l

L = l: 4(OB)' - (AB)'

Br-rt

(on1z - r /2[(Oq,), + (o$),1- (Bq2)2

2056

We get

K. VISWANATIIAN

(Oqn)2 : 2(93.0 + 253.0) - 68.5 - r24.o: 499.5

Thus 499.5 is either the unit Q-value along the third axis or a higher(odd) order of it. In this case it happens to be the third Q-value(9X55.5:499.5). This example shows that such systematic differences

indicate the presence of an axis and that it is possible to calculate tte

Iength of the axis after finding the value of R.

DrscussroN

For triclinic KBOB'HzOz the assumed axes hy hz and /23 happen to be

the shortest axes and hence the reciprocal lattice is directly obtained.

For the second example, both planes Ar and 81 indicate the presence

of three binary axes and hence the reciprocal lattice can be easily con-

structed (Fig. 2). By drawing the planes parallel to the principal planes,

2t9.0 161.5 181.0 337.sFrc. 2. Reconstruction of the reciprocal lattice if three binary axes are exhibited

by the parallel planes (Example II. progesterone).

4 99.5

I N DNX I NG POW DIiR PATTERNS 2057

Frc. 3. Reconstruction of the reciprocal lattice, assuming only one binary

axis has been observed on plane ,41 of progesterone.

it is found that the unit Q-values along the three axial directions are55.5, 37.5 and 31.0. The following method is suggested to reconstructthe reciprocal lattice, when only one binary axis is observed (a.g. in thecase of a monoclinic crystal).

Taking example II, i t is assumed that only the binary axis (withp-value 337.5) is found on plane ,41. The axis is at right angles to thevector with p-value 86.5 (424.0-337.5)-a vector common to plane ,4and passing through the origin. Figure 3 shows an orthogonal arraywith 337.5 and 86.5 as axes and also a portion of the centrosymmetricalplane ,4 intersecting the former along the common axis O-86.5. TheIattice points of 41 are connected to the corresponding points of ,4 byvectors parallel to one with Q-value 124.0. Hence the pairs of points337.5, 236.5 and 424.0, 274.0 are connected in Figure 3. Plane ,4 is in-clined to the binary axis. The rows 93.0-68.5 and 236.5-274.0 are lowerthan the row 337.5-424.0. As there should be a symmetry plane at thefoot of the binary axis, planes drawn parallel to it along the rows93.0-68.5 and 236.5-274.0 should cut the binary axis at two points below337.5. Therefore the third Q-value 337.5 should be nine times the unit

K. VISWANATHAN

p-value 37.5. With these data the basal plane can be constructed. Butthe situations may not always be identical. It must be borne in mindthat if a binary axis is located on one of the parallel planes, the followingconditions exist to reconstruct the reciprocal lattice.

1. There is a vector lying at right angles to the axis and passingthrough the origin and the value of this vector can be obtained by sub-tracting the Q-value of the axis from one of the two adjacent equal

Q-values.2. This vector is also common to the centrosymmetrical plane parallel

to which the plane, on which the binary axis is located, has been drawn.A consideration of the three reciprocal lattices, orthorhombic, mono-

clinic and triclinic, reveals that the distribution of Q-values in theparallel planes (,41 and Br) will exhibit one of the following symmetries:(1) two mirror planes at right angles to each other, (2) a single mirrorplane, (3) three binary axes or one binary axis, or (4) absence of all theabove-mentioned symmetries.

Sometimes, the derived planes may show rows with pairs of equalp-values (plane ,41, Example I) which are not symmetrically situatedwith reference to a central lattice point. Such rows show equal valuesonly accidentally and are not due to a symmetry axis. Oniy when thepoint of intersection of the axis is observed as a lattice point, does itgive rise to pairs of equal values on its sides. Plane -B1 (Example II)shows this phenomenon. ft does not show the presence of the third axisbecause the point of intersection is not a lattice point. But it can beseen that the mid-points between the pairs of p-values (86.5, 259.5)and (93.0, 253.0) are equidistant from the origin. (Fig. 1).

It may be noted that the cases (1) and (2) wil l be met with only ifthe centrosymmetrical plane, parallel to which the first plane is con-structed, is an orthogonal array. In such cases the plane parallel to theorthogonal array can be drawn very easily on the basis of the followingequations. Applying the Eq. (4) to the lattice points Q(hrlhrth') and

Q(hr+h2-h), the following Eq. (5) valid for all types of lattices canbe derived.

Q(h+ hz t he ) - Q (h ' l ha ) - Q@,+ h ): Q(hr I h, - h') - Q&r - h') - Q(h, - h) (s)

If it is assumed that the orthogonal plane ,4 has the axes hs and hzand the second centrosymmetrical plane B the axes hr and h, (h, beingthe axis of intersection) and if a plane ,41 is drawn through the firstrow of B parallel to the orthogonal array A, the Eq. (5) becomes

Q ( h r * h z * h s ) - Q ( h * h a ): Q (h r * hz ) _ Q(h r ) : Q@,* h ,_ h , ) _ Q( fu_ ha ) (6 )

INDEXING POWDDR PATTERNS 2059

If the Eq. (6) is generalized to all other lattice points of plane ,41, itwould be found that the differences in Q-values of the horizontal rowsare equal to those between the corresponding Q-values of the first rowof plane B to which they are parallel. Hence it suffices to calculate the

Q-values along the central vertical row in terms of R, assuming Q(h-fh)to be R. AIso, the first row containing R is interchangeable with thelower first row containing Q(fu-ha). The value R can therefore beassumed to lie on the obtuse-angle side. This assumption means that Ris less than Q(ht-hr). But caution must be exercised while finding thevalue of R. By assuming R to lie on the obtuse-angle side the range ofR is set between O and lQ(h)+Q(h) l. But if R happens to be muchsmaller than Q(h), then the lattice point Q(hr*2h) from the secondupper row may also have a value between the above mentioned range.As both these rows show the same differences, the group of two or three

Q-values found with the help of their differences may belong to thefirst row containing R or the second upper row. Sometimes the differencebetween R and Q@r+2hs) wil l solve the problem. Otherwise it is ad-visable to find a second group of p-values having the expected differ-ences. These two groups belong to two different rows in the first plane,41, thus determining the value of R unequivocally. Similarly if none ofthe Q-values from the first upper row occurs in the list of observedp-values because of systematic extinctions or other causes, it may bedifficult to distinguish whether the identified row of Q-values belongsto the second upper row or the first lower row. Theoretically it maybelong to still higher rows; but as the selected Q-values are low such apossibility is considerably less.

The problem may appear to become more difficult if the axis of inter-section (hr) ol the two centrosymmetrical planes happens to be a binaryaxis, i.e. if both planes are orthogonal arrays. On the contrary, in suchcases, there is no necessity to construct the parallel planes at all. Onlythe plane (hQhr) has to be constructed using hr and Zs as axes andassuming R[i.e. Q(h1tfu)] to l ie on the obtuse-angle side. All other

Q-values in this reflection plane will be in terms of R. As in the previouscases a table containing the mutual differences between the first three

Q-values along the axis hz is prepared. From the list of unidentified(Iow) Q-values at least two groups of Q-values can be found with theexpected differences. These two groups will give two p-values which lienear the origin on the plane (htOht). Using these two values the valueof R can be determined unequivocally.

Before concluding it may be said that what De Wolff stated regardingthe "accuracy of measurementstt is also applicable to this method, be-cause the success depends upon the correctness of the Q-values of thefirst two centrosymmetrical planes.

2060 K. VISWANATHAN

AcKNowLEDGMINTS

The author wishes to thank Mr. H. Kroll for his comments on the manuscript andProf. H. U. Bambauer for the facilities orovided.

RnrnnuNcos

Az.trorr, L. V., aNo M. J. Bunncnn (1958) The powd.er ntethod in X-ray crystdlography.McGraw-Hill Book Co., New York, p. 117 123.

Dn Wor-rl, P. M. (1957) On the determination of unit-cell dimensions from powderdifiraction patterns. Acta Crystdlogz, f0, 590-595

Iro, T. (1949) A general powder X-ray photography. Nolure,164,755-756.-- (1950) X-ray stud,ies on pol.ym,orphismMantzen Co., Tokyo, 187 228.RuNcr, C. (1917) Die Bestimmung eines Kristallsystems durch Rontgenstrahlen. Physik.

2. . r8. 509-515.

Manwscript receireil, Ianuary 15, 1968; accepted. Jor publication, August 13, 1968.