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111 Acta Cryst. (1957). 10, 111 R e m a r k s on the Delaunay Reduction BY A. L. PATTERSON AND WARNER E. LOVE* Institute for Cancer Research and Lanlcenau Hospital Research Institute, Fox Chase, Philadelphia 11, Pennsylvania, U.S.A. (Received 22 June 1956) ])elatmay's reduction of a primitive cell to a standard form is summarized with emphasis on im- portant points which have been omitted in the current accounts of his work. A new reduction is introduced which has the same object as that of Delaunay but which is more rapidly convergent though less general. The counterpart of the Delaunay reduction in reciprocal space is also in- vestigated and comment is made on the relation of the Delaunay standard form to others which have been suggested. Introduction I n a very i m p o r t a n t paper in the field of lattice geometry, D e l a u n a y (1933, referred to as Delaunay) set up a m e t h o d for the reduction of the most general primitive cell of a n y lattice to a s t a n d a r d form in- volving four vectors which make obtuse angles with one another. A n y three of these four vectors form a primitive triplet of the lattice if the original cell is primitive. D e l a u n a y also presented tables which per- mit the recognition of the s y m m e t r y properties of the lattice in the reduced set. The power of D e l a u n a y ' s reduction and the degree of uniqueness which can be achieved with its aid m a k e it an ideal basis for the establishment of a cataloguing system for crystal lattice cell dimensions. The reduction m e t h o d has been described and used b y D o n n a y (in D o n n a y & Nowacki, 1954, referred to as Donnay) as a basis for a table of triclinic cell dimensions. Accounts of the m e t h o d have also been given b y Ito (1950, referred to as Ito) and in the International Tables for X-ray Crystallography (1952, * The w o r k of t h e a u t h o r s is s u p p o r t e d b y grants f r o m the National Cancer I n s t i t u t e , National I n s t i t u t e s of H e a l t h of t h e U.S. Public H e a l t h Service (C1253); from t h e Research Corporation; a n d b y a n I n s t i t u t i o n a l G r a n t f r o m t h e A m e r i c a n Cancer Society. referred to as ITXC). I t is unfortunate, however, t h a t no one of these descriptions of the m e t h o d is suffi, c ient to p e r m i t the completion of the D e l a u n a y reduction. I n the following section the m a i n points of Delau- n a y ' s paper are summarized. Special emphasis is laid on the ambiguities of the reduction, well recognized b y D e l a u n a y but omitted from the discussions of the later writers. I n later sections, a new reduction is introduced which is of less general applicability t h a n t h a t of D e l a u n a y b u t which increases the rate of con- vergence to the same final result. Comment is also m a d e on the counterpart of the D e l a u n a y reduction in reciprocal space and on the relationship of the D e l a u n a y reduced cell to t h a t having the shortest three non-coplanar translations. Description of the Delaunay reduction The reduction starts with a n y primitive triple bl, bR, b3 and adds to it a fourth vector b4 = - b l - b R - b 3 . It t h e n proceeds, b y a series of steps, to transform these four vectors to a set a 1, a~, %, a4 for which the six scalar products gij = (aiaj), i . j, are all negative or zero. A single step of the reduction is indicated in Fig. l(a, b), where the scalar products of the set bi are indicated b y hi/= (bib~), and are written sche- b3 b3+b, b, b2 -b, b~ (a) (b) f ,3 I g23 g,4 I g23 a, a 2 - a, az (c) (d) Fig. 1. (a, b) A single step in t h e D e l a u n a y reduction. N o t a t i o n : hii = (bib;). The scalar p r o d u c t hi2 -- (bib2) is assumed to be positive in (a). I t is m a d e negative in (b) b y changing the sign of b 1, leaving b 2 u n c h a n g e d , a n d adding b 1 to b 3 a n d b 4. The scalar p r o d u c t s for the n e w set are as indicated in (b): i.e. h12 is s u b t r a c t e d f r o m h34 a n d a d d e d to h23 a n d h24 a n d to t h e i n t e r c h a n g e d pair h14 a n d h13. (c, d) Two a m b i g u o u s cases for g12 ---- 0 in t h e triclinic s y s t e m obtained b y 'changing the sign' of t h e zero product.
112 REMARKS O1~ T H E D E L A U N A Y REDUCTION matically on the edges of a tetrahedral symbol. Note that hii : -- (hi/+ hik + hil), i, j , k, 1 = 1, 2, 3, 4 . (1) An i m p o r t a n t check is provided b y D e l a u n a y ' s remark t h a t the negative sum of the scalar products decreases at each step b y the magnitude of t h a t pro- duct whose sign is changed. Thus in Fig. 1 (a), the nega- tive sum of the scalar products is the positive n u m b e r • ~ 1 = ½ - ( h i l + h 2 2 + h a s + h 4 4 ) = -(h12+h2a+h31+h14+h24+h34 ) , (2) while in Fig. l(b) this sum has decreased to X 2 = -- ( 2 h 1 2 + h ~ 3 + h a 1 + h 1 ~ + h 9 4 + h 3 4 ) . (3) Thus the decrease in Z is Z 1 - Z ~ = h19. (4) The reduction is repeated until all scalar products are either negative or zero. D e l a u n a y shows t h a t there are twenty-four basic forms which the reduced set m a y take and exhibits a table of these forms indicating their relationship to the fourteen Bravais lattices. Modified and amplified forms of this table have been presented b y Ito and I T X C * . D e l a u n a y presents an a r g u m e n t b y exhaustion to show that, in the case in which all scalar products are non-zero, the reduced form is unique. He also points out t h a t in the cases in which zero values occur the reduction is ambiguous, since one can then alter the a r r a n g e m e n t of the products in the tetrahedron, and hence the vectors in the reduced set, without changing the q u a n t i t y (2). This operation is of course carried out b y 'changing the sign' of the zero terms. There are two ways in which the presence of a zero m a y be i m p o r t a n t in the final reduction: (i) It m a y lead to an a m b i g u i t y in the choice of cell. (ii) It m a y lead to a s y m m e t r y p a t t e r n which is not recognized as one of the twenty-four t a b u l a t e d b y Delaunay, Ito, and I T X C . The first difficulty is most i m p o r t a n t in the triclinic system and is illustrated in Fig. 1 (c, d). If we assume t h a t Fig. 1 (c) (or one of its permutations under the s y m m e t r y of the reduction) is the first reduced con- figuration, t h e n b y changing the sign of the zero pro- duct we obtain the configuration of Fig. 1 (d). I n the first case, the possible base vectors have m a g n i t u d e s t h a t are the square-roots of the quantities: gll = -- (gla+g~a); g2~. = -- (g~a+g24); ga3 = -- (gla +g2a +g3a) ; g44 = -- (g14 +g24 +ga4). I n the second case, the corresponding values are: h l l = g11; h22 = g~.~.; haa = - ( g 1 4 + g 2 a + g a 4 ) ; h44 = - (g13 + g3~ + g24) * Note that in I T X C , Table 5.1.1 (pp. 534-5), the figures for the second and sixth monoclinie cases should be inter- changed. The limiting ranges for several of the cases are incorrectly stated, but this does not affect the use of the table. and we note t h a t : h33 - - ga3+gll+2g13; and h44 = g 4 4 + g 1 1 + 2 g 1 4 . It is thus possible t h a t the shortest three non-coplanar translations with a tri-obtuse cell m a y not occur in the first reduced quadruple which is obtained after a D e l a u n a y reduction, but m a y occur only after one or more changes of the sign of zero. Practical examples of this situation are readily obtained from the D o n n a y tables. The two entries for wollastonite (T:0-773 and T:0.923) are related by the transformation which we have just described. Clearly the second of these has the smaller primitive translations. There is an alter- native cell for pectolite (T:0-757) which has a shorter b translation t h a n t h a t of the setting given b y Donnay. There are alternative settings for cyclohexane nitro- site (T: 0.705), kaolinite (T: 0.825), and a r a m a y o i t e (T:0.949), but in each case the cell presented in the tables is t h a t with the shorter translations. I n those cases in which two scalar products are zero, there are three alternative quadruples which can easily be found by the reader. We have found a case of this type in D o n n a y ' s tabulation: i.e. parahilgardite (T: 0.504). There are clearly other entries in the table which are subject to the same general type of am- biguity, but we have not a t t e m p t e d a complete discus- sion of them. I n the monoclinic primitive cases there are two zeros on adjacent edges of the tetrahedron and no equalities among the remaining products. There are t h e n three reduced cases and only in one of these will the least two translations in the ac net be exhibited. No a t t e m p t has been m a d e to search for such cases in the D o n n a y t a b u l a t i o n and none is expected because a special reduction for the monoclinic system was used. I n the monoclinic centered lattices caution m u s t be exercised in using Table 5.1.1 of I T X C in the choice of cell translations since in addition to the ambiguities now under discussion there are also possible ambi- guities in allotting the symbols in the formulae and diagrams of the table to a given numerical example. I n these cases and in the primitive and centered lattices of s y m m e t r y higher t h a n monoclinic (where the choice of cell translations should offer no problem) difficulty m a y arise for the second reason cited above, i.e. t h a t one or more changes of sign of zero m a y be necessary before the s y m m e t r y pattern is exhibited in the form in which D e l a u n a y t a b u l a t e d it. This, of course, was recognized b y D e l a u n a y but was not discussed b y Ito and I T X C . The ambiguous cases are exhibited in Fig. 2(a), which is essentially an amplification of the two text figures at the top of p. 136 of D e l a u n a y ' s paper. A practical example of the situation envisaged in the present discussion is presented b y the lattice worked out b y Ito. He obtains a presentation of the lattice C m m m in one of its forms. The other configura- tion exhibited in Fig. 2(a) could equally well have been obtained b y a different m a n i p u l a t i o n of the
A. L. P A T T E R S O N AND W A R N E R E. L O V E 113 original data, and a difficulty would then have been encountered in its interpretation. In all lattices, other than those referred to above, the alternative settings produced by change of sign of zero will correspond merely to a change among the equivalent primitive cells to which the lattice may be referred. The example discussed in I T X C is of this type. Another point made by Delaunay which has not been emphasized by the later writers is that the sym- metry of the twenty-four standard forms is minimal. Extra equalities between products may occur without changing the standard form provided that the added symmetry is not that required by another standard form. This situation is illustrated by Fig. 2(b) in which all the possible equalities for two adjacent zero products are exhibited. We have investigated all possibilities and have found that there are 72 cases which exhibit zeros and equalities in arrangements which are different under C2/m Immm Pmmm AA AA AA C2/m C m mrn Pm3m AA A.A AA (a) 3 1 2 1 1 Cmmm A,A A-A S k -A 1 1 1 1 1 1 (b) Fig. 2. (a) A l t e r n a t i v e p r e s e n t a t i o n s of t h e s ~ n m e t r i c a l cases w h i c h m a y a p p e a r as a result of a D e l a u n a y reduction. These alternatives t r a n s f o r m into one a n o t h e r b y changes of sign of zero. The space-group symbols correspond to the L a u e groups of t h e lattices in question. (b) E x a m p l e s (for t h e cases w i t h two a d j a c e n t zeros) of a d d e d s y m m e t r y which does n o t change t h e s t a n d a r d form. I n each box t h e t a b u l a t e d s t a n d a r d f o r m is given on the left. F o r m s w h i c h are related b y a zero t r a n s f o r m a t i o n are c o n n e c t e d by double arrows. The n u m b e r b e n e a t h each symbol indicates the n u m b e r of n o n - e q u i v a l e n t cells of t h e s a m e s y m m e t r y which m a y be o b t a i n e d f r o m it b y a zero t r a n s f o r m a t i o n . the tetrahedral group. All of them can be shown to be specializations of the 24 cases of Delaunay. These need not be presented in detail here since they can be easily derived and are of little importance provided that the point mentioned in the first two sentences of the preceding paragraph is kept in mind. It is worthwhile mentioning that when the initial cell is badly misset the Delaunay reduction may take many steps. It is therefore important to choose a reasonable cell in preparing the original data, and teachers of crystallography should exercise caution in devising examples for reduction by their students. It is difficult to formulate rules for the best procedure in the reduction. The form of the expression (2) would suggest that it is always best to work with the largest term and this is undoubtedly correct in the earlier stages of the reduction. In the later stages, it is perhaps better to choose a move which does not unduly increase the magnitude of the terms which are already negative. Since the order in which steps are taken in the reduction will not affect the end result, except as to setting, and will merely concern the rate of conver- gence, the routine seems ideally adapted for machine computation. An alternative reduction The approach to the end point of a Delaunay reduction may often be accelerated by using the alternative reduction shown in Fig. 3. In this reduction two b3 -bs A h34+2h23+2h13~ bl bz bl b2 (a) (b) Fig. 3. A n a l t e r n a t i v e r e d u c t i o n (the n o t a t i o n is t h a t of Fig. l(a, b)). The s u m h23+h31 i s a s s u m e d to be positive in (a). The signs of these two t e r m s are c h a n g e d in (b) b y c h a n g i n g t h e sign of b 3 a n d a d d i n g twice b 3 to b4, b 1 a n d b 2 being u n c h a n g e d . The scalar p r o d u c t s for t h e new set are as i n d i c a t e d in (b): i.e. h31 a n d h23 are c h a n g e d in sign, while hi2 is u n c h a n g e d . The t h r e e c h a n g e d p r o d u c t s are hz4+ 2h31, h214- 2h23, h34q- 2h23 q - 2h13. vectors are kept constant and the sign of a third is changed, with the addition of twice the changed vector to the fourth. The changes in the scalar prod- ucts are given in the figure. With this reduction, the change in 2: (cf. equation (4)) is 2:1-2:2 = 2(h23+h31 ) . (5) Thus this transformation is very much more effective than that of Fig. l(a, b). It should always be used when two adjacent products are both positive. It can
114 REMARKS ON THE DELAUNAY REDUCTION Case 1 2 3 Table 1. Tri-acute cell with shortest three non-coplanar translations Vectors Condition Transformation al~-a~ replaces a~ --gl~ < --(glad-g14) A1 = --al; A2---- --al--a2; Aa ~ aa a2-i-a 3 replaces a 3 --g23 < --(g21-~-g24) A1 = a l ; A2---- - - a 2 ; A3 = --a~---a3 aa-~a 1 replaces a a --g13 < --(g12~g14) A1 = --al; A2 ---- a2; Aa ---- --al--aa be used with profit when only one (say h~3 ) is positive provided t h a t the sum h23+hal is greater i h a n one- half of h23. We have called the reduction of Fig. 3 a 'double reduction'. T h e cell w i t h t h e s h o r t e s t t h r e e n o n - c o p l a n a r translations As D e l a u n a y pointed out, the reduced cell is not necessarily the cell with the shortest three non- coplanar translations. If the cell with the shortest three translations is tri-obtuse (cf. Donnay, 1943), it will be given directly by the D e l a u n a y reduction. If t h a t cell is tri-acute, at least two of the shortest three translations will a p p e a r i n ' t h e reduced quadruple and the third will be one of the diagonals a l + a 2, a~.+aa, a 3 + a l in the ordered quadruple (a I _< a 2 _< a 3 _< a4). The three cases which can arise are indicated in Table l, which includes the conditions under which the diagonal vectors replace vectors of the original quadruple in forming the tri-acute cell. I t m a y interest the reader to convince himself t h a t the transforma- tions of Column 4 of the table m a y be obtained on the D e l a u n a y t e t r a h e d r a l diagram by first changing the sign of a~ in such a m a n n e r as to retain a 2 or a a as required and at the same time to exhibit the required diagonal. The second step then consists of a double reduction to m a k e the three scalar products positive, and possibly an inversion if it is necessary to correct the h a n d of the axial system. I t is obvious t h a t if there are zero scalar products in the reduced set, the cell with the shortest three translations can be presented as either non-acute or non-obtuse. Another more serious a m b i g u i t y arises when all scalar products are non-zero, and equalities appear in the Column 3 of the table. I n such cases, there will be a tri-acute and a tri-obtuse cell with the shortest three translations. T h e c o u n t e r p a r t of t h e D e l a u n a y r e d u c t i o n in r e c i p r o c a l s p a c e If we form the triple reciprocal to a n y three of the four vectors of a D e l a u n a y reduced set, then this triple will be non-obtuse since we know t h a t the angles in the reciprocal lattice for a non-acute cell in crystal space cannot be obtuse (cf. Buerger, 1942). The tetra- hedron defined by these three reciprocal vectors then has the property t h a t every one of its vertices projects orthogonally within or on the b o u n d a r y of its opposite face, and the same t e t r a h e d r o n or its inverse is ob- tained when a n y three of the four D e l a u n a y reduced vectors are chosen. We have called such a t e t r a h e d r o n 'compact'. I t is clear t h a t there is a one-to-one corre- spondence between the D e l a u n a y reduced vectors in crystal-lattice space and the faces of the compact t e t r a h e d r o n in reciprocal space. If the D e l a u n a y reduc- tion is unique, as it is when there are no right angles between the reduced vectors, the compact t e t r a h e d r o n in reciprocal space will also be unique. If there are several D e l a u n a y reduced sets in the crystal lattice corresponding to zero values of scalar products there will be corresponding compact t e t r a h e d r a in the recip- rocal lattice. The compact t e t r a h e d r o n is the Voronoi t e t r a h e d r o n L 1 referred to by D e l a u n a y (p. 146). I t m a y be, but it is not necessarily, the principal tetra- hedron as defined b y Bravais (1850, pp. 54--7). The importance of the idea of the compact tetra- hedron lies in the fact t h a t t h e l a t t e r can often be recognized directly in a stack of layers obtained from a series of reciprocal-lattice photographs m a d e on the de J o n g - B o u m a n (1938) retigraph or the Buerger (1942, 1944) precession camera; or in a set of sketches of reciprocal-lattice nets m a d e from d a t a obtained b y other methods. I n a stack of photographs of the layers of a promi- nent zone, one t e t r a h e d r o n will be a p p a r e n t , as for example the t e t r a h e d r o n ABCD of Fig. 4, in which O "',, .i . ......................... L A B Fig. 4. Discussion of compact tetrahedron in reciprocal space. the base ABC is one of the two congruent principal triangles of the net on which the stack is based a n d the v e r t e x D projects inside or on t h e edge of the base and is a lattice point of the net n e x t above t h a t con- taining the base*. This t e t r a h e d r o n will be a compact t e t r a h e d r o n if the dihedral angles DA, DB, DC are all non-obtuse. This fact can be verified by calculation, but if, as is usual, a second stack based, say, on the D A B net is t a k e n and the vertex C falls within the base, it only remains possible t h a t the CD edge could be obtuse. If C does not fall within D A B in the new stack, obviously a new choice C' which does will give * Note that the present discussion refers only to a general region of the reciprocal lattice in which no systematic absences due to glide planes or screw axes occur.
A. L. P A T T E R S O ~ AI~D W A R N E R E. L O V E 115 a tetrahedron which is more n e a r l y compact t h a n the one first chosen. I t is easy to see t h a t this process of 'rolling' a tetrahedron in reciprocal space is equivalent to one or more D e l a u n a y reductions in crystal space. W e do not advocate the use of this process as a replacement of the D e l a u n a y reduction but merely suggest t h a t even a partial use of the concept of the compact tetrahedron in the choice of suitable recip- rocal vectors will lead to a set of lattice vectors which are either D e l a u n a y reduced or only a few steps re- moved from a D e l a u n a y reduced set. T h e t w o - d i m e n s i o n a l a n a l o g u e of t h e D e l a u n a y reduction The reduction of a two dimensional net to the obtuse- angled cell h a v i n g the shortest two translations is simple enough when a scale drawing is made, but from numerical d a t a the reduction m a y be cumbersome. D o n n a y has provided a n o m o g r a m for this reduction. There is, however, a numerical method for the reduc- tion for the two-dimensional net which is strictly analogous to the D e l a u n a y reduction for three-dimen- sional lattices. The scheme for this reduction is given in the caption of Fig. 5. and need not be repeated in b 3 b3 + 2 b l h31 h23 h31+2h12 h23+2h12 b, b~ -b, b2 Fig. 5. Two-dimensional analogue of the Delaunay reduction. The sign of hz2 is reversed by changing the sign of the vector bz, keeping b2, and adding 2b 1 to b 8. The new products are obtained b y adding twice the product which is reversed to the other two. the text. The squared m a g n i t u d e of a n y D e l a u n a y vector is again given b y the negative sum of the prod- ucts which meet at a corner of the triangle. Note also t h a t the q u a n t i t y (~ ~ 2 2 2 ~-- (al +a2-~a3) ---- _ (g12~g23-t-g31) is m i n i m i z e d b y this reduction. Since only one of the three quantities gij can be negative, there-is no choice to be m a d e except for the two equivalent possibilities of choosing either of the two vectors entering into a product for reversal. Discussion There are two possible purposes to be served b y a conventional 'setting' for crystal cell dimensions. The first is to provide a basis for a catalogue of crystal data. The second is to provide a s t a n d a r d framework for the presentation of the results of crystallographic investigations. W h e n the only possible result of a crystal s t u d y was a description and a sketch of its external forms a s t a n d a r d presentation seemed pos- sible. We now know t h a t to present the results of a structure s t u d y we need various sections and projec- tions, and t h a t the choice of these sections and projec- tions m u s t be dictated b y the actual structure rather t h a n by convention. Granted, therefore, t h a t no one s t a n d a r d setting will be adequate for the presentation of structural data, the question of the choice of a set of criteria for catalogue purposes should not be biased b y pictorial considerations. For catalogue purposes alone a set of rules is re- quired which will identify a unique e n t r y for a crystal cell. This e n t r y m a y be accompanied b y a small n u m b e r of cross-references to take care of differences in the accuracy of the d a t a possessed by the user and the compiler of the tables, and to take care of the fact t h a t a crystal of low s y m m e t r y m a y h a v e the lattice dimensions of one of a higher s y m m e t r y . Two proposals of criteria for a conventional cell have received the most discussion: (a) the cell with the shortest three non-coplanar translations; (b) the cell which has as its translations the shortest three translations of the D e l a u n a y reduced quadruple*. Clearly either of these criteria as stated will lead to ambiguous cases, but under either criterion an in- vestigation of all possible ambiguities can be m~de to lead to a set of rules for their resolution. D e l a u n a y has provided a simple algorithm for reaching his reduced quadruple and has given a complete account of the ambiguities involved in this procedure. The only other a m b i g u i t y which can arise in applying the second criterion appears when two or more of the translations of the reduced quadruple are equal in length, and this a m b i g u i t y can readily be removed b y appropriate rules t. No complete investigation of the ambiguities for the first criterion has as yet been made, and no algorithm has been presented for the reduction of data. We have made a partial s t u d y of this problem and have shown how the cells appropriate to the first classification can be obtained from the D e l a u n a y quadruple. We have also considered some of the ambiguities which arise, but our investigation is far from complete. Until such a s t u d y is completed the first criterion cannot be seriously supported as a basis for the classification of crystal data. References BALASHOV, V. (1956). Acta Cryst. 9, 319. BRAVAIS, A. (1850). J. l~c. polyt., Paris, 19, 1. BUERGER, M. J. (1942). X-ray Crystallography. New York: Wiley. * The recent proposal of Balashov (1956) is a mixture of the two. He suggests the use of the tri-obtuse cell with the shortest three non-coplanar translations, but his criteria for the tri-acute cases do not always lead to the tri-acute cell with the shortest three translations. t We are indebted to Prof. Jtirg Waser for pointing out to us the necessity for this remark.
116 REMARKS ON T H E DELAUNAY REDUCTION BUERGEI~, M. J. (1944). The Photography of the Reciprocal Lattice. A.S.X.R.E.D. Monograph No. 1. DELAUNAY, B. (1933). Z. KristaUogr. 84, 109. DOI~NAY, J. D. H. (1943). Amer. Min. 28, 313, 470, 507. (See also Amer. Min. (1952). 37, 1063.) DOENAY, J. D. S . & NOWACKI, W. (1954). Crystal Data. Geological Society of America, Memoir No. 60. International Tables for X-ray Crystallography, (1952), vol. 1, pp. 530-5. Birmingham: Kynoch Press. ITO, T.-I. (1950). X-ray Studies on Polymorphism. Tokyo: Maruzen. JONG, ~V. F. DE ~5 BOUMAN, J. (1938). Z. KristaUogr. 08, 456. Acta Cryst. (1957). 10, 116 An Efficient P r o c e s s for Solvinp, Crystal Structures by Sip, n Relationships :BY M. M. WOOLFSON Physics Department, College of Science and Technology, Manchester 1, England (Received 24 July 1956) I t is shown that, by expressing sign relationships in a suitable tabular form, it is possible to deduce signs for other structure factors if the signs are known for the structure invariants (the structure factors of even indices whose signs do not vary with the choice of origin of the unit cell). The economy introduced by considering only the structure invariants as unknowns, and not all the struc- ture factors, makes possible the determination of reasonably complex structures by hand calcula- tion. The application of the method to the structure of azo-benzene 2-sulphenyl chloride is described. 1. I n t r o d u c t i o n The role of direct methods, which m a k e use of sign relationships for solving crystal structures, is becom- ing progressively more i m p o r t a n t in crystallography but it would be fair to say t h a t , at the present time, t h e y do not seriously challenge older methods in achievement. Almost all the structures which have been solved by direct methods would have yielded to some other method, but the converse is certainly not true. However, direct methods do have the im- p o r t a n t virtue t h a t t h e y m a y often be tried fairly quickly, and failure is not too costly in time. The present p a p e r describes an efficient, systematic process by which sign relationships m a y be applied to the solution of crystal structures. W i t h the aid of electronic computers the process should be capable of application to structures outside the scope of other direct methods hitherto reported. 2, The basis of the method The discussion of this section will, for simplicity, be restricted to two dimensions, although the m e t h o d can be applied in three dimensions. The hk0 projection of salicylic acid (two-dimensional space group pgg) will be used as an example of the application of the method. The reflexions considered are those of large u n i t a r v structure factor which were used by Cochran & Douglas (1954). The reflexions are divided into the following four groups- (a) h even, k even; (b) h odd, k odd; (c) h odd, k even; (d) heven, /todd. The signs of the reflexions of group (a) are indicated by the symbols al, a2, etc., those of group (b) by bl, b2, etc., and so on. For salicylic acid we shall first con- sider only groups (a) and (c), for which the symbols and their significance are listed below. a 1 = s(10, 4) c 1 = s(3, 12) a 2 = s (4, 6) c 2 = s ( 3 , 2 ) a s = s (2, 12) c a = s(1, 6) a 4 = s (2, 10) c 4 = s(7, 2) a 5 = s (6, 2) c 5 = s ( 3 , 6 ) a 6 = s (2, 4) c 6 = 8 ( 5 , 4 ) a T = s (2, 6) c 7 = s ( 9 , 4 ) a s = s (4, 0) a S = s (6, 0) where s(h, k) denotes the sign of the reflexion hk0. The signs of a s and a 9 are found from inequalities to be + and - respectively. All the sign relationships are now sought which relate one m e m b e r of group (a) with two of group (c) and it is found t h a t the following products are prob- ably positive : -alc4c 5 -a7c4c 7 -a5c3c 6 alc2c4 aTClC 3 -a7c4c 6 --a2ClC 3 asc2c 4 --a4c5c 6 --a3c3c5 a6C2C 3 a5c5c 7 a5c2c7 --aTc2e 6 asc6c 7