International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 2.2, pp. 217225
Section 2.2.5. Phasedetermining formulae ^{a}Dipartimento Geomineralogico, Campus Universitario, I70125 Bari, Italy 
From the earliest periods of Xray structure analysis several authors (Ott, 1927; Banerjee, 1933; Avrami, 1938) have tried to determine atomic positions directly from diffraction intensities. Significant developments are the derivation of inequalities and the introduction of probabilistic techniques via the use of joint probability distribution methods (Hauptman & Karle, 1953).
An extensive system of inequalities exists for the coefficients of a Fourier series which represents a positive function. This can restrict the allowed values for the phases of the s.f.'s in terms of measured structurefactor magnitudes. Harker & Kasper (1948) derived two types of inequalities:
Type 1. A modulus is bound by a combination of structure factors: where m is the order of the point group and .
Applied to loworder space groups, (2.2.5.1) gives The meaning of each inequality is easily understandable: in , for example, must be positive if is large enough.
Type 2. The modulus of the sum or of the difference of two structure factors is bound by a combination of structure factors: where stands for `real part of'. Equation (2.2.5.2) applied to P1 gives
A variant of (2.2.5.2) valid for cs. space groups is After Harker & Kasper's contributions, several other inequalities were discovered (Gillis, 1948; Goedkoop, 1950; Okaya & Nitta, 1952; de Wolff & Bouman, 1954; Bouman, 1956; Oda et al., 1961). The most general are the Karle–Hauptman inequalities (Karle & Hauptman, 1950): The determinant can be of any order but the leading column (or row) must consist of U's with different indices, although, within the column, symmetryrelated U's may occur. For and , equation (2.2.5.3) reduces to which, for cs. structures, gives the Harker & Kasper inequality For , equation (2.2.5.3) becomes from which where If the moduli , , are large enough, (2.2.5.4) is not satisfied for all values of . In cs. structures the eventual check that one of the two values of does not satisfy (2.2.5.4) brings about the unambiguous identification of the sign of the product .
It was observed (Gillis, 1948) that `there was a number of cases in which both signs satisfied the inequality, one of them by a comfortable margin and the other by only a relatively small margin. In almost all such cases it was the former sign which was the correct one. That suggests that the method may have some power in reserve in the sense that there are still fundamentally stronger inequalities to be discovered'. Today we identify this power in reserve in the use of probability theory.
For any space group (see Section 2.2.3) there are linear combinations of phases with cosines that are, in principle, fixed by the magnitudes alone (s.i.'s) or by the values and the trigonometric form of the structure factor (s.s.'s). This result greatly stimulated the calculation of conditional distribution functions where , is an s.i. or an s.s. and is a suitable set of diffraction magnitudes. The method was first proposed by Hauptman & Karle (1953) and was developed further by several authors (Bertaut, 1955a,b, 1960; Klug, 1958; Naya et al., 1964, 1965; Giacovazzo, 1980a). From a probabilistic point ofview the crystallographic problem is clear: the joint distribution , from which the conditional distributions (2.2.5.5) can be derived, involves a number of normalized structure factors each of which is a linear sum of random variables (the atomic contributions to the structure factors). So, for the probabilistic interpretation of the phase problem, the atomic positions and the reciprocal vectors may be considered as random variables. A further problem is that of identifying, for a given Φ, a suitable set of magnitudes on which Φ primarily depends. The formulation of the nested neighbourhood principle first (Hauptman, 1975) fixed the idea of defining a sequence of sets of reflections each contained in the succeeding one and having the property that any s.i. or s.s. may be estimated via the magnitudes constituting the various neighbourhoods. A subsequent more general theory, the representation method (Giacovazzo, 1977a, 1980b), arranges for any Φ the set of intensities in a sequence of subsets in order of their expected effectiveness (in the statistical sense) for the estimation of Φ.
In the following sections the main formulae estimating loworder invariants and seminvariants or relating phases to other phases and diffraction magnitudes are given.
The basic formula for the estimation of the triplet phase given the parameter is Cochran's (1955) formula where , is the atomic number of the jth atom and is the modified Bessel function of order n. In Fig. 2.2.5.1 the distribution is shown for different values of G.
The conditional probability distribution for , given a set of and , is given (Karle & Hauptman, 1956; Karle & Karle, 1966) by where is the most probable value for . The variance of may be obtained from (2.2.5.7) and is given by which is plotted in Fig. 2.2.5.2.
Equation (2.2.5.9) is the socalled tangent formula. According to (2.2.5.10), the larger is α the more reliable is the relation .
For an equalatom structure .
The basic conditional formula for sign determination of in cs. crystals is Cochran & Woolfson's (1955) formula where is the probability that is positive and k ranges over the set of known values . The larger the absolute value of the argument of tanh, the more reliable is the phase indication.
An auxiliary formula exploiting all the 's in reciprocal space in order to estimate a single Φ is the formula (Hauptman & Karle, 1958; Karle & Hauptman, 1958) given by where C is a constant which differs for cs. and ncs. crystals, is the average value of and p is normally chosen to be some small number. Several modifications of (2.2.5.12) have been proposed (Hauptman, 1964, 1970; Karle, 1970a; Giacovazzo, 1977b).
A recent formula (Cascarano, Giacovazzo, Camalli et al., 1984) exploits information contained within the second representation of Φ, that is to say, within the collection of special quintets (see Section 2.2.5.6): where k is a free vector. The formula retains the same algebraic form as (2.2.5.6), but where , is assumed to be zero if it is experimentally negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplications in the contributions.
G may be positive or negative. In particular, if the triplet is estimated negative.
The accuracy with which the value of Φ is estimated strongly depends on . Thus, in practice, only a subset of reciprocal space (the reflections k with large values of ) may be used for estimating Φ.
(2.2.5.13) proved to be quite useful in practice. Positive triplet cosines are ranked in order of reliability by (2.2.5.13) markedly better than by Cochran's parameters. Negative estimated triplet cosines may be excluded from the phasing process and may be used as a figure of merit for finding the correct solution in a multisolution procedure.
A strength of direct methods is that no knowledge of structure is required for their application. However, when some a priori information is available, it should certainly be a weakness of the methods not to make use of this knowledge. The conditional distribution of Φ given and the first three of the five kinds of a priori information described in Section 2.2.4.1 is (Main, 1976; Heinermann, 1977a) where stand for h, , , and for . The quantities have been calculated in Section 2.2.4.1 according to different categories: is a suitable average of the product of three scattering factors for the ith atomic group, p is the number of atomic groups in the cell including those related by symmetry elements. We have the following categories.
In early papers (Hauptman & Karle, 1953; Simerska, 1956) the phase was always expected to be zero. Schenk (1973a,b) [see also Hauptman (1974)] suggested that Φ primarily depends on the seven magnitudes: , called basis magnitudes, and , called cross magnitudes.
The conditional probability of Φ in P1 given seven magnitudes according to Hauptman (1975) is where L is a suitable normalizing constant which can be derived numerically, For equal atoms . Denoting gives Fig. 2.2.5.3 shows the distribution (2.2.5.18) for three typical cases. It is clear from the figure that the cosine estimated near π or in the middle range will be in poorer agreement with the true values than the cosine near 0 because of the relatively larger values of the variance. In principle, however, the formula is able to estimate negative or enantiomorphsensitive quartet cosines from the seven magnitudes.

Distributions (2.2.5.18) (solid curve) and (2.2.5.20) (dashed curve) for the indicated values in three typical cases. 
In the cs. case (2.2.5.18) is replaced (Hauptman & Green, 1976) by where is the probability that the sign of is positive or negative, and The normalized probability may be derived by . More simple probabilistic formulae were derived independently by Giacovazzo (1975, 1976): where and . Q is never allowed to be negative.
According to (2.2.5.20) is expected to be positive or negative according to whether is positive or negative: the larger is C, the more reliable is the phase indication. For , (2.2.5.18) and (2.2.5.20) are practically equivalent in all cases. If N is small, (2.2.5.20) is in good agreement with (2.2.5.18) for quartets strongly defined as positive or negative, but in poor agreement for enantiomorphsensitive quartets (see Fig. 2.2.5.3).
In cs. cases the sign probability for is where G is defined by (2.2.5.21).
All three cross magnitudes are not always in the set of measured reflections. From marginal distributions the following formulae arise (Giacovazzo, 1977c; Heinermann, 1977b):
Equations (2.2.5.20) and (2.2.5.23) are easily modifiable when some cross magnitudes are not in the measurements. If is not measured then (2.2.5.20) or (2.2.5.23) are still valid provided that in G it is assumed that . For example, if and are not in the data then (2.2.5.21) and (2.2.5.22) become In space groups with symmetry higher than more symmetryequivalent quartets can exist of the type where are rotation matrices of the space group. The set is called the first representation of Φ. In this case Φ primarily depends on more than seven magnitudes. For example, let us consider in Pmmm the quartet Quartets symmetry equivalent to Φ and respective cross terms are given in Table 2.2.5.1.

Experimental tests on the application of the representation concept to quartets have recently been made (Busetta et al., 1980). It was shown that quartets with more than three cross magnitudes are more accurately estimated than other quartets. Also, quartets with a cross reflection which is systematically absent were shown to be of significant importance in direct methods. In this context it is noted that systematically absent reflections are not usually included in the set of diffraction data. This custom, not exceptionable when only triplet relations are used, can give rise to a loss of information when quartets are used. In fact the usual programs of direct methods discard quartets as soon as one of the cross reflections is not measured, so that systematic absences are dealt with in the same manner as those reflections which are outside the sphere of measurements.
A quintet phase may be considered as the sum of three suitable triplets or the sum of a triplet and a quartet, i.e. or It depends primarily on 15 magnitudes: the five basis magnitudes and the ten cross magnitudes In the following we will denote Conditional distributions of Φ in P1 and given the 15 magnitudes have been derived by several authors and allow in favourable circumstances in ncs. space groups the quintets having Φ near 0 or near π or near to be identified. Among others, we remember:
For cs. cases (2.2.5.24) reduces to Positive or negative quintets may be identified according to whether G is larger or smaller than zero.
If is not measured then (2.2.5.24) and (2.2.5.25) are still valid provided that in (2.2.5.25) .
If the symmetry is higher than in then more symmetryequivalent quintets can exist of the type where are rotation matrices of the space groups. The set is called the first representation of Φ. In this case Φ primarily depends on more than 15 magnitudes which all have to be taken into account for a careful estimation of Φ (Giacovazzo, 1980a).
A wide use of quintet invariants in direct methods procedures is prevented for two reasons: (a) the large correlation of positive quintet cosines with positive triplets; (b) the large computing time necessary for their estimation [quintets are phase relationships of order , so a large number of quintets have to be estimated in order to pick up a sufficient percentage of reliable ones].
In a crystal structure with N identical atoms the joint probability distribution of n normalized s.f.'s under the following conditions:
Advantages, limitations and applications of determinantal formulae can be found in the literature (Heinermann et al., 1979; de Rango et al., 1975, 1985). Taylor et al. (1978) combined K–H determinants with a magicinteger approach. The computing time, however, was larger than that required by standard computing techniques. The use of K–H matrices has been made faster and more effective by de Gelder et al. (1990) (see also de Gelder, 1992). They developed a phasing procedure (CRUNCH) which uses random phases as starting points for the maximization of the K–H determinants.
According to the representations method (Giacovazzo, 1977a, 1980a,b):
The more general expressions for the s.s.'s of first rank are
In other words:
The set of special quartets (2.2.5.35a) and (2.2.5.35b) constitutes the first representations of Φ.
Structure seminvariants of the second rank can be characterized as follows: suppose that, for a given seminvariant Φ, it is not possible to find a vectorial index h and a rotation matrix such that is a structure invariant. Then Φ is a structure seminvariant of the second rank and a set of structure invariants ψ can certainly be formed, of type by means of suitable indices h and l and rotation matrices and . As an example, for symmetry class 222, or or are s.s.'s of the first rank while is an s.s. of the second rank.
The procedure may easily be generalized to s.s.'s of any order of the first and of the second rank. So far only the role of onephase and twophase s.s.'s of the first rank in direct procedures is well documented (see references quoted in Sections 2.2.5.9 and 2.2.5.10).
Let be our onephase s.s. of the first rank, where In general, more than one rotation matrix and more than one vector h are compatible with (2.2.5.36). The set of special triplets is the first representation of . In cs. space groups the probability that , given and the set , may be estimated (Hauptman & Karle, 1953; Naya et al., 1964; Cochran & Woolfson, 1955) by where In (2.2.5.37), the summation over n goes within the set of matrices for which (2.2.5.35a,b) is compatible, and h varies within the set of vectors which satisfy (2.2.5.36) for each . Equation (2.2.5.36) is actually a generalized way of writing the socalled relationships (Hauptman & Karle, 1953).
If is a phase restricted by symmetry to and in an ncs. space group then (Giacovazzo, 1978) If is a general phase then is distributed according to where with a reliability measured by The second representation of is the set of special quintets provided that h and vary over the vectors and matrices for which (2.2.5.36) is compatible, k over the asymmetric region of the reciprocal space, and over the rotation matrices in the space group. Formulae estimating via the second representation in all the space groups [all the base and cross magnitudes of the quintets (2.2.5.40) now constitute the a priori information] have recently been secured (Giacovazzo, 1978; Cascarano & Giacovazzo, 1983; Cascarano, Giacovazzo, Calabrese et al., 1984). Such formulae contain, besides the contribution of order provided by the first representation, a supplementary (not negligible) contribution of order arising from quintets.
Denoting formulae (2.2.5.37), (2.2.5.38), (2.2.5.39) still hold provided that is replaced by where m is the number of symmetry operators and is the Hermite polynomial of order four.
is assumed to be zero if it is computed negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplication in the contributions.
Twophase s.s.'s of the first rank were first evaluated in some cs. space groups by the method of coincidence by Grant et al. (1957); the idea was extended to ncs. space groups by Debaerdemaeker & Woolfson (1972), and in a more general way by Giacovazzo (1977e,f).
The technique was based on the combination of the two triplets which, subtracted from one another, give If all four 's are sufficiently large, an estimate of the twophase seminvariant is available.
Probability distributions valid in according to the neighbourhood principle have been given by Hauptman & Green (1978). Finally, the theory of representations was combined by Giacovazzo (1979a) with the joint probability distribution method in order to estimate twophase s.s.'s in all the space groups.
According to representation theory, the problem is that of evaluating via the special quartets (2.2.5.35a) and (2.2.5.35b). Thus, contributions of order will appear in the probabilistic formulae, which will be functions of the basis and of the cross magnitudes of the quartets (2.2.5.35) . Since more pairs of matrices and can be compatible with (2.2.5.34), and for each pair more pairs of vectors and may satisfy (2.2.5.34), several quartets can in general be exploited for estimating Φ. The simplest case occurs in where the two quartets (2.2.5.35) suggest the calculation of the sixvariate distribution function which leads to the probability formula where is the probability that the product is positive, and It may be seen that in favourable cases .
For the sake of brevity, the probabilistic formulae for the general case are not given and the reader is referred to the original papers.
References
Argos, P. & Rossmann, M. G. (1980). Molecular replacement method. In Theory and practice of direct methods in crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 381–389. New York: Plenum.Avrami, M. (1938). Direct determination of crystal structure from Xray data. Phys. Rev. 54, 300–303.
Banerjee, K. (1933). Determination of the signs of the Fourier terms in complete crystal structure analysis. Proc. R. Soc. London Ser. A, 141, 188–193.
Bertaut, E. F. (1955a). La méthode statistique en cristallographie. I. Acta Cryst. 8, 537–543.
Bertaut, E. F. (1955b). La méthode statistique en cristallographie. II. Quelques applications. Acta Cryst. 8, 544–548.
Bertaut, E. F. (1960). Ordre logarithmique des densités de répartition. I. Acta Cryst. 13, 546–552.
Beurskens, P. T., Gould, R. O., Bruins Slot, H. J. & Bosman, W. P. (1987). Translation functions for the positioning of a well oriented molecular fragment. Z. Kristallogr. 179, 127–159.
Beurskens, P. T., Prick, A. J., Doesburg, H. M. & Gould, R. O. (1979). Statistical properties of normalized differencestructure factors for noncentrosymmetric structures. Acta Cryst. A35, 765–772.
Bouman, J. (1956). A general theory of inequalities. Acta Cryst. 9, 777–780.
Buerger, M. J. (1959). Vector space and its applications in crystal structure investigation. New York: John Wiley.
Busetta, B., Giacovazzo, C., Burla, M. C., Nunzi, A., Polidori, G. & Viterbo, D. (1980). The SIR program. I. Use of negative quartets. Acta Cryst. A36, 68–74.
Böhme, R. (1982). Direkte Methoden für Strukturen mit Uberstruktureffekten. Acta Cryst. A38, 318–326.
Camalli, M., Giacovazzo, C. & Spagna, R. (1985). From a partial to the complete crystal structure. II. The procedure and its applications. Acta Cryst. A41, 605–613.
Cascarano, G. & Giacovazzo, C. (1983). Onephase seminvariants of first rank. I. Algebraic considerations. Z. Kristallogr. 165, 169–174.
Cascarano, G., Giacovazzo, C., Calabrese, G., Burla, M. C., Nunzi, A., Polidori, G. & Viterbo, D. (1984). Onephase seminvariants of first rank. II. Probabilistic considerations. Z. Kristallogr. 167, 37–47.
Cascarano, G., Giacovazzo, C., Camalli, M., Spagna, R., Burla, M. C., Nunzi, A. & Polidori, G. (1984). The method of representations of structure seminvariants. The strengthening of triplet relationships. Acta Cryst. A40, 278–283.
Cascarano, G., Giacovazzo, C. & Luić, M. (1985b). Direct methods and superstructures. I. Effects of the pseudotranslation on the reciprocal space. Acta Cryst. A41, 544–551.
Cascarano, G., Giacovazzo, C. & Luić, M. (1987). Direct methods and structures showing superstructure effects. II. A probabilistic theory of triplet invariants. Acta Cryst. A43, 14–22.
Cascarano, G., Giacovazzo, C. & Luić, M. (1988b). Direct methods and structures showing superstructure effects. IV. A new approach for phase solution. Acta Cryst. A44, 183–188.
Castellano, E. E., Podjarny, A. D. & Navaza, J. (1973). A multivariate joint probability distribution of phase determination. Acta Cryst. A29, 609–615.
Cochran, W. (1955). Relations between the phases of structure factors. Acta Cryst. 8, 473–478.
Cochran, W. & Woolfson, M. M. (1955). The theory of sign relations between structure factors. Acta Cryst. 8, 1–12.
Crowther, R. A. & Blow, D. M. (1967). A method of positioning of a known molecule in an unknown crystal structure. Acta Cryst. 23, 544–548.
Debaerdemaeker, T. & Woolfson, M. M. (1972). On the application of phase relationships to complex structures. IV. The coincidence method applied to general phases. Acta Cryst. A28, 477–481.
Fan, H.F., Yao, J.X., Main, P. & Woolfson, M. M. (1983). On the application of phase relationships to complex structures. XXIII. Automatic determination of crystal structures having pseudotranslational symmetry by a modified MULTAN procedure. Acta Cryst. A39, 566–569.
Fortier, S. & Hauptman, H. (1977). Quintets in : probabilistic theory of the fivephase structure invariant in the space group . Acta Cryst. A33, 829–833.
Gelder, R. de (1992). Thesis. University of Leiden, The Netherlands.
Gelder, R. de, de Graaff, R. A. G. & Schenk, H. (1990). On the construction of Karle–Hauptman matrices. Acta Cryst. A46, 688–692.
Giacovazzo, C. (1975). A probabilistic theory in of the invariant . Acta Cryst. A31, 252–259.
Giacovazzo, C. (1976). A probabilistic theory of the cosine invariant . Acta Cryst. A32, 91–99.
Giacovazzo, C. (1977a). A general approach to phase relationships: the method of representations. Acta Cryst. A33, 933–944.
Giacovazzo, C. (1977b). Strengthening of the triplet relationships. II. A new probabilistic approach in . Acta Cryst. A33, 527–531.
Giacovazzo, C. (1977c). On different probabilistic approaches to quartet theory. Acta Cryst. A33, 50–54.
Giacovazzo, C. (1977d). Quintets in and related phase relationships: a probabilistic approach. Acta Cryst. A33, 944–948.
Giacovazzo, C. (1977e). A probabilistic theory of the coincidence method. I. Centrosymmetric space groups. Acta Cryst. A33, 531–538.
Giacovazzo, C. (1977f). A probabilistic theory of the coincidence method. II. Noncentrosymmetric space groups. Acta Cryst. A33, 539–547.
Giacovazzo, C. (1978). The estimation of the onephase structure seminvariants of first rank by means of their first and second representation. Acta Cryst. A34, 562–574.
Giacovazzo, C. (1979a). A probabilistic theory of twophase seminvariants of first rank via the method of representations. III. Acta Cryst. A35, 296–305.
Giacovazzo, C. (1980a). Direct methods in crystallography. London: Academic Press.
Giacovazzo, C. (1980b). The method of representations of structure seminvariants. II. New theoretical and practical aspects. Acta Cryst. A36, 362–372.
Giacovazzo, C. (1983a). From a partial to the complete crystal structure. Acta Cryst. A39, 685–692.
Giacovazzo, C. (1988a). New probabilistic formulas for finding the positions of correctly oriented atomic groups. Acta Cryst. A44, 294–300.
Gillis, J. (1948). Structure factor relations and phase determination. Acta Cryst. 1, 76–80.
Goedkoop, J. A. (1950). Remarks on the theory of phase limiting inequalities and equalities. Acta Cryst. 3, 374–378.
Gramlich, V. (1984). The influence of rational dependence on the probability distribution of structure factors. Acta Cryst. A40, 610–616.
Grant, D. F., Howells, R. G. & Rogers, D. (1957). A method for the systematic application of sign relations. Acta Cryst. 10, 489–497.
Harker, D. & Kasper, J. S. (1948). Phases of Fourier coefficients directly from crystal diffraction data. Acta Cryst. 1, 70–75.
Hauptman, H. (1964). The role of molecular structure in the direct determination of phase. Acta Cryst. 17, 1421–1433.
Hauptman, H. (1965). The average value of . Z. Kristallogr. 121, 1–8.
Hauptman, H. (1970). Communication at New Orleans Meeting of Am. Crystallogr. Assoc.
Hauptman, H. (1974). On the identity and estimation of those cosine invariants, , which are probably negative. Acta Cryst. A30, 472–476.
Hauptman, H. (1975). A new method in the probabilistic theory of the structure invariants. Acta Cryst. A31, 680–687.
Hauptman, H. & Green, E. A. (1976). Conditional probability distributions of the fourphase structure invariant in . Acta Cryst. A32, 45–49.
Hauptman, H. & Green, E. A. (1978). Pairs in : probability distributions which lead to estimates of the twophase structure seminvariants in the vicinity of π/2. Acta Cryst. A34, 224–229.
Hauptman, H. & Karle, J. (1953). Solution of the phase problem. I. The centrosymmetric crystal. Am. Crystallogr. Assoc. Monograph No. 3. Dayton, Ohio: Polycrystal Book Service.
Hauptman, H. & Karle, J. (1958). Phase determination from new joint probability distributions: space group . Acta Cryst. 11, 149–157.
Heinermann, J. J. L. (1977a). The use of structural information in the phase probability of a triple product. Acta Cryst. A33, 100–106.
Heinermann, J. J. L. (1977b). Thesis. University of Utrecht.
Heinermann, J. J. L., Krabbendam, H. & Kroon, J. (1979). The joint probability distribution of the structure factors in a Karle–Hauptman matrix. Acta Cryst. A35, 101–105.
Karle, J. (1970a). An alternative form for , a phase determining formula. Acta Cryst. B26, 1614–1617.
Karle, J. (1970b). Partial structures and use of the tangent formula and translation functions. In Crystallographic computing, pp. 155–164. Copenhagen: Munksgaard.
Karle, J. (1972). Translation functions and direct methods. Acta Cryst. B28, 820–824.
Karle, J. (1979). Triple phase invariants: formula for centric case from fourthorder determinantal joint probability distributions. Proc. Natl Acad. Sci. USA, 76, 2089–2093.
Karle, J. (1980). Triplet phase invariants: formula for acentric case from fourthorder determinantal joint probability distributions. Proc. Natl Acad. Sci. USA, 77, 5–9.
Karle, J. & Hauptman, H. (1950). The phases and magnitudes of the structure factors. Acta Cryst. 3, 181–187.
Karle, J. & Hauptman, H. (1956). A theory of phase determination for the four types of noncentrosymmetric space groups . Acta Cryst. 9, 635–651.
Karle, J. & Hauptman, H. (1958). Phase determination from new joint probability distributions: space group . Acta Cryst. 11, 264–269.
Karle, J. & Karle, I. L. (1966). The symbolic addition procedure for phase determination for centrosymmetric and noncentrosymmetric crystals. Acta Cryst. 21, 849–859.
Klug, A. (1958). Joint probability distributions of structure factors and the phase problem. Acta Cryst. 11, 515–543.
Lajzérowicz, J. & Lajzérowicz, J. (1966). Loi de distribution des facteurs de structure pour un répartition non uniforme des atomes. Acta Cryst. 21, 8–12.
Langs, D. A. (1985). Translation functions: the elimination of structuredependent spurious maxima. Acta Cryst. A41, 305–308.
Main, P. (1976). Recent developments in the MULTAN system. The use of molecular structure. In Crystallographic computing techniques, edited by F. R. Ahmed, pp. 97–105. Copenhagen: Munksgaard.
Naya, S., Nitta, I. & Oda, T. (1964). A study on the statistical method for determination of signs of structure factors. Acta Cryst. 17, 421–433.
Naya, S., Nitta, I. & Oda, T. (1965). Affinement tridimensional du sulfanilamide β. Acta Cryst. 19, 734–747.
Nordman, C. E. (1985). Introduction to Patterson search methods. In Crystallographic computing 3. Data collection, structure determination, proteins and databases, edited by G. M. Sheldrick, G. Kruger & R. Goddard, pp. 232–244. Oxford: Clarendon Press.
Oda, T., Naya, S. & Taguchi, I. (1961). Matrix theoretical derivation of inequalities. II. Acta Cryst. 14, 456–458.
Okaya, J. & Nitta, I. (1952). Linear structure factor inequalities and the application to the structure determination of tetragonal ethylenediamine sulphate. Acta Cryst. 5, 564–570.
Ott, H. (1927). Zur Methodik der Struckturanalyse. Z. Kristallogr. 66, 136–153.
Podjarny, A. D., Yonath, A. & Traub, W. (1976). Application of multivariate distribution theory to phase extension for a crystalline protein. Acta Cryst. A32, 281–292.
Rae, A. D. (1977). The use of structure factors to find the origin of an oriented molecular fragment. Acta Cryst. A33, 423–425.
Rango, C. de, Mauguen, Y. & Tsoucaris, G. (1975). Use of highorder probability laws in phase refinement and extension of protein structures. Acta Cryst. A31, 227–233.
Rango, C. de, Mauguen, Y., Tsoucaris, G., Dodson, E. J., Dodson, G. G. & Taylor, D. J. (1985). The extension and refinement of the 1.9 Å spacing isomorphous phases to 1.5 Å spacing in 2Zn insulin by determinantal methods. Acta Cryst. A41, 3–17.
Rango, C. de, Tsoucaris, G. & Zelwer, C. (1974). Phase determination from the Karle–Hauptman determinant. II. Connexion between inequalities and probabilities. Acta Cryst. A30, 342–353.
Rossmann, M. G., Blow, D. M., Harding, M. M. & Coller, E. (1964). The relative positions of independent molecules within the same asymmetric unit. Acta Cryst. 17, 338–342.
Schenk, H. (1973a). Direct structure determination in and other noncentrosymmetric symmorphic space groups. Acta Cryst. A29, 480–481.
Schenk, H. (1973b). The use of phase relationships between quartets of reflexions. Acta Cryst. A29, 77–82.
Sim, G. A. (1959). The distribution of phase angles for structures containing heavy atoms. II. A modification of the normal heavyatoms method for noncentrosymmetrical structures. Acta Cryst. 12, 813–815.
Simerska, M. (1956). Czech. J. Phys. 6, 1.
Taylor, D. J., Woolfson, M. M. & Main, P. (1978). On the application of phase relationships to complex structures. XV. Magic determinants. Acta Cryst. A34, 870–883.
Tsoucaris, G. (1970). A new method for phase determination. The maximum determinant rule. Acta Cryst. A26, 492–499.
Van der Putten, N. & Schenk, H. (1977). On the conditional probability of quintets. Acta Cryst. A33, 856–858.
Wolff, P. M. de & Bouman, J. (1954). A fundamental set of structure factor inequalities. Acta Cryst. 7, 328–333.