International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 221230
Section 2.2.5. Phasedetermining formulae ^{a}Dipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 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 of view 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) bywhere 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 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 magnitudesand the ten cross magnitudesIn the following we will denoteConditional 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 directmethods 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:
is given (Tsoucaris, 1970) [see also Castellano et al. (1973) and Heinermann et al. (1979)] by for cs. structures and for ncs. structures. In (2.2.5.27) and (2.2.5.28) we have denoted is an element of , and is the covariance matrix with elements is a K–H determinant: therefore . Let us call the K–H determinant obtained by adding to the last column and line formed by , and , respectively. Then (2.2.5.27) and (2.2.5.28) may be written and respectively. Because is a constant, the maximum values of the conditional joint probabilities (2.2.5.29) and (2.2.5.30) are obtained when is a maximum. Thus the maximum determinant rule may be stated (Tsoucaris, 1970; Lajzérowicz & Lajzérowicz, 1966): among all sets of phases which are compatible with the inequality the most probable one is that which leads to a maximum value of .
If only one phase, i.e. , is unknown whereas all other phases and moduli are known then (de Rango et al., 1974; Podjarny et al., 1976) for cs. crystals and for ncs. crystals where Equations (2.2.5.31) and (2.2.5.32) generalize (2.2.5.11) and (2.2.5.7), respectively, and reduce to them for . Fourthorder determinantal formulae estimating triplet invariants in cs. and ncs. crystals, and making use of the entire data set, have recently been secured (Karle, 1979, 1980a).
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 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.
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