InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 223-224
## Section 2.2.5.3. Triplet relationships |

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 *j*th 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 so-called *tangent formula*. According to (2.2.5.10), the larger is α the more reliable is the relation .

For an equal-atom 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, 1970*a*; Giacovazzo, 1977*b*).

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.

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