research papers 700 Giacovazzo and Siliqi  Joint probability distribution functions Acta Cryst. (2001). A57, 700±707 Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Received 2 April 2001 Accepted 3 September 2001 # 2001 International Union of Crystallography Printed in Great Britain ± all rights reserved The method of joint probability distribution func- tions applied to MAD techniques. The two-wave- length case for acentric crystals Carmelo Giacovazzo a,b * and Dritan Siliqi b,c a IRMEC c/o Dipartimento Geomineralogico, Universita Á di Bari, Campus Universitario, Via Orabona 4, 70125 Bari, Italy, b Dipartimento Geomineralogico, Universita Á di Bari, Campus Universitario, Via Orabona 4, 70125 Bari, Italy, and c Laboratory of X-ray Diffraction, Department of Inorganic Chemistry, Faculty of Natural Sciences, Tirana, Albania. Correspondence e-mail: c.giacovazzo@area.ba.cnr.it MAD (multiple-wavelength anomalous dispersion) techniques are often considered as a special MIR (multiple isomorphous replacement) case. The rigorous method of the joint probability distribution functions is applied to solve the phase problem for acentric crystals, on the assumption that the anomalous scatterer's substructure is a priori known. The two-wavelength case is considered: errors in measurements and in the model substructure are handled. The probabilistic approach provides a very simple and ef®cient formula for estimating structure-factor phases. 1. Notation N: number of atoms in the unit cell. a: number of anomalous scatterers in the unit cell. na  N  a: number of non-anomalous scatterers. f j  f 0 j  f j  if 00 j  f 0 j  if 00 j : scattering factor of the jth atom. f 0 is its real, f 00 is its imaginary part. The thermal factor is included.  a ;  na ;  N  P  f 02 j  f 002 j , where the summation is extended to a, na and N atoms. F  jF  j expi'   F h  P N j1 f j exp2hr j  F  a jF  a j expi'  a  P a f j exp2ihr j  F  jF  j expi'   F h  P N j1 f j exp2ihr j  F  a jF  a j expi'  a  P a f j exp2ihr j   ano jF  jjF  j: 2. Introduction The increasing power and tunability of synchrotron beamlines have strongly improved the ef®ciency of the MAD (multiple- wavelength anomalous dispersion) method for solving the phase problem in protein crystallography. The technique exploits the differences among structure-factor moduli generated, at wavelengths around the absorption edges, by the anomalously scattering atoms present in the unit cell (Hendrickson & Ogata, 1997; Smith, 1997). The ®rst step of the procedure aims at locating the anomalously scattering atoms (Terwilliger et al., 1987; Miller et al., 1994; Sheldrick & Gould, 1995). The second step tries to determine the phase values on assuming the partial structure of the anomalously scattering atoms as prior information. Previous probabilistic approaches consider MAD data as special MIR (multiple isomorphous replacement) cases (Blow & Crick, 1959; Terwilliger & Eisenberg, 1987) or adapt Karle's (1980) alge- braic analysis to a probabilistic description (Pa È hler et al., 1990; Chiadmi et al., 1993). This paper applies the rigorous method of the joint probability distribution function to the two- wavelength case on assuming that the anomalously scattering atoms are located. The paper follows: (a) a contribution by Giacovazzo & Siliqi (2001a), from now on paper I, where the joint probability distribution method has been applied to the SAD (single-wavelength anomalous dispersion) case, on the assumption that the positions of all or a part of the anomalous scatterers have been found via one of the current methods; (b) a contribution by Giacovazzo & Siliqi (2001b), from now on paper II, where the MAD case has been treated for symmetry-restricted re¯ections. The two-wavelength case is crucial for MAD data treat- ment: the algebraic aspects have been studied by several authors (i.e. Singh & Ramaseshan, 1968; Unangst et al., 1967; Bartunik, 1978; Cascarano et al. , 1982; Klop et al., 1989). The probabilistic aspects of this case are here investigated: the joint probability distribution functions PF  1 ; F  2 ; F  1 ; F  2 jF  a ; F  a  will be derived, from which the