Phasing by tangent formula

Phasing by the tangent formula (%phase command)

In the EXPO2014 program the most reliable one-phase structure seminvariants (s.s.) are treated as known phases. Besides triplets, also the most reliable negative quartets and two-phase  s.s. may be actively used.
Each relationship is used with its proper weight: the concentration parameter of the first representation for quartets and two-phase s.s., and C or G for triplets.


The convergence procedure (Germain et al. 1970) is a convenient way of defining an optimum starting set of phases to be expanded by the tangent formula or by any other algorithm.
When the P10 formula is used, as a default, a special convergence process is devised which chooses the starting set according to

\langle \alpha \rangle = \sum_j [G_j \cdot D_1(G_j)] \cdot D_1 (\langle \alpha_{h-k} \rangle)                                                                                               (1)

as suggested by Giacovazzo (1979) and by Burla et al. (1987), with

D_1(G_j) = \frac{I_1(G_j)}{I_0(G_j)}

I1 and I0 represent modified Bessel functions of order one and zero respectively. The summation in (1) is over all relationships defining the reflection h. If P3 formula is used the default choice is

\langle \alpha \rangle = \sum_j [C_j \cdot D_1(C_j)]

Once the starting set has been defined, a good pathway for phase expansion is determined by a divergence procedure. In the divergence map, starting from the reflections in the starting set, each new reflection is linked to the preceding ones with the highest value of \langle \alpha \rangle.


The starting set defined by the preceding step is usually formed by the origin (and enantiomorph) fixing reflections, a few one phase s.s. and a number of other phases which may be obtained:
a) by magic integer permutation (White et al. 1975; Main, 1978),
b) by a random approach (Baggio et al. 1978; Burla et al. 1992).
The option a) is the default, b) runs if the directive RANDOM is used. In this last case a large number (depending on the available computer time) of trials can be requested.
Directives SYMBOLS, SPECIALS, MAXTRIAL may be used to change default values.
Phase expansion and refinement are carried out by means of a tangent formula using triplets, negative quartets, psi-zero triplets and the most reliable two-phase structure seminvariants. In the weighting scheme the experimental distributions of the alpha parameters are forced to match with the theoretical ones (Burla et al. 1987).
For each phase set, several FOM’s are computed using all invariants and seminvariants estimated by means of the representation method. Their meaning and an optimized way of combining all the computed FOM’s to give a highly selective combined figure of merit (CFOM) is described in the papers by Cascarano et al. (1987, 1992).
All FOM’s, as well as the combined CFOM, are expected to be equal to 1.0 for correct solutions. CFOM larger than 0.5 can be considered encouraging.
If pseudotranslational symmetry is present, CFOM>0.3 may characterize the correct solution.

Directives in the %phase command

The following directives must be added after the command %phase in the input file to activate specific non-default procedures:

enantiomorph n
Code of the reflection chosen by the user to fix the enantiomorph (by default the enantiomorph is chosen by the program ).

list n
The number of reflections at the top of the divergence map for which a list of the map is obtained. Default = no printout of the map.

maxtrials n
Maximum number of trials when random approach is used (default value = 100). Only if RANDOM directive is on. (See Example 1 for its use).

minfom x n
x: the program automatically stops when a solution is found with CFOM > x . Default value = 1.00.
n: is the maximum number of phase sets (with the largest combined figure of merit – CFOM) to retain (default value = 20). (See Example 1 for its use).

Used to retain equivalent sets of phases.

origin n(i) phi(i)
Codes and phases of the reflections chosen by the user to fix the origin. The total number of reflections (max 3) needed to fix the origin must be given (by default the origin is chosen by the program).

permute n(i)
Codes of the permuted reflections chosen by the user (i.e. included in the number of required symbols, max 12).

phase n(i) phi(i) wt(i)
Codes, phases and weights of reflections with known phase (max 200). If one card is not sufficient, the directive phase must be repeated in the other card(s).

random n
To use random phases starting sets. (See Example 1 for its use).
n: is the number of random phases. If n is omitted, it is equal to one half of the strong reflections (nstrong).
Maximum value allowed for n = nstrong – 4
Minimum value allowed for n = nstrong / 4
Default value for random generation seed is 67543.
(Not consistent with SYMBOLS directive)

specials n
The number of reflections with restricted phase to be permuted (i.e. included in the number of required symbols, max 12).

symbols n
The number of permuted phases (max 12). Default value = 5.
(Not consistent with RANDOM directive)

To print the table of alpha values (at the beginning of PHASE procedure).

trial n
To print the final phases for the specified trial
n .

To actively use negative triplets in the phasing process (default does not use them).


Example 1
A random approach is used and the best 100 sets of phases, over 250 trials, are retained.

%structure nizr
%job nizr structure
pattern nizr.pow
cell 12.388  8.9269  8.8449  90.0  90.550  90.0
cont zr 8 p 12 o 48 ni 4
space p 21/n 
wave 1.0302
    maxtrials 250
    minfom 1.0 100

Baggio R., Woolfson M.M., Declerq J.P. & Germain G. (1978). Acta Cryst. A34, 883-892.
Burla M.C., Cascarano G., Giacovazzo C., Nunzi A. & Polidori G. (1987).
Acta Cryst. A43, 370-374.
Burla M.C., Cascarano G. & Giacovazzo C. (1992).
Acta Cryst. A48, 906-912.
Cascarano G., Giacovazzo C. & Viterbo D. (1987).
Acta Cryst. A43, 22-29.
Cascarano G., Giacovazzo C. & Guagliardi A. (1992).
Acta Cryst. A48, 859-865.
Germain G., Main P. & Woolfson M.M. (1970). Acta Cryst. B26, 274-285.
Giacovazzo C., (1979). Acta Cryst. A35, 757-764.
Main P. (1978).
Acta Cryst. A34, 31-38.
White P.S. & Woolfson M.M. (1975).
Acta Cryst. A31, 53-56.