The main modules of the
program are:
EXPO, LIST, DATA, N-TREOR, EXTRACTION, NORMAL, INVARIANTS, PHASE, FOURIER/LEAST-SQUARES, DIRECT, PATTERSON, RIETVELD.
EXPO module
It interprets commands and calls desired routines.
LIST module
It is the software interface between EXPO2004 and the direct access file on which data and results are stored.
DATA module
This routine reads the
basic crystallographic information like cell parameters, space group symbol
(it may be also found by the program, using "findspace" directive),
wavelength, unit cell content and the pattern (or reflections). It includes
a modified version of the subroutine SYMM Burzlaff & Hountas (1982). Symmetry
operators are directly derived from the space group symbol.
An external file (containing counts or reflections) is supplied by the user.
If an external reflection file is supplied (avoiding the extraction step),diffraction
data are checked in order to find out equivalent reflections or systematically
absent reflections (which are then excluded from the data set).
Using the "peaksearch" directive, a peak search analysis is applied. The
procedure exploits both profile fitting and the derivative method.
N-TREOR module
This routine performs the indexing step if the peak search procedure has been applied or an external peak positions file has been supplied. It retains the basic search algorithm of the indexing program TREOR90 (Werner, Eriksson & Westdahl, 1985) and has been strengthened (Altomare, Giacovazzo, Guagliardi, Moliterni, Rizzi & Werner, 2000).
EXTRACTION module
In this module diffraction
integrated intensities are extracted from the powder diffraction pattern.
To calculate the integrated intensities, the program uses the Le Bail algorithm
(Le Bail, Duroy & Fourquet, 1988) and combines it with a least squares
procedure minimizing the quantity:
|
å wi (yoi - yci) 2 |
(1) |
The summation in (1) goes
over all the data points,
wi = 1 /yoi2
yoi = observed intensity at the i-th step
yci = calculated intensity at the i-th step
 |
(2) |
The summation over h includes neighbouring Bragg h reflections,
|
s
|
=
|
scale factor |
|
LPh
|
=
|
Lorentz - polarization factor |
|
mh
|
=
|
reflection multiplicity |
|
Fh
|
=
|
structure factor |
|
P
|
=
|
profile function |
|
A
|
=
|
asymmetry function |
|
ybi
|
=
|
background intensity at the i-th step. |
The background in (2) is automatically estimated and is described by the polynomial
function (Young, 1993)
N, the polynomial degree,
is automatically defined by the program as the number which provides the best
fit of the background interpolation points.
2qM is the 2q
associated to the minimum intensity in the range in which the background is
estimated.
From the pattern a standard peak is automatically selected. The standard peak
fitting is performed to provide the starting values of the variables to refine
in the least squares procedure.
The full pattern is automatically divided into intervals. The refined parameters
derived by the fitting of each interval will be taken into account for the
final full pattern fitting.
The following profile shape functions are available for user choice:
 |
Pearson VII |
 |
Gaussian |
 |
Lorentzian |
 |
Pseudo-Voigt |
where
 |
 |
 |
 |
 |
 |
 |
The default is the Pearson
VII function. In this case
 |
otherwise
 |
The peak shape, the asymmetry,
the cell and the background parameters are all refined by the program. To
overcome the tendency of the Le Bail algorithm to equipartition the intensities
of strongly overlapping reflections, a new approach was introduced. It is
based on the random variation of the reflection intensities belonging to a
given cluster and on the subsequent selection of the good fit feasible trials.
After a first default run of EXTRA, it is possible to run it again starting
from the refined variables in the previous run. In this case the preliminary
standard peak procedure and the interval by interval extraction procedure
are skipped. The related command is %RECYCLE.
NORMAL module
In this module diffraction
intensities are normalized using the Wilson method. Statistical analysis of
the intensities is made in order to suggest the presence or absence of the
inversion centre, to identify the possible presence and type of pseudotranslational
symmetry (Cascarano, Giacovazzo & Luic', 1988 a,b; Fan, Yao & Qian,
1988) and to detect preferred orientation effects (Altomare, Cascarano, Giacovazzo
& Guagliardi, 1994; Altomare, Burla, Cascarano, Giacovazzo, Guagliardi,
Moliterni & Polidori,1996).
When some additional prior information, besides positivity and atomicity of
electron density, is available, then a suitable renormalization of structure
factors is made.
INVARIANTS module
Up to 10000 triplets relating
reflections with normalized E values greater than a given threshold (strong
triplets) are stored for active use in the phasing process. Also triplets
(PSI-ZERO triplets) relating two reflections with large E and one with E close
to zero are generated: they are used to define a special figure of merit (PSCOMB).
Negative quartets are generated by combining the psi-zero triplets in pairs,
and those with cross-magnitudes smaller than a given threshold are estimated
by means of their first representation, as described by Giacovazzo (1976).
These quartets are used to provide an important contribution to the CPHASE
FOM.
Active triplets may be estimated according to Cochran's (1955) P3 distribution:
the concentration parameter of the von Mises distribution is then:
 |
(3) |
Triplets can also be estimated according to their second representation (i.e. P10 formula, as described by Cascarano, Giacovazzo, Camalli, Spagna, Burla, Nunzi and Polidori, 1984). The concentration parameter of the new von Mises distribution is given by
| G = C (1 + q) | (4) |
where q is a function
(positive or negative) of all the magnitudes in the second representation
of the triplet. The G values are rescaled on the C values and the triplets
are ranked in decreasing order of G. The top relationships represent a better
selection of triplets with phase value close to zero than that obtained when
ranking according to C. These triplets will be actively used in the phase
determination process.
Triplets characterized by a negative G represent a sufficiently good selection
of relationships close to 180 degrees, to be used for the calculation of a
powerful FOM (CPHASE). Triplets with G close to zero are expected to have
values widely dispersed around 90 or 270 degrees and are used to compute an
enantiomorph sensitive FOM. A similar FOM is also computed using quartets
estimated with a very small concentration parameter.
As a default, triplets are estimated according to P10 formula.
The parameter C of the Cochran distribution (say P3) is suitably modified
when the pseudotranslational information in the section 'NORMAL module' is
available. Then triplet phases are no longer expected to be around zero (see
quoted references) and may lie anywhere between 0 and 2p.
The method of the joint probability distribution functions (Cascarano, Giacovazzo,
Guagliardi & Steadman, 1991; Carrozzini, Giacovazzo, Guagliardi, Rizzi,
Burla & Polidori, 1997) can be applied to obtain an estimate of normalized
structure factors, that will be used as a source of supplementary prior information
in the extraction process (if directive ESTIMATION is used).
PHASE module
Triplet relationships are actively used in the phasing process.
- CONVERGENCE/DIVERGENCE PROCEDURE - The convergence procedure
(Germain, Main & Woolfson, 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
 |
(5) |
as suggested by Giacovazzo (1979) and by Burla, Cascarano, Giacovazzo, Nunzi & Polidori (1987), with
 |
(6) |
I1 and I0 represent modified Bessel functions of order one and zero, respectively. The summation in (5) is over all relationships defining the reflection h. If P3 formula is used the default choice is
 |
(7) |
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 <a>.
- PHASE EXTENSION AND REFINEMENT -
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 & Woolfson, 1975; Main, 1978),
b) by a random approach (Baggio, Woolfson, Declercq & Germain, 1978; Burla,
Cascarano & Giacovazzo, 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, Cascarano, Giacovazzo, Nunzi & Polidori, 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, Giacovazzo
& Viterbo (1987) and by Cascarano, Giacovazzo & Guagliardi (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 then CFOM > 0.3 may characterize
the correct solution.
FOURIER/LEAST-SQUARES module
The highest CFOM figure
of merit set of phases generated by the tangent routine is used in the fast
fourier transform routine written by L.F.Ten Eick (1977) and subsequently
modified by the MULTAN team (Main, Fiske, Hull, Lessinger, Germain, Declercq
& Woolfson, 1980).
Cycles of structure factor calculation -least squares- (2Fo -Fc) Fourier synthesis
are automatically performed in order to provide a more complete and refined
crystal structure. Due to reflection overlap the results of the refinement
must carefully be considered by the user taking into account the ratio 'number
of parameters to refine/number of independent observations'.
The program provides the following R-factors which can be used as reliability
parameters:
 |
(8) |
 |
(9) |
 |
(10) |
An alternative approach, based on real space techniques (Altomare, Cuocci, Giacovazzo, Guagliardi, Moliterni & Rizzi, 2002) originally used for phase extension and refinement of proteins (Giacovazzo & Siliqi, 1997; Burla, Camalli, Carrozzini, Cascarano, Giacovazzo, Polidori & Spagna, 2000), has been introduced in EXPO2004 and modified to improve the phasing process from powder data. It combines an Electron Density Modification procedure with Fourier map calculations.
DIRECT module
In this module some "direct-space" approches for solving crystal structures from powder diffraction data, can be used. The basic idea is to actively use structural and geometrical information, available a priori, during the structure solution process.
Labelling procedure
The atomic species assigned to the peaks of an electron density map obtained at the end of a direct method procedure, can be modified using chemical information about the structure. The procedure (Altomare, Giacovazzo, Ianigro, Moliterni & Rizzi, 2001) is activated using "%changelab" command. The prior information on the number and coordination of the heavy atoms (the procedure can only handle tetrahedral and octahedral coordinations) as well as the range of the typical distances between heavy atoms and between heavy and light atoms are required.
POLPO procedures
Using the "%polyhedra" command, two procedures can be activated. Both of them use a Monte Carlo approach and exploit the experimental information about the heavy - atom connectivity. The first one (POLPO1) (Altomare, Giacovazzo, Guagliardi, Moliterni & Rizzi, 2000) is suggested when all the heavy atoms are rightly positioned and labelled. It is able to correctly locate the light atoms. The second one (POLPO2) (Giacovazzo, Altomare, Cuocci, Moliterni & Rizzi, 2002) is able to position one or more missing cations and surrounding anions.
PATTERSON module
In EXPO program it is
possible to compute a PATTERSON map using various coefficients.
By using a special directive (INVERSION) it is possible to exploit the positivity
of the electron density (in the direct space) in the extraction routine. The
so-modified Patterson map is inverted providing a set of structure factor
values to use in a new extraction process.
RIETVELD module
In this module the Rietveld
refinement is performed starting from the profile model supplied by EXTRA
and the structural model given by SIRPOW (eventually modified by the user)
or supplied to the program by an external file (using the directive FRAGMENT
in the FOURIER command).
The procedure can be activated by means of the corresponding option of the
graphic menu.
As soon as the Rietveld option is selected the observed, calculated and difference
patterns are shown on the screen.
The routine adopts the same profile shape function used by EXTRA. An automatic
least-squares refinement can be performed in which the profile, background
parameters, the atomic and cell parameters are refined. After each cycle the
minimized quantity
S = å wi (yi - yci)2
is calculated using the refined parameters.
At the end of the refinement cycles, the following agreement indices are printed on the output file (Hill & Fischer, 1990; Young, 1993):
 |
 |
 |
 |
 |
 |
 |
 |
 |
where:
N is the total number of counts in the pattern
Np is the number of counts belonging to the peak region
wi is 1/yi (default) or 1/yi2