Quanto
General Information
Work supported by the Italian CNR, Progetto Finalizzato "Beni Culturali".
Quanto is a Rietveld package for quantitative phase analysis (QPA).
It is devoted to automatically estimate the weight fraction of each crystalline phase in a mixture by means of its powder diffraction pattern. Data can be collected with X-Ray or constant wavelength neutron sources.
The maximum number of phases managed by the program is 10; the maximum number of counts is 10000; the maximum number of refinable parameters is 700.
The program is available for Microsoft Windows and Unix (HP, Compaq and Linux) platforms (see the Notes on the implementation).
Quanto Main features
The program has been designed to:
·
require a minimal information in input·
work automatically·
reduce the user intervention and facilitate his interaction by means of a useful graphic interface·
allow the user management of the Quanto Data Bank containing structural phase information·
model preferred orientation (PO) effects by means of the March-Dollase function·
correct microabsorption effects by means of the Brindley correction·
estimate the amorphous content by means of the internal standard method·
handle an external reflection intensities file for phases of unknown crystal structure·
extract integrated intensity of reflections by means of the Le Bail algorithm·
fit anisotropic broadening of peaks
.
Rietveld Method for QPA: theoretical background
The Rietveld method (Rietveld, 1969) applied to QPA requires, for each phase in the mixture, the prior knowledge of the structural model; the mixture pattern is calculated by taking into account the contributing pattern of each phase according to
where the summation is over the number of phases in the mixture and, for each step i,
Yc(i) is the total calculated pattern,
Yb(i) is the calculated background,
Ycj(i) is the calculated pattern for the phase j, according to
where
Sj is the scale factor,
takes into account the contribution of each reflection to
the value of Ycj at the point i,
Ih = mh|Fhc|2Lph , that is the integrated intensity of reflection h, including the multiciplicity (mh), the structure factor modulus (|Fhc|) calculated from the model and the Lorentz-polarization correction (Lph),
P(i,h) is the profile shape function,
A(i,h) is the asymmetry function,
O(h) is the preferred orientation correction.
Each contributing pattern is a function of the phase weight fraction in the mixture; this information is derived from the refined scale parameter Sj (Hill and Howard, 1987; Bish and Howard, 1988), according to
where r j is the density of the phase j.
The quantity minimized in the least squares refinement is
where:
the summation is over the number of counts in the pattern
Yo(i) is the measured count at the step i
wi = 1/ sigma2(i) [sigma2(i) is the variance of the i-th observation]
The degree of fit between observed and calculated patterns is valued by means of suitable Agreement Indices (Hill and Fischer, 1990):
Where:
Np is the total number of counts,
P is the number of refined parameters.
To assess the goodness of structural models employed for the analysis, the RF index is computed for each phase in the mixture:
where:
|Fo| is the "observed" structure amplitude extracted from the experimental pattern
|Fc| is structure amplitude calculated from the model.
As soon as the input data become available, the following steps are carried out:
1) Input Data Processing
Checks correctness of input file and completeness of input data. Absorption correction for data collected in trasmission mode (Bragg-Brentano flat sample, Debye-Sherrer capillary sample) can be applied. For each phase in the mixture, the symmetry operators are directly derived from the space group symbol, the Miller indices of reflections are generated and, reflections multiplicity, angular position and intensity value are calculated from the specified information. The mass absorption coefficient of each phase is computed from the corresponding values of all the atoms in the cell; these values are stored in the SirWare.xen file for Copper and Molibdenum K-alpha radiations; for different wavelengths they should be supplied by the scafact.dat file, edited by the user in the working directory.
2) Preliminary Procedure
Provides :
a) starting values of scale and profile shape parameters for each phase in the mixture by means of the automatic selection and profile fit of a "single peak" or "reference range"
b) a starting value for the 2q zero-shift
c) a background model
3) Rietveld Procedure
Includes:
The whole pattern calculation and an automatic refinement strategy in which refinable parameters are progressively involved in least squares minimization; numerical restrictions to the parameters and a dynamic variation of the damping factors are applied; the c 2 value is checked cycle-by-cycle in order to control the convergence; parameters getting out of allowed ranges are blocked and a suitable message is displayed on the screen; phase weight fractions are calculated after each cycle. The program stops when:
a) the convergence is reached, that is when
for all refined parameters,
are the shift and the estimated standard deviation of the parameter j respectively, as obtained by inverting the normal equations matrix, and
(default value);
b) the automatic refinement strategy is not able to converge to a minimum after 500 l.s.q. cycles;
c) the automatic refinement strategy definitely diverges.
The user's analysis should try to make out the possible causes (many tools are supplied for helping); after that, if any model has been modified or any new information has been introduced, the program can be restarted from the Preliminary or Rietveld procedures. Alternatively, an interactive non automatic refinement can be accomplished.
Besides displaying the preliminary steps of the procedure and the progress of the Rietveld process, the graphic interface is useful for many additional options:
·
to guide the access operations to the Data Bank;·
to write the input file or modify an existing one;·
to modify parameter values;·
to modify the automatic choices of the program (single peak, background modelling, parameters restrictions);·
to apply the user refinement strategy (e.g., at the end of a default run or in a new interactive one);·
to set the window options (size, colours, etc.);·
to have a zoom and/or detailed information about a selected region of the pattern (counts, 2q or d values, Miller indices of reflections), to display the calculated pattern of each phase, the FWHM (Full Width at Half Maximun) and Mixing parameter (for Pearson VII or Pseudo-Voigt function) trend versus 2q , etc.;·
to restart the program from the Preliminary or Rietveld procedure;·
to have a look of the output file or the short results;·
to require a phase reflections list (hkl Miller indices, angular positions, observed and calculated structure factors moduli or intensities, applied corrections) or a bond distances and angles table;·
to require the application of the Le Bail algorithm for the extraction of integrated intensities from the experimental pattern;·
to require the application of the "Find PO" procedure.
A Data Bank containing several organic and inorganic phases is available with the program. The structural information for each phase is stored into an ascii file with extension .pha. It includes: a "Title" line for user notes, cell parameters, space group and, for each atom in the asymmetric unit, atomic specie, fractional coordinates (x, y, z) , isotropic thermal factor (Biso), occupancy factor (for partially occupied atomic sites). An example of .pha file is shown for corundum ( the symbol ">" identifies a comment line)
Title Corundum - Round Robin data
Cell 4.7592 4.7592 12.992 90.0 90.0 120.0
Space R -3 c
> Specie x y z Biso
Coord Al 0.0000 0.0 0.35228 0.32
Coord O 0.3064 0.0 0.250 0.33
The user can have access to the Quanto Data Bank by graphics in order to add new phases or modify the stored information. The program is also able to read a .CIF file downloaded from CSD (Cambridge Structural Database, http://www.ccdc.cam.ac.uk/) or ICSD (Inorganic Crystal Structure Database, http://barns.ill.fr./dif/icsd/), to extract the structural information and to store the corresponding .pha file into the Quanto Data Bank.
To model the background, 3 options are available in Quanto:
1) Young’s Polynomial (default choice)
N = number of coefficients (max 11)
Cj = refinable polynomial coefficients
2) Chebyshev approximation
3) Linear interpolation
Interpolating points are automatically located by the program. In this case the background will be subtracted and not refined in the subsequent procedures.
For each option, a model automatically defined by the program is proposed to the user and can be modified by means of the graphic options.
Models 2) and 3) are suggested when the mixture contains a certain percentage of amorphous material.
Structure factors are computed for each reflection of each phase from the atomic coordinates, isotropic thermal factor and site occupancy read from the .pha file of the phase and from the atomic scattering factors whose coefficients are stored into the SirWare.xen file (for all the species up to Cf). Anomalous dispersion is taken into account by default; anomalous scattering factors for Copper and Molibdenum K-alpha radiations are stored; for different wavelenghts they should be supplied by the scafact.dat file, edited by the user in the working directory.
For QPA only the amplitude of structure factor needs to be calculated:
|F|2 = A2h,j + B2h,j + C2h,j + D2h,j
where, for the phase j
fi,h = (f 0i,h + f 'i) · exp - (Bish2)
f "i,h = f "i · exp - (Bish2)
f 0i = scattering factor of the atom i-th at rest
f 'i = real part of the anomalous dispersion factor of the atom i-th
f "i = imaginary part of the anomalous dispersion factor of the atom i-th
Bi = isotropic thermal factor of the atom i-th
ni = atomic site fraction
t = number of atoms in the asymmetric unit
m = number of symmetry operators
Cs = symmetry operator; s runs from 1 to m
ri = atomic positional vector
Lorentz-Polarization Correction
The Lorentz-Polarization (Lp) correction is applied to the reflection h as a unique correction, computed by combining the Lorentz factor with the appropriate Polarization correction, according to the fact that a crystal monochromator (on the incident or on the diffracted beam) is used or not and according to the radiation source.
Lorentz factor
Polarization correction
K must be supplied by the user
4) neutron data:
Ph = 1.0
5) synchrotron data (incident beam totally polarized in the vertical plane):
Ph = 1.0
To model the profile shape and width of peaks, the following analytical functions are available in Quanto:
Pearson VII (default choice)
To model the asymmetry of peaks, the following analytical functions are available in Quanto:
For Pearson VII profile shape function
For all the other profile shape functions
where a is a refinable asymmetry parameter; for the other symbols see the corresponding profile shape function.
To correct preferred orientation effects of cilindrical symmetry samples, the March-Dollase function (March, 1932 and Dollase, 1986) has been implemented in Quanto:
where G is a refinable parameter and a is the angle between the preferred orientation plane and the vector h.
Max 3 PO planes can be supplied; if more than one plane is supplied, the total correction for each vector h is
where fi is the preferred orientation
fraction of the plane i-th, according to the restraint
fi are refinable parameters.
For data collected in trasmission mode, the following absorption corrections are available to correct the raw data:
1) Debye-Scherrer geometry (capillary sample)
2) Bragg-Brentano (flat sample)
To take into account eventual microabsorption effects, the average particle radius of each phase should be supplied. The values are automatically corrected according to Taylor and Matulis (1992)
For mixtures including an amorphous material, the corresponding percentage can be estimated by means of the internal standard method. The phase added in known quantity to the original mixture should be specified in the input file as well as its percentage. The amorphous content is obtained according to:
where wk is the known fraction of the standard added to the mixture and ws is the corresponding fraction estimated by the Rietveld method.
The weight fraction of the crystalline phases in the mixture (wi’ ) are rescaled by the standard according to:
The weight fraction of each crystalline phase and the amorphous content in the original mixture (wi’' ) are automatically derived according to:
Owing to lattice defects, disorder, compositional variability, etc., the published structural models could not perfectly represent one or more of the actual phases in the mixture. In these situations structure refinement cannot converge to a satisfactory fit with the experimental data and inaccurate quantitative determinations can result. An alternative way to treat the problem has been implemented in Quanto: the intensity values extracted from the powder pattern of the pure phase (as well as from measurements by single crystal experiments) can be used instead of those calculated from the structural model. The intensity values, supplied by an external file, are put on the absolute scale by:
a) comparing them with the calculated intensities (no matter if the structural model is inadequate for single intensities) provided that the unit cell content does not change;
b)
a Wilson-plot (Wilson 1942) ; in this case a modified
.pha file is required,
incuding the unit cell content instead of the atomic positions, as it is
shown in the example:
Title Corundum - Round Robin data
Cell 4.7592 4.7592 12.992 90.0 90.0 120.0
Space R -3 c
Content Al 12 O 18
The Le Bail algorithm (Le Bail et al., 1988) allows to extract the integrated intensity of each reflection of a given phase from the experimental diffraction pattern, according to the following calculation:
The alogorithm is automatically applied by the program to obtain "observed" intensity and structure amplitude values to be written in the reflections list produced by the program; the |Fo| values are used also to compute the RF agreement index for each phase. For these applications I0h = mh|Fhc|2Lph , |Fhc| being the structure amplitude calculated from the model (see the Theoretical background section for description of all symbols).
An additional application of the Le Bail algorithm can be required in Quanto by means of the Le Bail option of the Tools sub-menu. This option allows to use Quanto as an integrated intensities extraction program; it can be useful to produce an intensity reflections file from the experimental pattern of a pure phase with unknown (or partially known) crystal structure (see External Intensity file option). It is not advisable to apply the option for QPA analysis: wrong estimates can derive. For this application, in order to obtain the starting intensity values I0h, is necessary to supply the .pha file of the corresponding phase, including cell parameters and space group; atomic coordinates are suggested if a structural model is known (even if partial or not properly correct for single intensities), otherwise only one atom in random position is required to start the procedure.
Anisotropic
broadening of peaks
The
empirical function proposed by Le Bail and Jouanneaux (1997) to model variations
of both FWHM and peaks shape versus 2q
and hkl Miller indices has been implemented in Quanto. This allows to
improve the full-profile fit by qualitatively taking into account the
anisotropic line broadening and shape, without any reference to a model based on
physical parameters. About the FWHM, a modified Caglioti formula is applied:
Analogous expressions are derived for Vh and Uh and for mixing parameters of Pearson VII and Pseudo-Voigt shape functions.
Wij, Vij and Uij are the elements of a second-rank symmetric tensor; the number is limited by the crystal symmetry.
To install the program Quanto, the following steps should be carried out:
On Unix platforms
1) move the files SirWareQ.pal, SirWare.xen and Quanto.htm and the directory "help" in your HOME directory, i.e.
move SirWareQ.pal $HOME
move SirWareQ.xen $HOME
move Quanto.htm $HOME
move help $HOME
2) create the executable file using the command
make -f file_name
where file_name must be one of the following:
makefile.hp for Hewlett Packard
makefile.axp for Digital Alpha
makefile.linux for Linux
WARNING! All the above makefiles require:
the f90 compiler (except makefile.hp requiring the f77 compiler);
the C compiler;
the X11 Library.
WARNING! The first time the program will take some minutes to create the configuration file SirWareQ.cfg@Local; if errors occur or you are not satisfied of the fonts used by the program, you have to create the SirWareQ.cfg@Local file:
- edit the sirwareQ.cfg.all and extract the part you are interested in;
- assign to this part the name SirWareQ.cfg@Local (upper and lower cases are mandatory)
3) run the test:
quanto cpd1a
or
quanto
then, by using the graphic interface
- click on "File" --> "LOad" --> "Browse" and select cpd1a.qua in data working directory, then click OK;
- click OK.
WARNING! Some Browsers are not able to open the help on line from the Quanto window; in this case it is necessary to directly access the file Quanto.htm using your Internet Browser.
On Microsoft Windows platforms
1) Install Quanto on your DeskTop using the standard link mechanism of windows
2) to run the program, click on Quanto icon,
then, by using the graphic interface
- click on "File" --> "LOad" --> "Browse" and select cpd1a.qua in data working directory, then click OK;
- click OK.
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Altomare, A., Burla, M.C., Cascarano, G., Giacovazzo, C., Guagliardi, A., Moliterni, A.G.G., Polidori, P. (1996). J.Appl.Cryst, 29, 341-345
Azaroff, L.V. (1955). Acta Cryst., 8, 701
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Brindley, G.W (1945). Philos. Mag., 36, 347-369
Dollase, W.A. (1986). J.Appl.Cryst, 19, 267-272
Hill, R.J. and Howard, C.J. (1987). J.Appl.Cryst, 20, 467-474
Hill, R.J. and Fischer, R.X. (1987). J.Appl.Cryst, 23, 462-468
Kahn, R., Fourme, R., Gadet, A., Janin, J., Dumas, C. and André, D. (1982). J. Appl. Cryst., 15, 330-337
Le Bail, A., Duroy, H., Fourquet, J.L. (1988). Math.Res.Bull, 23, 447-452
Le Bail, A., Jouanneaux, A. (1997). J.Appl.Cryst, 30, 265-271
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Rietveld, H.M. (1969). J.Appl.Cryst, 2, 65-71
K.D. Rouse, M.J. Cooper, E.J.York & A. Chakera (1970). Acta Cryst., A26, 682-691
Taylor, J.C., Matulis, C.E. (1991). J.Appl.Cryst, 24, 14-17
Wilson, A.J.C. (1942). Nature, 150-152
Recommended Books:
Klug, H., Alexander, L.E. (1974). X-Ray Diffraction Procedures for Policrystalline and Amorphous Materials. 2° ed., New York: Wiley
Zevin, L.S., Kimmel, G. (1995) Quantitative X-Ray Diffractometry. Springer-Verlag New York, Inc.
Young, R.A. (1993). The Rietveld Method. IUCr Monographs in Crystallography – 5, Oxford Science Publications.