Extraction of the integrated intensities

Extraction of the integrated intensities (%extraction command)

The extraction process is devoted to the decomposition of the measured powder diffraction pattern for extracting the single integrated intensities associated to each reflection in the pattern (Altomare et al., 1995). From the integrated intensities the experimental structure factor moduli are derived. To calculate the integrated intensities, EXPO2014 uses the Le Bail algorithm (Le Bail et al., 1988), suitably combined with a least squares procedure minimizing the quantity:

\sum_{i}w_i(y_{oi} - y_{ci})^2


The summation in (1) goes over all the data points,
w_i= \frac {1}{y_i}
y_oi = observed intensity at the i-th step
y_ci = calculated intensity at the i-th step

y_{ci} = s\sum_{h}LP_hm_h |{F_h}|^2P(2\theta_i - 2\theta_h)A(2\theta_i - 2\theta_h) + y_{bi}


The summation over h includes neighbouring Bragg h reflections,



scale factor



Lorentz – polarization factor



reflection multiplicity



structure factor



profile function



asymmetry function



background intensity at the i-th step.

The background in (2) is automatically estimated and described by the polynomial function (Young, 1993)

y_{bi} = \sum_{m=0}bkg_m[\frac{2\theta_i}{2\theta_M} - 1]^m

m, the polynomial degree, is automatically defined by the program as the number which provides the best fit of the background interpolation points.

From the pattern a standard reference 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:

\frac{\Gamma(\beta_h)}{\Gamma(\beta_h-0.5)} (\frac{C_2}{M})^{\frac{1}{2}}\frac{2}{H_b}(1+4C_2X_{ih}^2)^{\beta_h}

Pearson VII

\frac{C_o^{1/2}}{\sqrt{\pi H_h}}exp(-C_oX_{ih}^2)


\frac{C_1^{1/2}}{\pi H_h}(1+C_1X_{ih}^2)^{-1}


\eta \frac{C_1^{1/2}}{\pi H_h}(1+C_1X_{ih}^2)^{-1} + (1-\eta)\frac{C_o^{1/2}}{\sqrt{\pi H_h}}exp(-C_oX_{ih}^2)





X_{ih} = (\Delta\theta_{ih}/H_h)

H_h = [w + v\cdot \tan{\theta_h} + u\cdot \tan^2{\theta_h}]^{1/2}

H_h^{'} = Z \tan{\theta_h} + T/\cos{\theta}

\eta = \eta_0 + \eta_1 \cdot \tan{\theta_h} + \eta_2 \cdot \tan^2{\theta_h}

\beta_h = \beta_0 + \frac{\beta_1}{2\theta_h} + \frac{\beta_2}{(2\theta_h)^2}

The default choice is the Pearson VII function. In this case

A(i,\textbf{h}) = \frac{a \cdot (2\theta_i -2\theta_\textbf{h})^3}{[\frac{H_\textbf{h}^2}{C_3}+(2\theta_i-2\theta_\textbf{h})^2]^{3/2}} + 1


A(i,\textbf{h}) = 1 - \frac{a \cdot sign(2\theta_i - 2\theta_\textbf{h})\cdot(2\theta_i 2\theta_\textbf{h})^2}{tg\theta_\textbf{h}}

The peak shape, the asymmetry, the cell and the background parameters are all refined by the program.

Directives in the %extraction command

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

To activate the procedure of selection of the profile function type (among Pearson VII, Gaussian, Lorentzian, Pseudo-Voigt) from the best fit of each interval.

To activate the procedure of selection of the profile function type (among Pearson VII, Gaussian, Lorentzian, Pseudo-Voigt) from the best fit of the first interval.

To activate the procedure of selection of the profile function type (among Pearson VII, Gaussian, Lorentzian, Pseudo-Voigt) from the best fit of the standard peak.

To select the gaussian function.

To select the lorentzian function.

To select the Pseudo-Voigt function.

interval n
n is the number of intervals in which the experimental powder pattern is divided (max 8). n is only given if the user wants to change the automatic division of the pattern.

ncycle n
n is the number of cycles for each interval (default value is 60).

To omit the preliminary fitting of the standard peak (automatically selected by the program). In this case a preliminary fitting of the first interval is carried out.

To reduce the background by a constant factor equal to the minimum value  of the observed counts in the experimental pattern.

To skip the background coefficients refinement.

polynomial k
k is the background polynomial degree (max 10). k is given if the user wants to use a fixed degree in each interval and not the degree automatically evaluated by the program.

To activate the smoothing procedure described by Brückner (Brückner, 2000) for estimating the background. In this case the background coefficients are not refined.

To skip the cell parameters refinement.

random p
To activate the random procedure (Altomare et al., 2001) in order to improve the quality of the extracted structure factor moduli values. p is the coefficient for the reflection overlapping definition (default value is 0.5).

frecycle filename.hkl
If the file filename.hkl, containing hkl and the corresponding F-squared values, is available from a previous run of EXPO2014, or from any other source, the Le Bail algorithm starts, for a new decomposition process, by using the provided F-squared values as starting values.

damping x
x is the damping factor used in the least squares routine (default value 0.1). (It is convenient to use a small x value).

width x
x is the factor which, multiplied by the FWHM of each peak in the experimental pattern, gives the total peak width. (In this case the automatic calculation of x is not taken into account).

zeroprofile x
x is the 2θ zero shift value supplied by the user (it is used to correct the 2θ observed positions of the experimental counts). In this case the automatic calculation of the 2θ zero shift is skipped.

To stop the run of EXPO2014 after some relevant calculations and wait for the user graphic interaction to go on. This directive can be used if the graphic interface is active. 

To have a more detailed print in the output file. In this case, the Rp and Rwp values and the values of the refined variables are printed interval by interval.


Example 1
A Pseudo-Voigt function is used and the background in each interval is described by a constant, the graphic window is suppressed.

 %structure mes
 %job MES - data from home diffractometer
 pattern mes.pow
 cell 8.588 9.931 11.105 90.0 93.754 90.0
 content C 24 N 4 O 20 S 4 H 52
 spacegroup p 21/c
 wavelength 1.5406
 polynomial 0

Example 2
The starting values of the integrated intensities in the extraction step by the Le Bail formula are supplied by the user in the file “yono.hkl” and the number of least-square cycles is set to 40.

 %structure yono
 %job YONO - Synchrotron data 
 pattern yono.pow
 cell 9.385 16.394 3.627 90.000 101.067 90.000
 content Y 8 O 26 N 2 H 18
 spacegroup p 21
 wavelength 1.2323
 frecycle yono.hkl
 ncycle 40

Example 3
The preliminary fitting of the standard peak is skipped and the starting 2
θ zero shift value is supplied by the user (the program refines it).

%structure agpz
%job AGPZ - data from home diffractometer
pattern agpz.pow
cell 6.526 20.059 6.464 90.000 90.000 90.000
spacegroup p b c a
content Ag 8 N 16 C 24 H 24
wavelength 1.54056
zeroprof 0.02

Altomare A., Burla M.C., Cascarano G., Giacovazzo C., Guagliardi A., Moliterni A.G.G. and Polidori G. (1995). J. Appl. Cryst. 28, 842-846.
Altomare A. , Giacovazzo C., Moliterni A. G. G. and Rizzi R. (2001). J. Appl. Cryst. 46, 476-482.
Brückner S. (2000). J. Appl. Cryst. 33, 977-979.
Le Bail, A., Duroy, H., Fourquet, J.L. (1988).  Math. Res. Bull. 23, 447-452.