The programs allows the refinement of modulated structures (4-D): it is possible to describe both displacive and compositional modulation. The only limitation is that polarisation vectors may be expanded only to the first order Fourier term. The Fourier term is expressed in exponential form giving two parameters: the amplitude and the phase.
To simplify the treatment of data collected with linear counters possessing unequally spaced detectors and to treat variable step scans, the functions describing the profile are expressed using the measured angles. It should be noticed that the program will not calculate large peaks having the center falling at more than 1 degree in 2theta outside the collected region. The allowed Miller indexes hkl depend on the implementation of the short integer on the computer but run atleast between -127 and +127.
Actually, the quantity which the refinement algorithm tries to minimize is
S = SIGMA(m) SIGMA(n) SIGMA(i) (W(m,n,i) (Y_obs(m,n,i)-Y_cal(m,n,i))^2 + Penalwhere m describes the number of experiences, n is the number of diagrams for each experience and i the number of points on the diagram. The intensities of each diagram Y_calc(m,n,i) are obtained from the formula
Y_calc(m,n,i)= SIGMA(p) SIGMA(hkl) I(m,n,p,hkl) G(theta(m,n,i)-theta_(m,n,p,hkl} ) + Y_bkg(m,n,i)where I(m,n,p, hkl) represents the intensity of the hkl reflection of the p phase in the the n diagram of the m experiment, properly corrected of instrumentation geometry (by scale, Lorentz-polarization factor, multiplicity, preferred orientation ...); on the other hand, G is the function defining the global lineshape. This function takes care of the description of the peak width and may vary from one diagram to another and from phase to phase. It depends on theta, on the asymmetry corrections and, eventually, on the Miller indexes of the peak. The intensity of the peak is distributed on an interval centered on the peak and whose amplitude is at least twice the full width at half maximum (FWHM). The amplitude of this interval can be changed acting on a parameter (EpsProf): the profile lineshape is calculated until the calculated intensity of the peak is lesser than sigma(Y_min(m,n,i)).EpsProf, where sigma(Y_min(m,n,i)) is the standard deviation of the least significant point of the diagram (commonly EpsProf=0.5).
The penalty term, Penal, is a penality function which is calculated on some choosen bonds. To this purpose it is necessary to precise the bond lenght length_o for the bond b in a defined phase p at a certain Temp. The penalty algorithm xi increases the minimized sum S of a quantity proportional to the square of the difference between ideal length_o and actual length_cbond length. The exact form of the penality function is in which d_length can be seen as the standard deviation on the bond length
Penal= SIGMA(n) SIGMA(p) SIGMA(b)((length_o(m,n,p,b) - length_o(m,n,p,b)) /(d_length(m,n,p,b))^2