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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
In xnd a very simple implemantation of the spherical harmonic is use as it is it not obvious to remind the shape of complex hybritation and to choose the rigth one to describe the crystal properties. That is why xnd only uses the simpliest polar function of each degrees, that is the p orbitals and allow to define for each function its polar axis. In the case in which we need a term which does not depend on the direction, we use the non polar s function.
The polar axis is defined with respect to the
orthogonal repear (X, Y, Z) in which Z is
parallel to c* and Y to b. Its orientation is characterized by the angles
of the unit vector
Each function is repeared by an index which the simply
related to the degrre of the associated Legendre polynomial.
Their argument z is the cosine of the angle
of the hkl direction with the polar axis define above.
0 : P1 = 1 1 : P2 = (3 z2-1)/2 2 : P4 = (35 z4 - 30 z2 + 3)/8 3 : P6 = (231 z6 -315 z4 + 105 z2 -5)/16 4 : P8 = (6435 z8 -12012 z6 + 6930 z4 - 1260 z2 + 35)/128
When they share the same axis, all this functions are orthogonal. To construct
a sharp function in a direction, you have to had they up to a degree which
give you the wished sharpness.
For a given polynomial, there is only a restricted number of possible
independant functions. One for P0 which is isotropic, three for P2, five for P4 ...
Then it is not usefull to introduce more than two orientations for P2 or
four for P4. User have to respect the crystallographic symmetry, in a cubic
phase there is no P2 functions, as their sum is a trivial isotropic function.
Orien(hkl) = Coef[0] * P[0,hkl] + Coef[1] * P[1,hkl] + ...The scale factor can be regarded as an isotropic prefered orientation function, therefor we can not introduce the isotropic P0 function to define a prefered orientation function.
Voigt(hkl) = Voigt_Instru(hkl) * Voigt_Sample(hkl)in which the line widths are added according to
Width_Gauss(hkl)^2 = Width_Gauss_Instru(hkl)^2 + Width_Gauss_Sample(hkl)^2 Width_Lorentz(hkl) = Width_Lorentz_Instru(hkl) + Width_Lorentz_Sample(hkl)
Width_Gauss_Sample(hkl)^2 = (Width_G[0, hkl]* P[0, hkl])^2 + (Width_G[1, hkl]* P[1, hkl])^2 + ... Width_Lorentz_Sample(hkl) = Width_L[0, hkl]* P[0, hkl] + Width_L[1, hkl]* P[1, hkl]) + ...
Assym = ( 1 + a . H1(dtheta/width) + b . H3(dtheta/width) )in which H1 and H3 are the odd Hermite functions with polynomial of order 1 and 3; and a and b are functions with different angular dependencies. (Berar, Baldinozzi, J. Appl. Cryst. 1993).
F(h,k,l) = A + i B + i ( a + i b )in which a and b contain all the imaginary part related to f". We will have
I(h,k,l) = k( A^2 + B^2 + a^2 + b^2 + 2 (aB-Ab)) I(-h,-k,-l) = k( A^2 + B^2 + a^2 + b^2 - 2 (aB-Ab)) Ihkl = k( A^2 + B^2 + a^2 + b^2 )
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