LSQINT

Introduction

The purpose of LSQINT is to provide an automatic method for the integration of intensities for fibre diffraction data. The program will handle patterns which are largely crystalline in nature, or patterns which have continuous layerlines, sampling only occurring parallel to the Z axis. More than one lattice can be fitted in a single pattern.

The approach used is to generate profiles and then to fit these profiles to the observed pixel intensities by a robust linear least-squares method or using the maximum entropy algorithm of Skilling and Bryan, Mon. Not. R. astr. Soc. (1984)..

Four background subtraction options are available:

• A global background can be fitted simultaneously with the intensities;
  
• A "roving-aperture" method can be used to estimate the background;
  
• background can be fitted locally by a least-squares method, the process being iterated so that the shape of the background can evolve.
  
• Alternatively, background can be removed by a specialist program (eg. XFIX) before LSQINT is run and no background fitting will be done.

Refinement of the cell and profile parameters is also available by a downhill simplex method. This can be cycled with the intensity fitting procedure in order to provide an overall refinement procedure.

FileName: This section enable to select input and output filenames and output option.The path and filename can be entered directly in the text window or selected by using the File Open Dialog Tool, which is opened by clicking the Browse button. If no path is specified, the current directory is searched.For multi-frame BSL image user is able to select start, last and increment and the binary file type if there are more than one binary file exist.

Input Header FileName: Input to LSQINT is provided by the program FTOREC which maps flat area detector data into reciprocal space coordinates, either cartesian or polar.

Standard deviation Input Header FileName: The remapped image is in standard BSL format with the addition of the second header record being used to store extra image information. Optionally, FTOREC produces an image of the standard deviation of pixel values between the four quadrants of the original pattern. If desired, LSQINT can use this image to weight pixel values.

Input Intensity FileName: input intensity file (the format of this file should be similar to the LSQINT output file)to simulate the image file for the intensities.

Output Image Header FileName:A 2-frame BSL file is also output providing a visible indication of the goodness of fit.
Frame1: Data - background
Frame2: Fit

Output Intensity Ascii FileName:an ascii file containing 7 columns. The first three columns contain integers defining h,k,l, the fourth column gives the reciprocal space radius, R of the sampling point and the fifth, the fibre multiplicity of the reflection.

NoFit :force output of equally weighted profiles in the calculated image

SetZero: set negative intensities to zero between least-squares fitting and R-factor calculation

Xplor output: output composite structure factor amplitudes in a format suitable for the version of X-PLOR modified for polycrystalline fibre diffraction. Amplitudes and estimated standard deviations will be calculated according to the method of Sivia & David Acta Cryst. (1995). A50, 703-714 for non-overlapping peaks.

NoCalculate: by default, LSQINT calculates and outputs the calculated image and background. This prevents this behaviour. Background will be in the first frame corresponding to the current image, the total calculated image will be in the second.

BACK -  This section enable to select Background subtraction methods can be used successively. GLOBAL background fitting is not allowed with MAXENT intensity fitting. 

Global - 8 parameters defining a global background function will be fitted

Circularly symmetric - a background will be formed by radial binning of pixel values, averaging those in the range Lpixel to Hpixel and then fitting a smoothing spline under tension to these values.

Window - Paul Langan's roving window background subtraction method.

Flat plane - local least-squares background fitting and spline fitting.

• Lpixel - The starting position in the sorted pixel list for selecting pixels for the background, expressed as a percentage of the total number of pixels in the window
• Hpixel - The final position in the sorted pixel list for selecting pixels for the background, expressed as a percentage of the total number of pixels in the window
• Smoothing -The smoothing factor for the spline fitting
• Tension -The tension factor for the spline fitting
• Xwidth -The window will be 2*Xwidth + 1 in X
• Ywidth - The window will be 2*Ywidth + 1 in Y
• Xseparation - The separation of the centres of the windows in the X direction
• Yseparation-The separation of the centres of the windows in the Y direction (values in pixel units)
• Flat - The plane will be held horizontal
• Incl - The plane can be inclined (defaults: Xwidth 10, Ywidth 10, Lpixel 0, Hpixel 25 Smoothing 1.0, Tension 1.0)

LIMITS - This section enable to select  limits for profile calculation

Layerlines - The first and last layerlines
Radii- The min. and max. radii (image units)
Dlimits - requires two values to follow, the min. and max. d* values
(defaults Dlimits 0.0 Max(d*) Layerlines 0 (Dmax/c*) Radii first pixel last pixel)

Helix - signals use of helical selection rule for Bessel function orders

• Turn -The number of turns in the c repeat
• Unit - The number of subunits in the c repeat
• Start -The rotational symmetry of the structure
• Stack -The number of vertically stacked helices
• Bessel -The maximum bessel function order to be considered in invoking the selection rule
• Radi -The maximum radius of the structure (defaults: Turn 1, Unit 1, Start 1, Stack 1, Bessel 10)

Mixture - up to five values to follow, corresponding to the c/P(i) where P(i) is the pitch of the i'th component of a multi-component system. The layerline is allowed if any of the c/P(i) = an integer
(default behaviour - no selection rules, all layerlines allowed)

parameter profile parameter refinement

Data type (BRAGG/CONTINUOUS) - Fitting will be performed for Bragg data or for continuous layerline intensity
Cell - 6 cell parameters (a,b,c ,alpha,beta,gamma) for Bragg data, 1(c) for continuous.
Cstar - Defines the initial cell orientation with c* as the axis parallel to the Z axis of the particle (for monoclinic cells, etc). By default, the initial orientation has c parallel to Z. a* always starts in the Z-X plane.

Spacegroup - Space group symbol for Bragg data (default P1)

Missetting - The user to specify the following misetting angles

• Phiz - Rotation of the initial orientation around the reciprocal space particle Z axis.
• Phix - Rotation (after application of Phiz) of the cell around the reciprocal space particle X axis.(defaults Phiz 0.0 Phix 0.0)

profile parameter

• Awidth - The width of the orientation distribution function
• Shape - The shape of the orientation distribution function (1->Lorentzian peak profiles, 10->Gaussian)
• Zwidth - 1/particle length
• R0width -1/particle width
• R1width - distribution of a*,b*
• R2width - Paracrystallinity (N.B. Rwidth = R0width + R1width*R + R2width*(R**2))

(defaults Awidth 0.001, Shape 3.0, Zwidth 0.001, R0width 0.001, R1width 0.0, R2width 0.0)
The initial shifts for cell and profile parameter refinement (N.B. Parameters with initial shifts equal to zero will not be refined)

Add - allow to specify more than one lattice positions and profiles.

REFINE/FITTING

Refine - cell and profile refinement

• Number of iterations for refinement
• Number of cycles with intensity fitting
• R-factor shift tolerance (defaults Itamx 20, Ifit 1, Tolerance 0.01)

Signals Calculating the standard deviations of pixel values

• Minimum- Te base standard deviation for every pixel
• Factor - Te coefficient of Sqrt(I) (Defaults Minimum 1.0, Factor 0.0) (N.B. SD(I) = Minimum + Factor*Sqrt(I) )

Signals fitting of intensities by maximum entropy

• Default -Fixed default pixel value for entropy calculation
• Rate -The maximum step size for each iteration
• Cycles-The maximum number of iterations
• Test-Convergence criterion for fit
• Chifactor -Factor determining target chi-squared (defaults Default 1.0, Rate 0.3, Cycles 100, Test 0.1, Chifactor 3.29)

Run - Signals end of input and causes the final lattice positions and profiles to be calculated.

Hints:
If the pattern contains a mixture of crystalline sampling and continuous transform, LSQINT, will not deal with a mixture of the two. However LSQINT still can be used to measure the two components (crystalline and continuous) separately - you will have to relate/scale them in some way later on.

LSQINT effectively deals with the Lorentz factor. It also deals with the "normal" polarisation correction. However it does not deal with the extra (very much smaller) horizontal polarisation correction that is required for SR data. This is something that should be taken care of after background subtraction and before the data goes into FTOREC (which does quadrant averaging), since it will depend on sample rotation relative to the orbital plane.

Output

A log file, LSQINT.LOG, is opened when the program is executed. This contains information about the input file, consequences of the input, the parameters for which the profiles are calculated, R-factors and any error messages or warnings. The R-factors which are reported are calculated by the formula,

R'' = sqrt(sum(Po-Pc)**2/sum(Po**2))

where the Po are the observed pixel values minus background and the Pc are the calculated pixel values.

Output is in the form of an ascii file containing 7 columns. The first three columns contain integers defining h,k,l, the fourth column gives the reciprocal space radius, R of the sampling point and the fifth, the fibre multiplicity of the reflection. A non-zero fibre multiplicity is assigned only to the last reflection in the list constituting a multiplet. A reflection is considered to be part of a multiplet with the previous reflection if their positions fall within the same pixel on the input image. The last two columns contain real numbers describing the integrated intensity of the spot and its standard deviation. The standard deviation is calculated directly from the differences between the calculated and observed pixel values over the predicted profile of the peak. A file, DRAGON.OUT, is also created containing a full list of the reflections and their reciprocal space coordinates.

A 2-frame BSL file is also output providing a visible indication of the goodness of fit.
Frame1: Data - background
Frame2: Fit

when the least squares option is used (as opposed to the maximum entropy option) a file called "cov.dat" is output, which gives the covariance matrix