TEMStrain    

Strain tensor in the crystal coordinate system

In order to call the program determining strain with respect the reference lattice parameters, select "CCS" option and click on “Strain” command button (on main interface). This activates "Strain determination CCS" dialog box. There is a possibility to select strain components which are subject of determination. These are components of the strain tensor given in the Cartesian crystal coordinate system. One can also request fitting of voltage and/or camera length. Camera length is then calculated separately for each diffraction pattern. Voltage and is either calculated separately for each diffraction pattern or the same value is fitted to all patterns of the project (if "=" is checked). 

Strain components (and other fitted parameters) are calculated by locating a (supposedly global) minimum  of a goodness-of-fit function. Besides displayed strain componensts, one can also fit their combinations by modifying the "Transformation matrix". Bounds limiting the fitted parameters can be modified in text boxes on the "Strain determination CCS" dialog panel . The bounds are in units of strain*10000 in the case of strain, (relative wavelength)*10000 in the case of  wavelength, and (relative camera length)*10000 in the case of  camera length.

A number of distinct goodness-of-fit functions are available by selecting “Strategy” in “Strain determination CCS”. (CCS stands for crystal coordinate system). 

The first two strategies  (referred to as "Kinematic equation of the line") are based directly on the equation of the diffraction line written in the form f(parameters)=0. The goodness-of-fit is the squared deviation of f(parameters) from zero if "Exact form" is selected. The option "Linearized" uses an approximation of  f(parameters) linear with respect to parameters.

The second group of strategies (reffered to as "Line intersections") uses goodness-of-fit functions based on distances between intersections of diffraction lines. The domain of the intersections can be modified by selecting "Domain" command button on "Strain determination CCS".

In the case, referred to as “Distances”, the function is a sum of squared differences of the type (d-d’) where d represents a distance between HOLZ line intersections in an experimental patterns and d’ is a distance between corresponding intersections in a (kinematically) simulated HOLZ line diagram. 

In the case referred to as “Distance ratios”, the function is a sum of squared differences of the type (d - r d’) where d represents a distance between HOLZ line intersections in an experimental patterns, d’ is a distance between corresponding intersections in a (kinematically) simulated HOLZ line diagram, and r is a magnification factor constant for a given pattern. The magnification factor is a ratio between the average (largest) intersection distances if  “Aver” (“Max”) is selected. The domain of the intersection points (minimal and maximal intersection angles and the radius of a circle encompassing the intersections) can be modified by selecting the "Domain" command button in the "Strain determination" dialog boxes.

The third group of strategies allows for a further refinement by matching elements of experimental and dynamically simulated patterns. This approach can be attempted for high quality experimental patterns and only if the starting values of optimization parameters are already close to their true values. Results can be improved in cases with small number of parameters. Moreover, such "dynamical" refinement is a very slow process, and with multiple runs and multiple patterns, the execution times may easily become prohibitive. Summarizing, the fitting based on dynamical simulation is recommended for a small number of parameters and only after the best possible job was done using the combination of strategies utilizing the kinematical approach.

In some cases, it makes sense to examine "1D scans" of the objective function.

After completion of the calculation, results are displayed in the “Results” dialog box. The box allows for displaying three different groups of results.

• The first one contains values (times 10000) of strain and/or relative values of voltages and/or relative values of camera lengths. If the optimization bounds are reached they are considered to  be incorrect.
• The second group are estimated errors (times 10000) of strain and/or relative values of voltages and/or relative values of camera lengths.
• The third group contains absolute values (times 10000) of lattice parameters and/or voltages and/or camera lengths.

The “Update” in the “Results” dialog box command button will change the current settings for all patterns in terms of  orientations,  lattice parameters and/or voltages and/or camera lengths, whichever were subject of fitting. In the case of strategies based on HOLZ line intersections, the user may select whether to update or not the rotation about the optical axis of the scope. (With the method used in this case (eigenproblem with large differences between eigenvectors), multiple updating may lead to accumulation of errors.)