Darkfield Position-Frequency Localization

Extended here to the Local Analysis of Fourier Phase

Operator's First Choices...

Begin by defining the number of pixels (n) along the side of each square image, the illumination convergence half-angle (α) in radians, plus in meters: the lateral width of each pixel (mppx), the microscope's vertical focus instability (Δz), the spherical aberration coefficient (Cs), the electron wavelength (λ), and the objective lens defocus setting (Δf)...

(Other user definable parameters are in blue, below.  Note: Display size 558x558 yields 512x512 cut & paste.)

Declarations

Getting the Image...

To load in an image from disk, first convert it into a greyvalue GIF file, with n pixels on a side, stored in c:/test1kc.gif.  Then execute the following two commands...

A point is grabbed by selecting the image, holding down CTRL, clicking on the target location, selecting Edit/Copy from the menu, and the pasting the result after an equals sign below like that below.  This defines the variables apcenter[[1]] and apcenter[[2]] values representing x and y coordinates of the selected aperture location.

Big single darkfield images

Displacement and Strain Maps

Comparisons to Theory

Radial Displacements

Here's a plot along the displacement map diagonal, from lower left to upper right...

Here's a plot of displacement from the center of the lower left "precipitate" proceeding from there to the nearest corner.  The origin of displacement in this plot is of course arbitrary...

Displacement Model:  In the model of displacement used to generate the starting image (following Ashby & Brown, Phil Mag. 8 (1963) 1083-1103), d is the "lattice spacing", epsilon is the fractional misfit, and ro is the inclusion radius.  Here d and ro are in pixels, epsilon is dimensionless.

Note: Part of the displacement problem here is that the jump in displacement from the top of the curve to the bottom may have been more than half of a lattice spacing and therefore in the experimental case subject to a "branch cut reduction".  We'll see in future tests if this improves.

Radial Displacement Gradients

Here's a plot of the gradient of displacement along the LL to UR diagonal, per unit length in that same direction...

Here's a plot of that same gradient as a function of distance from the center of the lower left precipitate...

Strain Model:  Strain, the rate of change of displacement (from the ideal location) per unit length, is in the above model isotropic with no shear components.  When this displacement gradient is positive the strain is tensional (displacement from ideal increases with distance), and when negative the strain is compressive.  Hence in the experimental plot above, one sees that the precipitate bodies show tensional strain, while the lattice immediately around them is compressed.

Note:  The experimental strains above (dots) may be reduced relative to the theoretical value (line) because displacements of less than one pixel were often ignored in the discrete representation on which the analysis was done.  This probably cannot explain all of the difference between theory and experiment.  Insight may be gained with use of larger images having better statistics (and better spatial resolution) in this regard.

monochrome: specimen, image / diffpat, imagePS

complex color: exit wave, photowave / backfocal wave, imageFT

3D plots: imaginary and real parts of contrast transfer/point spread