Strain fitting of 2D data in GSAS-II

For this demo, data were collected with a MAR2300 area detector at APS 1-ID-C with a wavelength 0.12398 Å (100mm diameter beam) from a 1.5mm square hot rolled steel test specimen; the front surface has been dusted with some CeO₂ to aid in calibration. Three images are used here; one is with no load on the sample and the other two are with the sample under tension along the y-axis (vertical). The loading will distort the steel (ferrite & austenite) diffraction rings and our purpose here is to characterize this distortion via a strain model by B. B. He & K. L. Smith (Adv. in X-ray Anal. 41, 501, 1997). Note that menu entries are listed in bold face below as Help/About GSAS-II, which lists first the name of the menu (here Help) and second the name of the entry in the menu (here About GSAS-II).

Step 1: Start GSAS-II and read the images

Use the Data/Read image data menu item to read the data file into a new GSAS-II project. A file selection dialog will be shown; its appearance will depend on your OS. Change the search directory to /Exercises/strain-texture/nx09_strain_data and then select the files nx09_strain_001.mar2300, nx09_strain_006.mar2300 and nx09_strain_011.mar2300 and press Open. Select the 1^st image (001) from the GSAS-II data tree and select Image Controls; adjust the Max intensity so that the CeO2 rings are visible (~1500 will suffice). You should see:

For the image and

for the controls.

Step 2: Calibration of the detector position

This image is a superposition of three diffraction patterns; CeO₂ calibrant, ferrite (BCC iron) and austenite (FCC iron). Consequently the calibration needs to be done with some care; the steps are as follows:

Set the wavelength to 0.12398

Set the calibrant to CeO2 SRM674b

Set Min calib d-spacing to 1.09; this avoids picking a CeO₂ ring that overlaps one of the steel ones.

Pick Show ring picks? check box

Set Pixel search range to 5; the rings are fairly widely separated will little chance for overlap.

Use Operations/Calibrate, select 5 points on the inner CeO₂ ring with left mouse button and start calibration with right mouse button. The result should be something like

I’ve zoomed in to show that the CeO2 rings are well fitted and don’t overlap any of the more intense steel ones. The calibration results show on the data window

and the console

You should now copy these results to the other image. Use Operations/Copy controls; a popup dialog will appear; select All IMG and OK.

Step 3: Fitting unstrained steel

In this step you will select the diffraction rings to be analyzed for strain; it is best to pick a few CeO₂ ones as well to confirm that the sample position does not shift during loading. Select the first image (001) and the Stress/Strain tree item. An empty plot (“Strain”) will appear and the initial Stress/Strain data window

Change Sample delta-z to -0.75 since center of the steel sample is that much closer to the detector than the CeO₂ calibrant. Then select 2D Powder Image and follow the instruction to add a ring; press a and use the mouse left button to select a ring. Do as many as you like but be sure to select some CeO₂ rings as well; it is useful to zoom in to help with the selection. You can control the image display with the Image Controls. Each time you pick a ring, GSAS-II will display (briefly) the Strain plot with the points it found and then show the 2D Powder Image zoomed in where you were looking before your selection. I chose 12 rings. My Strain plot showed

Notice that each ring is plotted in different colors (they are somewhat mixed for the two lines ~2Å! We will fix this next.)

The 2D Powder Image shows the picks as well

And the rings are listed on the Stress/Strain data window; a scroll bar allows easy access to all the entries.

We can now fix the mixing of the two 2Å lines; change the Pixel search range to 5 for both. Each change will cause a refresh of the ring selections; after you are done the Strain plot should look like

We are now ready for the first strain fit; do Operations/Fit stress/strain. Each ring will be fitted to the three strain tensor elements (ε₁₁, ε₁₂ & ε₂₂) that can be determined from a single 2D diffraction ring. The resulting Strain plot now shows curves calculated from the fitted tensor elements; these are very nearly straight lines as expected from an unloaded sample. In particular the austenite 111 line (~2.08Å) is much flatter than the ferrite 200 line (~2.03Å); this reflects the sample history (hot rolled steel). The tensor elements with esds are shown on the console and in the data window

The d-zero ave is the average d-spacing around each ring; this can be used as the reference d-spacing. Each d-zero value for this no load sample can be set to this value; this will suppress any residual stress offset. To update all the d-zero values to the calculated ones do Operations/Update d-zero then repeat Operations/Fit stress/strain. Repeat as needed to make d-zero match d-zero ave for all lines. I got for the no load sample

Finally you want this set of d-zero values to be applied to the images from the loaded sample; do Operations/Copy stress/strain to apply them to the other images.

Step 4: Sequential fitting of steel under load

For the result of the sequential fitting to make sense, we need to enter the applied load for each image. Unfortunately, this must be done by hand here because the load value is not the image header. The loads are: 001: - 2.17425, 006: 527.917 and 011: 1004.41 all in neutons (N). Each needs to be divided by the cross section area (2.25mm²) of the sample; GSAS-II will do this bit of math for you, and this gives the values in MPa. For image 001 enter -2.17425/2.25 in the Sample load (MPa) box, 527.917/2.25 for image 006 and 1004.41/2.25 for image 011. When done the data window for image 011 should look like

Note that if the beam line software has prepared the appropriate file containing this information; it could be loaded via the Operations/Load sample data menu item.

Next, we do the sequential refinement. Do Operations/All image fit from the Stress/Strain data window menu from any of the images. A popup window will show the selection of images

Do Set All; all will show a check box. Then do OK; the sequential refinement will proceed doing each image in turn. The console will show detailed results of the fitting for each image with values and esds for each of the tensor values.

I only show the bottom of the listing; ignore the ‘no phases” messages. The data window shows the results of the sequential fit.

This table is the results of the sequential fit to the three images. There are columns for each parameter; if you double click on a column heading (try a steel e11 or e22 – I selected 1;e22 from my list). The plot shows the variation of this parameter with image sequence; press ‘s’ and select Sample load and the plot will show

We can next fit this to a straight line. Select Parametric fit/Add equation, select a variable (Global) in the 1^st pull down box (I chose 1;e22) and enter m*x+b in the equation box; the frame will change to give selections for m, x & b. Use Free for m & b and Global/Sample load for x. Then press Fit. The Expression Editor window should look like; press OK to exit this window.

And the console shows the parameters with esds

The plot will show the best fit line

You can explore other features of this facility; pick more than one column (single click on the label) and the Selected cols/Plot selected menu item to plot them together.

To see the effect of stress on one diffraction line, select the Strain plot and zoom in on one line (I zoomed in on just one line; ferrite 211) from image 011.

When you look at the other lines, you should notice that the ones at ca. 3.13Å, 1.91Å, and 1.63Å remain straight lines; these are from CeO₂ and should be unchanged under load.

Again you could save the project with File/Save project on the main GSAS-II data tree menu although it is not needed for any further tutorials. The strain results from each image can be exported as a comma delimited set of values which can easily be read by Microsoft Excel and other spread sheet programs; see Export/Image data as/Strain CSV file.