# Changeset 4962 for Tutorials/2DTexture

Ignore:
Timestamp:
Jun 18, 2021 8:44:17 AM (20 months ago)
Message:

revise equations to match code

Location:
Tutorials/2DTexture
Files:
69 edited

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Removed
• ## Tutorials/2DTexture/Texture analysis of 2D data in GSAS-II.htm

 r4961 Von DreeleVon Dreele, Robert B.829531030352021-06-08T17:35:00Z2021-06-17T16:44:00Z14709268442021-06-18T13:41:00Z40473827010Argonne National Laboratory2236231491225633168516.00 font-weight:bold;} h2 {mso-style-priority:9; {mso-style-noshow:yes; mso-style-priority:9; mso-style-qformat:yes; mso-style-link:"Heading 2 Char"; span.Heading2Char {mso-style-name:"Heading 2 Char"; mso-style-noshow:yes; mso-style-priority:9; mso-style-unhide:no; style='mso-bidi-font-style:normal'>Lm(h)(y)Ln(y)(h)hy and (h)GSAS-II according to the space group symmetry and the user chosen sample symmetry. The available sample symmetries are cylindrical symmetry, 2/m, mmm and no symmetry. The choice of sample symmetry profoundly affects the selection of harmonic coefficients. For example, in the case of cylindrical sample symmetry (fiber texture) only  terms are nonzero so the rest are excluded normal'>CLm0oncoefficients is sufficient to describe the effect on the diffraction pattern due to texture. The crystal normal'>kLmn(h), is defined for each reflection, h, via polar and azimuthal coordinates (kLnm(y), is defined according to polar and azimuthal coordinates (y, φ,βψ,γLn(ψ,γ)(φ,β)
top:15.0pt;mso-text-raise:-15.0pt;mso-ansi-language:EN-US;mso-fareast-language: EN-US;mso-bidi-language:AR-SA'>
and   tan(

style='mso-bidi-font-style:normal'>Lm
(φ,β)(ψ,γ) and , are developed from (those for , are developed from

are similar)

kLmneimβinβL m(

style='mso-bidi-font-style:normal'>PLmn(x), are defined via a Fourier expansion as

style='mso-bidi-font-style:normal'>L mnL msns

for m even and

margin-left:0in;line-height:115%'>for n even and

L msnsL msns

for m odd. Each sum is only over either the even or odd values of s, respectively, because of the properties of the Fourier coefficients, ns.  These Fourier coefficients are determined so style='mso-bidi-font-style:normal'>L mnL mnL+mL+n!L-mL-n!(-1)L-mL-n-m-n2dL-mL-ndxL-mL-n

style='font-family:"Cambria Math",serif'>cos
()(nβ)L m(cosnβ) and  are combined depending on the symmetry and the value of m along with appropriate normalization coefficients to give the value of n along with appropriate normalization coefficients to give the harmonic terms  and .  For cubic crystal symmetry, the term  is obtained directly from the Fourier normal'>kLmnn=0j=0LBLnmnj

normal'>BLnm
nj, as tabulated by Bunge (1982).

Note that this version of the general axis equation differs from that shown in Von Dreele (1997) in that the assignment of m and n are reversed.

The Rietveld refinement of texture then proceeds by constructing derivatives of the profile intensities with respect to the allowed harmonic coefficients, , and the three sample orientation angles, Ws, Cs, Fs, all of which can be adjustable parameters of the refinement. Once the refinement is complete, pole figures for any reflection may be constructed by use of the general axis equation, the refined values for  and the sample orientation angles

if looking from the x-ray source. The sample is assumed to be a capillary (which may be spun to impose cylindrical symmetry), although other sample shapes may be used, and is aligned with the cylinder axis horizontal. Integration of the image from a series of caked slices gives a set of powder patterns, each assigned an azimuthal angle where zero is along the X-axis. Thus, this diffraction plane is horizontal and contains the cylinder axis so W, C, F = 0.

shapes may be used, and is aligned with the cylinder axis horizontal. Integration of the image from a series of caked slices gives a set of powder patterns, each assigned an azimuthal angle where zero is along the X-axis. Thus, at azimuth=0 the diffraction plane is horizontal and contains the cylinder axis so W, C, F = 0.

In this tutorial we will use both of these descriptions to determine the texture of the two phases in a NiTi shape memory alloy sample with cylindrical symmetry (wire texture) as collected at APS on beam line 1ID-C (data kindly provided by Paul Paradise & Aaron Stebner of Colo. School of Mines) with the sample mounted so the cylinder axis was vertical. Thus, there are three ways within GSAS-II that can be used for this texture analysis all beginning with the same 2D area detector image. Each will be described in turn after the initial setup of the GSAS-II project, image input & integration.

margin-left:0in;line-height:115%'>In this tutorial we will use both of these descriptions to determine the texture of the two phases in a NiTi shape memory alloy sample with cylindrical symmetry (wire texture) as collected at APS on beam line 1ID-C (data kindly provided by Paul Paradise & Aaron Stebner of Colo. School of Mines) with the sample mounted so the cylinder axis was vertical. Thus, there are three ways within GSAS-II that can be used for this texture analysis all beginning with the same 2D area detector image. Each will be described in turn after the initial setup of the GSAS-II project, image input & integration.

src="Texture%20analysis%20of%202D%20data%20in%20GSAS-II_files/image118.png" alt="GSAS-II plots: <unnamed project>" v:shapes="Picture_x0020_1">

and the

o:title=" <unnamed project>"/>

o:title=" <unnamed project>"/>

o:title=" <unnamed project>"/>

o:title=" <unnamed project>"/>

o:title=" <unnamed project>"/>

o:title=" NiTi"/>

o:title=" <unnamed project>"/>

o:title=" NiTi"/>

o:title=" <unnamed project>"/>

o:title=" <unnamed project>"/>

o:title=" <unnamed project>"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

style='mso-no-proof:yes'>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

This shows the probability of reflection vectors coinciding with the sample wire axis; the high spots are the 111 family of reflections. We should try the next higher harmonic order (10).

high spots are the 111 family of reflections. We should try the next higher harmonic order (10).

Next go to the o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

o:title=" NiTi-A"/>

the tutorial on Method A for doing texture analysis. It is useful for a case like this one where there are very few data sets required for the texture analysis. However, for the case of lower sample symmetry one must several dozen or even a few hundred histograms and then the suite of parameters can easily be > 1000 of which only a few dozen describe the texture. This leads to the next Methods for texture analysis in GSAS-II.

analysis. However, for the case of lower sample symmetry one must have several dozen or even a few hundred histograms and then the suite of parameters can easily be > 1000 of which only a few dozen describe the texture. This leads to the next Methods for texture analysis in GSAS-II.

This begins the texture analysis in much the same way as Method A except that all the initial refinements are done sequentially, that is refinements are done for the parameters associated with each powder pattern to convergence in a serial the texture analysis in much the same way as Method A except that all the initial refinements are done sequentially, that is refinements are done for the parameters associated with each powder pattern to convergence in a serial fashion. In this case where there are 10 PWDR data sets, there will be 10 refinements done in sequence. Parameters that span all the data (e.g. lattice mso-hansi-theme-font:minor-latin'>Phase fraction box and change the Preferred orientation model to Preferred orientation model to Spherical harmonics. Then do

o:title=" NiTi-B"/>

You have the choice of starting at the last one and for copying results from one to the next. We wont do either here. Each refinement will use the other controls (e.g. Max cycles) as controls. We are now ready to do the 1st sequential refinement. o:title=" NiTi-B"/>

o:title=" NiTi-B"/>

o:title=" NiTi-B"/>

minor-latin'>Harmonic order
to 6 and check the Refine box for it (Preferred orientation). Then in two steps, first do 6 and check the Edit Phase/Copy flags and Set All for the file selection. Then do Edit Phase/Copy selected data; that will bring up a new popup

minor-latin;mso-bidi-theme-font:minor-latin'>Refine box for it (Preferred orientation). Then in two steps, first do Edit Phase/Copy flags and Set All for the file selection. Then do Edit Phase/Copy selected data; that will bring up a new popup

This allows you to select which parameters to copy data and flags. Select Pref. Ori. and press This allows you to select which parameters to copy data and flags. Select OK; minor-latin;mso-hansi-theme-font:minor-latin;mso-bidi-theme-font:minor-latin'>Pref. Ori. and press OK; the file selection is next. Do Showing that perhaps the B19 phase needs high order spherical harmonics. Select the Showing that perhaps the B19 phase needs high order spherical harmonics. Select the Data o:title=" NiTi-B"/>

o:title=" NiTi-B"/>

o:title=" NiTi-B"/>

The preferred orientation correction is Prfo; notice a few entries in red. These are nonphysical correction values (physically, the correction cant be negative) but by in large these are small. This next step uses these

The preferred orientation correction is Prfo; notice a few entries in red. These are nonphysical correction values (physically, the correction cant be negative) but by in large these are small. This next step uses these corrections as input data for a texture refinement. To start select the Texture/Refine texture from the menu; the window will be repainted, and a bullseye pole figure will appear. Change the Texture/Refine texture from the menu; the window will be repainted, and a bullseye pole figure will appear. Change the

o:title=" NiTi-C"/>

o:title=" NiTi-C"/>

o:title=" NiTi-C"/>

o:title=" NiTi-C"/>

o:title=" NiTi-C"/>

o:title=" NiTi-C"/>

o:title=" NiTi-C"/>

• ## Tutorials/2DTexture/Texture analysis of 2D data in GSAS-II_files/filelist.xml

 r4961
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