# Changeset 1856 for Tutorials/2DTexture/Texture analysis of 2D data in GSAS-II.htm

Ignore:
Timestamp:
May 14, 2015 12:41:07 PM (8 years ago)
Message:

put spaces around paragraphs

File:
1 edited

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

 r1855 Von DreeleVon Dreele82140921422015-05-14T17:06:00Z2015-05-14T17:30:00Z2015-05-14T17:39:00Z14340 mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} .MsoPapDefault {mso-style-type:export-only; margin-bottom:4.0pt;} @page WordSection1 {size:8.5in 11.0in; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:4.0pt; mso-para-margin-left:0in; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt;

Texture analysis of 2D data in GSAS-II

Introduction

Texture analysis using GSAS-II employs spherical harmonics modeling, as described by Bunge, "Texture Analysis in Materials Science" (1982), and implemented by Von Dreele, J. Appl. Cryst., Texture analysis of 2D data in GSAS-II

Introduction

Texture analysis using GSAS-II employs spherical harmonics modeling, as described by Bunge, "Texture Analysis in Materials Science" (1982), and implemented by Von Dreele, J. Appl. Cryst., 30, 517-525 (1997) in GSAS. The even part of the orientation distribution function (ODF) via the general axis equation

is used to give the intensity corrections due to texture. The two harmonic terms,  and , take on values according to the crystal and sample symmetries, respectively, and thus the two inner o:title="" chromakey="white"/>  terms are nonzero so the rest are excluded o:title="" chromakey="white"/> coefficients is sufficient to describe the effect on the diffraction patterns due to texture. The crystal harmonic factor, , is defined for each reflection, h, via polar and azimuthal coordinates ( , is defined according to polar and azimuthal coordinates (y,

In a diffraction experiment the crystal reflection coordinates (In a diffraction experiment the crystal reflection coordinates (f, b) are determined by the choice of reflection index (hkl) on the diffractometer.

To define the sample coordinates (y, g), we have defined an instrument coordinate system (I, J, K) such that K is normal to the diffraction plane and J is coincident with the direction of the incident radiation beam toward the source. We further define a standard set of right-handed goniometer eulerian angles (W, C, F) so that W and F are rotations about K and C is a rotation about J when W  = 0.  Finally, as the sample may be mounted so that the sample coordinate system (Is, Js, Ks) does not coincide with the instrument coordinate system (I, J, K), we define three eulerian sample rotation offset angles (Ws, Cs, Fs) that describe the rotation from (Is, Js, Ks) to (I, J, K).  The sample rotation angles are defined so that with the goniometer angles at zero Ws and Fs are rotations about I and Cs is a rotation about J.  The zeros of these three sample rotation angles can be refined as part of the Rietveld analysis to accommodate any angular offset in sample mounting. After including the diffraction angle, Q, and a detector azimuthal angle, A, the full rotation matrix, M, is

M = -QAWC(F+To define the sample coordinates (y, g), we have defined an instrument coordinate system (I, J, K) such that K is normal to the diffraction plane and J is coincident with the direction of the incident radiation beam toward the source. We further define a standard set of right-handed goniometer eulerian angles (W, C, F) so that W and F are rotations about K and C is a rotation about J when W  = 0. Finally, as the sample may be mounted so that the sample coordinate system (Is, Js, Ks) does not coincide with the instrument coordinate system (I, J, K), we define three eulerian sample rotation offset angles (Ws, Cs, Fs) that describe the rotation from (Is, Js, Ks) to (I, J, K).  The sample rotation angles are defined so that with the goniometer angles at zero Ws and Fs are rotations about I and Cs is a rotation about J.  The zeros of these three sample rotation angles can be refined as part of the Rietveld analysis to accommodate any angular offset in sample mounting. After including the diffraction angle, Q, and a detector azimuthal angle, A, the full rotation matrix, M, is

M = -QAWC(F+Fs)CsWs

By transformation of unit Cartesian vectors (100, 010 and 001) with this rotation matrix, the sample orientation coordinates (y, g) are given by

cos(y) =  and   tan(

The harmonic terms,  and , are developed from

where the normalized associated Legendre functions, , are defined via a Fourier expansion as

for m even and

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, .  These Fourier coefficients are determined so that the definition

is satisfied.  Terms of the form  and  are combined depending on the symmetry and the o:title="" chromakey="white"/>  and .  For cubic crystal symmetry, the term  is obtained directly from the Fourier expansion

using the coefficients, , as tabulated by Bunge (1982).

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,  and the sample orientation angles

The magnitude of the texture is evaluated from the texture index by

If the texture is random then J = 1, otherwise J > 1; for a single crystal J = ¥.

In GSAS-II the texture is defined in two ways to accommodate the two possible uses of this correction. In one a suite of spherical harmonics coefficients is defined for the texture of a phase in the sample; this can have any of the possible sample symmetries and is the usual result desired for texture analysis. The other is the suite of spherical harmonics terms for cylindrical sample symmetry for each phase in each powder pattern (histogram) and is usually used to accommodate preferred orientation effects in a Rietveld refinement. The former description allows refinement of the sample orientation zeros, Ws, Cs, Fs, but the latter description does not (they are assumed to be zero and not refinable). The sample orientation angles, (W, C, F) are specified in the Sample Parameters table in the GSAS-II data tree structure and are applied for either description.

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). 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.

Step 1. Image input & integration

If you have not done so already, If the texture is random then J = 1, otherwise J > 1; for a single crystal J = ¥.

In GSAS-II the texture is defined in two ways to accommodate the two possible uses of this correction. In one a suite of spherical harmonics coefficients is defined for the texture of a phase in the sample; this can have any of the possible sample symmetries and is the usual result desired for texture analysis. The other is the suite of spherical harmonics terms for cylindrical sample symmetry for each phase in each powder pattern (histogram) and is usually used to accommodate preferred orientation effects in a Rietveld refinement. The former description allows refinement of the sample orientation zeros, Ws, Cs, Fs, but the latter description does not (they are assumed to be zero and not refinable). The sample orientation angles, (W, C, F) are specified in the Sample Parameters table in the GSAS-II data tree structure and are applied for either description.

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). 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.

Step 1. Image input & integration