Homework 6: Calculation of Misorientation; A.D. Rollett, , Texture, Microstructure and Anisotropy

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1 Homework 6: Calculation of Misorientation; A.D. Rollett, 27-75, Texture, Microstructure and Anisotropy Due date: 8 th November, 211 Corrected 8 th Nov. 211 Q1. [6 points] (a) You are given a list of orientations and your task is to calculate the misorientations between them, being careful to find the smallest misorientation possible. You may use any method that is convenient to you. You must calculate both the magnitude of the misorientation and the rotation axis. Also give the values of the Rodrigues vectors that correspond to each axis-angle set. (b) Also compute how near each misorientation is to each CSL type up to Σ=29 and record how near it is in terms of the Brandon criterion. Remember that the angle that you quote must be the minimum misorientation angle, which is what we generally take as the physically most meaningful quantity. To make the latter easier, you may use the axis that is associated with the minimum misorientation matrix even though it may have some negative values [sorting out which symmetry operators place the all the axes in the same stereographic triangle is harder to do]. Remember also to include switching symmetry in addition to the crystal symmetry operators. Finally, each of these components has several positions, based on the orthorhombic sample symmetry. You must calculate the misorientations between each possible pair of component+ because each is a physically distinct orientation (try sketching pole figures for the s to convince yourself about this, if it is not immediately obvious). This means that you will have to generate the set of Euler angles (or whichever representation that you use) for the s before you compute the misorientations [suggestion: you can use mill2eul.f to compute all the equivalent Euler angles for a specific texture component]. You must include a copy of your code or spreadsheet with your submission to show how you performed the calculation. No. Name Indices Bunge (ϕ 1,Φ,ϕ 2 ) RD= 1 1 Copper {112} 111 4, 65, S3 {213} , 37, Brass {11}<112> 35, 45, 2 Number of Variants (sample symmetry)

2 To aid you in confirming that your calculations are coming out correctly, the table below lists the misorientation angles (to two significant figures) that can arise between each combination of components. Copper S3 Brass Copper, 6 19, S3 [you tell me!] 19, 54 Brass, 6 Your final submission should look like this. Note that each entry has only one angle and axis, so be sure to sort your answers properly! If the misorientation angle is zero (no grain boundary!) then the axis is irrelevant, obviously. 1 st 2nd Brass, 1 st Brass, 2 nd Euler angles Brass, 1st Brass, 2 nd Table of Misorientation Angles and Axes S3, 1 st S3, 2 nd S3, 3 rd S3, 4 th 1 st 2 nd 6, <111> {1/ 3,1/ 3,1/ 3} S3, 1 st

3 S3, 2nd S3, 3 rd S3, 4 th Q2. [2 points] The maximum misorientation angle that is possible between two cubic crystals has been given (in L18). (a) Explain how the value is obtained (hint: refer to Rodrigues space) in relation to crystal symmetry elements. (b) Now repeat the calculation for hexagonal crystals and determine the maximum possible misorientation between, e.g., a pair of zirconium crystals.

4 Q3. [2 points] The shape of the FZ for orientations of hexagonal materials is a dodecagonal prism, see the diagram (d) reproduced from the book by Sutton & Balluffi below. Explain the shape in terms of what you learned about the action of symmetry operators for determining the shape of the FZ for cubic orientations, i.e. the truncated cube (shown in (f)). Q4. [extra credit, 2 points] Explain the shape of the FZ for hexagonal-hexagonal grain boundaries, which is a 3 sector cut out of the dodecagonal prism for orientations of hexagonal crystals. Remember that you need to explain how symmetry is used to arrive at the shape of the FZ. You may have to read some papers in order to be sure of the shape that you are dealing with. Q5. [extra credit, 25 points]

5 The effect of any symmetry operator in Rodrigues space is to insert a dividing plane in the space. If R (= tan(θ/2)v) is the vector that represents the symmetry operator (v is a unit vector), then the dividing plane is y + tan(±θ/4)v, where y is an arbitrary vector perpendicular to v. This arises from the geometrical properties of the space. Prove this property of the Rodrigues-Frank vector.