24 lines
1.4 KiB
TeX
24 lines
1.4 KiB
TeX
% ########################
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\chapter{Crystallographic orientations}
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% ########################
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\section{Bunge Euler angles}
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\label{bunges}
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Euler angles $(\varphi_1, \phi, \varphi_2)$---following the Bunge convention---rotate the sample coordinate system ($X$, $Y$, $Z$ or RD, TD, ND) into the crystal coordinate system ($x_\text c$, $y_\text c$, $z_\text c$).
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Three successive rotations are carried out in the following way \citep[p.~4]{Bunge1982}:
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\begin{enumerate}
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\item Rotate by $\varphi_1$ around Z, to bring X into the $x_\text c$--$y_\text c$-plane. The new intermediate axes are $X^\prime$, $Y^\prime$ and $Z$ (unchanged).
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\item Now rotate by $\phi$ around $X^\prime$, to make $Z$ parallel with $z_\text c$. The intermediate axes are $X^\prime$, $Y^{\prime\prime}$, $Z^\prime$.
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\item A final rotation by $\varphi_2$ around $Z^\prime \equiv z_\text c$ makes the rotated axes then identical to the crystal axes.
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\end{enumerate}
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The rotation matrix can be calculated as
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\[% Gottstein pg 55
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\tnsr{g}=\left(\begin{array}{ccc}
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\cos{\varphi_1}\cos{\varphi_2}-\sin{\varphi_1}\sin{\varphi_2}\cos{\phi} & \sin{\varphi_1}\cos{\varphi_2}+\cos{\varphi_1}\sin{\varphi_2}\cos{\phi}& \sin{\varphi_2}\sin{\phi}\\
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-\cos{\varphi_1}\sin{\varphi_2}-\sin{\varphi_1}\cos{\varphi_2}\cos{\phi} & -\sin{\varphi_1}\cos{\varphi_2}+\cos{\varphi_1}\cos{\varphi_2}\cos{\phi}& \cos{\varphi_2}\sin{\phi}\\
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\sin{\varphi_1}\sin{\phi} & -\cos{\varphi_1}\sin{\phi}& \cos{\phi}
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\end{array}\right)
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\]
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