OOF2: The Manual

`Abg (Abg)`
	6.5.2. Subclasses

Name

Abg (Abg) — Euler angles (alpha, beta, gamma) are applied: first beta about the z axis, then alpha about the y, and finally gamma about z. This operation brings the crystal axes into coincidence with the lab axes.

Synopsis

Abg(alpha,beta,gamma)

Details

Base class: Orientation
Parameters:

alpha

second rotation, about the y-axis, in degrees. Type: A real number in the range [0, 180].

beta

first rotation, about the z-axis, in degrees. Type: A real number in the range [-180, 180].

gamma

third rotation, about the z-axis, in degrees. Type: A real number in the range [-180, 180].

Description

An Abg object represents the orientation of a three dimensional object (assumed to be a crystal in this discussion) in three dimensional space in terms of the Euler angles $(\alpha, \beta, \gamma)$ . The literature contains different ways of interpreting the angles. Here we present four different, but equivalent, prescriptions for determining the actual rotation described by the three angles, as used in the Abg class.

We define $(\hat{\mathrm{\bf x}}, \hat{\mathrm{\bf y}}, \hat{\mathrm{\bf z}})$ to be the screen coordinate system, with $\hat{\mathrm{\bf x}}$ pointing to the right, $\hat{\mathrm{\bf y}}$ upwards, and $\hat{\mathrm{\bf z}}$ out of the screen. We will call the crystal axes $(\hat{\mathrm{\bf a}}, \hat{\mathrm{\bf b}}, \hat{\mathrm{\bf c}})$ . As usual, all rotations are right handed -- a positive rotation about an axis is counterclockwise if the axis is pointed towards you.

One Method of Finding the Euler Angles: This method describes the rotation that must be applied to the crystal axes to bring them into alignment with the screen axes, assuming that the crystal starts in its desired orientation with respect to the screen. First, rotate the axes by $\beta$ around the $\hat{\mathrm{\bf c}}$ axis. This defines new axes $(\hat{\mathrm{\bf a}}', \hat{\mathrm{\bf b}'}, \hat{\mathrm{\bf c}})$ . Next rotate by $\alpha$ about the $\hat{\mathrm{\bf b}'}$ axis, defining another coordinate system $(\hat{\mathrm{\bf a}}'', \hat{\mathrm{\bf b}'}, \hat{\mathrm{\bf c}}'')$ . Finally, rotate by $\gamma$ about the $\hat{\mathrm{\bf c}}''$ axis, bringing the axes into agreement with $(\hat{\mathrm{\bf x}}, \hat{\mathrm{\bf y}}, \hat{\mathrm{\bf z}})$ .

An equivalent method: Consider the material oriented with a globe, with the origin at the center of the earth and $\hat{\mathrm{\bf c}}$ -axis pointing towards the north pole, the $\hat{\mathrm{\bf a}}$ -axis pointing at the Greenwich Meridian where it intersects the equator and the $\hat{\mathrm{\bf b}}$ -axis pointing towards the Indian ocean somewhere southeast of Sri Lanka. The unrotated globe has its $\hat{\mathrm{\bf a}}$ , $\hat{\mathrm{\bf b}}$ , and $\hat{\mathrm{\bf c}}$ axes aligned with the $\hat{\mathrm{\bf x}}$ , $\hat{\mathrm{\bf y}}$ , and $\hat{\mathrm{\bf z}}$ axes of the screen, respectively. The rotation triplet describes how to rotate the material into its desired orientation (in contrast to the previous method, which started with the material in its desired orientation). The first number of the rotation triplet tilts the point on the north pole southward along the Greenwich Meridian by - $\alpha$ degrees to a new latitude. The second number spins the globe to the east by $-\beta$ degrees about its (tilted) $\hat{\mathrm{\bf c}}$ -axis. The third number rotates the tilted globe to the east by $-\gamma$ degrees about the screen's $\hat{\mathrm{\bf z}}$ axis.

A third method: The order of rotations in the previous definition can be rearranged. Rotate the globe to the east by $-\beta$ degrees about the $\hat{\mathrm{\bf z}}$ -axis, tilt the north pole by $-\alpha$ about the $\hat{\mathrm{\bf y}}$ -axis, and rotate to the east by $-\gamma$ again about the $\hat{\mathrm{\bf z}}$ -axis.

The fourth method: If you know the components in the crystal coordinate system of the screen vectors $\hat{\mathrm{\bf x}}$ , $\hat{\mathrm{\bf y}}$ , and $\hat{\mathrm{\bf z}}$ , you can compute the Euler angles as follows. Let $(x_1, x_2, x_3)$ be the components of the screen's $\hat{\mathrm{\bf x}}$ -axis in the crystal coordinate system. Let $(y_1, y_2, y_3)$ and $(z_1, z_2, z_3)$ similarly be the components of the $\hat{\mathrm{\bf y}}$ and $\hat{\mathrm{\bf z}}$ -axes. Then

$\begin{align*} \sin(\alpha) &= \sqrt{z_1^2 + z_2^2} \\ \cos(\alpha) &= z_3 \\ \\ \sin(\beta) &= z_2/\sqrt{z_1^2 + z_2^2} \\ \cos(\beta) &= z_1/\sqrt{z_1^2 + z_2^2} \\ \\ \sin(\gamma) &= y_3/\sqrt{z_1^2 + z_2^2} \\ \cos(\gamma) &= -x_3/\sqrt{z_1^2 + z_2^2} \\ \end{align*}$


`TwoVectorFieldInit`		`Absolute (AbsoluteErrorScaling)`