Rotations and Euler angles: an EPR example

Suppose you have a g tensor with three principal values g1, g2, g3 along three principal axis g1, g2 and g3. The axes of the reference frame are labelled x, y and z.

Now suppose that the g3-axis is located in the first octant of the xyz system, with θ=30° and φ=70°. This means that the angle between the g3 and the z axis is θ=30°, and the angle between the x axis and the projection of g3 onto the xy plane is φ=70°.

The direction of g3 in the xyz frame can be easily computed using ang2vec:

phi = 70*pi/180; theta = 30*pi/180;
g3 = ang2vec(phi,theta);

g3 =
    0.1710
    0.4698
    0.8660	

All the three numbers are positive. They consitute the cosines of the angles between g3 and x, y, and z respectively. They are called direction cosines and denoted l_3x, l_3y, and l_3z, respectively. The subscript 3 stands for the g3 axis.

We can now take this vector and transform it from the xyz representation to the 123 representations using

R = erot([phi theta 0])
R*g3

R =
    0.2962    0.8138   -0.5000
   -0.9397    0.3420         0
    0.1710    0.4698    0.8660
ans =
   -0.0000
         0
    1.0000

% Rotation matrices to be used with erot:
%   l_x1  l_x2 l_x3
%   l_y1  l_y2 l_y3
%   l_z1  l_z2 l_z3
% xyz is the reference frame (e.g. crystal frame), and
% 123 is the tensor frame (e.g. g or A frame).
% Columns are principal tensor axes in reference frame
% Rows are reference axes in tensor frame.
%
% The transformation of a diagonal tensor in its eigenframe 123
% to the reference frame xyz is then
%   g = R*g_diag*R';
% where is the rotation matrix from above.