Module frame_order_matrix_ops

Module for the handling of Frame Order.

Functions

compile_1st_matrix_pseudo_ellipse(matrix, theta_x, theta_y, sigma_max)
Generate the 1st degree Frame Order matrix for the pseudo-ellipse.

compile_2nd_matrix_free_rotor(matrix, R, z_axis, axis, theta_axis, phi_axis)
Generate the rotated 2nd degree Frame Order matrix for the free rotor model.

compile_2nd_matrix_iso_cone(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, cone_theta, sigma_max)
Generate the rotated 2nd degree Frame Order matrix for the isotropic cone.

compile_2nd_matrix_iso_cone_free_rotor(matrix, R, z_axis, cone_axis, theta_axis, phi_axis, s1)
Generate the rotated 2nd degree Frame Order matrix for the free rotor isotropic cone.

compile_2nd_matrix_iso_cone_torsionless(matrix, R, z_axis, cone_axis, theta_axis, phi_axis, cone_theta)
Generate the rotated 2nd degree Frame Order matrix for the torsionless isotropic cone.

compile_2nd_matrix_pseudo_ellipse(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, theta_x, theta_y, sigma_max)
Generate the 2nd degree Frame Order matrix for the pseudo-ellipse.

compile_2nd_matrix_pseudo_ellipse_free_rotor(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, theta_x, theta_y)
Generate the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

compile_2nd_matrix_pseudo_ellipse_torsionless(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, theta_x, theta_y)
Generate the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

compile_2nd_matrix_rotor(matrix, R, z_axis, axis, theta_axis, phi_axis, smax)
Generate the rotated 2nd degree Frame Order matrix for the rotor model.

daeg_to_rotational_superoperator(daeg, Rsuper)
Convert the frame order matrix (daeg) to the rotational superoperator.

float part_int_daeg1_pseudo_ellipse_xx(phi, x, y, smax)
The theta-sigma partial integral of the 1st degree Frame Order matrix element xx for the pseudo-ellipse.

float part_int_daeg1_pseudo_ellipse_yy(phi, x, y, smax)
The theta-sigma partial integral of the 1st degree Frame Order matrix element yy for the pseudo-ellipse.

float part_int_daeg1_pseudo_ellipse_zz(phi, x, y, smax)
The theta-sigma partial integral of the 1st degree Frame Order matrix element zz for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_00(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix element 11 for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_04(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix element 22 for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_08(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix element 33 for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_11(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_13(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_22(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_26(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_40(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_44(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_48(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_55(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_57(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_80(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_84(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_88(phi, x, y, smax)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_00(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_08(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_11(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_44(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_48(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_80(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_free_rotor_88(phi, x, y)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_00(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_04(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_08(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_11(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_22(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_44(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_48(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_55(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

float part_int_daeg2_pseudo_ellipse_torsionless_88(phi, x, y)
The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

populate_1st_eigenframe_iso_cone(matrix, angle)
Populate the 1st degree Frame Order matrix in the eigenframe for an isotropic cone.

populate_2nd_eigenframe_iso_cone(matrix, tmax, smax)
Populate the 2nd degree Frame Order matrix in the eigenframe for the isotropic cone.

populate_2nd_eigenframe_iso_cone_free_rotor(matrix, s1)
Populate the 2nd degree Frame Order matrix in the eigenframe for the free rotor isotropic cone.

reduce_alignment_tensor(D, A, red_tensor)
Calculate the reduction in the alignment tensor caused by the Frame Order matrix.

reduce_alignment_tensor_symmetric(D, A, red_tensor)
Calculate the reduction in the alignment tensor caused by the Frame Order matrix.

rotate_daeg(matrix, R)
Rotate the given frame order matrix.

float tmax_pseudo_ellipse(phi, theta_x, theta_y)
The pseudo-ellipse tilt-torsion polar angle.

Variables

__package__ = 'maths_fns'

Imports: dep_check, cos, pi, sin, sqrt, cross, dot, sinc, transpose, norm, quad, isNaN, order_parameters, spherical_to_cartesian, kron_prod, transpose_23, pec, euler_to_R_zyz, two_vect_to_R

Function Details

compile_1st_matrix_pseudo_ellipse(matrix, theta_x, theta_y, sigma_max)

Generate the 1st degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• matrix (numpy 3D, rank-2 array) - The Frame Order matrix, 1st degree to be populated.
• theta_x (float) - The cone opening angle along x.
• theta_y (float) - The cone opening angle along y.
• sigma_max (float) - The maximum torsion angle.

compile_2nd_matrix_free_rotor(matrix, R, z_axis, axis, theta_axis, phi_axis)

Generate the rotated 2nd degree Frame Order matrix for the free rotor model.

The rotor axis is assumed to be parallel to the z-axis in the eigenframe.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• z_axis (numpy 3D, rank-1 array) - The molecular frame z-axis from which the rotor axis is rotated from.
• axis (numpy 3D, rank-1 array) - The storage structure for the axis.
• theta_axis (float) - The axis polar angle.
• phi_axis (float) - The axis azimuthal angle.

compile_2nd_matrix_iso_cone(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, cone_theta, sigma_max)

Generate the rotated 2nd degree Frame Order matrix for the isotropic cone.

The cone axis is assumed to be parallel to the z-axis in the eigenframe.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• eigen_alpha (float) - The eigenframe rotation alpha Euler angle.
• eigen_beta (float) - The eigenframe rotation beta Euler angle.
• eigen_gamma (float) - The eigenframe rotation gamma Euler angle.
• cone_theta (float) - The cone opening angle.
• sigma_max (float) - The maximum torsion angle.

compile_2nd_matrix_iso_cone_free_rotor(matrix, R, z_axis, cone_axis, theta_axis, phi_axis, s1)

Generate the rotated 2nd degree Frame Order matrix for the free rotor isotropic cone.

The cone axis is assumed to be parallel to the z-axis in the eigenframe. In this model, the three order parameters are defined as:

   S1 = S2,
   S3 = 0

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• z_axis (numpy 3D, rank-1 array) - The molecular frame z-axis from which the cone axis is rotated from.
• cone_axis (numpy 3D, rank-1 array) - The storage structure for the cone axis.
• theta_axis (float) - The cone axis polar angle.
• phi_axis (float) - The cone axis azimuthal angle.
• s1 (float) - The cone order parameter.

compile_2nd_matrix_iso_cone_torsionless(matrix, R, z_axis, cone_axis, theta_axis, phi_axis, cone_theta)

Generate the rotated 2nd degree Frame Order matrix for the torsionless isotropic cone.

The cone axis is assumed to be parallel to the z-axis in the eigenframe.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• z_axis (numpy 3D, rank-1 array) - The molecular frame z-axis from which the cone axis is rotated from.
• cone_axis (numpy 3D, rank-1 array) - The storage structure for the cone axis.
• theta_axis (float) - The cone axis polar angle.
• phi_axis (float) - The cone axis azimuthal angle.
• cone_theta (float) - The cone opening angle.

compile_2nd_matrix_pseudo_ellipse(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, theta_x, theta_y, sigma_max)

Generate the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• eigen_alpha (float) - The eigenframe rotation alpha Euler angle.
• eigen_beta (float) - The eigenframe rotation beta Euler angle.
• eigen_gamma (float) - The eigenframe rotation gamma Euler angle.
• theta_x (float) - The cone opening angle along x.
• theta_y (float) - The cone opening angle along y.
• sigma_max (float) - The maximum torsion angle.

compile_2nd_matrix_pseudo_ellipse_free_rotor(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, theta_x, theta_y)

Generate the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• eigen_alpha (float) - The eigenframe rotation alpha Euler angle.
• eigen_beta (float) - The eigenframe rotation beta Euler angle.
• eigen_gamma (float) - The eigenframe rotation gamma Euler angle.
• theta_x (float) - The cone opening angle along x.
• theta_y (float) - The cone opening angle along y.

compile_2nd_matrix_pseudo_ellipse_torsionless(matrix, R, eigen_alpha, eigen_beta, eigen_gamma, theta_x, theta_y)

Generate the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• eigen_alpha (float) - The eigenframe rotation alpha Euler angle.
• eigen_beta (float) - The eigenframe rotation beta Euler angle.
• eigen_gamma (float) - The eigenframe rotation gamma Euler angle.
• theta_x (float) - The cone opening angle along x.
• theta_y (float) - The cone opening angle along y.

compile_2nd_matrix_rotor(matrix, R, z_axis, axis, theta_axis, phi_axis, smax)

Generate the rotated 2nd degree Frame Order matrix for the rotor model.

The cone axis is assumed to be parallel to the z-axis in the eigenframe.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.
• z_axis (numpy 3D, rank-1 array) - The molecular frame z-axis from which the rotor axis is rotated from.
• axis (numpy 3D, rank-1 array) - The storage structure for the axis.
• theta_axis (float) - The axis polar angle.
• phi_axis (float) - The axis azimuthal angle.
• smax (float) - The maximum torsion angle.

daeg_to_rotational_superoperator(daeg, Rsuper)

Convert the frame order matrix (daeg) to the rotational superoperator.

Parameters:

• daeg (numpy 9D, rank-2 array or numpy 3D, rank-4 array) - The second degree frame order matrix, daeg. This must be in the Kronecker product layout.
• Rsuper (numpy 5D, rank-2 array) - The rotational superoperator structure to be populated.

part_int_daeg1_pseudo_ellipse_xx(phi, x, y, smax)

The theta-sigma partial integral of the 1st degree Frame Order matrix element xx for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg1_pseudo_ellipse_yy(phi, x, y, smax)

The theta-sigma partial integral of the 1st degree Frame Order matrix element yy for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg1_pseudo_ellipse_zz(phi, x, y, smax)

The theta-sigma partial integral of the 1st degree Frame Order matrix element zz for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_00(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix element 11 for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_04(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix element 22 for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_08(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix element 33 for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_11(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_13(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_22(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_26(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_40(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_44(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_48(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_55(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_57(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_80(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_84(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_88(phi, x, y, smax)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.
• smax (float) - The maximum torsion angle.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_00(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_08(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_11(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_44(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_48(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_80(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_free_rotor_88(phi, x, y)

The theta-sigma partial integral of the 2nd degree Frame Order matrix for the free rotor pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_00(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_04(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_08(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_11(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_22(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_44(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_48(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_55(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

part_int_daeg2_pseudo_ellipse_torsionless_88(phi, x, y)

The theta partial integral of the 2nd degree Frame Order matrix for the torsionless pseudo-ellipse.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• x (float) - The cone opening angle along x.
• y (float) - The cone opening angle along y.

Returns: float

The theta partial integral.

populate_1st_eigenframe_iso_cone(matrix, angle)

Populate the 1st degree Frame Order matrix in the eigenframe for an isotropic cone.

The cone axis is assumed to be parallel to the z-axis in the eigenframe.

Parameters:

• matrix (numpy 3D, rank-2 array) - The Frame Order matrix, 1st degree.
• angle (float) - The cone angle.

populate_2nd_eigenframe_iso_cone(matrix, tmax, smax)

Populate the 2nd degree Frame Order matrix in the eigenframe for the isotropic cone.

The cone axis is assumed to be parallel to the z-axis in the eigenframe.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree.
• tmax (float) - The cone opening angle.
• smax (float) - The maximum torsion angle.

populate_2nd_eigenframe_iso_cone_free_rotor(matrix, s1)

Populate the 2nd degree Frame Order matrix in the eigenframe for the free rotor isotropic cone.

The cone axis is assumed to be parallel to the z-axis in the eigenframe. In this model, the three order parameters are defined as:

   S1 = S2,
   S3 = 0

This is in the Kronecker product form.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree.
• s1 (float) - The cone order parameter.

reduce_alignment_tensor(D, A, red_tensor)

Calculate the reduction in the alignment tensor caused by the Frame Order matrix.

This is both the forward rotation notation and Kronecker product arrangement.

Parameters:

• D (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• A (numpy 5D, rank-1 array) - The full alignment tensor in {Axx, Ayy, Axy, Axz, Ayz} notation.
• red_tensor (numpy 5D, rank-1 array) - The structure in {Axx, Ayy, Axy, Axz, Ayz} notation to place the reduced alignment tensor.

reduce_alignment_tensor_symmetric(D, A, red_tensor)

Calculate the reduction in the alignment tensor caused by the Frame Order matrix.

This is both the forward rotation notation and Kronecker product arrangement. This simplification is due to the symmetry in motion of the pseudo-elliptic and isotropic cones. All element of the frame order matrix where an index appears only once are zero.

Parameters:

• D (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• A (numpy 5D, rank-1 array) - The full alignment tensor in {Axx, Ayy, Axy, Axz, Ayz} notation.
• red_tensor (numpy 5D, rank-1 array) - The structure in {Axx, Ayy, Axy, Axz, Ayz} notation to place the reduced alignment tensor.

rotate_daeg(matrix, R)

Rotate the given frame order matrix.

It is assumed that the frame order matrix is in the Kronecker product form.

Parameters:

• matrix (numpy 9D, rank-2 array) - The Frame Order matrix, 2nd degree to be populated.
• R (numpy 3D, rank-2 array) - The rotation matrix to be populated.

tmax_pseudo_ellipse(phi, theta_x, theta_y)

The pseudo-ellipse tilt-torsion polar angle.

Parameters:

• phi (float) - The azimuthal tilt-torsion angle.
• theta_x (float) - The cone opening angle along x.
• theta_y (float) - The cone opening angle along y.

Returns: float

The theta max angle for the given phi angle.