Module pseudo_ellipse

Module for the pseudo-ellipse frame order model.

Functions

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

compile_2nd_matrix_pseudo_ellipse(matrix, Rx2_eigen, theta_x, theta_y, sigma_max)
Generate the 2nd degree Frame Order matrix for the pseudo-ellipse.

float part_int_daeg1_pseudo_ellipse_00(phi, x, y)
The theta-sigma partial integral of the 1st degree Frame Order matrix element xx for the pseudo-ellipse.

float part_int_daeg1_pseudo_ellipse_11(phi, x, y)
The theta-sigma partial integral of the 1st degree Frame Order matrix element yy for the pseudo-ellipse.

float part_int_daeg1_pseudo_ellipse_22(phi, x, y)
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, sinc_2smax)
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, sinc_2smax)
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)
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, sinc_2smax)
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, sinc_2smax)
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)
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)
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, sinc_2smax)
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, sinc_2smax)
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)
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)
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)
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, sinc_2smax)
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, sinc_2smax)
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)
The theta-sigma partial integral of the 2nd degree Frame Order matrix for the pseudo-ellipse.

pcs_numeric_qr_int_pseudo_ellipse(points=None, max_points=None, theta_x=None, theta_y=None, sigma_max=None, c=None, full_in_ref_frame=None, r_pivot_atom=None, r_pivot_atom_rev=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None, pcs_theta=None, pcs_theta_err=None, missing_pcs=None)
Determine the averaged PCS value via numerical integration.

float pcs_numeric_quad_int_pseudo_ellipse(theta_x=None, theta_y=None, sigma_max=None, c=None, r_pivot_atom=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None)
Determine the averaged PCS value via numerical integration.

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

numpy rank-1 float64 array tmax_pseudo_ellipse_array(phi, theta_x, theta_y)
The pseudo-ellipse tilt-torsion polar angle for numpy arrays.

Variables

__package__ = 'lib.frame_order'

Imports: cos, pi, sin, sqrt, divide, dot, eye, float64, multiply, sinc, swapaxes, tensordot, np_cos, np_sin, np_sqrt, quad, tplquad, pec, pcs_pivot_motion_full_qr_int, pcs_pivot_motion_full_quad_int, rotate_daeg

Function Details

compile_1st_matrix_pseudo_ellipse(matrix, R_eigen, 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.
• R_eigen (numpy 3D, rank-2 array) - The eigenframe rotation matrix.
• 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(matrix, Rx2_eigen, 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.
• Rx2_eigen (numpy 9D, rank-2 array) - The Kronecker product of the eigenframe rotation matrix with itself.
• 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.

part_int_daeg1_pseudo_ellipse_00(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg1_pseudo_ellipse_11(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg1_pseudo_ellipse_22(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_00(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_04(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_08(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_11(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_13(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_22(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_26(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_40(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_44(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_48(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_55(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_57(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_80(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_84(phi, x, y, sinc_2smax)

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.
• sinc_2smax (float) - The pre-calculated value of sinc(2*sigma_max).

Returns: float

The theta-sigma partial integral.

part_int_daeg2_pseudo_ellipse_88(phi, x, y)

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.

Returns: float

The theta-sigma partial integral.

pcs_numeric_qr_int_pseudo_ellipse(points=None, max_points=None, theta_x=None, theta_y=None, sigma_max=None, c=None, full_in_ref_frame=None, r_pivot_atom=None, r_pivot_atom_rev=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None, pcs_theta=None, pcs_theta_err=None, missing_pcs=None)

Determine the averaged PCS value via numerical integration.

Parameters:

• points (numpy rank-2, 3D array) - The Sobol points in the torsion-tilt angle space.
• max_points (int) - The maximum number of Sobol' points to use. Once this number is reached, the loop over the Sobol' torsion-tilt angles is terminated.
• theta_x (float) - The x-axis half cone angle.
• theta_y (float) - The y-axis half cone angle.
• sigma_max (float) - The maximum torsion angle.
• c (float) - The PCS constant (without the interatomic distance and in Angstrom units).
• full_in_ref_frame (numpy rank-1 array) - An array of flags specifying if the tensor in the reference frame is the full or reduced tensor.
• r_pivot_atom (numpy rank-1, 3D array) - The pivot point to atom vector.
• r_pivot_atom_rev (numpy rank-2, 3D array) - The reversed pivot point to atom vector.
• r_ln_pivot (numpy rank-1, 3D array) - The lanthanide position to pivot point vector.
• A (numpy rank-2, 3D array) - The full alignment tensor of the non-moving domain.
• R_eigen (numpy rank-2, 3D array) - The eigenframe rotation matrix.
• RT_eigen (numpy rank-2, 3D array) - The transpose of the eigenframe rotation matrix (for faster calculations).
• Ri_prime (numpy rank-3, array of 3D arrays) - The array of pre-calculated rotation matrices for the in-frame pseudo-elliptic cone motion, used to calculate the PCS for each state i in the numerical integration.
• pcs_theta (numpy rank-2 array) - The storage structure for the back-calculated PCS values.
• pcs_theta_err (numpy rank-2 array) - The storage structure for the back-calculated PCS errors.
• missing_pcs (numpy rank-2 array) - A structure used to indicate which PCS values are missing.

pcs_numeric_quad_int_pseudo_ellipse(theta_x=None, theta_y=None, sigma_max=None, c=None, r_pivot_atom=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None)

Determine the averaged PCS value via numerical integration.

Parameters:

• theta_x (float) - The x-axis half cone angle.
• theta_y (float) - The y-axis half cone angle.
• sigma_max (float) - The maximum torsion angle.
• c (float) - The PCS constant (without the interatomic distance and in Angstrom units).
• r_pivot_atom (numpy rank-1, 3D array) - The pivot point to atom vector.
• r_ln_pivot (numpy rank-1, 3D array) - The lanthanide position to pivot point vector.
• A (numpy rank-2, 3D array) - The full alignment tensor of the non-moving domain.
• R_eigen (numpy rank-2, 3D array) - The eigenframe rotation matrix.
• RT_eigen (numpy rank-2, 3D array) - The transpose of the eigenframe rotation matrix (for faster calculations).
• Ri_prime (numpy rank-2, 3D array) - The empty rotation matrix for the in-frame isotropic cone motion, used to calculate the PCS for each state i in the numerical integration.

Returns: float

The averaged PCS value.

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.

tmax_pseudo_ellipse_array(phi, theta_x, theta_y)

The pseudo-ellipse tilt-torsion polar angle for numpy arrays.

Parameters:

• phi (numpy rank-1 float64 array) - The azimuthal tilt-torsion angle.
• theta_x (float) - The cone opening angle along x.
• theta_y (float) - The cone opening angle along y.

Returns: numpy rank-1 float64 array

The array theta max angles for the given phi angle array.