Free rotor pseudo-ellipse equations

Figure 16.23: The free rotor pseudo-ellipse model simulated and calculated in-frame Daeg⁽¹⁾ frame order matrix elements. In these plots, θ_X corresponds to the cone opening half-angle θ_x and θ_Y to the cone opening half-angle θ_y. When the half-angle is not varied, the angle is fixed to either θ_x = π/4 or θ_y = 3π/8. Frame order matrix values have been calculated every 10 degrees. The first angle for the calculated elements is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_in_frame_theta_x_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_in_frame_theta_x_calc.eps}$

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_in_frame_theta_y_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_in_frame_theta_y_calc.eps}$

Figure 16.24: The free rotor pseudo-ellipse model simulated and calculated in-frame Daeg⁽²⁾ frame order matrix elements. In these plots, θ_X corresponds to the cone opening half-angle θ_x and θ_Y to the cone opening half-angle θ_y. When the half-angle is not varied, the angle is fixed to either θ_x = π/4 or θ_y = 3π/8. Frame order matrix values have been calculated every 10 degrees. The first angle for the calculated elements is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_in_frame_theta_x_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_in_frame_theta_x_calc.eps}$

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_in_frame_theta_y_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_in_frame_theta_y_calc.eps}$

Figure 16.25: The free rotor pseudo-ellipse model simulated and calculated out-of-frame Daeg⁽¹⁾ frame order matrix elements. In these plots, θ_X corresponds to the cone opening half-angle θ_x and θ_Y to the cone opening half-angle θ_y. When the half-angle is not varied, the angle is fixed to either θ_x = π/4 or θ_y = 3π/8. Frame order matrix values have been calculated every 10 degrees. The first angle for the calculated elements is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_out_of_frame_theta_x_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_out_of_frame_theta_x_calc.eps}$

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_out_of_frame_theta_y_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sij_pseudo-ellipse_free_rotor_out_of_frame_theta_y_calc.eps}$

Figure 16.26: The free rotor pseudo-ellipse model simulated and calculated out-of-frame Daeg⁽²⁾ frame order matrix elements. In these plots, θ_X corresponds to the cone opening half-angle θ_x and θ_Y to the cone opening half-angle θ_y. When the half-angle is not varied, the angle is fixed to either θ_x = π/4 or θ_y = 3π/8. Frame order matrix values have been calculated every 10 degrees. The first angle for the calculated elements is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_out_of_frame_theta_x_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_out_of_frame_theta_x_calc.eps}$

	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_out_of_frame_theta_y_ens1000000.eps}$	$\includegraphics[width=.5\textwidth]{images/frame_order_matrix/Sijkl_pseudo-ellipse_free_rotor_out_of_frame_theta_y_calc.eps}$

The rotation matrix is the full torsion-tilt rotation matrix of equation 12.74c on page

Free rotor pseudo-ellipse frame order matrix

The frame order matrix is

$\begin{subequations}\begin{align} \textrm{Daeg}^{(n)}&= \left. \int_S R^{\otimes... ... \,\mathrm{d}\sigma \right / \int_S \,\mathrm{d}S. \end{align}\end{subequations}$

The surface normalisation factor is

$\begin{subequations}\begin{align} \int_S \,\mathrm{d}S &= \int_{-\pi}^{\pi} \int... ...\sigma , \\ &= 2\pi \pec (\theta_x , \theta_y ) . \end{align}\end{subequations}$

Free rotor pseudo-ellipse 1 degree frame order

The 1 degree frame order matrix with tensor rank-2 is

$\begin{subequations}\begin{align} \textrm{Daeg}^{(1)}&= \left. \int_S R^{\otimes... ..._{\textrm{max}}\\ \end{pmatrix}\,\mathrm{d}\phi . \end{align}\end{subequations}$

Free rotor pseudo-ellipse 2 degree frame order

The 2 degree frame order matrix with tensor rank-4 consists of the following elements, using Kronecker product double indices from 0 to 8

$\begin{subequations}\begin{flalign} &\textrm{Daeg}_{00} = \frac{1}{6\pec (\theta... ...theta_{\textrm{max}} \,\mathrm{d}\phi \Bigg] , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{11} = \frac{1}{4\pec (\theta... ...\sin^2\theta_{\textrm{max}} \,\mathrm{d}\phi , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{22} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{33} = \textrm{Daeg}_{11} , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{44} = \frac{1}{6\pec (\theta... ...theta_{\textrm{max}} \,\mathrm{d}\phi \Bigg] , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{55} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{66} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{77} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{88} = \frac{1}{3\pec (\theta... ...theta_{\textrm{max}} \,\mathrm{d}\phi \Bigg] , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{04} = \textrm{Daeg}_{00} , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{40} = \textrm{Daeg}_{44} , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{08} = \frac{1}{3\pec (\theta... ...\textrm{max}}\right) \,\mathrm{d}\phi \Bigg] , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{80} = \frac{1}{6\pec (\theta... ...theta_{\textrm{max}} \,\mathrm{d}\phi \Bigg] , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{48} = \frac{1}{3\pec (\theta... ...\textrm{max}}\right) \,\mathrm{d}\phi \Bigg] , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{84} = \textrm{Daeg}_{80} , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{13} = -\textrm{Daeg}_{11} , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{31} = -\textrm{Daeg}_{11} , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{26} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{62} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{57} = 0 , & \end{flalign}\end{subequations}$ $\begin{subequations}\begin{flalign} &\textrm{Daeg}_{75} = 0 . & \end{flalign}\end{subequations}$

Free rotor pseudo-ellipse frame order matrix simulation and calculation

The frame order matrix element simulation script from Section 16.2, page was used to compare the implementation of equations 16.63 and 16.64 above. Frame order matrix Daeg⁽¹⁾ and Daeg⁽²⁾ values were both simulated and calculated, both within and out of the motional eigenframe. The in-frame Daeg⁽¹⁾ values are shown in figure 16.23 and Daeg⁽²⁾ in figure 16.24. The out-of-frame Daeg⁽¹⁾ values are shown in figure 16.25 and Daeg⁽²⁾ in figure 16.26.

The relax user manual (PDF), created 2020-08-26.