Subsections
Figure 16.15:
The pseudo-ellipse model simulated and calculated in-frame
Daeg(1) frame order matrix elements.
In these plots,
θX corresponds to the cone opening half-angle θx,
θY to the cone opening half-angle θy and
θZ to torsion half-angle
σmax.
When the half-angle is not varied, the angle is fixed to either
θx = π/4,
θy = 3π/8 or
σmax = π/6.
Frame order matrix values have been calculated every 10 degrees.
The first angle for the calculated
θX and
θY graphs is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.
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Figure 16.16:
The pseudo-ellipse model simulated and calculated in-frame
Daeg(2) frame order matrix elements.
In these plots,
θX corresponds to the cone opening half-angle θx,
θY to the cone opening half-angle θy and
θZ to torsion half-angle
σmax.
When the half-angle is not varied, the angle is fixed to either
θx = π/4,
θy = 3π/8 or
σmax = π/6.
Frame order matrix values have been calculated every 10 degrees.
The first angle for the calculated
θX and
θY graphs is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.
|
Figure 16.17:
The pseudo-ellipse model simulated and calculated out-of-frame
Daeg(1) frame order matrix elements.
In these plots,
θX corresponds to the cone opening half-angle θx,
θY to the cone opening half-angle θy and
θZ to torsion half-angle
σmax.
When the half-angle is not varied, the angle is fixed to either
θx = π/4,
θy = 3π/8 or
σmax = π/6.
Frame order matrix values have been calculated every 10 degrees.
The first angle for the calculated
θX and
θY graphs is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.
|
Figure 16.18:
The pseudo-ellipse model simulated and calculated out-of-frame
Daeg(2) frame order matrix elements.
In these plots,
θX corresponds to the cone opening half-angle θx,
θY to the cone opening half-angle θy and
θZ to torsion half-angle
σmax.
When the half-angle is not varied, the angle is fixed to either
θx = π/4,
θy = 3π/8 or
σmax = π/6.
Frame order matrix values have been calculated every 10 degrees.
The first angle for the calculated
θX and
θY graphs is set to 0.01 degrees as a pseudo-ellipse cone opening angle of 0.0 cannot be correctly handled by the numerical integration.
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For the pseudo-ellipse model, the full torsion-tilt system is used.
The full rotation matrix is given in equation 12.74c on page .
Pseudo-ellipse frame order matrix
The frame order matrix is
The surface normalisation factor is
The 1 degree frame order matrix with tensor rank-2 consists of the following elements
As the trigonometric functions of
θmax cannot be integrated, these components must be numerically integrated.
The 2 degree frame order matrix with tensor rank-4 consists of the following elements, using Kronecker product double indices from 0 to 8
As the trigonometric functions of
θmax cannot be integrated, these components must be numerically integrated.
All other frame order matrix elements can be numerically shown to be zero.
The frame order matrix element simulation script from Section 16.2, page was used to compare the implementation of equations 16.52 and 16.53 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 16.15 and
Daeg(2) in figure 16.16.
The out-of-frame
Daeg(1) values are shown in figure 16.17 and
Daeg(2) in figure 16.18.
The relax user manual (PDF), created 2020-08-26.