Subsections
The double rotor model consists of two standard rotations, the first about the xaxis and the second about the yaxis.
Hence the frame order matrix is simply the integration over both torsion angles of the Kronecker product of the product of the R_{x} and R_{y} rotation matrices, divided by the surface area normalisation factor.
Figure 16.27:
The double rotor model simulated and calculated inframe
Daeg^{(1)} frame order matrix elements.
In these plots,
θ_{X} corresponds to the torsion halfangle
σ_{max, 1} and
θ_{Y} to the torsion halfangle
σ_{max, 2}.
When the halfangle is not varied, the angle is fixed to either
σ_{max, 1} = π/4 or
σ_{max, 2} = 3π/8.
Frame order matrix values have been calculated every 10 degrees.

Figure 16.28:
The double rotor model simulated and calculated inframe
Daeg^{(2)} frame order matrix elements.
In these plots,
θ_{X} corresponds to the torsion halfangle
σ_{max, 1} and
θ_{Y} to the torsion halfangle
σ_{max, 2}.
When the halfangle is not varied, the angle is fixed to either
σ_{max, 1} = π/4 or
σ_{max, 2} = 3π/8.
Frame order matrix values have been calculated every 10 degrees.

Figure 16.29:
The double rotor model simulated and calculated outofframe
Daeg^{(1)} frame order matrix elements.
In these plots,
θ_{X} corresponds to the torsion halfangle
σ_{max, 1} and
θ_{Y} to the torsion halfangle
σ_{max, 2}.
When the halfangle is not varied, the angle is fixed to either
σ_{max, 1} = π/4 or
σ_{max, 2} = 3π/8.
Frame order matrix values have been calculated every 10 degrees.

Figure 16.30:
The double rotor model simulated and calculated outofframe
Daeg^{(2)} frame order matrix elements.
In these plots,
θ_{X} corresponds to the torsion halfangle
σ_{max, 1} and
θ_{Y} to the torsion halfangle
σ_{max, 2}.
When the halfangle is not varied, the angle is fixed to either
σ_{max, 1} = π/4 or
σ_{max, 2} = 3π/8.
Frame order matrix values have been calculated every 10 degrees.

The individual rotations are
The full rotation is then
Double rotor frame order matrix
The frame order matrix is
Daeg^{(n)} 
= R(σ_{1}, σ_{2})^{⊗n} dS dS, 
(16.68) 

= R(σ_{1}, σ_{2})^{⊗n} dσ_{max, 1} dσ_{max, 2} dS. 
(16.69) 
The surface normalisation factor is
The unnormalised 1 degress frame order matrix with tensor rank2 is
After normalisation, the full frame order matrix is
Daeg^{(1)} = . 
(16.72) 
The 2 degree frame order matrix with tensor rank4 consists of the following elements, using Kronecker product double indices from 0 to 8
The frame order matrix element simulation script from Section 16.2, page was used to compare the implementation of equations 16.72 and 16.73 above.
Frame order matrix
Daeg^{(1)} and
Daeg^{(2)} values were both simulated and calculated, both within and out of the motional eigenframe.
The inframe
Daeg^{(1)} values are shown in figure 16.27 and
Daeg^{(2)} in figure 16.28.
The outofframe
Daeg^{(1)} values are shown in figure 16.29 and
Daeg^{(2)} in figure 16.30.
The relax user manual (PDF), created 20200826.