2.1.1.3. Core shell cylinder¶

Right circular cylinder with a core-shell scattering length density profile.

Parameter	Description	Units	Default value
scale	Source intensity	None	1
background	Source background	cm^-1	0
core_sld	Cylinder core scattering length density	10^-6Å^-2	4
shell_sld	Cylinder shell scattering length density	10^-6Å^-2	4
solvent_sld	Solvent scattering length density	10^-6Å^-2	1
radius	Cylinder core radius	Å	20
thickness	Cylinder shell thickness	Å	20
length	Cylinder length	Å	400
theta	In plane angle	degree	60
phi	Out of plane angle	degree	60

The returned value is scaled to units of cm^-1.

The form factor is normalized by the particle volume.

Definition¶

The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006)

\[P(Q,\alpha) = {\text{scale} \over V_s} F^2(Q) + \text{background}\]

where

\[\begin{split}F(Q) = &\ (\rho_c - \rho_s) V_c {\sin \left( Q \tfrac12 L\cos\alpha \right) \over Q \tfrac12 L\cos\alpha } {2 J_1 \left( QR\sin\alpha \right) \over QR\sin\alpha } \\ &\ + (\rho_s - \rho_\text{solv}) V_s {\sin \left( Q \left(\tfrac12 L+T\right) \cos\alpha \right) \over Q \left(\tfrac12 L +T \right) \cos\alpha } { 2 J_1 \left( Q(R+T)\sin\alpha \right) \over Q(R+T)\sin\alpha }\end{split}\]

and

\[V_s = \pi (R + T)^2 (L + 2T)\]

and \(\alpha\) is the angle between the axis of the cylinder and \(\vec q\), \(V_s\) is the volume of the outer shell (i.e. the total volume, including the shell), \(V_c\) is the volume of the core, \(L\) is the length of the core, \(R\) is the radius of the core, \(T\) is the thickness of the shell, \(\rho_c\) is the scattering length density of the core, \(\rho_s\) is the scattering length density of the shell, \(\rho_\text{solv}\) is the scattering length density of the solvent, and background is the background level. The outer radius of the shell is given by \(R+T\) and the total length of the outer shell is given by \(L+2T\). \(J1\) is the first order Bessel function.

../_images/core_shell_cylinder_geometry.jpg

Figure 1: Core shell cylinder schematic.

To provide easy access to the orientation of the core-shell cylinder, we define the axis of the cylinder using two angles \(\theta\) and \(\phi\). As for the case of the cylinder, those angles are defined in .

NB: The 2nd virial coefficient of the cylinder is calculated based on the radius and 2 length values, and used as the effective radius for \(S(Q)\) when \(P(Q) \cdot S(Q)\) is applied.

The \(\theta\) and \(\phi\) parameters are not used for the 1D output. Our implementation of the scattering kernel and the 1D scattering intensity use the c-library from NIST.

Validation¶

Validation of our code was done by comparing the output of the 1D model to the output of the software provided by the NIST (Kline, 2006). Figure 2 shows a comparison of the 1D output of our model and the output of the NIST software.

Figure 2: Comparison of the SasView scattering intensity for a core-shell cylinder with the output of the NIST SANS analysis software. The parameters were set to: scale = 1.0 Å, radius = 20 Å, thickness = 10 Å, length =400 Å, core_sld =1e-6 Å^-2, shell_sld = 4e-6 Å^-2, solvent_sld = 1e-6 Å^-2, and background = 0.01 cm^-1.

Averaging over a distribution of orientation is done by evaluating the equation above. Since we have no other software to compare the implementation of the intensity for fully oriented cylinders, we can compare the result of averaging our 2D output using a uniform distribution \(p(\theta,\phi) = 1.0\). Figure 3 shows the result of such a cross-check.

Figure 3: Comparison of the intensity for uniformly distributed core-shell cylinders calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: scale = 1.0, radius = 20 Å, thickness = 10 Å, length = 400 Å, core_sld = 1e-6 Å^-2, shell_sld = 4e-6 Å^-2, solvent_sld = 1e-6 Å^-2, and background = 0.0 cm^-1.

2013/11/26 - Description reviewed by Heenan, R.

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