1.1.5. 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 0.001
sld_core Cylinder core scattering length density 10-6Å-2 4
sld_shell Cylinder shell scattering length density 10-6Å-2 4
sld_solvent 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 cylinder axis to beam angle degree 60
phi rotation about beam degree 60

The returned value is scaled to units of cm-1 sr-1, absolute scale.

Definition

The output of the 2D scattering intensity function for oriented core-shell cylinders is given by (Kline, 2006 [2]). The form factor is normalized by the particle volume.

\[I(q,\alpha) = \frac{\text{scale}}{V_s} F^2(q,\alpha).sin(\alpha) + \text{background}\]

where

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

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

Fig. 1.15 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 \(\phi\). (see cylinder model)

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.

../_images/core_shell_cylinder_autogenfig.png

Fig. 1.16 Figure 2: 1D and 2D plots corresponding to the default parameters of the model.

Reference

[1]see, for example, Ian Livsey J. Chem. Soc., Faraday Trans. 2, 1987,83, 1445-1452
[2]S R Kline, J Appl. Cryst., 39 (2006) 895

Authorship and Verification