hollow_cylinder

Parameter Description Units Default value
scale Source intensity None 1
background Source background cm-1 0.001
radius Cylinder core radius Å 20
thickness Cylinder wall thickness Å 10
length Cylinder total length Å 400
sld Cylinder sld Å-2 6.3
sld_solvent Solvent sld Å-2 1
theta Theta angle degree 90
phi Phi angle degree 0

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

This model provides the form factor, \(P(q)\), for a monodisperse hollow right angle circular cylinder (rigid tube) where the form factor is normalized by the volume of the tube (i.e. not by the external volume).

\[P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background}\]

where the averaging \(\left<\ldots\right>\) is applied only for the 1D calculation.

The inside and outside of the hollow cylinder are assumed have the same SLD.

Definition

The 1D scattering intensity is calculated in the following way (Guinier, 1955)

\[\begin{split}P(q) &= (\text{scale})V_\text{shell}\Delta\rho^2 \int_0^{1}\Psi^2 \left[q_z, R_\text{outer}(1-x^2)^{1/2}, R_\text{core}(1-x^2)^{1/2}\right] \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\ \Psi[q,y,z] &= \frac{1}{1-\gamma^2} \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\ \Lambda(a) &= 2 J_1(a) / a \\ \gamma &= R_\text{core} / R_\text{outer} \\ V_\text{shell} &= \pi \left(R_\text{outer}^2 - R_\text{core}^2 \right)L \\ J_1(x) &= (\sin(x)-x\cdot \cos(x)) / x^2\end{split}\]

where scale is a scale factor, \(H = L/2\) and \(J_1\) is the 1st order Bessel function.

NB: The 2nd virial coefficient of the cylinder is calculated based on the outer radius and full length, which give an the effective radius for structure factor \(S(q)\) when \(P(q) \cdot S(q)\) is applied.

In the parameters,the radius is \(R_\text{core}\) while thickness is \(R_\text{outer} - R_\text{core}\).

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).

../../_images/hollow_cylinder_autogenfig.png

Fig. 26 1D and 2D plots corresponding to the default parameters of the model.

References

L A Feigin and D I Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, New York, (1987)

Authorship and Verification

  • Author: NIST IGOR/DANSE Date: pre 2010

  • Last Modified by: Richard Heenan Date: October 06, 2016

    (reparametrised to use thickness, not outer radius)

  • Last Reviewed by: Richard Heenan Date: October 06, 2016