9.5  The Sans_spheres McStas Component

Sample for Small Angle Neutron Scattering - hard spheres in thin solution, mono disperse.

Identification

• Author: P. Willendrup, K. Lefmann, L. Arleth

• Origin: Risoe

• Date: 19.12.2003

Description

Sample for use in a SANS instrument, models hard, mono disperse spheres in thin solution. The shape of the sample may be a filled box with dimensions xwidth, yheight, zdepth, a cylinder with dimensions radius and yheight, a filled sphere with radius.

Example: Sans_spheres(R = 100, Phi = 1e-3, Delta_rho = 0.6, sigma_abs = 50, xwidth=0.01, yheight=0.01, zdepth=0.005)

Input parameters

Parameters in boldface are required; the others are optional.

Links

• Component source code found in file Sans_spheres.comp.

• The test/example instrument SANS.instr.

A sample of hard spheres for small-angle scattering

The component Sans_spheres models a sample of small independent spheres of radius \(R\), which are uniformly distributed in a rectangular volume \(x_w \times y_h \times z_t\) with a volume fraction \(\phi \). The absorption cross section density for the spheres is \(\sigma _a\) (in units of m\(^{-1}\)), specified for neutrons at 2200 m/s. Absorption and incoherent scattering from the medium is neglected. The difference in scattering length density (the contrast) between the hard spheres and the medium is called \(\Delta \rho \). \(d\) denotes the distance to the (presumed circular) SANS detector of radius \(R\).

A usage example of this component can be found in the Neutron site/tests/SANS instrument from the mcgui.

9.5.1  Small-angle scattering cross section

The neutron intensity scattered into a solid angle \(\Delta \Omega \) for a flat isotropic SANS sample in transmission geometry is given by [DL03]: \begin {equation} I_s(q) = \Psi \Delta \Omega T A z_\textrm {max} \frac {d\sigma _v}{d\Omega }(q) , \end {equation} where \(\Psi \) is the neutron flux, \(T\) is the sample transmission, \(A\) is the illuminated sample area, and \(z_\textrm {max}\) the length of the neutron path through the sample.

In this component, we consider only scattering from a thin solution of monodisperse hard spheres of radius \(R\), where the volume-specific scattering cross section is given by [DL03] \begin {equation} \frac {d\sigma _v}{d\Omega }(q) = n (\Delta \rho )^2 V^2 f(q) , \end {equation} where \(f(q) = \left ( 3\frac {\sin (qR)-qR\cos (qR)}{(qR)^3} \right )^2\), \(n\) is the number density of spheres, and \(V = 4 / 3 \pi R^3\) is the sphere volume. (The density is thus \(n = \phi /V\).)

Multiple scattering is ignored.

9.5.2  Algorithm

All neutrons, which hit the sample volume, are scattered. (Hence, no direct beam is simulated.) For scattered neutrons, the following steps are taken:

1. Choose a value of \(q\) uniformly in the interval \([0;q_\textrm {max}]\).

2. Choose a polar angle, \(\alpha \), for the q-vector uniformly in \([0;\pi ]\).

3. Scatter the neutron according to \((q,\alpha )\).

4. Calculate and apply the correct weight factor correction.

9.5.3  Calculating the weight factor

The scattering position is found by a Monte Carlo choice uniformly along the whole (unscattered) beam path with the sample, length \(l_\textrm {full}\), giving \(f_l = 1/l_\textrm {full}\). The direction focusing on the detector gives (in an small angle approximation) \(f_\Omega = d^2 / (\pi R_\textrm {det}^2)\).

Hence, the total weight tranformation factor becomes \begin {equation} \pi _j = l_\textrm {full} (\pi R_\textrm {det}^2 / d^2)/(4 \pi ) n (\Delta \rho )^2 V^2 f(q) \exp (-\mu _a l) , \end {equation} where \(\mu _a\) is the linear attenuation factor due to absorption and \(l\) is the total neutron path length within the sample.

This component does NOT simulate absolute intensities. This latter depends on the detector parameters.

Some alternative implementations exist as contributed components.

The SANS test/example instrument exists in the distribution for this component.