Generic SANS Calculator Tool

Description

This tool attempts to simulate the SANS expected from a specified shape/structure or scattering length density profile. The tool can handle both nuclear and magnetic contributions to the scattering.

Theory

In general, a particle with a volume \(V\) can be described by an ensemble containing \(N\) 3-dimensional rectangular pixels where each pixel is much smaller than \(V\).

Assuming that all the pixel sizes are the same, the elastic scattering intensity from the particle is

\[I(\vec Q) = \frac{1}{V}\left| \sum_j^N v_j \beta_j \exp(i\vec Q \cdot \vec r_j)\right|^2\]

Equation 1.

where \(\beta_j\) and \(r_j\) are the scattering length density and the position of the \(j^\text{th}\) pixel respectively.

The total volume \(V\)

\[V = \sum_j^N v_j\]

for \(\beta_j \ne 0\) where \(v_j\) is the volume of the \(j^\text{th}\) pixel (or the \(j^\text{th}\) natural atomic volume (= atomic mass / (natural molar density * Avogadro number) for the atomic structures).

\(V\) can be corrected by users. This correction is useful especially for an atomic structure (such as taken from a PDB file) to get the right normalization.

NOTE! \(\beta_j\) displayed in the GUI may be incorrect but this will not affect the scattering computation if the correction of the total volume V is made.

The scattering length density (SLD) of each pixel, where the SLD is uniform, is a combination of the nuclear and magnetic SLDs and depends on the spin states of the neutrons as follows.

Magnetic Scattering

For magnetic scattering, only the magnetization component, \(\mathbf{M}_\perp\), perpendicular to the scattering vector \(\vec Q\) contributes to the magnetic scattering length.

The magnetic scattering length density is then

\[\beta_M = \frac{\gamma r_0}{2 \mu_B}\sigma \cdot \mathbf{M}_\perp = D_M\sigma \cdot \mathbf{M}_\perp\]

where the gyromagnetic ratio is \(\gamma = -1.913\), \(\mu_B\) is the Bohr magneton, \(r_0\) is the classical radius of electron, and \(\sigma\) is the Pauli spin.

For a polarized neutron, the magnetic scattering is depending on the spin states.

Let us consider that the incident neutrons are polarised both parallel (+) and anti-parallel (-) to the x’ axis (see below). The possible states after scattering from the sample are then

• Non-spin flips: (+ +) and (- -)
• Spin flips: (+ -) and (- +)

Now let us assume that the angles of the \(\vec Q\) vector and the spin-axis (\(x'\)) to the \(x\)-axis are \(\phi\) and \(\theta_\mathrm{up}\) respectively (see above). Then, depending upon the polarization (spin) state of neutrons, the scattering length densities, including the nuclear scattering length density (\(\beta_N\)) are given as

• for non-spin-flips

\[\beta_{\pm\pm} = \beta_N \mp D_M M_{\perp x'}\]

• for spin-flips

\[\beta_{\pm\mp} = - D_M(M_{\perp y'} \pm i M_{\perp z'})\]

where

\[\begin{split}M_{\perp x'} &= M_{0q_x}\cos\theta_\mathrm{up} + M_{0q_y}\sin\theta_\mathrm{up} \\ M_{\perp y'} &= M_{0q_y}\cos\theta_\mathrm{up} - M_{0q_x}\sin\theta_\mathrm{up} \\ M_{\perp z'} &= M_{0z} \\ M_{0q_x} &= (M_{0x}\cos\phi - M_{0y}\sin\phi)\cos\phi \\ M_{0q_y} &= (M_{0y}\sin\phi - M_{0y}\cos\phi)\sin\phi\end{split}\]

Here the \(M_{0x}\), \(M_{0y}\) and \(M_{0z}\) are the \(x\), \(y\) and \(z\) components of the magnetisation vector in the laboratory \(x\)-\(y\)-\(z\) frame.

Using the tool

After computation the result will appear in the Theory box in the SasView Data Explorer panel.

Up_frac_in and Up_frac_out are the ratio

(spin up) / (spin up + spin down)

of neutrons before the sample and at the analyzer, respectively.

NOTE 1. The values of Up_frac_in and Up_frac_out must be in the range 0.0 to 1.0. Both values are 0.5 for unpolarized neutrons.

NOTE 2. This computation is totally based on the pixel (or atomic) data fixed in xyz coordinates. No angular orientational averaging is considered.

NOTE 3. For the nuclear scattering length density, only the real component is taken account.

Using PDB/OMF or SLD files

The SANS Calculator tool can read some PDB, OMF or SLD files but ignores polarized/magnetic scattering when doing so, thus related parameters such as Up_frac_in, etc, will be ignored.

The calculation for fixed orientation uses Equation 1 above resulting in a 2D output, whereas the scattering calculation averaged over all the orientations uses the Debye equation below providing a 1D output

\[I(|\vec Q|) = \frac{1}{V}\sum_j^N v_j\beta_j \sum_k^N v_k \beta_k \frac{\sin(|\vec Q||\vec r_j - \vec r_k|)}{|\vec Q||\vec r_j - \vec r_k|}\]

where \(v_j \beta_j \equiv b_j\) is the scattering length of the \(j^\text{th}\) atom. The calculation output is passed to the Data Explorer for further use.

Note

This help document was last changed by Steve King, 01May2015