Nuclear Scattering

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

Equation 1.

where β_j and r_j are the scattering length density and the position of the j'th pixel respectively.

The total volume V

for β_j ≠0 where v_j is the volume of the j'th pixel (or the j'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! β_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 & Polarisation

For magnetic scattering, only the magnetization component, M_perp, perpendicular to the scattering vector Q contributes to the magnetic scattering length.

The magnetic scattering length density is then

where the gyromagnetic ratio γ = -1.913, μ_B is the Bohr magneton, r₀ is the classical radius of electron, and σ 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 Q vector and the spin-axis (x') to the x-axis are φ and θ_up respectively (see above). Then, depending upon the polarization (spin) state of neutrons, the scattering length densities, including the nuclear scattering length density (β_N) are given as

• for non-spin-flips

  

• for spin-flips

  

where

Here the M0_x, M0_y and M0_z are the x, y and z components of the magnetisation vector in the laboratory xyz frame.

Using the SAS Calculator GUI

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 Data

The SAS 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

where v_j β_j ≡ b_j is the scattering length of the j'th atom. The calculation output is passed to the Data Explorer for further use.