mag2exp.sans.cross_section

mag2exp.sans.cross_section(field, /, method, polarisation=(0, 0, 1))

Calculation of scattering cross sections.

The scattering cross sections can be calculated using

\[\frac{d\sum}{d\Omega} \sim |{\bf Q} \cdot {\bf \sigma}|^2,\]

where \({\bf \sigma}\) is the Pauli vector

\[\begin{split}{\bf \sigma} = \begin{bmatrix} \sigma_x \\ \sigma_y \\ \sigma_z \end{bmatrix},\end{split}\]

and

\[\begin{split}\begin{align} \sigma_x &= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \\ \sigma_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \\ \sigma_z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \end{align}\end{split}\]

\({\bf Q}\) is the magnetic interaction vector given by

\[\begin{equation} {\bf Q} = \hat{\bf q} \times \widetilde{\bf M} \times \hat{\bf q} \end{equation}\]

\(\hat{\bf q}\) is the unit scattering vector and \(\widetilde{\bf M}\) is the Fourier transform of the magnetisation. The magnetic interaction vector is is dependent on the scattering geometry and the scattering vector is defined as

\[\begin{equation} {\bf q} = {\bf k}_1 - {\bf k}_0. \end{equation}\]

The spin-flip and non-spin-flip neutron scattering cross sections can be calculated for specific scattering geometries using

\[\begin{split}\begin{equation} \frac{d\sum}{d\Omega} = \begin{pmatrix} \frac{d\sum^{++}}{d\Omega} & \frac{d\sum^{-+}}{d\Omega}\\ \frac{d\sum^{+-}}{d\Omega} & \frac{d\sum^{--}}{d\Omega} \end{pmatrix}. \end{equation}\end{split}\]

The spin based cross sections then be combined in order to get the half polarised cross sections

\[\begin{split}\begin{align} \frac{d\sum^{+}}{d\Omega} &= \frac{d\sum^{++}}{d\Omega} + \frac{d\sum^{+-}}{d\Omega}, \\ \frac{d\sum^{-}}{d\Omega} &= \frac{d\sum^{--}}{d\Omega} + \frac{d\sum^{-+}}{d\Omega}. \end{align}\end{split}\]

These can further be combined to get the unpolarised cross section

\[\begin{split}\begin{align} \frac{d\sum}{d\Omega} &= \frac{1}{2} \left( \frac{d\sum^{+}}{d\Omega} + \frac{d\sum^{-}}{d\Omega} \right), \\ \frac{d\sum}{d\Omega} &= \frac{1}{2} \left( \frac{d\sum^{++}}{d\Omega} + \frac{d\sum^{+-}}{d\Omega} + \frac{d\sum^{--}}{d\Omega} + \frac{d\sum^{-+}}{d\Omega} \right). \end{align}\end{split}\]

Parameters:

• field (discretisedfield.field) – Magnetisation field.

• method (str) –

  Used to select the relevant cross section and can take the value of

  • pp - positive positive non-spin-flip cross section,

  • nn - negative negative non-spin-flip cross section,

  • pn - positive negative spin-flip cross section,

  • np - negative positive spin-flip cross section,

  • p - positive half polarised cross section,

  • n - negitive half polarised cross section,

  • unpol - unpolarised cross section.

• polarisation (turple) – Defines the polarisation direction of the incoming reutron beam with respect to the sample reference frame.

Returns:

Scattering cross section in arbitary units.

Return type:

discretisedfield.Field

Examples

1. Visualising unpolarised cross section with matplotlib.

>>> import discretisedfield as df
>>> import micromagneticmodel as mm
>>> import numpy as np
>>> import mag2exp
>>> mesh = df.Mesh(p1=(-25e-9, -25e-9, -2e-9),
...                p2=(25e-9, 25e-9, 50e-9),
...                cell=(1e-9, 1e-9, 2e-9))
>>> def v_fun(point):
...     x, y, z = point
...     q = 10e-9
...     return (0,
...             np.sin(2 * np.pi * x / q),
...             np.cos(2 * np.pi * x / q))
>>> field = df.Field(mesh, nvdim=3, value=v_fun, norm=1e5)
>>> field.sel('z').mpl()
>>> cs = mag2exp.sans.cross_section(field, method='unpol',
...                                 polarisation=(0, 0, 1))
>>> cs.sel(k_z=0).real.mpl.scalar()

2. Visualising spin-flip cross section with matplotlib.

>>> import discretisedfield as df
>>> import micromagneticmodel as mm
>>> import numpy as np
>>> import mag2exp
>>> mesh = df.Mesh(p1=(-25e-9, -25e-9, -2e-9),
...                p2=(25e-9, 25e-9, 50e-9),
...                cell=(1e-9, 1e-9, 2e-9))
>>> def v_fun(point):
...     x, y, z = point
...     q = 10e-9
...     return (0,
...             np.sin(2 * np.pi * x / q),
...             np.cos(2 * np.pi * x / q))
>>> field = df.Field(mesh, nvdim=3, value=v_fun, norm=1e5)
>>> field.sel('z').mpl()
>>> cs = mag2exp.sans.cross_section(field, method='pn',
...                                 polarisation=(0, 0, 1))
>>> cs.sel(k_z=0).real.mpl.scalar()