What Nexus calculates

Wavefield propagation

Nexus uses a scattering approach - a polarization-dependent transfer matrix formalism - to calculate the interaction of the wavefield with objects in the beam path. For that, the beam properties and the scattering properties of all objects in the beam path have to be specified or calculated.

The beam is described by a complex 2x2 coherency matrix in the general case or a Jones vector for calculations of amplitude parameters for fully coherent beams. The Jones vector \(\vec{E}\) describes a fully coherent beam, while the coherency matrix \(J\) also includes incoherent light. The Jones vector is

\[\begin{split}\vec{E} = \begin{pmatrix} E_{\sigma} \\ E_{\pi} \\ \end{pmatrix}\end{split}\]

and the coherency matrix is related to the Jones vector via \(J_{ij} = \braket{E_i E_j^*}\), such that

\[\begin{split}J = \begin{pmatrix} J_{\sigma\sigma} & J_{\sigma\pi} \\ J_{\pi\sigma} & J_{\pi\pi} \end{pmatrix} = \begin{pmatrix} \braket{E_\sigma E_\sigma^*} & \braket{E_\sigma E_\pi^*} \\ \braket{E_\pi E_\sigma^*} & \braket{E_\pi E_\pi^*} \end{pmatrix}.\end{split}\]

Polarization properties are described along two orthogonal polarization vectors, here the linear polarization components along \(\sigma\) and \(\pi\) directions. For example, the normalized Jones vector of a fully polarized beam along \(\sigma\) direction is given by

\[\begin{split}E = \left( \begin{matrix} 1 \\ 0 \end{matrix} \right)\end{split}\]

while the coherency matrix is

\[\begin{split}J_p = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}.\end{split}\]

For unpolarized light the coherency matrix is

\[\begin{split}J_u = \frac{1}{2} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}.\end{split}\]

All intermediate polarization states are described by

\[J = (1-P) J_u + P J_p.\]

The coherency matrix is Hermitian

\[J^{\dagger} = J.\]

and its trace

\[I = \operatorname{Tr}(J) = |E_{\sigma}|^2 + |E_{\pi}|^2,\]

corresponds to the intensity \(I\) of beam. Nexus always uses a normalized intensity input.

With an optical element \(M\) affecting the Jones vector \(\vec{E}\) the output vector is

\[\vec{E}_{out} = M * \vec{E}_{in},\]

while for \(J\) the relation is

\[J_{out} = M * J_{in} * M^{\dagger}.\]

The output field holds all the information of the scattering properties of the objects. In fact, the \(M(E)\) is actually energy dependent such that the spectra can be calculated.

The matrix \(M\) is the transfer matrix of the whole experiment and this is calculated from the parameters specified in all of the Nexus objects (like a sample or an analyzer). The transfer matrix \(M = \prod_i m_i\) can consist of many different object matrices \(m_i\). For a sample, each layer has its own layer matrix \(L_i\) and the sample matrix is \(m_s = \prod_i L_i\). These layers are described by a refractive index. This refractive index is determined by the electronic and nuclear scattering cross sections which are calculated by Nexus.

Nuclear response

By default, the complete response of the system is calculated by Nexus. The incoming photon field as well as electronic and nuclear scattering are considered. The transmitted field \(\vec{E}_{out}\) consist of two contributions [Smirnov]: electronic scattering \(\vec{E}_{el}\) and coherent nuclear resonant scattering \(\vec{E}_{nuc}\)

\[\vec{E}_{out} = \vec{E}_{el} + \vec{E}_{nuc} = M * \vec{E}_{in} = (M_{el} + M_{nuc}) * \vec{E}_{in},\]

where \(M_{el}\) is the total scattering response, \(M_{el}\) is the pure electronic scattering response, \(M_{nuc}\) is the nuclear response.

Sometimes one is only interested in the nuclear response, meaning that the transmitted pure electronic scattering should be excluded. Some of the methods that calculate nuclear properties provide the parameter electronic. When set to False only the nuclear response is calculated. Nexus will then calculate the output field \(\vec{E}_{out} = M * \vec{E}_{in}\) from the total response and subtract the pure electronic response \(M_{el} * \vec{E}_{in}\)

\[\vec{E}_{nuc} = (M - M_{el}) * \vec{E}_{in}.\]

For the intensity Nexus uses

\[I_{nuc} = \operatorname{Tr}((M - M_{el}) * J_{in} * (M - M_{el})^{\dagger}).\]

Transmission integral

The transmission integral formalism is essential in describing the shape of Moessbauer spectra properly. It describes the fraction of photons transmitted through a resonant absorber as a function of energy. Importantly, it leads to saturation effects for thick absorbers distorting the spectrum from the Lorentzian shape. The absorption for a single line absorber at zero detuning is given by [Wegener] and [Guetlich]

\[\frac{I(\infty) - I(0)}{I(\infty)}\left(t_{eff}\right) = \varepsilon \left(t_{eff}\right) = f_S \left[ 1 - e^{-\frac{t_{eff}}{2}} J_0\left(i \frac{t_{eff}}{2}\right) \right]\]

where \(f_S\) is the Lamb-Moessbauer factor of the source, \(t_{eff}\) is the effective thickness of the absorber and \(J_0\) is the zeroth order Bessel function of first kind.

In Nexus the energy response is calculated from propagation of the incoming wavefield through the sample. This spectrum is then convoluted with an instrument resolution function (specified by the user). For a Moessbauer spectrum the resolution function is a Lorentzian with a energetic width given by source. Via this formalism the transmission integral is automatically included in the calculation of Moessbauer spectra. However, the standard instrument function does not take into account the Lamb-Moessbauer factor of the source \(f_S\) (for fitting this is just a scaling factor and not relevant). In order to get the correct theoretical absorption of a Moessbauer spectrum the Lamb-Moessbauer factor of the source can be included in the instrument function, e.g. \(f_S=0.8\) for 57-Fe.

The plot shows this absorption behavior calculated from Nexus and the theoretical curves for the thin absorber limit and the general thickness dependence. Please note that the curve will approach the Lamb-Moessbauer factor of the source (\(f_S=0.8\) here).

The example on how to calculate this theoretical transmission behaviour with Nexus can be found in the following notebook.

Notebooks

transmission_integral - nb_transmission_integral.ipynb.