correlogram-tools

This project includes tools to simulate 1D and 2D dark field interferometry data in real space for the INFER project and simultaneously compare with small-angle scattering data in Fourier space. It utilizes existing form factor and structure factor models as well as the existing numerical implementation of the Hankel transform in the sasmodels package located at https://github.com/SasView/sasmodels in order to:

• generate simulated dark field and attenuation images that include optical corrections for specific instrumentation
• interactively and simultaneously simulate SANS and dark field 1D spectra

Theory and Definitions

One of the goals of the INFER project is to develop a grating-based far field neutron interferometer to collect spatially-resolved structural information at the same length scales as small-angle neutron scattering (SANS) from 1 nm to 10 $\mu$m. A simplified schematic diagram of a 2-grating far field interferometer is shown in the image below. The neutron source starts at the source slit location and passes through two phase gratings with the same period, $P_{G1}=P_{G2}$ before reaching the detector position. The gratings result in a Moire pattern across the detector with a period of $P_D$ which can be determined by the phase grating period, the source-to-detector distance, $L$, and the distance between phase gratings, $D$. The instrument probes structural features within the sample at a length scale of $\xi$, which is defined as the autocorrelation length and can be determined by the sample-to-detector distance, $Z$, the neutron wavelength, $\lambda$, and the period of the Moire pattern, $P_D$.

When a sample is placed in the beam path, small-angle scattering due to structural features at the probed autocorrelation length, $\xi$, result in a loss in visibility of the sinusoidal Moire pattern at the detector. This loss in visibility can be calculated from the mean, $(A)$, and amplitude, $(B)$, of the Moire pattern from a sample measurement and open beam measurement. The mean values can be used to first reconstruct a normalized transmission image, $H_0$:

$$H_0=\frac{A_{sample}}{A_{open}}$$

Similarly, the amplitude values can be used to reconstruct a normalized amplitude image, $H_1$:

$$H_1=\frac{B_{sample}}{B_{open}}$$

Finally, $H_0$ and $H_1$ can be used to reconstruct the loss in visibility, $\frac{V_s}{V_o}$:

$$\frac{V_s}{V_o} = \frac{H_1}{H_0}$$

This dark field intensity, $DF$, is related to the and structural features in the sample by:

$$DF(\xi) = -ln(\frac{V_s}{V_o}) = -\lambda^2 t(G(\xi)-G_0)$$

where $t$ is the sample thickness and $G$ is the projection function of the autocorrelation function, which is related to the scattering cross-section, $I(q)$, through the following Hankel transformation:

$$G(\xi) = \frac{1}{2\pi}\int^\infty_0 J_0(q\xi)I(q)qdq$$ $$G_0 = \frac{1}{2\pi}\int^\infty_0 I(q)qdq$$

where $J_0$ is the zeroth order Bessel function of the first kind and $q$ is the scattering vector. The scattering cross-section is defined as:

$$I(q) = \phi\Delta\rho^2P(q)S(q)$$

where $\phi$ is the volume fraction of scatterers, $\Delta\rho^2$ is the contrast, or difference in scattering length density squared between the scatterer and background/solvent, $P(q)$ is the form factor which includes information about the size and shape of the scatterer, and $S(q)$ is the structure factor which includes information about the correlations or interactions between scatterers.

Installation

A Python installation is required to utilize correlogram-tools. If you do not yet have Python installed on your system, you can do so by installing a package manager such as Anaconda. Create a new Python environment. If using conda, one can use the following command to create and activate the environment:

conda create -n correlograms python=3.10
conda activate correlograms

Navigate to the correlogram-tools directory and install the package and required dependencies with:

pip install .

Usage

Read more about different ways to use correlogram-tools in the documentation.

Contributors

• Caitlyn Wolf, NIST Center for Neutron Research, NIST, caitlyn.wolf@nist.gov
• Paul Kienzle, NIST Center for Neutron Research, NIST
• Youngju Kim, Physical Measurement Laboratory, NIST
• Pushkar Sathe, Information Technology Laboratory, NIST
• M. Cyrus Daugherty, Physical Measurement Laboratory, NIST
• Peter Bajcsy, Information Technology Laboratory, NIST
• Daniel Hussey, Physical Measurement Laboratory, NIST
• Katie Weigandt, NIST Center for Neutron Research, NIST

Please contact Caitlyn Wolf at the email above with any questions or comments regarding this code.