Overview#

The main functionality of TRINIDI is reconstructing areal density values, \(Z \in \mathbb{R}^{N_{\mathrm{P}} \times N_{\mathrm{m}}}\), from the neutron time-of-flight transmission measurements, \(Y_{\mathrm{s}} \in \mathbb{R}^{N_{\mathrm{P}} \times N_{\mathrm{A}}}\) (sample measurement) and \(Y_{\mathrm{o}} \in \mathbb{R}^{N_{\mathrm{P}} \times N_{\mathrm{A}}}\) (open beam measurement), where

\[\begin{split}N_{\mathrm{P}} &= \text{Number of projections} \\ N_{\mathrm{A}} &= \text{Number of measured time bins} \\ N_{\mathrm{m}} &= \text{Number of isotopes}\end{split}\]

We strongly advise the reader to review our manuscript [2] make yourself familiar with

  • assumptions made of the measurement system

  • time-of-flight (TOF) and how it relates to the neutron energy

  • resolution function, \(R\), and how it relates the time-of-flight (TOF) and the time-of-arrival (TOA)

  • TOF vector, \(t_F\), and TOA vector, \(t_A\).

  • cross section dictionary, \(D\)

Array Shapes#

In TRINIDI the measurements are handled using the nd-arrays Y_s and Y_o and

Y_s.shape == Y_o.shape == projection_shape + (N_A,).

The tuple projection_shape is typically the shape of the detector and np.prod(projection_shape) == N_P. For example, given a measurement with a \(64 \times 64\) pixel detector and \(1000\) TOA bins we get

Y_s.shape == projection_shape + (N_A,) == (64, 64) + (1000,) == (64, 64, 1000).

The array element Y_s[i_y, i_x, i_t] corresponds to the pixel with index (i_x, i_y) and TOA bin t_A[i_t], where t_A has size N_A.

Similarly, for the areal density array we have

Z.shape == projection_shape + (N_m,).

Below you can find a summary of the most important arrays and operators in TRINIDI.

Cross Section Dictionary#

The cross section dictionary, \(D \in \mathbb{R}^{N_{\mathrm{m}} \times N_{\mathrm{F}}}\), maps areal densities to hyperspectral attenuation or transmission values. Ignoring the effects of the resolution function, the transmission is expressed as

\[Q_0 = \exp(-ZD)\]

so that \((Q_0)_{i, j}\) is the fraction of flux neutrons that make it to the detector at projection \(i\) and TOF bin index \(j\).

In TRINIDI, the cross section data structure, D, is created using the XSDict class in the trinidi.cross_section module. This class has features including reading, merging and plotting cross section entries. The actual values of \(D\) are stored in D.values so that the expression of the transmission from above can be computed as

Q_0 = np.exp(- Z @ D.values).

In TRINIDI, D.values has units of \(\mathrm{mol}/\mathrm{cm}^2\), which is the reciprocal of the units of Z.

To make yourself familiar of the cross section dictionary data structure We recommend to check out the trinidi.cross_section Module Demo script which is also available as Jupyter notebook.

Time-of-Flight Calibration#

In practice, the first step will usually be to estimate the t_A array. We assume the TOF (TOA) bins are equidistant and the time vector t_F (t_A) is in increasing order. Mostly, we leave the calibration to the liberty of the user, however we provide an example illustration that we recommend you read first. For this check out the Time-Energy Calibration Demo script which is also available as Jupyter notebook.

Resolution Operator#

Summary of Array and Operator Shapes and Units#

Data Structure    |  Shape
------------------+--------------
t_F               |  (N_F,)
t_A               |  (N_A,)
R                 |  (N_F, N_A) (implied)
D.values          |  (N_m, N_F)
Z                 |  (N_P, N_m)
Y_o, Y_s, Φ, B    |  (N_P, N_A)

Quantity          |  Unit       |  Data Structure
------------------+-------------+--------------------
lengths           |  m          |  flight_path_lenght
times             |  μs         |  t_A, t_F, Δt, t_0
neutron energies  |  eV         |  E
cross sections    |  mol/cm²    |  D
areal densities   |  cm²/mol    |  Z