CANDOR detector calibration

The goal is to characterize the response of the CANDOR detector to input wavelengths

If a crystal with tight enough wavelength selection can be placed on the sample table, we can treat it as a monochromatic source

Then constructing a transfer matrix as

$$  \bigg(\:L\:\bigg) \bigg(\lambda\bigg) = \bigg(d\bigg) $$
where input vectors \(\lambda\) are of the form (for a good monochromator) e.g.
$$
    \bigg(\lambda\bigg) = \left( \begin{array}{@{}c@{}}
    0 \\
    0 \\
    I_\lambda \\
    0 \\
    \vdots \\
    0
    \end{array} \right)
$$

The first dimension of \(m\) of \(L\) will be defined by the \(\lambda\) steps taken for the calibration, and the second dimension \(n\) is equal to the number of detectors.

Then after \(m\) independent measurements of the response to a single wavelength, the \(L\) matrix is simply the stacking of the \(m\) measured measured \(d\) vectors, each with dimension \((1\times n)\), after normalizing each column to the (independently measured) input intensity \(I_\lambda\) for that wavelength with that monochromator.

Then in general use, any input with a known distribution of wavelengths \((\lambda)\) should produce a predictable output \((d)\) on the detector.

Of course, for interpreting the data, we need the reverse operation:  given a measured \((d)\), we can extract \((\lambda)\) as

$$    \bigg(\lambda\bigg) = \bigg(\:L\:\bigg)^{-1} \bigg(d\bigg) $$

Probability-based approach

Bayesian Back-Propagation

Again measuring the \(n\) detectors at \(m\) different wavelengths, the probability that any detector \(d_\nu\) will detect a neutron of a given wavelength \(\lambda_\mu\) is

$$
    P\left(d_\nu|\lambda_\mu\right) = \frac{P\left(\lambda_\mu|d_\nu\right) P(d_\nu)}{P(\lambda_\mu)}
$$

but what we want is the probability of a given wavelength, given a measured neutron in detector
\begin{equation}
P(\lambda_\mu|d_\nu) =  \frac{P\left(d_\nu|\lambda_\mu\right) P(\lambda_\mu)}{P(d_\nu)} \tag{4}
\end{equation}

The total probability \(P(\lambda)\) of an incident neutron having a given wavelength \(\lambda\) can be calculated by measuring the incident spectrum of all \(m\) wavelengths using a \({}^3\)He detector and normalizing to the integrated intensity at all wavelengths:
$$ P(\lambda_\mu) = \frac{I_{\lambda_\mu}}{\sum_m I_{\lambda_m}} $$

The total probability \(P(d_\nu)\) of detecting a neutron in detector \(d_\nu\) is the sum of that probability over all wavelengths:

$$
P(d_\nu) = \sum_m P(d_\nu|\lambda_m) P(\lambda_m)
$$
using the the \(P(\lambda_\mu)\) values from above, and the \(P(d|\lambda)\) calculated by normalizing to the sum over all \(n\) detectors at one wavelength:

$$
P(d_\nu|\lambda_\mu) = \frac{I_{d_{\mu,\nu}}}{\sum_n I_{d_{\mu,n}}}
$$

Noting that the sum over all \(I_{\lambda_m}\) occurs in the numerator and every term of the denominator of the fraction in equation 4, we can rewrite it as
$$
P(\lambda_\mu|d_\nu) = \frac{I_{d_{\mu,\nu}} I_{\lambda_\mu} /\sum_n I_{d_{\mu,n}}}{\sum_m (I_{d_{m,\nu}} I_{\lambda_{m}} / \sum_n I_{d_{m,n}})}
$$
In the case where we can treat the energy-dispersive detectors as perfectly efficient, this simplifies because then \(I_{\lambda_m} \equiv \sum_n I_{d_{m,n}}\) for all \(m\) and
$$
P(\lambda_\mu|d_\nu) = \frac{I_{d_{\mu,\nu}}}{\sum_m I_{d_{m,\nu}}}
$$

Matrix Inversion

Another approach would be to directly invert the measured intensity matrix.  In order to invert, the matrix has to be square and so we have to select 54 wavelength regions to integrate over.

One logical way to break up the wavelengths is to segment according to the most active detector for any given wavelength - then the total intensity of wavelengths in that band (let's call it \(I_{\lambda_\nu}\)) is given by the sum of all \(I_{\lambda_m}\) for all \(m\) where \(I_{d_{m,\nu}} > I_{d_{m,n \neq \nu}}\)

This will result in \(n\) integrated wavelength bands, measured at \(n\) detectors, giving a square matrix \(L\).  Given the small overlap between detector channels, it is also almost a diagonal matrix and should be easy to invert.

Then the number of neutrons in a wavelength band corresponding to a later measurement is given by
$$
\bigg( I_\lambda \bigg) = \bigg( \: L \: \bigg)^{-1} \bigg( I_d \bigg)
$$
This approach has the advantages that

• Each step has known error propagation
• Correlations between detector measurements can be preserved