DPDFreduction v1¶

Detailed Parameters description¶

• Vanadium: a preprocessed white-beam vanadium file, meaning that all events within a particular wide wavelength range have been rebinned into a single histogram, this for every detector.
• EnergyBins: user can input a triad \([E_{min}, E_{bin}, E_{max}]\) or only the energy binning \(E_{bin}\). If this is the case, \(E_{min}\) and \(E_{max}\) are estimated: \(E_{min}=-0.5E_i\) and \(E_{max} = 0.95E_i\) where \(E_i\) is the incident energy.
• MomentumTransferBins: user has three options. User can input a triad \([Q_{min}, Q_{bin}, Q_{max}]\), just \(Q_{bin}\), or leave this parameter empty. If left empty, \(Q_{bin}\) is estimated as the momentum gained when the neutron gained an amount of energy equal to \(E_{bin}\). If only \(Q_{bin}\) is at hand, \(Q_{min}\) and \(Q_{max}\) are estimated with algorithm ConvertToMDMinMaxLocal.
• NormalizeSlices: This option will normalize the final \(S(Q,E)\) for each energy bin, independent of each other.
• CleanWorkspaces: a series of intermediate workspaces are generated during the reduction process. If this option is set to False, the user may want to inspect these (listed in the order they are produced):
  • vanadium: vanadium events workspace
  • reduced: sample runs after DgsReduction algorithm
  • ec_reduced: empty can after DgsReduction algorithm
  • vanadium_S_theta: vanadium after GroupDetectors algorithm.
  • ec_S_theta_E: reduced empty can after GroupDetectors algorithm.
  • S_theta_E: reduced sample runs after GroupDetectors algorithm and after subtraction of empty can ec_S_theta_E.
  • S_theta_E_normalized: S_theta_E divided by vanadium vanadium_S_theta
  • S_theta_E_normalized_interp: S_theta_E_normalized with interpolation over the detector gaps.
  • S_Q_E: S_theta_E_normalized_interp converted to an events MDWorkspace with momentum transfer instead of angles.
  • S_Q_E_binned: Histogram MDWorkspace after application of BinMD algorithm to S_Q_E.
  • S_Q_E_sliced: conversion of S_Q_E_binned to a MatrixWorkspace with ConvertMDHistoToMatrixWorkspace algorithm.
  • S_Q_E_sliced_norm: S_Q_E_sliced where each E-bin has been normalized.
• OutputWorkspace: The final \(S(Q,E)\) which can serve as input for the Dynamic PDF interface.

Interpolation¶

The ARCS instrument has two gaps at particular \(\theta\) angles due to arrangement of the banks

The gaps lead to empty bins in the \(S(\theta,E)\) histogram which in turn generate significant errors in the final \(S(Q,E)\) for certain values of \(Q\). To prevent this we carry out a linear interpolation in \(S(\theta,E)\) at the blind-strip \(\theta\) angles.

Normalization by number of events¶

If user desires to plot the OutputWorkspace with Mantid’s slice viewer, user should choose the “# Events Normalization” view. The last step in the reduction is performed by executing ConvertMDHistoToMatrixWorkspace, which requires NumEventsNormalization. Our input workspace has as many spectra as instrument detectors. Each detector has a 2D binning in \(Q\) and \(E\). Each detector is at a particular \(\theta\) angle, thus \(E\) and \(Q\) are related by:

\(E(Q) \rightarrow \frac{\hbar Q^2}{2m} = 2E_i + E -2\sqrt{(E_i+E)E_i} \ \ \cos\theta\)

That means that only \((Q,E)\) bins satisfying the above condition have counts. Thus for detector \(i\) we have number of counts \(N_i(Q_j,E_k) \neq 0\) if the \((Q_j, E_k)\) pair satisfy the above condition. This represents a trajectory in \(Q-E\) space.

When we execute ConvertMDHistoToMatrixWorkspace with \(Q\) binning \(\Delta Q\) and E binning \(\Delta E\), we go detector by detectory and we look at the fragment of the \(Q(E)\) trajectory enclosed in the cell of Q-E phase space denoted by the corners \((Q,E)\), \((Q+\Delta Q,E)\), \((Q,E+\Delta E)\) and \((Q+\Delta Q,E+\Delta E)\). Thus we have for detector \(i\) to look at the \((Q_j, E_k)\) pairs within this cell for detector \(i\), with associated \(N_i(Q_j,E_k)\) counts and associated scattering cross-section:

\((\frac{d\sigma^2}{dE d\Omega})_{i,j,k} \ \ (Q_j,E_k) = \frac{N_i(Q_j,E_k)}{d\Omega \delta E}\)

The scattering cross-section in the aforementioned cell of dimensions \(\Delta Q\) x \(\Delta E\) is the average of all the scattering cross sections:

\(\frac{d\sigma^2}{\Delta E d\Omega}(Q,E) = \sum\limits_{i,j,k}(\frac{d\sigma^2}{\delta E d\Omega})_{i,j,k} \ \ (Q_j,E_k) \cdot \Pi_{Q,Q+\Delta Q} \ \ \ (Q_j) \cdot \Pi_{E,E+\Delta E} \ \ \ (E_k) / \sum\limits_{i,j,k} \Pi_{Q,Q+\Delta Q} \ \ \ (Q_j) \cdot \Pi_{E,E+\Delta E} \ \ \ (E_k)\)

where \(\Pi_{a,b} \ (x)\) is the boxcar function

Categories: AlgorithmIndex | Inelastic\Reduction