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Multi dimensional neutron scattering data normalization

Introduction to normalization

In any experiment, a measurement consists of a raw quantity of interest and a statistical significance of the measurement itself. For neutron diffraction, the differential scattering cross section at some point \(\mathbf{Q}\) in the reciprocal space, measured with a single detector with a solid angle \(d \Omega\), is given by:

(1)\[\frac{d\sigma}{d\Omega}=\frac{N}{\Phi \times d\Omega}\]

where \(N\) is the number of scattered neutrons in a small volume \(d \mathbf{Q}\) around \(\mathbf{Q}\), and \(\Phi\) is the time integrated incident flux that contribute to the scattering in the given volume. \(N\) is the raw quantity, while \(\Phi \times d \Omega\) is the statistical significance, or norm.

If there are multiple detectors, or multiple experiments contributing to the scattering in the \(d \mathbf{Q}\) volume, one needs to add together the raw data, add together the norms, and then divide

(2)\[\frac{d\sigma}{d\Omega}=\frac{\sum_i N_i}{\sum_i \Phi_i \times d\Omega_i}\]

The summation index \(i\) represents every detector and sample orientation or repeated measurement that contribute to the scattering in the desired region of the reciprocal space. In a similar fashion, for inelastic scattering, the double differential cross section must be written as:

(3)\[\frac{d^2 \sigma}{dE d\Omega}=\frac{\sum_i N_i}{\sum_i \Phi_i \times d\Omega_i \times dE_i}\]

What this means is that, in the triple axes type of measurements for example, where we have a single detector (solid angle \(d \Omega_i\) is a constant), we should not normalize data by monitor counts and then add different experiments together. The monitor count is a proxy for the incident flux. We should instead add raw data together, add monitors together, and only then divide.

Detector trajectories in reciprocal space for single crystal experiments

For direct geometry inelastic scattering, for any given experiment, all the incident flux \(\Phi_i\) contributes to the scattering, and it is just a number. For diffraction and indirect geometry inelastic experiments one has to account only for the flux that contribute to the scattering in the \(d \mathbf{Q}\) region, which is detector and momentum dependent. Similarly, \(dE_i\) is the length along energy transfer axis of the detector trajectory inside the \(d \mathbf{Q}\) region. It is therefore important to understand where is the scattering in reciprocal space for each of the detectors. In this section we describe the case of single crystal experiments. We assume that the regions \(d \mathbf{Q}\) are given by a regular gridding of the data in reciprocal space.

For a scattering event in a particular detector, the momentum transfer in the laboratory frame is related to the momentum transfer in the sample frame by the rotation of the sample goniometer. This is further related to the crystallographic \(HKL\) frame by the \(UB\) matrix. In Mantid notation this can be written as

(4)\[\begin{split}\left(\begin{array}{r} -k_F \sin(\theta) \cos(\phi)\\ -k_F \sin(\theta) \sin(\phi)\\ k_I - k_F \cos(\theta) \end{array}\right) = R \left(\begin{array}{c} Q^{sample}_x \\ Q^{sample}_y \\ Q^{sample}_z \end{array}\right) = 2 \pi R \cdot U \cdot B \left(\begin{array}{c} H \\ K \\ L \end{array}\right)\end{split}\]

where \(k_I\) is the momentum of the incident neutron and \(k_F\) is the one of the scattered neutron. R is the rotation matrix of the goniometer. For diffraction case, \(k_I = k_F =k\). For direct geometry inelastic \(k_I\) is fixed in a particular experiment, while for indirect geometry inelastic \(k_F\) is fixed for the detector. From equation (4) one can see that the trajectories in the reciprocal space are simply straight lines, parametrized by \(k\) for diffraction, \(k_I\) for direct geometry, or \(k_F\) for indirect geometry. If we calculate what the \(H, K, L\) coordinates are for two points, say at \(k_{min}\) and \(k_{max}\), we can then write:

(5)\[\frac{H-H_{min}}{H_{max}-H_{min}}=\frac{K-K_{min}}{K_{max}-K_{min}}= \frac{L-L_{min}}{L_{max}-L_{min}}=\frac{k-k_{min}}{k_{max}-k_{min}}\]

Thus, if we know for example that we want to calculate the intersection of the trajectory with a plane at \(H=H_i\), we can just plug in \(H_i\) in the above equation and get the corresponding \(K_i, L_i, k_i\).

Any trajectory can miss a particular box in \(HKL\) space, can be along one of the faces (say if \(H_{max}=H_{min}\) then all \(H\) points have the same value), or it can intersect the box in exactly two points. If we know the momentum corresponding to the intersections, say \(k_1\) and \(k_2\), all we need is to integrate the incident flux between these two values, and then multiply with the solid angle of the detector, in order to obtain the statistical weight of this detector’s contribution to this particular region in the \(HKL\) space.

A similar equation to (5) can be obtained for inelastic scattering, by replacing \(k\) with \(k_F\) for direct geometry or with \(k_I\) for the indirect case. We can then relate \(k_I\) or \(k_F\) with the energy transfer \(\Delta E\), to get the intersections along the energy transfer direction.

It is important to note that even if we calculate the intersections of the trajectory with a particular box the norm might still be zero, since we could have no incident neutron flux corresponding to that box in \(HKL\) space.

The way to account for excluded data is algorithm dependent. See the documentation for each particular implementation.

Symmetrization

To improve statistics in a certain region, one can use data from different regions of the reciprocal space that are related by the symmetry of the physics in the material that is being studied. A simple way to correctly estimate the statistical weight of the symmetrized data is to apply the symmetry operation on the detector trajectories (apply to \(H, K, L\) at \(k_{min}\) and \(k_{max}\)) and recalculate the normalization.

Current implementation

As of release 3.3, the normalization can be calculated for single crystal diffraction (MDNormSCD) and single crystal direct geometry inelastic scattering (MDNormDirectSC).

References

The source for how this calculation is done is

1. T.M. Michels-Clark, A.T. Savici, V.E. Lynch, X.P. Wang and C.M. Hoffmann Expanding Lorentz and spectrum corrections to large volumes of reciprocal space for single-crystal time-of-flight neutron diffraction. J Appl Crystallogr 49.2 (2016) doi: 10.1107/S1600576716001369

Category: Concepts