8. Theoretical background

8.1. Resolution functions
 8.1.1. Incident intensity (monitor)
 8.1.2. Intensity at the detector
 8.1.3. Normalization of R(Q,E)
 8.1.4. Norm factors reported by RESTRAX
8.2. Convolution in (Q,E) space
 8.2.1. Excitations — zero width in energy
 8.2.2. Diffuse dispersion — finite width in energy

8.1. Resolution functions

The resolution function R(Q,E) of a neutron spectrometer is defined by the relationship between the scattering function S(Q,E) as a material property of the sample, and the measured function I(Q0,E0), which is smeared by the instrument resolution:

(1)    i  .

In this formalism, R(Q,E) is a purely instrumental property independent of the sample material.

In a quasi-classical approximation we can express the resolution function by inspecting neutron trajectories and their transmission probabilities through the spectrometer. Let PM(r,ki) and PA(r,kf) be the probabilities of neutron transmission through the monochromator and analyzer parts of the spectrometer with (r,ki) and (r,kf) being the phase-space coordinates of incident and scattered neutrons at the point r in the sample, where the scattering event takes place. Using these probabilities, we can express the intensities at the sample and at the detector.

8.1.1. Incident intensity (monitor)

For an isotropic homogeneous source with wave-vector distribution , the flux distribution at a point r in the sample is

(2)     .

The incident flux is usually monitored using a detector with low efficiency, inversely proportional to the neutron velocity (or wave-number), η(ki) = η0ki-1. The monitor count-rate per unit area, M, averaged over the sample volume is then

(3)     ,

where the integral ranges over the full momentum space of incident neutrons and the sample volume.

8.1.2. Intensity at the detector

The structure factor S(Q,E) returned by the EXCI module of RESTRAX is related to the differential scattering cross-section by the equation

(4)     .

In order to simplify notation, our definition of S(Q,E) already includes the density of unit cells. Neutron current registered by the detector of an instrument set to (Qj,Ej) is then

(5)     ,

where , and .

We have introduced the resolution function defined by the integral expression

(6)     .

Our definition is similar to the one of Dorner [1], except for the additional integration over sample volume, V. This permits to take into account possible correlations between ki and kf and the dependence of detected count-rate on the irradiated volume of the sample. The dependence of R'(Q,E) on (Qj,Ej) is implicitly included in the transport probabilities PM, PA.

8.1.3. Normalization of R(Q,E)

In RESTRAX, we normalize the resolution function so that the detected count-rate, Cj, can be written as a simple convolution integral,

(7)     .

The definition of R(Q,E) then follows from (5) and (7), which imply that

(8)     .

The integral (8) is evaluated by two methods:

Integral norm of the resolution function is

(9)     .

For small samples, we can assume that does not depend on r and arrive thus to

(10)     .

Comparison with (3) gives

(11)     .

We can thus normalize intensities on monitor counts:

In RESTRAX, the count-rates are normalized to 106 monitor counts per unit area (M = 106 cm-2) at η0 = 1 Å-1.

8.1.4. Norm factors reported by RESTRAX

The commands RES 1 and MRES 1 report several norm factors related to the resolution function, R(Q,E). In Gaussian approximation,

(12)    R(Q,E) ≡ R0 exp(-0.5 XTMX)  ,

where M is the resolution matrix and X≡(Q-Q0,E-E0).

The resolution volume, Vol(QE) is then defined as

Vol(QE)TRAX ≡ (2π)2|M|-1/2

and is expressed in units [Å-3meV].

The resolution matrix corresponds to the inverse of the covariance matrix <XTX>. Therefore, we can calculate the resolution volume also from Xi events obtained by ray-tracing method,

Vol(QE)MC ≡ (2π)2|<XTX>|1/2 .

Similarly, the resolution volumes Vol(ki), Vol(kf) (equivalent to VI,VF notation in this paper) are calculated in [Å-3] as

Vol(ki)  ≡  (2π)3/2|<kiTki>|1/2

Vol(kf)  ≡  (2π)3/2|<kfTkf>|1/2

The other norm reported by the [M]RES command is derived from the resolution volumes:

R0  ≡  Vol(ki) Vol(kf) / Vol(QE)

It corresponds to the commonly used normalization by Dorner [1] and Popovici [3], who followed (apart of the kf/ki factor) the concept introduced by Cooper & Nathans [4].

The norm RN from (11) is calculated in Gaussian approximation as

(13)     ,

where <X> is the covariance matrix of a random vector variable X and |<X>| denotes its determinant. The factor 24/π is used to compensate for the ratio between the cylindrical sample volume and its Gaussian approximation. This expression takes into account possible correlations between ki, r and kf.


8.2. Convolution in (Q,E) space

Scan curves are simulated as a 4-dimensional convolution of the resolution function R(Q,E) with model scattering function S(Q,E). There are two kinds of the scattering function available in RESTRAX:

  1. Planar dispersion — a plane in (Q,E) space defined by the parameters GH,GK,GL and GMOD. The scan is simulated by the commands PHON, MPHON
  2. User-defined — anything defined in the EXCI module. Simulation of the scan curves and fitting to experimental data is performed using the commands in the FIT (or MFIT) menu.

For the purpose of convolution with S(Q,E), the resolution function is always represented as a set of points (Qα,Eα) with weights pα. Such representation is either obtained directly by the ray-tracing method or by generation of random numbers with a Gaussian distribution corresponding to the resolution matrix. The 4-dimensional convolution integral is then evaluated by the Monte Carlo method described below. To preserve the normalization of the resolution function, we set where RN is calculated according to (13).

8.2.1. Excitations — zero width in energy

The resolution function — a cloud of events (Qα,Eα,pα) — is swept through the dispersion surface Eexc(Q) along the scan. The events are sorted into a histogram, weighted by the value of scattering function. Intensity in the j-th step can be thus expressed as

(14)     .

The δ-function in the sum is approximated by counting only those events, which cross the dispersion branch within the step interval (j-0.5 ;j+0.5 ). Normalization of the delta function is ensured by weighting each such event by the factor 1/ΔEj, where ΔEj is the step width in energy with respect to the dispersion surface,
ΔEj = |  Eexc(Qj) -  Eexc(Qj-1) -  Ej +  Ej-1|.

8.2.2. Diffuse dispersion — finite width in energy

The procedure is similar, but the events are added in each step, weighted by the value of S(Q,E):
(15)    ij .

Note: In both cases, the convolution method is correct under the assumption that changes of shape or size of the resolution function are negligible within the scan range. This condition is always fulfilled for a3-scans (sample rotation).


 

References

  1. B. Dorner, Acta Cryst. (1972), A28, 319-327.
  2. M. Popovici, A. D. Stoica and I. Ionita, J. Appl. Cryst. (1987) 20, 90-101.
  3. M. Popovici, Acta Cryst. (1975), A31, 507-513.
  4. M. J. Cooper, R. Nathans, Acta Cryst. (1967), 23, 357-367.