How is the beam-sample convolution factor fC in Equation 1 of the paper calculated? How its uncertainty estimated?

First of all we recall that, as explained in the paper, the cross section of the 7Be(n,p) reaction is extracted relative to that of 6Li(n,t). To this end, we calculate fC for both samples, using the sample distributions determined experimentally (as described in Ref. [10] of the paper). The factor fC has been calculated both analytically and by means of a Monte Carlo technique. The ratio of these factors for the two samples (the only value that enters in the cross section determination), Rf=fC(Be)/fC(Li), resulted to be practically identical using the two methods and it is very close to unity. For both methods of calculation, the difference between the factors fC for the two samples is around 2%. By the way, this justifies the standard procedure adopted in the present work (i.e. the ratio method) used to deduce the cross section.

As for the uncertainty estimate, the main source of error is related to the centering of the two samples on the beam. In the measurement, the geometrical center of both the 6Li and 7Be samples (more precisely the center of the sample backings) were aligned to +/-1 mm relative to the beam center position, identified by means of gafchromic foils. Therefore, the maximum possible displacement between the two samples was +/- 2 mm. In the Monte Carlo simulation approach, one can vary the relative position between the centers of the samples according to a Gaussian distribution with σ = 2 mm. Note that the center of 7Be distribution is shifted, relative to the center of the sample backing, (see Tab. 1 in Ref. [10]), but clearly this displacement is taken into account in the simulations. The resulting distribution of Rf is approximately Gaussian, with a sigma of 8%.

An alternative way to estimate the uncertainty on Rf is to use a uniform distribution of a displacement of the sample, with the steps and direction varied independently. Such a choice avoids unrealistic large deviation that would instead be generated by a Gauss distribution. Starting from this assumption, the beam-sample convolution factor can be calculated using a 2-dimensional Irvin-Hall distribution i.e. the convolution of a series of uniform distributions. In this case one ends up with a triangular distribution in each direction leading to a somewhat tent shaped distribution displayed in the figure.

Picture1.png

This distribution represents the probability for a displacement of the target with respect to the expected beam center. The probability density of fC is displayed in the next figure.

Picture2.png

The center of gravity of the distribution is at 1.59 mm and -4.75 mm in x- and y-direction respectively, slightly shifted towards the beam center because fC is monotonously increasing in this direction. The standard deviation of the distribution is in both directions almost equal to 0.82 mm. To estimate the uncertainty, one can consider the region covering 68.3% (1 sigma region). This can be reached by a square of 2.34 mm centered on the 7Be distribution. Within this region the fC varies from +9.2% to 11.7% with respect to the nominal target position, consistently to the estimate obtained by Monte Carlo simulations.

(contributed by: N Colonna, R Dressler and A Mengoni - 2018-06-18)

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Topic revision: r12 - 2018-06-18 - AlbertoMengoni
 
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