How is the beam-sample convolution factor f_C 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 f_C for both samples, using the sample distributions determined experimentally (as described in Ref. [10] of the paper). The factor f_C 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), R_f=f_C(Be)/f_C(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 f_C 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 R_f is approximately Gaussian, with a sigma of 8%.

An alternative way to estimate the uncertainty on R_f 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.

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

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 f_C 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 f_C 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)