Impact of crystallite size on the performance of a beryllium reflector

1. Introduction

The usage of beryllium as an effective neutron reflector at a spallation neutron source has been known since the early days of spallation neutron source development [3,4,8]. The European Spallation Source [6] (ESS), currently under construction in Lund, Sweden, will also employ such a reflector material. The reflector will aide in the enhancement of the brightness of the novel cold and thermal neutron moderators [19]. To predict the behavior of such a system, it is common practice to carry out detailed Monte-Carlo simulation studies, using for example such codes such as MCNP6 [17] or PHITS [15], where the beryllium is treated as a perfect polycrystalline material. However, inhomogeneities and impurities in the beryllium could lead to discrepancies between the performance of the actual system when compared to the modeled system. Such a discrepancy was observed for example in a previous study which compared the experimentally measured performance of a nitrogen-cooled beryllium reflector-filter to the Monte-Carlo simulated performance [11]. It was found in that study, that by manually adjusting the beryllium thermal neutron cross-section library used by the simulations to account for crystallite effects, good agreement between simulation and experiment could be achieved.

The effect of crystallites, in particular their sizes, on the neutron transport properties of a material was reported in early neutron transmission measurements [5,18]. A crystallite referes to one of the many blocks of perfect crystal, each with varying orientations and dimensions, that make up a polycrystalline material. This particular effect, known as extinction [12], refers to the re-scattering of the Bragg reflected beam back into the direction of the transmitted beam. This results in an increased observed transmission of the primary beam and the effect is proportional to the size of the crystallites. An interesting recent development is that it has been possible to combine the theoretical treatment of extinction presented in [12] with Bragg-edge transmission imaging [13,14] in order to deduce information regarding the crystallite sizes in a material.

The current study builds upon the work first presented in [11] where the crystallite effects were proposed as one of the origins for the discrepancy between the simulations and measurements. We aim to quantify the effect of extinction due to crystallites in beryllium in this work. The effect of impurities on the cross-section of beryllium was discussed previously in [11], which helped motivate the selection of high-purity beryllium at ESS and thus we do not investigate it further in this work. Additionally, we do not investigate the effects of preferred orientation here. In the following, we first give an introduction of how Bragg neutron scattering is included in thermal neutron scattering simulation libraries, then present how we introduce the extinction effects into this process, and finally show some preliminary results on the impact of the calculated brightness of the moderators at ESS.

2. Methodology

For a material such as beryllium, the Bragg scattering from the planes in the crystal results in a jagged neutron cross-section as a function of energy. In thermal neutron scattering cross-section calculation codes, such as NJOY [10] and NCrystal [2] this effect is included by assuming a powder average, i.e. small crystallites of random orientation, and the coherent elastic scattering cross-section is given by the following, (1) σ el coh ( λ ) = λ 2 2 V ∑ h k l d h k l | F h k k | 2 , where V is the crystal structure unit cell volume, d h k l is the lattice plane spacing, h k l are the Miller indices, λ is the wavelength, and F h k l is the structure factor. The extension of this formula to include extinction effects is rather straightforward, as shown in [14], and results in, (2) σ el coh ( λ ) = λ 2 2 V ∑ h k l d h k l | F h k k | 2 E h k l ( λ , F h k l ) , and E h k l ( λ , F h k l ) is the extinction factor, given by, (3) E h k l ( λ , F h k l ) = E B sin 2 θ h k l + E L cos 2 θ h k l , with (4) θ h k l = arcsin ( λ 2 d h k l ) , and where E B and E L represent the Bragg and Laue components of the scattering and are given by, (5) E B = 1 1 + x , (6) E L = 1 − x 2 + x 2 4 − 5 x 3 48 + ⋯ for x ⩽ 1 , (7) E L = 2 x π [ 1 − 1 8 x − 3 128 x 2 − 15 1024 x 3 − ⋯ ] for x > 1 , and (8) x = S 2 ( λ F h k l V ) 2 , where S is the crystallite size with units of length. The overall effect is that as S increases, the coherent elastic scattering cross-section σ el coh ( λ ) decreases, thus increasing the mean free path of a neutron in the material.

Nuclear data libraries were then generated for different crystallite sizes using a modified NJOY code system and based on the beryllium input file data given in [9]. The calculation of the coherent elastic scattering component was carried out using a NCrystal-NJOY framework developed by Jose Ignacio Marquez Damian and available online at [1]. The calculation was modified to include the extinction effect described above.

3. Results and discussion

OPEN IN VIEWER

Fig. 1.

Neutron total cross-section for beryllium as a function of different crystallite sizes.

OPEN IN VIEWER

Fig. 2.

Vertical cut of the MCNP6 ESS model used for the calculation of the cold neutron brightness. Some specific components are highlighted and the colors represent different materials.

The total cross-sections as a function of energy for various crystallite sizes are shown in Fig. 1. As seen in the figure, the increase in crystallite size reduces the total cross-section in the Bragg scattering energy regime. The observed effect agrees with the trend reported in earlier studies on the crystallite effect in beryllium [7].

After creation of the neutron scattering libraries, we used them to calculate the effect of the crystallite size in the beryllium reflector on the cold neutron brightness at ESS. Figure 2 shows the model that was used for these calculations, which is an early design of the butterfly 1 concept [19]. The procedure for calculating the brightness followed the description as given in Ref. [19].

Table 1 shows the average relative effects on the cold neutron brightness when using the extinction corrections for the 42 beam ports at ESS. The data are presented relative to the case when using the neutron scattering library with no extinction corrections, as shown in Fig. 1. The overall trend is that an increasing crystallite size results in a lower average brightness. The impact was rather uniform across all beamports, with the simulations indicating that the difference between the brightness maximum and minimum was less than about half of a percent for each of the three cases.

It can be interesting to compare the impact of the crystallite effect to other effects, such as the para-hydrogen fraction for example. The impact of this effect on the brightness of a butterfly moderator was studied in Ref. [19]. At an operational temperature of 20 K, theoretically 99.8% of the H₂ molecules in the liquid hydrogen should be in the para state. Previous studies have shown that with a catalyzer in place, this fraction could be achieved during beam operation [16]. Referring to Fig. 20 in [19], such a fraction results in about a couple percent impact on the cold neutron brightness when compared to the ideal case.

In addition to having the possibility to more accurately predict the neutronic performance of the beryllium with different crystallite sizes, the ability to link these effects to the performance of the reflector could be used as valuable input in the beryllium selection process. For example, it could be possible to relax restrictions on the crystallite size if a small impact on the neutronic performance is deemed acceptable, which could translate into a cost savings for the facility. An important point to also mention is that the crystallite sizes indicated in this work may not reflect the grain sizes in the material. The possible presence of a sub-grain structure would mean that the crystallites would have smaller dimensions than the grain sizes. For the above mentioned reasons, a detailed experimental characterization of the beryllium would provide a valuable benchmark to the work presented here.

In summary, we have demonstrated how to include extinction effects directly into the thermal neutron data libraries used for Monte-Carlo simulations. We have used this method to study the effect of crystallite sizes on the performance of a beryllium reflector at a spallation neutron source. Future work will entail comparisons with detailed neutron transmission measurements and looking into the possibilities for including preferred orientation effects into the calculation process.