Thermal neutron scattering data for 7 LiF and BeF 2

. Based on the coherent elastic, incoherent elastic, coherent inelastic and incoherent inelastic scattering processes, a code named SIRIUS is developed to produce thermal neutron scattering data for crystals in ENDF-6 format. The phonon band structures and projected phonon densities of states of 7 LiF and BeF 2 crystals were calculated by Hellman-Feynman Theorem combined with a lattice dynamics direct method. Finally the thermal neutron scattering data for 7 LiF and BeF 2 crystals are given.


Introduction
Thermal neutron scattering data are widely used in nuclear engineering applications, such as reactor design, radiation shielding, long-lived nuclear waste transmutation, Boron Neutron Capture Therapy.
The study of thermal neutron scattering started from 1950s, which is due to the needs of the reactor design. Subsequently, GASKET code was developed at General Atomic [1,2], and was used to generate early thermal neutron scattering data. Now, LEAPR module in NJOY [3,4] code is widely used to calculate ENDF-6 format thermal neutron scattering law (TSL) data.
At present, some evaluated nuclear data libraries such as ENDF/B library [5], JENDL library [6], JEFF library [7] and IAEA nuclear data library contains the sublibrary of TSL data for about 23 moderators. However, new requirements for the development of current nuclear engineering, especially for the fourth generation reactor design need more TSL data. For example, in molten salt reactor, TSL data for Beryllium fluoride and Lithium fluoride are needed. Mei and Cai applied the CASTEP code and the modified NJOY code to generate the thermal neutron scattering data for LiF and BeF 2 crystals [8] and investigated the thermal neutron scattering data for molten salt Flibe based on the dynamic performance for Flibe [9]. To date, no thermal neutron scattering data in the ENDF format are available for Beryllium fluoride and Lithium fluoride.
Recently, a code named SIRIUS [10] is developed at IAPCM to generate the thermal neutron scattering data for solid in ENDF-6 format. In this paper, SIRIUS code is utilized to calculate the thermal neutron scattering data of 7 LiF and BeF 2 crystals.

Theoretical models
The theory of scattering of thermal neutron by crystals is exposed in details in many textbook [11,12] and is briefly considered here. There are four types of scattering, coherent elastic scattering, incoherent elastic a e-mail: sun weili@iapcm.ac.cn scattering, coherent inelastic scattering and incoherent inelastic scattering.
The expression of the coherent elastic scattering cross section for polycrystalline material is The expression of the differential incoherent inelastic scattering cross section is Where W (T ) = 1 β s coth β s 2 ρ(β s )dβ s is the Debye-Waller integral divided by the atomic mass, T is the temperature of the scattering medium, N and V 0 are the number of basis atoms and the volume of the primitive unit cell, respectively. m is the mass of the neutron, |F(N )| is the crystallographic structure form factors, τ is the length of reciprocal vectors, E i are the "Bragg edges", E is the incident neutron energies in the laboratory system, µ is the cosine of the scattering angle, the subscript coh means coherent and inc means incoherent. σ inc and σ coh are the incoherent and coherent bound scattering cross section for the material, respectively. For mixed moderator, the coherent elastic scattering cross section is more complicated and can be written as Where the subscript jmeans the jth atom in the unit cell. τ is the reciprocal lattice vector, d is atomic position of the ND2016 jth atom. The calculation procedure for coherent elastic scattering of the mixed material goes as follows. Firstly, the W j (T ) of all atoms are calculated separately. Secondly, the Eq. (3) is used to calculate the whole coherent elastic scattering cross section for the mixed moderator.
The expression of the double differential inelastic scattering cross section is means the momentum transfer and β = E − E k B T means the energy transfer, E is the scattering neutron energies in the laboratory system, S(α, β, T ) is the thermal neutron scattering law.
is know as the incoherent scattering law, and S d (α, β, T ) accounts for interference effects. Generally, the interference effects in inelastic are considered so small that can be neglected, and the incoherent approximation is used [13]. The incoherent thermal neutron scattering law is given as This expansion is known as the phonon expansion, subscript p represent the creation (or annihilation) of p phonons.where ρ(β) is the phonon frequency distribution, For strong coherent scatters such as graphite and beryllium, which means that the bound coherent scattering cross section is almost equal to the total bound scattering cross section, the incoherent approximation used in inelastic is known to introduce noticeable biases [14][15][16][17].
In these cases, the interference effects can't be neglected, so the thermal neutron scattering law is composed with coherent and incoherent contributions as follow Here, only the one-phonon coherent is considered, the summation for the incoherent approximation starts from two phonons. The numerical formula of S coh,1 p (α, β, T ) is given as Finally, based on the theory, a code program SIRIUS is constructed to generate thermal neutrons scattering data in ENDF-6 format. In order to calculate coherent inelastic scattering, both the PDOSs and the phonon dispersion relation for materials are needed, otherwise, PDOSs for materials are enough.

The phonon density of states
Beryllium fluoride (BeF 2 ) has a α-quartz-type structure and belongs to space group P3121 [18] with three beryllium atoms and six fluoride atoms in the unit cell.
Lithium fluoride ( 7 LiF) has a face centred cubic structure and belongs to space group FM3M [19] with four lithium atoms and four fluoride atoms in the unit cell.
The computed lattice parameters of BeF 2 and 7 LiF are in well agreement with experimentally reported [20][21][22]. The lattice constants values for BeF2 are a = 4.85Å, and c = 5.335Å. The lattice constants value for 7 LiF is a = 4.026Å. The structure for BeF 2 crystal and the structure for 7 LiF are showed in Fig. 1.
The phonon band structures and projected phonon densities of states of 7 LiF and BeF 2 crystals were calculated by Hellman-Feynman Theorem combined with a lattice dynamics direct method using the Vienna Ab inition Simulation Package (VASP) and the PHONON code [23,24]. For 7 LiF crystal, the model used a 3 × 3 × 3 supercell composed of 216 atoms and for BeF 2    Figure 2 shows the calculated partial POSs of 7 Li and F atoms for 7 LiF, Fig. 3 shows the calculated partial POSs of Be and F atoms for BeF 2 .There both were compared with the result from Mei and Cai [8].
Obviously, the calculated partial PDOSs of LiF are similar as the result from Mei and Cai [8], but the calculated partial PDOSs of BeF2 are different. BeF 2 crystal is low symmetrical crystal, need more supercell in the calculation to get the reasonable result. The same method were used to calculate the PDOSs, however different codes and initial inputs were used. To validate these, more experimental data are needed.

Thermal neutron scattering data
Based on the PDOS given above, and the bound atom scattering cross section given in Table 1 [25], the thermal neutron scattering law data were calculated using SIRIUS code. Here the incoherent approximation was used in inelastic scattering.
According to the bound atom cross section, the incoherent cross section and coherent cross section are same important for 7 Li, So the thermal neutron scattering data for 7 LiF crystals were divided into four parts. The   first part was the coherent elastic scattering data for the whole 7 LiF crystal. The second part was the incoherent inelastic scattering data for F in 7 LiF. The third part was the incoherent inelastic scattering data for 7 Li in 7 LiF. The last part was the incoherent elastic scattering data for 7 Li in 7 LiF. In this work, the interference effects in inelastic didn't considered, and the same in BeF 2 . Figure 4 shows the thermal neutron scattering cross section for 7 LiF crystal compared with the result from Mei and Cai [8] at 300 K. Table 2 shows the result of effective temperatures and Debye-Wall integrals divided by the atomic mass for 7 LiF crystal.
For BeF 2 crystal, the incoherent cross section can be neglected, so the thermal neutron scattering law data for BeF 2 crystals were divided into three parts. The first part was the coherent elastic scattering data for the whole BeF 2 crystals. The second part and the third part were incoherent inelastic scattering data for Be in BeF 2 and F in BeF 2 , respectively. Figure 5 shows the thermal neutron scattering cross section for BeF 2 crystal compared with the result from Mei and Cai [8] at 300 K. Table 3 shows the result of effective temperatures and Debye-Wall integrals divided by the atomic mass for BeF2 crystal.
As showed in Fig. 4 and Fig. 5, the present scattering cross sections have significant difference with the Ref. [8]. For LiF crystal, the present PDOSs of Li and F are similar to the reference, but the present total inelastic scattering cross section is about twice as much as the reference. Maybe in the reference work, the input value of number

Conclusions
In summary, the thermal neutron scattering cross section for 7 LiF and BeF 2 crystals have been calculated in this work using a combination of ab-initio simulations, lattice dynamics, and SIRIUS code. Compared with the Ref. [8], obvious difference can be found in PDOSs and scattering cross sections, the possible cause which may lead these had been analysed in the previous section. However, to validate these calculated thermal neutron scattering data, experimental data are needed. More work should be done in future. The first is to improve SIRIUS code for liquid materials. The second is to generate more thermal neutron scattering data for other materials such as silicon carbide and molten salt Flibe and so on.