Calculation of Neutron Production Rates and Spectra from Compounds of Actinides and Light Elements

The code NEDIS allows the calculation of neutron production rate and continuous energy spectra due to (α,n) reaction on Li, Be, B, C, O, F, Ne, Na Mg, Al, Si, P, S, Cl, Ar, K, and Ca. It accounts for anisotropic angular distribution of neutrons of (α,n) reaction in centre-ofmass system and dimensions of alpha emitting source material particles. Spontaneous fission spectra are calculated with evaluated half-life, spontaneous fission branching, νaveraged per fission, and Watt spectrum parameters. The results of calculations by NEDIS can be used as input for Monte Carlo simulation for materials that will be used in radiation shielding and for underground neutron experiments


Introduction
The code NEDIS has been developed for simulation of neutron production rate and spectra due to (α,n)reactions and spontaneous fission for using in nondestructive assay of plutonium in technology process of nuclear fuel fabrication and reprocessing [1,2].The development has required the accumulation and evaluation of measured and calculated (α,n)-reaction cross-sections and angular neutron distributions for the individual levels of the residual nucleus, α-particle stopping cross-section data and the actinides decay constants.Today the code library includes (α,n)-reaction data for the light target nuclides in the Table 1 for alphaparticle energy less 9 MeV [3].The code library has αdecay information for 58 heavy nuclides from 210 Pb to 254 Cf and 36 actinides have the spontaneous fission mode.

Calculation of neutron yields and energy spectra
The heavy nuclides in materials produce MeV αparticles.These α-particles interact with the nucleus in a thick target of light elements and yield neutrons.The neutron yield is calculated by: where Eth -is the energy of threshold, Eα -is the initial energy of the α-particle, ε -is atomic stopping power [9], σtot total cross-section of the (α,n)-reaction.For an arbitrary primary α-particle spectrum the neutron yield is calculated by: where P(Eα) number of α-particles with energy more than Eα.

Method of (α,n) neutron spectrum calculation
The differential neutron yield in the reaction (α,n) within the solid angle Ω * in the centre-of-mass system (CMS) for an individual state of the residual nucleus is given by the expression is the differential reaction cross section, which is expediently expressed as an expansion in Legendre polynomials [ ] Putting (4) to the (3) and after transformation from solid angle in the CMS to energy En in laboratory system the differential neutron yield in the reaction (α,n) is obtained where I(Eα) Jacobian of the transformation is equal to as the energy of neutron and cos(θ * ) is coupled by the expression ( ) (7) where R(Eα)=En(Eα,0 ○ )-En(Eα,180 ○ ), function γ( E α ) is the ratio of velocity CMS in laboratory system to velocity of neutron in CMS.After all appropriate transformation and integration with respect to Eα, the neutron spectrum of the (α,n)-reaction is obtained The limits of integration X(En) and Y(En) are determined by the kinematics of the reaction [10].
The neutron spectrum of a mixture of elements may be determined from the formula for an arbitrary primary α-particle spectrum by where i is the level of the residual nucleus as a result of reaction with the j-th isotope of the k-th element; j is the isotope of the k-th element participating in the (α,n)reaction; k is the element of the j-th isotope which is involved in the (α,n)-reaction; β is a particular element in the mixture, 1≤β≤L; L is the number of elements in the mixture; Nβ, Nk are the numbers of atoms of the corresponding element in unit volume of mixture; RATj,k is the ratio of the number of nuclei of isotope j which involved in the (α,n)-reaction to the total number of nuclei of element k or atomic abundance [11]; P(Eα) is the number of α-particles with energy more than Eα in an arbitrary primary α-particle spectrum; fi,j,k(En,Eα) is the angle component expansion of cross-section in Legendre polynomials.The limits of integration are determined by the kinematics of the reaction to the ground state and each of the excited states of the residual nucleus.
Gauss method [12] with 20 knots is used with prescribed accuracy for the calculation equation ( 9) in the code NEDIS2.0.The code such as SOURCES4C [13] calculates the neutron yield of many sources assuming isotropy of neutron emission and sometimes homogeneity in constitution.SOURCES performs trapezoidal intergration in each alpha particle user discretised energy group to obtain the (α,n) spectrum contributions and total spectrum.It is restricted to alpha energies up to 6.5 MeV.
The code as USD [14] did not compute the continuous retardation of the α-particles correctly.

Method of neutron spectrum calculation of spontaneous fission (SF)
Spontaneous fission spectra are calculated with evaluated half-life, spontaneous fission branching, νaveraged per fission, and Watt spectrum parameters.The currently available nuclear data relevant to the spontaneous fission neutron yields are presented in [15].Neutron multiplicities were similarly taken from [15].
The energy spectrum of neutrons emitted the spontaneous fission of 36 actinides is described by a Watt formula: e -E/a •sinh(b•E) 1/2 , where E is the neutron energy, and a and b are the radionuclide-specific fission parameters obtained from the NEDIS2.0data library.The parameter a (temperature) deals with the neutron multiplicity by the Terrell formula [16] with our new coefficients for spontaneous fission a=0,48+0,2(1+ SF ν ) 1/2 (10) The parameter b is determined by this temperature and average kinetic energy per nucleon of spontaneus fission products which is equal to Ef=0,76 MeV in NEDIS2.0 data library b=4Ef/a 2 =3,04/a 2 (11) In this case the average energy of neuron spectrum of spontaneous fission is equal to The parameters of some nuclides are presented in the Tables 2 and 3 in comparison with [17][18][19].Neutron spectral measurements have been reported [6] for two mixtures, one contains 227 AcO2 with daughter products and the second contains dioxide 244 CmO2 mixed with 13 C powder target material and [7] for 210 Po- 13 C source of neutrons .The neutron yields calculated by NEDIS-2.0 are in satisfactory agreement with measurements for radius of 227 AcO2 ~1 µm and radius of 244 CmO2 ~12.8 µm.The spectra calculated by NEDIS-2.0 and measured in [6,7] are presented in Fig. 1 241 Am(Au)-13 C plane source of neutrons for calibration in Daya Bay experiment The NEDIS2.0 has been used to calculate the neutron yield and spectra from the source [8].P(Eα) -the number of α-particles with energy more than Eα was calculated in continuous-retardation approximation, neglecting fluctuations in the α-particle energy loss due to scattering and inelastic collision in Am, in 1.1 µm gold and 13 C. Results are presented in Fig. 4 and in Table 4, which are in agreement with an assay of [8].

Decay-induced neutrons
Table 5 presents earlier values not updated in a longtime of the neutron yield per 10 6 α-particle in the materials given by the experiments [21,22] and presentday ones by the NEDIS code which assumes secular equilibrium in the decay chains of 238 U and 232 Th.The results of calculation presented in the Table 5 are in satisfactory agreement with the measurements may be used to compare and benchmark new codes to estimate uncertainty of the (α,n) neutron yield calculations for a number of materials that are used and will be used for construction of dark matter detectors [14].

Plutonium oxide
Measured total neutron yields from five plutonium oxide samples were collected in paper [23].Two samples were measured at the Japan Atomic Energy Research Institute (JAERI) with using neutron coincidence counter and others at Oak Ridge National Laboratory (ORNL), Savannah River Plant (SRP), and at Pacific Northwest Laboratories (PNL).Results of calculations and measurements are given in Table 6.Measured and calculated neutron yields are in satisfactory agreement.

Borosilicate glass
The total neutron emission has been measured [24] at the Savanna River Laboratory from three glass samples doped with PuO2 where mass fraction of 238 Pu ~ 90%.The results of measurements are compared with calculated neutron yields using NEDIS-2.0 and ORIGEN-S [25] codes in Table 7.It was noted in [24] the uncertainty in the content of 238 Pu in the glass was about 5% and yields may be underestimated by as much as 10% due to the calibration with 252 Cf source (spectral differences).However, the dimensions of particles of PuO2 in glass may be ~ 0.1-5 µm [26].For 1.5 µm radius of particles NEDIS-2.0 outputs 4970, 474, 168 n/s and agreement with the measurements are ~ 3%.The predicted neutron source spectra from the glass sample 1 are illustrated in Fig. 6.

Conclusions
There is a growing need for high-accuracy (α,n)yields and spectra from compounds of actinides and light elements in different nuclear technology mediums (as an example LiCl+KCl) due to interest in using neutron measurements for nuclear fuel fabrication and reprocessing plant safeguards and nuclear material accountancy.Improving these calculations will ultimately help safeguards professionals make better measurements with fewer resources.More broadly, computer codes such as MCNP are now being used to simulate shielding, require accurate continuous neutron spectrum and magnitude as input.The availability of accurate data is critical in these scenarios.The code NEDIS-2.0 is a useful and flexible computational tool for the accurate analysis of neutron emission rates and energy spectra of different materials of nuclear fuel cycle.

Fig. 6 -
Fig. 6 -Energy dependent neutron source strength in glass sample 1 as calculated by 1-NEDIS-2.0with account for radius of PuO2 particles and 2-ORIGEN-S .

Table 1 .
Light nuclides data available in NEDIS-2.0 library

Table 3 .
Average energy of SF spectrum

Table 5 . Neutron yield (per 10 6 α-particles) due to presence of 238 U and 232 Th in secular equilibrium
Neutron spectra due to presence of 238 U in secular equilibrium for Li,F,O,B,C and Be are presented in Fig.5.