Neutron radiation damage studies in the structural materials of a 500 MWe fast breeder reactor using DPA cross-sections from ENDF / B-VII.1

Abstract

The radiation damage in the structural materials of a 500 MWe Indian prototype fast breeder reactor (PFBR) is re-assessed by computing the neutron displacement per atom (dpa) cross-sections from the recent nuclear data library evaluated by the USA, ENDF / B-VII.1, wherein revisions were taken place in the new evaluations of basic nuclear data because of using the state-of-the-art neutron cross-section experiments, nuclear model-based predictions and modern data evaluation techniques. An indigenous computer code, computation of radiation damage (CRaD), is developed at our centre to compute primary-knock-on atom (PKA) spectra and displacement cross-sections of materials both in point-wise and any chosen group structure from the evaluated nuclear data libraries. The new radiation damage model, athermal recombination-corrected displacement per atom (arc-dpa), developed based on molecular dynamics simulations is also incorporated in our study. This work is the result of our earlier initiatives to overcome some of the limitations experienced while using codes like RECOIL, SPECTER and NJOY 2016, to estimate radiation damage. Agreement of CRaD results with other codes and ASTM standard for Fe dpa cross-section is found good. The present estimate of total dpa in D-9 steel of PFBR necessitates renormalisation of experimental correlations of dpa and radiation damage to ensure consistency of damage prediction with ENDF / B-VII.1 library.

Appendix A

1.1 Group averaging point dpa cross-section

The point dpa cross-section is group-averaged into a neutron group structure using eq. (A.1). This expression shows the total multigrouped dpa cross-section in the gth neutron group.

$$\begin{aligned} {\sigma _{\mathrm{D}}^{t}},{}_{\mathrm{g}} =\frac{\displaystyle \int \nolimits _{E_{\mathrm{g}(\min )} }^{E_{\mathrm{g}(\max )} } {{\sigma _{\mathrm{D}}^{t}} (E)\varphi (E){\mathrm{d}}E} }{\displaystyle \int \nolimits _{E_{\mathrm{g}(\min )} }^{E_{\mathrm{g}(\max )} } {\varphi (E){\mathrm{d}}E} }. \end{aligned}$$

(A.1)

1.2 Reaction kinematics

The energy–cross-section data for different reactions i are given in different sections of file 3 of ENDF / B-VII.1 tape for the material. The kernel of energy transfer for each reaction depends on the kinematics of neutron–nucleus reaction. These are briefly given below [5, 18, 19, 21, 24, 34, 35].

1.2.1 Elastic and resolved level inelastic scattering

$$\begin{aligned}&E_\mathrm{R} =\frac{AE}{(A+1)^{2}}(1-2R\mu +R^{2}), \end{aligned}$$

(A.2)

$$\begin{aligned}&R=\sqrt{1-\frac{(A+1)(-Q)}{AE}},\nonumber \\&K(E,E_{\mathrm{R}} )=\frac{1}{2\pi }\sum _{l=0}^{NL} {\frac{2l+1}{2}a_l (E)P_l (\mu )} , \end{aligned}$$

(A.3)

where Q is the Q-value for discrete inelastic scattering to take place from a particular level. $\mu$ is the centre of mass cosine of angle of scattering of the neutron. The incident energy-secondary angular parameters, $a_{l}(E)$ or the full summation in eq. (A.3) are given in file 4 section 2 for elastic scattering. For discrete inelastic scattering $a_{l}(E)$ are given in sections 51–91 of either file 4 or file 6.

1.2.2 Continuum inelastic scattering and (n, 2n) reaction

The secondary particle energy distribution is either given in file 5 or file 6. File 6 may also contain the product energy-angle distribution and recoil energy distribution for these reactions. If the recoil energy distribution is available, then these data are used from the file. If such data are not available, evaporation model of nuclear reaction is adapted. In this model, the energy transfer kernel is given as follows:

A.2.2.1 Continuum inelastic scattering.

$$\begin{aligned} K(E, E_{\mathrm{R}} )=\int \nolimits _0^{E'^{\max }} {{\mathrm{d}}{E}'\frac{f(E,{E}')}{4\frac{1}{A+1}(E{E}')^{1/2}}} . \end{aligned}$$

(A.4)

The maximum value of $E'$ is

$$\begin{aligned} {E}'=\frac{A}{A+1}\left( Q_l +\frac{A}{A+1}E\right) , \end{aligned}$$

where $Q_{l}$ is the Q-value for the lowest level. The distribution function $f(E, E')$ represents the probability that a neutron of energy $E'$ in the centre of mass frame is evaporated from the compound nucleus. In the centre of mass frame it is given as the Maxwellian of nuclear temperature $E_{\mathrm{D}} = { kT}$:

$$\begin{aligned} f(E,{E}')=\frac{{E}'}{I(E)}\hbox {e}^{-E'/E_{\mathrm{D}} }, \end{aligned}$$

(A.5)

where

$$\begin{aligned} I(E)=E_{\mathrm{D}}^2 \left[ 1-\left( 1+\frac{{E}'^{\max }}{E_{\mathrm{D}} }\right) \hbox {e}^{-E'^{\max }/E_{\mathrm{D}} }\right] \end{aligned}$$

(A.6)

is a normalization factor such that

$$\begin{aligned} \int \nolimits _0^{E'^{\max }} {{\mathrm{d}}{E}'f(E,{E}'} )=1. \end{aligned}$$

(A.7)

A.2.2.2 (n, 2n) Reaction.

$$\begin{aligned} K(E,E_{\mathrm{R}} )= & {} \int \nolimits _0^{E' { }^{\max }} \frac{{E}'}{I(E)}\hbox {e}^{-E'/E_{\mathrm{D}} }\nonumber \\&\int \nolimits _0^{E-E'} \frac{E''}{I(E,E')}\hbox {e}^{-E''/E_{\mathrm{D}} }{\mathrm{d}}E'\hbox {d}E'' , \end{aligned}$$

(A.8)

where I(E) is given by eq. (A.6) with $E'^{\mathrm{max} }=E$ and $I(E, E')$ is given by eq. (A.6) with $E'^{\mathrm{max}}$ replaced by $E''^{\mathrm{max} }=E$–$E'$.

The recoil energies in these two reactions can be found by

$$\begin{aligned} E_{\mathrm{R}} (E,E',\mu )=\frac{1}{A}(E-2\mu \sqrt{EE'}+E'), \end{aligned}$$

(A.9)

where $\mu$ is the laboratory cosine of the angle of emission of secondary neutron.

1.2.3 Threshold n, particle reactions

File 6 of the ENDF / B-VII.1 tape for the material may contain the recoil energy distribution for these reactions in specific sections. These data are used when available. If such data are not available, then the recoil energy is found from the following equations:

$$\begin{aligned} E_{\mathrm{R}} =\frac{1}{A+1}(E^*-2\sqrt{aE^*E_{\mathrm{p}} }\cos \phi +aE_{\mathrm{p}} ), \end{aligned}$$

(A.10)

where a is ratio of the emitted particle mass to neutron mass,

$$\begin{aligned} E^*=\frac{(A+1-a)}{A+1}E \end{aligned}$$

and energy of the emitted particle $E_{\mathrm{p}}$ is approximately chosen as the smaller value between the available energy

$$\begin{aligned} Q+\frac{AE}{A+1} \end{aligned}$$

and the Coulomb barrier energy

$$\begin{aligned} \frac{1.029\times 10^{6}zZ}{a^{1/3}+A^{1/3}}\hbox { in eV} \end{aligned}$$

z and Z being charges of the emitted particle and the target respectively.

The energy transfer kernel in this case (considering isotropic emission of recoil atom) is

$$\begin{aligned} K(E,E_{\mathrm{R}} )=\frac{1}{E_{\mathrm{R}_{\max }} (E)-E_{\mathrm{R}_{\min }} (E)}. \end{aligned}$$

(A.11)

1.2.4 (n, $\gamma$) reaction

$$\begin{aligned} E_{\mathrm{R}}= & {} \frac{E}{A+1}-2\sqrt{\frac{E}{A+1}}\sqrt{\frac{E_\gamma ^2 }{2(A+1)mc^{2}}}\cos \phi \nonumber \\&+\,\frac{\overline{E_\gamma ^2 } }{2(A+1)mc^{2}}. \end{aligned}$$

(A.12)

The average of $E_{\gamma }^{2}$ is evaluated from the gamma yield data given in file 12 for discrete gamma and distribution given in file 15 for continuum gamma. The damage energy from this reaction is calculated from recoil energy considering isotropic gamma emission.