Extended Validity of the Energy Dependent Scattering Kernel within the Boltzmann Transport Equation

Figures & data

Figure 1. Comparison between total cross section based on the OM (Koning and Delaroche 2003) and ENDF/B-VIII library for Fe56.

Figure 2. Comparison between the OM (Koning-Delaroche) and ENDF/B-VIII (Brown et al. 2018) and JEFF (JEFF 2017) libraries for W182.

Figure 3. Elastic scattering cross section of the ENDF/B-VII library (Chadwick et al. 2006) for W182.

Figure 4. OM based angular scattering kernel for Fe56 at 25 keV for the original.

Potential (53 MeV) and two arbitraries at 50 and 56 MeV. The scattering is almost isotropic for all cases.

Figure 5. “Classical” momentum and energy dependent angular scattering kernel (Equation 1(1) $$\begin{array}{c}{\sigma }_s^T\left(E\to E\prime ,\overrightarrow{\Omega }\to \overrightarrow{\Omega }\prime \right)=\frac{1}{2\pi }{\sigma }_s^T\left(E\to E\prime ,{\mu }_0^{\mathit{lab}}\right)=\frac{1}{2\pi \mathrm{v}}{\left(\frac{A+1}{A}\right)}^4{\left(\frac{A}{\pi }\right)}^{3/2}\\ \int 2\pi u^{2}du\int d{\mu }_u\int c^{2}dc\int {(u\prime )}^{2}du\prime \int d{\mu }_{u\prime }\int \frac{2}{\hspace{0.17em}\text{sin}\hspace{0.17em}\phi }\frac{\delta (u\prime -u)}{{(u\prime )}^2}\text{exp}\hspace{0.17em}\left[{\mathrm{v}}^2-(A+1)\left(\frac{u^2}{A}+c^2\right)\right]\\ \frac{1}{u\mathrm{v}c}\delta \left[{\mu }_u-\frac{({\mathrm{v}}^2-c^2-u^2)}{2uc}\right]\frac{1}{2u\prime c{k}_{B}T}\delta \left[{\mu }_{u\prime }-\frac{{(\mathrm{v}\prime )}^2-{(u\prime )}^2-c^2}{2u\prime c}\right]\\ \frac{4\text{vv}\prime c^2}{B_0^{\prime }}\delta (\hspace{0.17em}\text{cos}\hspace{0.17em}\phi -\hspace{0.17em}\text{cos}\hspace{0.17em}\widehat{\phi })u{\sigma }_s(E_r)\frac{P(u,{\mu }_0^{cm})}{2\pi }d\hspace{0.17em}\text{cos}\hspace{0.17em}\phi \end{array}$$(1) ) at 25 keV for Fe56 with temperature of 1200 K the angle dependent anisotropy is well pronounced.

Figure 6. OM based angular scattering kernel forW182 at 10 keV. The scattering is quite isotropic. The difference between lowest and highest value is about 6%.

Figure 7. Momentum and energy dependent angular scattering kernel (Equation 1(1) $$\begin{array}{c}{\sigma }_s^T\left(E\to E\prime ,\overrightarrow{\Omega }\to \overrightarrow{\Omega }\prime \right)=\frac{1}{2\pi }{\sigma }_s^T\left(E\to E\prime ,{\mu }_0^{\mathit{lab}}\right)=\frac{1}{2\pi \mathrm{v}}{\left(\frac{A+1}{A}\right)}^4{\left(\frac{A}{\pi }\right)}^{3/2}\\ \int 2\pi u^{2}du\int d{\mu }_u\int c^{2}dc\int {(u\prime )}^{2}du\prime \int d{\mu }_{u\prime }\int \frac{2}{\hspace{0.17em}\text{sin}\hspace{0.17em}\phi }\frac{\delta (u\prime -u)}{{(u\prime )}^2}\text{exp}\hspace{0.17em}\left[{\mathrm{v}}^2-(A+1)\left(\frac{u^2}{A}+c^2\right)\right]\\ \frac{1}{u\mathrm{v}c}\delta \left[{\mu }_u-\frac{({\mathrm{v}}^2-c^2-u^2)}{2uc}\right]\frac{1}{2u\prime c{k}_{B}T}\delta \left[{\mu }_{u\prime }-\frac{{(\mathrm{v}\prime )}^2-{(u\prime )}^2-c^2}{2u\prime c}\right]\\ \frac{4\text{vv}\prime c^2}{B_0^{\prime }}\delta (\hspace{0.17em}\text{cos}\hspace{0.17em}\phi -\hspace{0.17em}\text{cos}\hspace{0.17em}\widehat{\phi })u{\sigma }_s(E_r)\frac{P(u,{\mu }_0^{cm})}{2\pi }d\hspace{0.17em}\text{cos}\hspace{0.17em}\phi \end{array}$$(1) ) at10 keV for W182 with temperature of 1000 K. The difference between lowest and highest value is almost factor 6.

Koning, A. J., and J. P. Delaroche. 2003. Local and global nucleon optical models from 1 keV to 200 MeV. Nucl. Phys. A. 713 (3–4):231–310. doi:https://doi.org/10.1016/S0375-9474(02)01321-0

 Web of Science ®Google Scholar

Brown, D. A., M. B. Chadwick, R. Capote, A. C. Kahler, A. Trkov, M. W. Herman, A. A. Sonzogni, Y. Danon, A. D. Carlson, M. Dunn, et al. 2018. ENDF/B-VIII.0: The 8th major release of the nuclear reaction data library with CIELO-project Cross sections, new standards and thermal scattering data. Nucl. Data Sheets 148:1–142. doi:https://doi.org/10.1016/j.nds.2018.02.001

 Google Scholar

JEFF. 2017. OECD- NEA - Nuclear Energy Agency. https://www.oecd-nea.org/dbdata/jeff/jeff33

 Google Scholar

Chadwick, M. B., P. Obložinský, M. Herman, M. Greene, R. D. McKnight, D. L. Smith, P. G. Young, R. E. MacFarlane, G.M. Hale, S. C. Frankle, et al. 2006. ENDF/B-VII.0: Next generation evaluated nuclear data library for nuclear science and technology. Nucl. Data Sheets 107 (12):2931.

 Google Scholar