Question 2.16: Using the macroscopic scattering cross sections in Appendix ......
Using the macroscopic scattering cross sections in Appendix Table II-3, calculate the slowing down decrement for UO_2, where U is natural uranium. Does the presence of oxygen have a significant effect on the slowing down decrement?
First note that \alpha^{25}=(235-1)^{2}/(235+1)^{2}=0.9831
\alpha^{28}=(238-1)^{2}/(238+1)^{2}=0.9833
Assume alpha for the uranium isotopes is that of uranium-238
\xi_{U}=1+{\frac{0.9833}{1-.9833}}\ln(0.9833)=0.00840
\alpha^{\circ}=(16-1)^{2}/(16+1)^{2}=0.0.8823
\xi_{O}=1+\frac{0.8823}{1-.8823}\ln(0.8823)=0.0613
Note these numbers can be obtained more quickly using the approximation of Eq. (2.57)
\xi_{U O_{2}}=\frac{\xi_{U}N_{U}\sigma_{s}^{U}+\xi_{O}N_{O}\sigma_{s}^{O}}{N_{U}\sigma_{s}^{U}+N_{O}\sigma_{s}^{o}} Since N_o = 2N_u:
\xi_{UO_{2}}=\frac{\xi_{U}\sigma_{s}^{U}+2\xi_{O}\sigma_{s}^{O}}{\sigma_{s}^{U}+2\sigma_{s}^{O}}= \frac{0.0084\cdot9.146+2\cdot0.0613\cdot3.761}{9.146+2\cdot3.761}=0.0322
Thus \xi_{U O_{2}} is significantly larger than ξ_U as a result of the oxygen