After studding this lecture, the students should be able to: define main approximations of the diffusion theory — Fick’s Law and the diffusion equation, explain each term in the diffusion equation. In this module we develop a one-speed diffusion theory. Such a relatively simple description has a great advantage of illustrating many of the important features of nuclear reactors without a complexity introduced by the treatment of important effects associated with the neutron energy spectrum and with highly directional neutron transport, This topics are the subjects of subsequent chapters. Moreover, diffusion theory is sufficiently accurate to provide a quantitative understanding of many physical features of nuclear reactors and is, in fact, the workhorse computational method of nuclear reactor physics. First, neutrons have the same energy, that is the case when energy of neutrons does not change in the interaction with nuclei. It is possible only if there are processes of absorption or second we considered potential scattering on heavy nuclei (the atomic mass more than 10). Third, there is a source of monoenergetic and isotropic neutrons and the scattering on heavy nuclei is isotropic in lab coordinates. Now our functions vary depending on only space and time positions. That means that we consider the global flux, neutron density and others. Let's consider volume unit at the point with radius vector r. Calculate the balance in the volume as a change of the neutron density per time unit. Gain means the quantity of neutrons to be gained in the volume by the process of generation by fission and by outer sources. Loss means the quantity of neutrons to be lost in the volume by the process of leakage and absorption. Thus, the balance equations looks like this. Consider all members of the equation separately. Take a small volume ΔV near the point r with the total surface area S and calculate the number of neutrons leaking from the volume in a moment t. The number of neutrons crossed through small area dS in any directionn: it's a product of neutron current and dS. It's by the definition of neutron current. For the whole surface it is needed to integrate over the area. By using the divergence theorem the leakage in unit volume equals the divergence of the current vector. How to find the reaction rate of x process (x — the absorption, scattering, fission etc.) Let imagine that v be the neutron velocity, λx — mean free path for the process x. Then the ratio λx/v is the mean time between two interactions; ratio v/λx is the average number of process x for one neutron in a volume unit. Multiplying by neutron density we obtain the number of process x in a volume unit. And, finally: multiplying by the volume we get the reaction rate in a volume ΔV. So, in the end, we got that the reaction rate of thr process x is thr product of macroscopic cross section and the neutron flux. Finally, the absorption rate is the product of the absorption cross section and the neutron flux. The fission rate is the product of the fission macroscopic cross section and the flux, and multiplying by the average number of neutrons per one fission (the nu f) we can get the neutron generation by fission. In a result, we can get the balance equation. The first member is the neutron leakage from the unit volume. The second is the loss by absorption, the third is the generation by fission and the last one is the outer source. The equation is obtained within the most general assumptions: all functions are statistical quantities, the neutron is considered as a point particle, neutron-neutron interactions aren't considered, the neutron is a stable particle, and the last — all neutrons have the same energy that is the same as we integrate the flux by energy — the global flux. The equation that is written down contains two unknown functions — the neutron flux and the current vector; therefore it is necessary to derive one more equation connecting these two functions. This relation is called Fick’s Law.
