After studying this lecture, the student should be able to: Derive the slowing down equation; explain the terms in the slowing down equation; find the solution in a non-absorbing medium. The last lecture was devoted to general description of slowing down. In this lecture we will find a dependency of the neutron flux on the energy¨— the neutron spectrum. First, we took an infinite, non-multiplying, homogeneous medium in which there are isotropic, stationary external uniformly distributed sources. Second, we suppose that Potential scattering is isotropic in CoM. In this case the phase space formed by the energy variable only. So, the flux depends only on the energy. The balance equation. We consider as mentioned above the infinite, non-multiplying, homogeneous medium. There are isotropic, stationary external uniformly distributed sources of emitted neutron with the energy E0. The balance equation in stationary situation means that there is a balance between gain and loss of neutrons in the unit interval dE at the energy E. The loss includes all processes in a unit of time when the neutrons disappeared from the balance. First, disappearance rate by elastic scattering. Second, disappearance rate by absorption. The gain includes all processes in unit of time when the neutrons appear in the balance. First, ppearance rate by elastic scattering from energies above the energy E. Appearance rate from external sources. At this energy range during scattering a neutron can only losses the energy. The first two processes of disappearance are simple the reaction rates relative to scattering and absorption. The external source term is a product of power of source to delta function and interval dE because the source neutrons have the energy E0. The third: the neutron appearance rate by elastic scattering from higher energies. Neutrons can appear in dE only by scattering from the interval [E, E/alpha] (the upper post collision energy range). Take any unit interval dE_prime from [E, E/alpha]. The number of neutrons scattered in the interval per unit of time is Sigma_s*flux*dE_prime. By definition of the reaction rate. The possibility for neutron to appear in dE after scattering in dE_prime is the ratio of dE and the post collision range because through the post collision range the probability density is constant. To find the rate of appearance we have to integrate over the post collision energy range from energy E to E/alpha or E0 (which is smaller). Finally we obtained the slowing down equation of balance. The first two terms are the loss of neutrons and last two terms are the gain of neutrons. Sometime the following two quantities are introduced for simplification of the equation: the scattering density and the collision density. The next very important function is the slowing down density. The function is denoted by the lowercase j. One describes the number of neutrons in the unit spatial volume whose energy during slowing down in the time unit changes from the value above, to value below the energy E, i.e. crosses the energy E per unit of time during slowing down. Let’s return to the slide. The final formula is presented. Try to develop one by yourself. In case of the infinite, homogeneous, non-absorbing medium the slowing down density is constant and equals to the power of outer monochromatic source. In case of the absorbing medium the slowing down density decreases during the process of slowing down. Derived corresponding slowing down densities at E1 and Е2 (E1 is bigger than Е2) describe the very important in nuclear reactor physics quantity as the probability to avoid the absorption during slowing down or simple Resonance Escape Probability from E1 to Е2. Ok. Now we are ready to find the flux in the simplest case — a non absorbing medium, the moderator is the hydrogen. Slowing down equation for the medium is presented here. We will find the solution as a sum of two functions. The first, denoted as F figured is the scattering density of moderated neutrons (the neutrons that suffered as minimum one collision). The second is the scattering density of neutrons of source (by other words the neutrons did not suffer any collision). Skip the solution of the integral equation. So we can obtained the final expression — the flux equals to the ratio of sources power and Sigma_S times E. This function is called the Fermi spectrum. In the lethargy variables the Fermi spectrum is constant. Slowing down density for non-absorbing medium is constant and equals to thesources power q. Checking of the correctness of the last statement can be suggested as your self-education work. The next more complicate case. Slowing down on nuclei with atomic mass bigger than 1 without absorption. Ok. The solution is too hard with respect to the hydrogen medium. In this case we should find the solution separately in every collision interval. Slowing down equation for the 1st collision energy range (collision interval) is presented there. For the 2st post collision energy range the solution is more complicated. But it is simple to show that there is discontinuity of the function of scattering density at the point αE. So for the third post collision interval the solution is complicated too. But we can separate out the, so called, asymptotic energy range. the energy range in which the neutron doesn’t feel influence of the source. For a moderator with atomic mass more than 10 all collision intervals after 3rd collision interval are in the asymptotic energy range. It is simple to prove that the solution at the asymptotic energy range is the Fermi spectrum too as we obtained for the hydrogen medium with the only exception ξ average logarithmic energy loss. The whole neutron spectrum for the medium is depicted there. In the point of first collision interval — the product alpha*E0 there is the function discontinuity. The second collision interval — discontinuity of the first derivation and so on. But in the asymptotic energy range it is clearly the Fermi spectrum. The whole neutron spectrum in the lethargy variable for the medium is depicted there. This function is called the Plachek function. So the next the slowing down in mixtures. For an infinite, non-multiplying, homogeneous, non-absorbing medium consisted of a mixture of isotopes in which there are isotropic, stationary external uniformly distributed sources of the power q [n/s] emitted neutrons of the energy E0. It is simple to prove that for the medium of mixture of isotopes in the asymptotic energy range is it to prove that the spectrum is the Fermi spectrum. The difference from the previous consideration is in the presence of quantity of average logarithmic energy loss for mixtures. The summary of this lecture — in a non-absorbing medium the Fermi spectrum is formed. It is true for hydrogen, for others moderators it is true for asymptotic energy range.