Thus, the whole slowing down range is divided into two types of intervals — the interval of weak absorption and the interval with the isolated and narrow resonances. Lets find the neutron spectrum and the function of resonance escape probability for the interval of weak absorption. The weak absorption means that the macroscopic scattering cross section of the moderator is much bigger than the macroscopic absorption cross section of heavy nuclei characteristic to nuclear fuel. Actually in this energy range the microscopic cross sections of scattering and absorption are the same, but as usual in a reactor core the number of nuclei of the moderator is much bigger than the number of nuclei of fuel in a unit of volume. Let's take an energy range from E2 to E1. In this range let's take an energy interval Delta E. The slowing down density through the range changes because of the absorption. The difference of slowing down densities at the energy E plus delta E and at the energy E equals to the absorption rate at the interval delta E. Because of the weak absorption we can assume that in this area the neutron spectrum is closer to the Fermi spectrum. The difference compared to the non-absorbing medium — in the numerator instead of power of outer source the slowing down density is given that task into account possible absorption in at energies indicated above. It turns out that moving from interval Delta E to differential dE we can obtain the expression for slowing down density. The expression will be the same as we got for the hydrogen. The second type of intervals — the resonance range characterized by the presence of narrow and isolated resonances. Let’s take the energy range from E2 to E1. In this range let’s take an interval energy Delta E. The slowing down density through the range changes because of the absorption. The difference of slowing down densities at the energy E plus delta E and at the energy E equals the absorption rate at the interval delta E. The difference to weak absorption range is that the spectrum is not the Fermi spectrum, because of the presence of the giant resonance of absorption. Let’s consider the neutron balance in delta E in two cases: the first the resonance is in the interval from E2 to E1. The second the imaginary case the absence of resonances in the interval from E2 to E1. Because of the stationary task we are considering, there is a balance between the process of gain and loss of neutrons in both cases. Examine the process of neutron gain. The neutrons can come to unit interval dE from the upper post collision energy range — from the energy E to the energy E divided by alpha. This interval can be divided into two parts: from the energy E to E1 and from E1 to E/alpha. The narrow resonance approximation allows to us to exclude or neglect the first interval from energy E to E1. Thus it doesn’t matter the presence or absence of a resonance at delta E, the number of neutrons coming to delta E will be the same in both cases. The next approximation — the resonances has to be isolated. That means that the neutron spectrum disturbed at the resonance range can be restored in the range between resonances. It turns out that the spectrum between resonances is the Fermi spectrum. Look at the slide. We can see that, if the gain processea are equal to each other, it means that the loss process in case of presence and absence of resonance are equal to each other too. The loss process in case of the presence of a resonance is the collision density at delta E. The loss process in case of absence of a resonance is the scattering density, because in this case there is a weak absorption and the flux is the Fermi spectrum. Thus by the last equation we can obtain the neutron spectrum in the resonance range. In this area the neutron spectrum is strongly different from the Fermi spectrum. It is called the Wigner spectrum. The Wigner spectrum characterized by eating away of neutron spectrum compared to the Fermi spectrum. The Wigner spectrum sets conditions for self-shielding on the resonance. That means the fast increase of the density of resonance absorber leads to slow increasing of absorption reaction rate, because of the accompanying decreasing of the neutron spectrum. Further, it is easy to prove that the expression of resonance escape probability is the same as we got earlier for weak absorption range and for the hydrogen. So finally the spectrum in slowing down range is the Fermi spectrum excluding the range of narrow and isolated resonances, where the Wigner spectrum is formed.