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Neutron Slowing-Down Spectrum. B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr. Contents.
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Neutron Slowing-Down Spectrum B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr.
Contents • We derive the most commonly referenced analytical form of the neutron slowing-down spectrum, the 1/E variation, subject to the appropriate approximations. • Reference: Duderstadt & Hamilton, Sections 8.I A &B
Probability Distribution of Neutron’s Post-Collision Energy in L • In the previous learning object, we established from the kinematics of neutron-nucleus collisions that the probability distribution of the neutron’s post-collision energy in the laboratory system is uniform in the allowed range:
Approximations • Let us deal with the simplest case first, hydrogen, i.e., = 0, for which the probability distribution becomes • We will also assume that the slowing down is in an infinite medium (or that leakage can be neglected during the slowing down, which is a fairly good approximation – leakage represents at most 2-3 %). This approximation also means that all space dependence disappears. • We will assume that absorption can be neglected with respect to scattering – this is a good approximation for most moderators, including hydrogen. • And we will assume that upscattering is negligible during neutron slowing down. This is a very good approximation for neutron energies higher than that corresponding to the thermal motion of nuclei. • Also, finally, there is no external neutron source, and fission neutrons are born at a single representative high energy, say Eh.
Neutron-Transport Equation in Energy • The neutron-transport equation as we know it is: • The assumptions we made r and disappear; the flux gradient 0; the only source term is the fission source term, which we rewrite as S(E-Eh) – we use a function since the source is assumed to be at a single energy. The equation then becomes effectively where s(E) is the total scattering cross section, the only component of t, since the absorption cross section has been assumed negligible. • Note that Eh is the upper limit on the integral, since its is the source energy, and we have assumed no upscattering. E is also the lower limit, since neutrons of lower energy cannot be upscattered.
Effective Neutron-Transport Equation • In the previous learning object, we derived, from the kinematics and the assumed isotropic scattering off a light moderator, the relationship • Then Eq. (4) becomes • If we write R(E) = s(E)(E) (7) we get the new form
Solving the Neutron-Transport Equation in E • Since there is a function on the r.h.s. of Eq. (8), the solution for R0(E) will have to have a function, i.e., we can write • If now we substitute the form (9) into Eq. (8), we get • The functions must be the same on both sides K = S (11) • And therefore also
Solving the Neutron-Transport Equation in E To solve Eq. (12), let’s first differentiate it with respect to E:
The Slowing-Down Spectrum in Hydrogen • Let’s put all the results together: Eq. (17) for R0(E), Eq. (9) for R(E), and Eq. (11) for the value of K, and finally Eq. (7) to evaluate . We get: • If we make the further approximation that the hydrogen total scattering cross section s(E) is independent of energy (or very weakly dependent on energy, a good approximation in most of the slowing-down range), we see right away from Eq. (19) that
The Slowing-Down Spectrum in Hydrogen • Another way of interpreting Eq. (20) is to say that the product E(E) is nearly constant with energy below the fission energy Eh. • Statement (20), that the slowing-down flux is proportional to 1/E [or its equivalent statement above] provides an important, simple, basic formula for the slowing-down spectrum – even if it is somewhat of an approximation.
Non-Hydrogen Moderator • The analysis of the slowing-down spectrum in a non-hydrogenous moderator is more complicated, but under similar approximations the same form of the slowing-down spectrum can be found: • This has the same form for the slowing-down spectrum, with an additional factor of (the average gain in lethargy per collision) in the denominator.
Slowing-Down Density • The “neutron slowing-down density” qsd(E) as a function of the energy E is defined as the number of neutrons which slow down past the energy E. • It can be shown that
Relaxed Approximations • The approximations which have been made in the above analysis can be relaxed one by one. • The analysis then becomes progressively more complicated; the 1/E factor remains, but other factors enter into the final formula. • For example: Distribution of fission-neutron energies. Relaxing the approximation that all neutrons are born at a single energy gives back the same 1/E form, for energies E well below the range of fission energies (cont’d)
Relaxed Approximations (cont.) • For another example: Non-negligible absorption cross section. • In this case, it can be shown that the 1/E spectrum is modified (below the source energies) to: • The exponential factor in Eq. (25) represents the probability that the neutron survives slowing down to energy E; i.e., it is the resonance-escape probability to energy E, denoted p(E).
Summary • Although the real situation in real reactors can become quite complicated, as a simple conceptual result, we should just remember that the slowing-down spectrum has approximately a 1/E form, modified by a number of other factors.
Flux Spectrum Over Full Energy Range • Now that we have derived the (approximate) slowing-down spectrum, we are able to “piece together” the neutron flux over the energy range from fission energies to the thermal range, using: • the fission spectrum at energies above about 50-100 keV • the slowing-down spectrum to about 1 eV • the Maxwellian spectrum at thermal energies, below about 1 eV[note that in the thermal energy range neutrons can gain as well as lose energy in collisions; to be consistent with the approximation of no upscattering in the derivation of the slowing-down spectrum, the “boundary” between thermal and epithermal energies should be selected sufficiently high to ensure negligible upscattering from the thermal region to the epithermal: in many applications and computer codes, this boundary is taken as 0.625 eV].
Flux Spectrum Over Full Energy Range • The piecing together of the neutron spectrum results in the sketch in the next slide. • Note that the thermal spectrum is not a perfect Maxwellian (which applies to a gas, without absorption); the Maxwellian is deformed somewhat by neutron absorption.
Numerical Spectrum Calculations • The analytical derivations above are useful to acquire a general understanding of the dependence of the neutron flux on energy. • However, since they do depend on some approximations, the analytical forms may not be sufficiently accurate for precise reactor design and analysis. • Therefore, most modern reactor analysis relies on numerical solutions of the neutron-transport equation in the basic lattice, either multigroup calculations using a very large number of energy groups (several dozen to several hundred), or Monte Carlo calculations in continuous energy.
