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Neutron Slowing-Down Spectrum B. Rouben McMaster University Course EP 6D03 – Nuclear Reactor Analysis (Reactor Physics) 2009 Jan.-Apr. 2009 January 1 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 2009 January 2 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: P EnL EnL 1 ' , EnL EnL EnL (1) ' 1 EnL 2009 January 3 Approximations Let us deal with the simplest case first, hydrogen, i.e., = 0, for which the probability distribution becomes P EnL EnL 1 ' , 0 EnL EnL ( 2) ' EnL 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. 2009 January 4 Neutron-Transport Equation in Energy The neutron-transport equation as we know it is: r, E , S r, E , S f r, E t r, E r, E , 1 4 s r, E ' E , ' r, E ' , ' dE ' d' (3) ' E ' 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 E h s E E s E ' E E ' dE ' S E Eh (4) E 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. 2009 January 5 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 s E ' s E ' s E E ' for hydrogen (5) 1 E ' E ' Then Eq. (4) becomes s E ' Eh s E E ' E 'dE ' S E Eh (6) 0 E If we write R(E) = s(E)(E) (7) we get the new form RE ' ' Eh R E ' dE S E Eh (8) E E 2009 January 6 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 RE R0 E K E Eh , where K is a cons tan t (9) If now we substitute the form (9) into Eq. (8), we get R0 E K E Eh Eh R E K E E dE S E E ' ' 0 h ' h E E' R0 E ' ' K Eh R0 E K E Eh ' dE S E Eh (10) E E Eh The functions must be the same on both sides K = S (11) And therefore also R0 E ' ' S E h R0 E ' dE (12) E E E 2009 January 7 Solving the Neutron-Transport Equation in E To solve Eq. (12), let’s first differentiate it with respect to E: dR0 E R E dR E dE 0 , which we can write 0 (13) dE E R0 E E and the general solution to this is immediate : ln R0 E ln E P, where P is a cons tan t (14) We can evaluate P by writing Eq.(12) at Eh , and taking log arithms : R0 Eh ln R0 Eh ln Eh ln S S (15) Eh and comparing this to Eq.(14) P ln S (16) and we can then evaluate R0 E from Eq.(14) for any E : R0 E S (17) E 2009 January 8 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: R E R0 E K E Eh S E Eh (18 ) S E E E Eh S S (19 ) E s E s E 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 Below Eh , the energy at which the fission neutrons are born, 1 the slowing down flux in hydrogen is proportional to ( 20 ) E 2009 January 9 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. 2009 January 10 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: E S ( 21) E s 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. 2009 January 11 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 E / E E ' qsd E s E ' dE ' ( 22) E E 'E ' and , without absorption : E / E E ' dE ' qsd E S E 1 E '2 ( 23) For E below the source, this gives qsd E S ( 24) i.e., the slowing down density is cons tan t below source energies, which makes perfect sense when assu min g no absorption. 2009 January 12 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) 2009 January 13 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: Eh a E ' dE ' E S exp (25) Et E E t E ' E ' 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). 2009 January 14 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. 2009 January 15 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]. 2009 January 16 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. 2009 January 17 Flux Spectrum Over Full Energy Range 2009 January 18 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. 2009 January 19 END 2009 January 20