Lesson 8: Slowing Down Spectra, p, Fermi Age

Transcription

1 Lesson 8: Slowing Down Spectra, p, Fermi Age Slowing Down Spectra in Infinite Homogeneous Media Resonance Escape Probability ( p ) Resonance Integral ( I, I eff ) p, for a Reactor Lattice Semi-empirical Relations for I eff Neutron Migration during Slowing Down Fermi Age Theory Physical Significance of Age ( τ ) p, τ.. 1

2 Slowing-Down Energy Spectra in Infinite, Homog. Media OK FROM HERE!!! In a reactor, there are sufficiently large, individual zones.. For an infinite, homogeneous medium Angular fluxes: isotropic, Scalar flux: uniform, Net current: zero ( ) Eq. (2) Neutron balance eq. for band E, E+dE (for all ) with q(e) to be obtained from Eq. (1) Slowing-down source eq. One may consider 2 different cases: I. Σ a 0 (Non-absorbing medium, e.g. moderator ) II. Σ a finite (Fuel / moderator mixture ) p, τ.. 2

3 Non-absorbing Medium Very simple neutron balance equation: In general, Q is the fission-source density (fission spectrum) Integrating, For For (total fission source) for all E < E s (constant slowing-down source) In absence of absorption (Σ a = 0) and of leakage ( = 0), no. of n s crossing each energy = Q f - No accumulation of n s at energy E p, τ.. 3

4 Non-absorbing Medium (contd.) Laboratory for Reactor Physics and Systems Behaviour One can show that the solution, with C a constant, gives: q(e) = constant, for E < E s Substituting for Φ in Thus, Since Σ s ~ constant in practice, the slowing-down spectrum is ~ 1/ E (Fermi) p, τ.. 4

5 Comment (1) For a mixture of isotopes, one needs to define ξ such that If one assumes and Σ si constant, Thus,, i.e. (as given before) p, τ.. 5

6 Comment (2) Derivation of the result was done in an approximate manner In general, for a source of n s of energy E 0 (e.g. ~ 2 MeV, on average, for fission n s) The result is the asymptotic solution (for E << E 0 ) There are transitions near E 0 (at αe 0, α 2 E 0, ) E.g., for the first collision, αe 0 E E 0, etc. For H 1, there are no transitions Detailed treatment: Ligou, Section Solution of Placzek : In practice, the transitions are not very important p, τ.. 6

7 Absorbing Medium For a fuel-moderator mixture, a few assumptions need to be made: even though q(e) constant For E < E s, Integrating, Considering Σ a 0 for E E s, Thus, in absence of absorptions probability of escaping absorption during slowing down from E s to E p, τ.. 7

8 Absorbing Medium (contd.) With Σ s >> Σ a, one may take Σ s ~ Σ t In spite of the assumptions made, Eq. (3) is valid in certain, very different situations: In hydrogeneous media (slowing down in hydrogen, even with Σ a > Σ s ) In the region of sharp (narrow), isolated resonances (representative of epithermal absorptions in the fertile isotopes, U 238, Th 232 ) Most important contribution to absorptions during slowing down p, τ.. 8

9 Resonance Escape Probability, p For a reactor, one has a mixture (moderator, fuel, structure, ) Reasonable approximation: epithermal absorptions only in fertile material For others, resonances are generally much less important than thermal absorptions For fissiles, resonances compensate partly (in terms of productions, absorptions) For p, reference energy is E = E t * Energy < first resonance, but > E th Limit rather arbitrary, e.g. ~ 4 ev p, τ.. 9

10 p (contd.) In, one sets Thus, with p, τ.. 10

11 Dilution Cross-section, I, I eff We have introduced in expression for p : dilution cross-section I eff : effective resonance integral (depends only on σ e ) In the limit σ e (N m >> N c ), I : infinite-dilution resonance integral (σ e depends on the dilution, N m / N c ) In practice, I eff << I, since σ e not that large One has the phenomenon of self-shielding Flux depressed within the resonances For N m (σ e ), self-shielding effect reduced (max. value) p, τ.. 11

12 p, for a Homogeneous Mixture We have: Laboratory for Reactor Physics and Systems Behaviour For N m / N c, I eff, but the denominator more strongly Effectively, p 1 for N m, but slowly An increase in the slowing-down power allows neutrons to jump over the traps (greater probability of having an energy loss >> width of the resonance) p, τ.. 12

13 p, for a Reactor Lattice In practice, fuel rods regularly spaced in the moderator: heterogeneous lattice Equivalence theorem: where N c /N m considered with respect to the volume of the core I eff defined as before, but with where characterises fuel-rod dimensions (diam. if cylindrical, otherwise) fuel density in the usual sense (i.e. per unit volume of the fuel) factor characteristic of the lattice (Bell factor) σ e, hence I eff, independent of moderator For a given ratio N c /N m, p when (σ e 0) For 0 (thin rods), σ e, p e min. (infinite dilution : I eff max.) Thus, heterogeneity of a lattice is needed, not only for technological reasons p, τ.. 13

14 Semi-empirical Relations for I eff In general, with Thus, Experimental measurements of I eff, for different lattices, yield semi-empirical of the form, e.g. Often it is which gives a better fit ) Qualitaively, (2) corresponds to resonance absorptions of the type: Per nucleus, n s absorbed in entire volume (σ a moderate) n s absorbed on rod surface (σ a very high) p, τ.. 14

15 Neutron Migration during Slowing Down Till now: infinite, homogeneous media Φ uniform (same for all ) In practice, one has a reactor of finite dimensions, non-homogeneous There is a relationship between Φ(E) and distance from the source Numerical approach (multigroup theory) allows treating n s in many energy groups One can then speak of, for example, a diffusion area for each group A simplified treatment allows one to obtain analytical solutions (Fermi s theory) Corresponding hypotheses: λ t does not vary strongly with energy ξ is small (slowing down almost continuous) Σ a ~ 0 Neutron spectrum not affected by differential leakage (greater leakage for fast n s) Diffusion theory valid p, τ.. 15

16 Fermi Age Theory One considers the neutron balance in The change is due to leakage In absence of absorptions, Thus, = Defining Fermi Age corresponding to energy E by, i.e. (Age Equation) p, τ.. 16

17 Solution of Age Equation The form is that of the time-dependent, heat conduction equation but has the dimensions of area, not of time One may use the the method of Fourier integrals to solve the Age Equation E.g. for a point source in an infinite medium : The distribution is Gaussian For τ = 0 (E = E 0 ) : δ-function at r = 0 For large τ : flat distribution Neutrons of large τ are scattered over large distances, distributed in a uniform manner For E = E th, one obtains the distribution of the thermal source (can be used in combination with the diffusion kernel for thermal n s) p, τ.. 17

18 Physical Significance of τ As for L, one may consider the average square of the distance travelled by a neutron for acquiring the age τ No. arriving with age τ in the shell between r, r+dr Thus, i.e. Age is proportional to the average squared distance travelled by a n between emission and its arrival at the corresponding energy E For E = E th, τ = τ th : slowing down length (important for calculating the leakage during slowing down) p, τ.. 18

19 Summary, Lesson 8 Slowing Down Spectrum in Infinite Non-absorbing Medium Consideration of Absorption during Slowing Down Resonance Escape Probability p and Effective Resonance Integral I eff Semi-empirical Relations for Reactor Lattices Neutron Migration during Slowing Down Fermi Age Equation Solution for a Point Source Physical Significance of Age ( τ ) p, τ.. 19