22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')

Transcription

1 22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966), hap 2. S. Yip, Leture Notes (1975), hap 7. Allan F. Henry, Nulear-Reator Analysis (MIT Press, 1975), hap 2. Having disussed the alulation of the angular differential sattering ross setion σ ( θ ) = dσ / dω, and its integral, the sattering ross setion σ ( E), we an now ask about the energy differential sattering ross setion dσ / de '. When properly normalized, this ross setion beomes the probability that given the neutron is sattered at E, it will have post-ollision energy in the interval de' about the energy E'. A similar normalization will give a orresponding distribution for dσ / dω. Sine the two differential ross setions are related in that their integrals are just the elasti sattering ross setion σ ( E ) = dω(dσ / dω) = de '(dσ / de ') (6.1) we an define two probability distributions, P( Ω ) and F (E E '), where and P( Ω)dΩ probability of sattering (at energy E) into solid angle element dω about Ω 1 dσ ) = ( dω (6.2) E σ ( ) d Ω F (E E ')de' probability of sattering (at E) into de' about E' = 1 dσ de ' (6.3) σ ( E ) de ' In view of their indiated onnetions to σ ( E), it is not surprising that the two distributions are diretly related to eah other suh that if one is known the other is readily obtained by a transformation. Indeed this is a general property of the transformation of distributions. Suppose g(y) and f(x) are both distributions and y=y(x), then g(y) an be obtained from f(x) by the relation, 1

2 g ( ydy ) = f ( x ) dx (6.4) or g( y ) = f ( x) dx / dy (6.5) To apply this argument to the angular and energy probability distributions, (6.2) and (6.3), we first need to redue the former quantity whih is a funtion of two variables, the angles θ and ϕ, sine the latter is a funtion of one variable. The redution is possible beause we are dealing with entral fore sattering, in whih ase the probability distribution P( Ω) does not depend on the azimuthal angle ϕ. To make this expliit we write heneforth P( Ω ) = P(θ ), and integrate (6.2) over ϕ, ϕ =0 P( θ ) sin d θ θd ϕ G(θ)dθ (6.6) The redued angular distribution is G( θ ), it is only a funtion of the polar angle θ, or the angle of sattering. We have previously derived a partiularly simple relation between the energy of the sattered neutron E' and the sattering angle in CMCS, see Eq.(3.10) in Le 3 (2003), E ' = (E / 2){(1 + α ) + (1 α )os θ ]. This result shows that there is a one-to-one orrespondene between E' and θ. Notie that this orrespondene also holds between E' and the sattering angle in LCS. On the other hand, for bona fide nulear reations (elasti sattering is not onsidered proper nulear reation), reall that one an doublevalued solutions to the Q-equation. In the example onsidered in Le 3 (2003) (p.7) we had two different values of E' for the same angle in LCS. Our interest here is to relate the two probability distributions, G( θ ) and F (E E '), using (3.10) to evaluate the Jaobian of transformation dx / dy in (6.5). Thus we write F (E E ')de ' = G(θ )dθ (6.7) The orresponding physial statement is that the probability of sattering into de' about E' is the same as the probability of sattering through an angle θ. If the angular distribution in CMCS were known, then the energy sattering kernel beomes From (3.10) we find F (E E ') = G(θ ) dθ / de ' (6.8) dθ / de ' = [(E / 2)(1 α )sin 1 θ ] (6.9) Eqs. (6.8) and (6.9) are as far as we an go without speifying the angular distribution. We now onfine our disussions to low-energy, s-wave sattering in whih ase the angular distribution in CMCS, P(Ω ), is spherially symmetri. This means or P( Ω ) = 1/ 4π (6.10) 2

3 G ( θ ) = (1/2)sin θ (6.11) Inserting (6.9) and (6.11) into (6.8) we obtain the energy transfer kernel F( E E ') = 1/ E(1 α ), α E E ' E 0. otherwise (6.12) Notie that the upper and lower bounds on E' orrespond respetively to forward sattering, θ = 0, where the neutron loses no energy, and to bakward sattering, θ = π, where the neutron suffers maximum energy loss. The student is advised to make sure to speify the kernel throughout the entire energy range by not forgetting to write out the seond line of (6.12) when asked to give the energy transfer kernel. A sketh of the energy transfer kernel is shown in Fig. 1. Fig.1. The energy transfer kernel F ( E E ') derived under the assumptions of elasti sattering, target nuleus at rest, and spherially symmetri sattering in CMCS, with 2 α = [( A 1) /( A + 1)] and A = M/m. The very simple form of F (E E ') makes it easy to understand all its features. The probability distribution is uniform in the range between (α EE),, where we reall 2 α = [( A 1) /( A +1)], with A = M/m, and zero outside this range. The existene of a utoff in the range of energy that an be transferred from the neutron to the target nuleus orresponds to the range of sattering angle that one an have, minimum value of θ is zero whih is forward sattering (no ollision) and maximum is π whih is bakward sattering. In the ase of sattering by hydrogen, α = 0, so the energy range extends down to zero. With neutron and hydrogen having the same mass (true only for the purpose of our alulation) it is not surprising that in a ollision the neutron an transfer all its energy to the hydrogen. Lastly we an ask why is the probability distribution uniform. The answer is again quite straightforward, namely the uniform distribution is a diret onsequene of the spherially symmetri form of the angular probability distribution, whih in turn arises beause of s-wave sattering. Another way to disuss F (E E ') is to examine the assumptions that have been made in deriving (6.12). There are three suh assumptions, (i) elasti sattering, (ii) target nuleus at rest, and (iii) spherially symmetri sattering in CMCS. Realling our disussion of kinematis of nulear reations (Le 3 (2003)) elasti sattering is the ase of Q = 0, and the assumption at the outset of target nuleus being at rest allows us to simplify the algebra onsiderably in the subsequent analysis. Relative to our disussion of ross setion alulation (Chap 4), the onditions of elasti sattering and stationary target are equivalent to our solution of the Shrodinger equation for an effetive partile 3

4 in CMCS. While the third assumption, spherially symmetri sattering in CMCS, has no ounterpart in any of the disussions in Le 3 (2003), we know from Chap 4 that s- wave sattering is spherially symmetri. Thus, the question is when an we ignore all the other partial-wave ontributions to the sattering. In Le 4 (2003) we have emphasized that this would be a good approximation under the ondition of low-energy sattering, that is, kr o < 1. Thus, the energy transfer kernel F (E E ') given in (6.12) is valid for neutrons at suffiiently low energy. We an estimate an upper limit on the energy by taking kr o ~ 0.1, with r o ~ 2 x m. This gives k ~ 5 x m -1, or E ~ 47 kev. Certainly for neutrons at thermal energies or 100 ev, or even 1 kev, (6.12) should be appliable. Suppose we wish to relax the assumption of spherially symmetri sattering in CMCS. What would F (E E ') look like if the sattering were biased either in the forward or in the bakward diretion? One an postulate simple forms of bias instead of (6.10) and arry through the transformation as before, along with using (3.10). One should then obtain non-uniform distributions in the final energy E'. The orrespondene between angular distribution and energy distribution is relatively simple to figure out. If the angular distribution favors the forward sattering diretion, then the energy distribution should show a bias toward smaller energy transfer; similarly a bakward sattering bias should translate into an energy distribution that favors larger energy transfer. Fig. 2 shows the harateristi behavior that is expeted. Fig.2. Shemati behavior of energy transfer kernel F (E E ') when the angular distribution in CMCS P(Ω ) is peaked in the forward (bakward) diretion. The dashed line is the result shown in Fig. 1 for spherially symmetri sattering (isotropi in CMCS). What if we wish to relax the other two assumptions? Let us onsider for a moment what one an say about other neutron sattering proesses besides elasti potential sattering. There are two suh proesses, one is elasti resonane sattering whih we have touhed on in Le 2 (2003) (f. (2.12)) with regard to the dependene of the sattering ross setion on the inoming neutron energy. In addition to elasti potential and resonane sattering, neutrons with suffiient energy an indue inelasti sattering, a reation involving the formation of a ompound nuleus whih deays to an exited state of the target nuleus with the emission of a neutron at onsiderably lower energy. If the need arises, our theoretial understanding of nulear reations is probably good enough to allow us to onstrut an energy transfer kernel for these proesses. On the other hand, suh analysis, to our knowledge, would be well beyond the sope of any nulear engineering ourse that is being taught. 4

5 In ontrast to elasti sattering, relaxing the assumption of target nuleus at rest is a very relevant issue, sientifially and tehnologially. When neutrons get down to thermal energies, it is no longer justified to assume that they move muh faster than the target nuleus. In that ase, target nuleus motion beomes a signifiant variable and one must take this effet into aount. It turns out that this not suh a simple issue, we will return to disuss it in some detail in the next leture (Chap 7). Before losing this leture we take up one final topi, that of the behavior of the angular differential ross setion dσ / dω = σθ) (, whih has played a entral role in the development in this leture. Given its importane, it would be instrutive to see an example or two of this distribution. In Fig. 3 we show the differential ross setion for a Fig. 3. Angular differential sattering ross setion of C 12 (a, left panel) and U 238 (b, right panel) at two inident neutron energies, 0.5 MeV and 14 MeV. light element, C12, and a heavy element, U238, eah at two inoming neutron energies, 0.5 MeV and 14 MeV. The distributions are plotted as funtions of µ = osθ. Based on what we have disussed above, one expets that at low energy the distribution should be independent of the sattering angle. Indeed this is learly seen in the result for C at 0.5 MeV. We an estimate what is kr o in this ase. Let r o ~1.2 x A 1/3 F = 2.75 x m. At 0.5 MeV, k = 2mE / = = 2 x1.67 x10 24 x0.5 x10 x1.6 x 10 /10 54 = 2.6 x 10 m -2 so kr o ~ 0.44, small enough to satisfy the s-wave sattering approximation. On the other hand, at 14 MeV, k = 8.5 x m -1, and kr o ~ In this ase one would expet that the p-wave and possibly the other higher-order partial wave ontributions to beome important. The breakdown of the s-wave approximation means that the angular ross setion should be peaked in the forward sattering diretion. This is seen in Fig. 3(a). Besides the forward bias, one should notie that the angular distribution is osillatory. This an be understood as evidene of neutron diffration by the nuleons, in the same manner as thermal neutron diffration in a sample of atoms and moleules. When the wavelength of the partile undergoing sattering beomes omparable to the spaing between a olletion of sattering enters, one an expet interferene effets. With thermal neutrons the wavelength is of the order of angstroms (10-8 m) whih are omparable to intermoleular distanes in ondensed matter - this leads to interferene 5

6 among the sattered waves and the appearane of osillations (the diffration pattern). In the present ase, the neutron wavelength is λ = 2π / k = 7.39 F, apparently short enough to begin to be sensitive to interferene effets among different nuleons. In view of the results for C 12, we an readily predit what should happen in the ase of U 238. Sine r o is now ~ 7.44 F, we see that at 0.5 MeV, kr o is now 1.66x0.744 = It is then not surprising that the harateristi forward peaking, signaling the breakdown of the s-wave sattering approximation, is seen in Fig. 3(b). That said, it is also to be expeted that at 14 MeV, with kr o ~ 6.3, quite pronouned diffration behavior should be observed. In the next leture we will ontinue to disuss the energy dependene of the sattering ross setion, fousing on the 'total' ross setion σ ( E) as well as F(E E ') when thermal motion and hemial binding effets ome into play. Together Letures 6 and 7 will provide the understanding of ross setion behavior that will form the basis of neutron interation in preparation for our disussion of neutron transport. 6