Kinematics of Elastic Neutron Scattering

Transcription

1 .05 Reator Physis - Part Fourteen Kineatis of Elasti Neutron Sattering. Multi-Group Theory: The next ethod that we will study for reator analysis and design is ulti-group theory. This approah entails dividing the range of possible neutron energies into sall regions, ΔE i and then defining a group ross-setion that is averaged over eah energy group i. Neutrons enter eah group as the result of either fission (reall the distribution funtion for fission neutrons) or sattering. Aordingly, we need a thorough understanding of the sattering proess before proeeding.. Types of Sattering: Fission neutrons are born at high energies (> MeV). However, the fission reation is best sustained by neutrons that are at theral energies with theral defined as 0.05 ev. It is only at these low energies that the ross-setion of U-35 is appreiable. So, a ajor hallenge in reator design is to oderate or slow down the fission neutrons. This an be ahieved via either elasti or inelasti sattering: a) Elasti Satter: Kineti energy is onserved. This ehanis works well for neutrons with energies 0 MeV or below. Light nulei, ones with low ass nuber, are best beause the lighter the nuleus, the larger the fration of energy lost per ollision. b) Inelasti Satter: The neutron fors an exited state with the target nuleus. This requires energy and the proess is only effetive for neutron energies above 0. MeV. Collisions with iron nulei are the preferred approah. Neutron sattering is iportant in two aspets of nulear engineering. One is the aforeentioned neutron oderation. The other is shielding. In both ases, for reators, one uses elasti ollisions with light nulei as the ehanis for energy loss. For aelerators, the situation is different beause neutrons with energies in exess of 0 MeV are produed. For these, shields are ade of iron so as to take advantage of the energy loss assoiated with inelasti sattering. 3. Methodology: Neutron sattering is ost easily analyzed in a enter-of-ass frae of referene (COM). Collisions in a COM frae appear as rotations with the speeds the sae

2 both before and after the ollision. However, the reator is designed in a laboratory frae of referene (LAB). So, we will analyze the sattering ollisions in a COM frae and then translate that result to a LAB frae. This is not diffiult provided one keeps the notations lear. 4. Fraes of Referene: The aterial in this setion follows that of Henry (p. 5). ν l ν l θ M M V l Before After Trajetories of a neutron and its target nuleus before and after a sattering event as seen fro the laboratory referene syste. ν ν V M V M ϕ Before After Trajetories of a neutron and its target nuleus before and after a sattering event as seen fro the enter-of-ass syste.

3 The notation used is as follows: Sall letters and v refer to the ass and speed of the neutron. Capital letters M and V refer to the ass and speed of the target nuleus A subsript l refers to the LAB syste. A subsript refers to the COM syste. A prie on v or V indiates a easureent after the ollision. V COM is the speed of the enter-of-ass in the LAB syste. Finally we note that the ass nuber of a neutron is l while that of the target nuleus is A. 5. Analysis of a Sattering Collision: The aterial in this setion follows that of Henry (pp ). We begin the analysis in the COM syste. Suh a syste is defined by the requireent that the total linear oentu of the neutron and the nuleus easured relative to that syste vanish. The viewpoint of a COM frae of referene is that of an observer who is loated at the enter-of-ass. The neutron and target nuleus both approah wi th equal and opposite oentus. They ollide and ove off at soe angle ϕ Given the requireent that the total oentu of the syste be zero, we have v V M 0, Before () v V M 0, After () B eause of the onservation of kineti energy (elasti ollision): v MV v + MV + (3) Using () and () to eliinate v and v, we get ( M / ) + M V ( M / ) + M V (4) Therefore, V V (5) 3

4 And insertion of this result into () and () yields: v v, (6) So, the speeds of the neutron and target nuleus are the sae, both before and after a ollision when viewed in the COM frae. Next we need to exaine the LAB frae and obtain an expression for the speed of the enter-of-ass in the LAB frae. The viewpoint of a LAB frae of referene is that of an external observer. The target nuleus is stationary. The neutron approahes with speed v. As a result of the ollision, the neutron oves off at soe angle relative to its original diretion of trave l. The target nuleus also oves off in a way so as to onserve oentu. The speed of the enter of ass in the LAB syste is that of the neutron and the target nulei weighted by their speeds. Thus, V COM v v + M v + A + + MV M beause the ass of the neutron is and the initial veloity of the target nuleus is zero. The above relation is also obtained fro a oentu balane. Naely, the oentu of the COM equals the su of the oentus of the individual partiles or ( + M) V v + MV COM Another way to obtain this relation is as follows: The target nuleus is at rest in the lab syste and oving to the left with speed V in the COM syste. Hene, the enter-of-ass itself ust be oving to the right in the LAB syste with the speed V. Thus, V COM V V (Note: This says that the agnitudes are equal; the diretions are opposite.) Fro this we obtain that the speed of the neutron in the COM is: v v V COM v V 4

5 Using (), we obtain: VM or v V V V v v + M + A We now draw a vetor diagra of the veloity of the neutron. The final veloity of the neutron in the LAB syste is obtained by adding the vetor that represents the oveent of the enter-of-ass in the LAB syste to the vetor that represents the veloity of the neutron after the ollision in the COM syste. V COM ϕ v v l ϕ θ Vetor relationship between the veloity of a neutron after a ollision easured in the laboratory syste and that easured in the COM syste. ϕ a C b Illustration of law of osines 5

6 The law of osines states that a +b - ab os C when C is opposite. In the above diagra, C would be (80º - ϕ ). Note that os (80º - ϕ ) -os ϕ. Thus, appliation of the law of osines to the above diagra yields: ( v ) ( v ) + ( V ) COM + v V COM os ϕ M M v + + osϕ M + Next let M/ be defined as A. Also note that the ratio of the neutron s kineti energies before and after the ollision is: Thus, we obtain: E / E v / v v / v E E A + + A os ϕ ( + A) This equation gives the energy of the neutron before and after the ollision in the laboratory frae of referene as a funtion of the sattering angle in the enter-ofass frae of referene. A further siplifiation in the forula is possible. Define: Then, A α A + E + α α + osϕ E Finally, we note without derivation, the relation between the sattering angles in the LAB and COM systes. It is: Cos θ + Aosϕ [ A + Aosϕ + ] / 6

7 Or, for hydrogen, + osϕ Cos θ / 6. Insights into Neutron Sattering: The aterial in this setion follows that of Henry (p. 56). The relation between E and E as well as that relating θ to ϕ insights: yields the following For glaning or straight-ahead ollisions, ϕ 0 and E E. Thus, there is no energy loss in suh ollisions. For head-on ollisions, ϕ 80, in whih the neutron reverses its diretion of travel, E / E takes on its iniu value whih is α. Therefore, Miniu possible energy of a neutron following an elasti ollision is α E. Maxiu energy loss is E( α). Given that α is zero for hydrogen and near unity for large ass nubers, it is apparent that lighter eleents are better oderators. That is, they result in greater energy losses per ollision. A neutron will lose all of its energy in a single head-on ollision with hydrogen beause hydrogen (a proton) is the sae ass as a neutron. The sattering angle ϕ in the COM is always greater than ( θ) in the LAB frae. For hydrogen, as ϕ goes fro 0 to π, θ goes fro 0 to π/. 7. Maxiu Energy Transfer: The axiu energy transfer ours for head-on ollisions. That is for os ϕ -. It is: 7

8 M E E E M E M + + M + M ( + M + M ) E ( ( + M) ( + M) M + E E ) M E or 4M Δ E ( + M) E This is the relation given earlier in the ourse without derivation. This relation is of fundaental iportane in the understanding of how radiation auses daage to tissue. For large partiles (proton or alpha) that slow down by ollisions with atoi eletrons, it shows that the energy loss per ollision is sall (~50 ev) and hene any thousands of ollisions are needed. These partiles travel in straight paths until stopping. For light partiles (eletrons) that also slow down by ollisions with atoi eletrons, oplete energy loss is possible in one ollision. These partiles zig zag. For neutrons, the energy loss per ollision is large and hene they would undergo only a few ollisions before stopping in the huan body. But, those ollisions will be with water and will result in free protons. The protons then slow down through any ollisions, eah of whih involves enough energy deposition to daage ellular strutures. 8. Consequenes of Sattering: For neutron energies below 0 MeV, whih is the upper liit for neutrons produed fro fission, the sattering is isotropi in the COM. That is, quoting fro Henry (p. 57), If we iagine the sattering enter to be surrounded by a sphere of unit radius so that the trajetories of the sattered neutrons lie along radii of that sphere, the probability that the sattered neutron will pass through any partiular area on the surfae of the unit sphere is just that area divided by 4 π, the total area of the unit sphere. All areas of equal size on the surfae of the sphere, no atter what their shape, have an equal probability that the sattered neutron will pass through the. In partiular, if we onsider a irular ring of 8

9 area π sin ϕ dϕ on the surfae of the sphere, the probability that the sattered neutron will pass through that ring is: πsin ϕdϕ 4π sin ϕdϕ d ( osϕ) dμ, where dμ is a negative nuber if d ϕ is positive. Henry goes on to show that the probability that a neutron having an initial energy E will, as a result of an elasti satter, eerge with energy E d is: P(E E )de de E( α) This is a ruial result beause, again quoting Henry (p. 58), it shows that, for the isotropi-elasti-sattering ase, the probability of sattering into any energy interval d E in the aessible range of energies E( α j) is unifor, i.e., does not depend on E and is, hene, equally likely for all equal intervals d E in the range. In other words, the energy loss spetru for neutrons sattering off hydrogen is flat. Consider a ono-energeti neutron soure with energy En. The probability of losing an aount of energy E is siply E/E n and is independent of where E is loated. P (E ) E n ΔE 0 0 E n This is a very pratial result for ulti-group theory beause it shows that for a hydrogen oderator, the nuber of neutrons that satter into a given group is a funtion of the group s width and not the absolute value of the energy of 9

10 that group. Thus, the probability that a 0 MeV neutron satters off hydrogen to a group defined as. to. MeV is the sae as for a group defined as 0.4 to 0.5 MeV beause both have the sae width (0. MeV). For oderators other than hydrogen, a siilar result holds. Speifially, quoting Henry (p. 58), for the isotropi-eleasti-sattering ase, the probability of sattering into an any energy interval d E in the aessible range of energies E( α j) is a unifor, i.e., does not depend on E and is, hene, equally likely for all equal intervals d E in the range. 9. Average of the Cosine of the Laboratory Sattering Angle: It will be realled fro the disussion of Fik s Law that the diffusion onstant D was defined in ters of a transport ean free path. Speifially, and D λ λtr /3 tr / Σtr / [ Σ μ ] Σ t 0 Σt refers to the total rate of whih neutrons are reoved by all proesses fro a defined volue and energy region. Σs refers to the sattering rate. μ 0 is the average of the osine of the laboratory sattering angle. If absorption is sall opared to Σs or at least weak, then s Σt ay be replaed by Σ s and we obtain: λ tr Σs( μ0) λ s ( μ0) and D 3Σs ( μ 0 ) 0

11 We are now in a position to opute μ 0 The sattering of neutrons with energies less than 0 MeV (i.e., fission neutrons) is isotropi in the enter-of-ass frae of referene. So, all angles fro 0 to π (0 to 80 ) are equally likely. The average is π/ (90 ) and the average of os ϕ is 0 where is the COM sattering angle. ϕ The sattering angle in the laboratory syste is always less than that of the COM syste. This eans that sattering in the LAB syste is in the forward diretion. For hydrogen, whih is the ost oon satterer, the LAB angle of satter ranges fro 0 to 90. To obtain the average value of the osine of the sattering angle, ultiply eah value of μ 0 by the fration of neutrons that are sattered between μ 0 and (μ 0 plus dμ 0 ) and then integrate over μ 0. The result (see Henry p. 58) is: μ 0 3A For a ixture of isotopes one would then average again, this tie weighting eah μ by the perent of the isotope present. 0 What does the above ean? If sattering in the LAB syste were isotropi (instead of forward direted), μ 0 would be zero and the transport ean free path would equal the sattering ean free path. Hene, the quantity /(- μ 0 ) serves as a orretion fator for anistropi sattering. Note that μ 0 dereases as the ass nuber inreases. Hene, sattering is essentially isotropi for neutrons that strike heavy nulei. 0. Average Loss of the Logarith of Energy: Another useful quantity in the study of the slowing down of neutrons is the average value of the derease in the natural logarith of the neutron energy per ollision. This quantity is denoted by the Greek sybol ζ. If E and E are the neutron energies before and after the ollision, then ζ is the average for all ollisions of the quantity (ln E - ln E ). The probability that a neutron with energy E satters into an interval defined by (E to E + de) is:

12 P(E E )de de E ( α) Note that this probability depends on the width (de ) and on the initial energy (E ) and it does NOT depend on the final energy (E ). Therefore, E de ζ ln E ln E ) αe ( E( α) A + / 3 (Note: See Henry p. 59 for the integration.) The above relation is very aurate for A > 0. For A, it is aurate to 3 %. Of iportane is that the value of ζ is independent of the neutron s energy. This allows us, as shown in the next setion, to use it to estiate the effetiveness of oderators.. Lethargy: The aterial in this setion follows that of Henry (p. 60). Lethargy, whih is also alled the logarithi energy dereent, is denoted by the sybol u. It is defined as: u ln(e0 / E) Where E 0 is the highest energy of a fission neutron, typially 0 MeV. Hene, for EE 0, u0. For E0, u. Consider a neutron that satters fro E to E. We an write:

13 ln E ln E E0 ln E u u E ln E [ ln E ln E (ln E ln E) ] We previously defined ζ as the average value of ln (E / E ). So, ζ is also the average hange in lethargy of a neutron per ollision. A useful appliation of the lethargy onept is for the alulation of the nuber of ollisions needed to theralize a neutron. The ost probable energy of a fission neutron is 0.85 MeV. Theral energy is defined as 0.05 ev. So the needed hange in lethargy to slow down fro 0.85 MeV to 0.5 ev is ln (.85 x 0 6 /.05) or Fro the relation /(A+/3), we obtain ζ for various eleents. For arbon, it is So, (7.34/0.58) 0 ollisions. The table below (Soure: Glasstone, p. 33) lists soe data for several oderators. Sattering Properties of Nulei Eleent Mass No. α ζ Collisions to Theralize Hydrogen Deuteriu Heliu Berylliu Carbon Uraniu The tie required for a neutron to attain theral energy is a funtion of the nuber of ollisions that it ust undergo to attain 0.05 ev. For oderators ade of Be or graphite, that tie will be signifiantly greater than for those ade of light water. Thus, leakage will be greater in reators that do NOT use light water as a oderator. That is, if it takes longer to slow down, the likelihood of neutron 3

14 leakage out of the ore goes up. So, reators that use Be or graphite oderators will have to be larger than those that use light water.. Moderating Ratio: The oderating ratio is a figure of erit for the evaluation of oderators. ζσ MR s Σa The desired properties are a high energy loss per ollision, a high sattering rosssetion, and a low absorption ross-setion. Soe values (Glasstone, p. 39) are: Moderator MR Light Water 58 Heavy Water 000 Heliu 45 Berylliu 30 Graphite 00 MR values should be taken as guidelines only. For exaple, heavy water has the highest oderating ratio. But, that is beause of its very, very low absorption ross-setion. Light water is a better satterer and hene would allow a ore opat ore. So, light water is the best hoie for a oderator and heavy water the best for a refletor where the objetive is not to absorb the already theralized neutrons. Also, while heliu has a good MR value, it is gaseous and hene its use would also ake for a large reator. 4