References --.54 Neutron Interctions nd Applictions (Spring 003) Lecture (/11/03) Neutron Rection Systemtics -- Energy Vritions of Cross Sections, Nucler Dt Enrico Fermi, Nucler Physics, Lecture Notes compiled by J. Orer, A. H. Rosenfeld, nd R. A. Schluter (Univ. Chicgo Press, 1949), revised edition, chp. VIII. Mrmier nd Sheldon, Physics of Nuclei nd Prticles (Acdemic Press, New York, 1969), vol. I, pp. 64 95. S. Perlstein, "Evluted Nucler Dt Files", in Advnces in Nucler Science nd Technology (Acdemic Press, New York, 1975), vol. 8. In this lecture we will survey the energy vritions of nucler rection cross sections, mking use of simple results from perturbtion theory in quntum mechnics, Fermi's Golden Rules. This is to provide some pprecition of the vriety of cross sections tht hve found useful pplictions in nucler science nd technology. We will lso give brief description of nucler dt compiltion nd evlution, the importnce of which cnnot be overemphsized since no serious nucler clcultions cn be performed without this bsic informtion. We consider the rection + A b + B, where is the incoming prticle (neutron), A the trget nucleus, b the outgoing prticle, nd B the product nucleus. First we re not concerned with rersonnce rection which is usully treted s two-step process, involving the formtion nd subsequent decy of the compound nucleus. This will be tken up lter in this lecture. We define the Q-vlue for the rection s Q [(m + M A ) (m + M B )]c (.1) b where m, M re tomic msses, nd c is the speed of light. For Q > 0, the rection is exothermic, with energy being given off. For Q < 0, the rection is endothermic, with energy being bsorbed. From conservtion of totl energy, Q lso cn be written in terms of the kinetic energy of the prticles nd nuclei, Q = T + T B (T + T A ) (.) b Notice tht despite the ppernce of (.) which my mke one think tht Q is dependent on the coordinte system used to mesure the kinetic energies (lbortory vs. center-of-mss), Q is independent of the reference frme s shown by (.1). In most cses, exception is in neutron thermliztion discussions, we cn ignore the motion of the trget nucleus nd set T A = 0, considerble simplifiction. In the following we will switch nottion nd use E insted of T for the kinetic energy. For the neturon the kinetic energy is then E (or E n ). In quntum mechnics the clcultion of cross section σ ( b), is usully clculted by pplying time-dependent perturbtion theory to the rection A(,b)B, obtining the generl expressions known s Fermi's Golden Rules. There re two Golden Rules; for our discussion we re concerned with Golden Rule no., which dels with so-clled first-order trnsitions nd gives the expression for the trnsition rte (number of trnsitions per unit time between initil stte nd finl stte b of the prticle) w b, w b π dn = H b (.3) = de 1
where H b is the mtrix element of the perturbtion H cusing the trnsition from to b, nd dn/de is the density of sttes, the number of sttes per unit finl energy of the prticle. For prticle in vcuum (free prticle) with momentum p, we cn redily clculte the density of sttes. Let the vcuum hve finite volume Ω (we tke the system volume to be finite for purpose of simple normliztion of the wve function), the number of sttes with momentum in dp bout p is 3 dn = 4π p dp Ω /(π =) (.4) If the outgoing prticle b nd the product nucleus B hve spins I b nd I B, respectively, then we need to include in Eq.(.4) sttisticl fctor, (I b +1)(I B +1) for the number of spin orienttions one cn hve in the finl stte. For n outgoing prticle with momentum p, b de = v dp (.5) b b Strictly speking, v b nd p b should be the velocity nd momentum of the finl stte (b+b) in the center of mss coordinte system (frme of reference), rther thn of the outgoing prticle b. The difference is smll when the product nucleus is hevy compred to the mss of the prticle. We cn rewrite (.1) by combining ll (.4), (.5) nd the sttisticl fctor, w = 1 b π = 4 p b vb H b b B Ω (I + 1)(I + 1) (.6) To go from the rte of trnsition to the cross section, we simply write w b = n v rel σ (, b ) (.7) which essentilly introduces nd therefore defines the cross section. Compre this rgument with Eq.(1.3) in Lec 1. In (.7), n is the density of incoming prticle nd v rel is the reltive velocity in the initil stte (+A). Agin, if the trget nucleus is mssive compred to the incoming prticle, then to good pproximtion v rel is given by v, the velocity of the incoming prticle in CMCS. Combining (.6) nd (.7) we obtin 1 p σ ( b), = ΩH b 4 b (I b + 1)(I B + 1) (.8) π = vv b This formul is useful for deducing the qulittive vrition of the cross section with energy for number of typicl rections, without knowing much bout the relly complicted prt of the clcultion, the mtrix element of the perturbtion H between initil nd finl sttes. We now ssume tht there re two prts to the trnsition mtrix element, one involving nucler interctions mong the nucleons nd the other pertins to electrosttic (Coulomb) interctions between the incoming or outgoing prticle, if it were chrged prticle, nd the nucleus (either the trget or the product s the cse my be). We further ssume tht the nucler interctions, however complicted, my be tken to be constnt for the purpose of estimting the energy vrition of σ ( b),. Wht is left in H b is the Coulomb interction. The effect of this cn be estimted by using the model of chrged prticle tunneling through Coulomb brrier. (Recll from.101 tht such model ws introduced to clculte the decy constnt for α decy.) For positively chrged prticle, chrge z with velocity v, to penetrte nucleus, chrge Z, the trnsmission fctor is exp(-g/), where [Fermi's Notes, p. 143]
π Zze G / (.9) =v where G is clled the Gmow fctor. Accordingly, if the incoming nd outgoing prticles re both chrged, we will write H b U exp( G G b ) (.10) where U is the nucler interction prt, which we will tke to be energy independent. Now we re redy to exmine the vrition with energy of incoming prticle of vrious neutron rections t low energies, sy ev rnge. Eq.(.8) shows tht there re two prts which cn vry with energy, the kinemticl fctor, p / vv, nd the trnsmission fctors, if present. We cn distinguish four b b types of neutron interctions of interest. 1. Elstic scttering (n,n) With incoming nd outgoing prticles being unchrged, the trnsmission fctor is unity. Also, with v = v b, the kinemticl fctor p / vv is constnt. Eq.(3.8) then predicts the cross section to be constnt, s b b shown in Fig. 3-1(). Notice tht elstic scttering is, strictly speking, not proper rection.. Chrged-Prticle Emission (exothermic) Since Q is typiclly few MeV, while the incoming neutron energy is of order ev, p / vv 1/ v. b b H b U exp( G b ) constnt. So σ 1/ v n ; this is the '1/v' behvior often seen in rections like ( n, α ), ( n, γ ), etc. 3. Inelstic scttering (n,n') Since the product nucleus is left in n excited stte, this is n endothermic rection tht requires threshold vlue for the neutron energy, E* = -Q ~ 1 MeV. Velocity of the outgoing neutron is v n', with v = (excess n ' energy bove threshold)/(m n /) ~ (E n - E*). Thus, σ ( nn'), v ~ (E E ) n ' n * 1/ 16 13 An exmple here is O ( n,α)c, with Q = -.15 MeV. 4. Chrged-Prticle Emission (endothermic) This rection is like the endothermic rection of inelstic neutron scttering, except there is now Coulomb fctor, σ ( nb), ~ e G b (E E ) n * 1/ with G b ~ 1/v b. 3
Fig. -1. Schemtic energy vritions of cross sections, the first four correspond to neutron elstic scttering, neutron-induced rection (exothermic), neutron inelstic scttering, nd neutron-induced rection (endothermic). The lst two rections re chrged prticle nd unchrged prticle, nd chrged prticles in nd out, both rections being exothermic. In ddition, we cn consider exothermic rections with incoming chrged prticles. If the outgoing prticle is unchrged, then with E << Q, p / vv ~1/v, pb ~ constnt, we hve b b σ ( b), ~ 1 e G v An exmple would be the inverse rection to tht lredy mentioned, C 13 ( α, n ) O 16, Q =.15 MeV. If the outgoing prticle is lso chrged, s in ( α, p), p / vv ~1/v, nd b b Both cses re lso shown in Fig. -1. Resonnce Rections b) ~ 1 ( G + G b ) σ (, e v The energy vritions we hve discussed re qulittive nd smooth behvior; they re not intended to pply to resonnces which re shrp fetures of the energy dependence of the cross sections. To describe resonnces one cn pply Golden Rule no. 1 which curiously dels with second-order trnsitions. Insted of going from initil to finl stte directly in trnsition, s in the cse of Golden Rule no., we consider two-step process where the incoming prticle intercts with the trget nucleus to form compound nucleus which exists for finite period nd then decys to the finl stte consisting of n outgoing prticle nd the product nucleus, + A C b+b (.11) Without going into detils, we will simply give the results for two neutron resonnces which re quite commonly encountered, elstic scttering ( nn), nd rditive cpture ( n, γ ). The cross sections for these two rections hve the form of so-clled Breit-Wigner resonnces. 4
For elstic scttering, σ ( nn), = π f (E E / 4 + 4π Γ f E E γ (E E ) +Γ Γ n o ) n o γ +Γ γ + σ p (.1) / 4 where = π / k, k being the neutron wve number, E = = k /m, is the scttering length which ppers in the definition of the potentil scttering cross section σ p = 4π (we will come bck to discuss wht we men by potentil scttering in more detil), nd f = (J +1) / ( I +1) is the sttisticl fctor o for spin orienttions, I being the spin of the trget nd J is the totl spin, I ±1/. The other quntities in (.1) re the resonnce energy E γ, nd the neutron nd totl resonnce width, Γ nd Γ=Γ +Γ +..., n n γ with Γ being the rdition width; these re typiclly known s resonnce prmeters, they re prt of the γ nucler dt nd re tbulted for given trget nucleus. Eq.(.1) shows there re three contributions to elstic scttering, the first term is the resonnce contribution, clled resonnce elstic scttering, the second term is n interference term between resonnce scttering nd potentil scttering, the ltter being the ordinry scttering in the bsence of ny resonnce. The third term is the potentil scttering which is typiclly tken to be constnt, depending only on the scttering length, bsic property of the nucleus (more discussion will be given in the lectures on cross section clcultion nd neutron-proton scttering). To see the cross section vrition with neutron energy in elstic scttering, we note tht ~ v or E, Γ ~ n E, Γ ~ constnt. Imgine there is resonnce t energy E γ, for neutron energy much less γ or much greter thn E γ the first nd second terms do not contribute, so the cross section is constnt. In the vicinity of E γ the cross section hs vrition shown in Fig. -. Notice tht the interference effect is destructive t energies just below the resonnce, nd constructive t energies bove. This is chrcteristic feture of resonnce rection which sometimes re observed in ctul cross section mesurements. Fig. -. Schemtic energy vrition of neutron elstic scttering showing interference effects between resonnt nd potentil scttering. 5
For rditive cpture the cross section hs the pure Breit-Wigner form, ΓΓ n γ σ ( n,γ ) = π f o (.13) (E E ) γ +Γ / 4 One cn see tht well below the resonnce the cross section behves like 1/v, nd the full width t hlf mximum of the resonce pek is Γ, s shown in Fig. -3. Fig. -3. Schemtic energy vrition of neutron cpture resonnce, showing chrcteristic 1/v behvior below the resonnce nd full width t hlf mximum of Γ. To conclude our brief survey of neutron cross sections we show the ctul cross sections for few nuclei which re of definite interest to nucler engineering students. The first exmple is U 35, n isotope of urnium which is fissile (cpble of undergoing fission when bombrded by therml neutrons). Fig. -4. Vrious cross sections of the urnium isotope 35, fission (n,f), rditive cpture ( n,γ ), elstic scttering (n,n), inelstic scttering (n,n'), nd stripping rection (n,n). In Fig. -4 one cn pick out some of the chrcteristic fetures discussed bove, energyindependent behvior for potentil scttering, 1/v vrition for cpture nd fission below the resonnces, nd threshold behvior for inelstic scttering. The student should lso py ttention to the mgnitudes of the cross sections nd the wide energy rnge covered. In the therml region, round 0.05 ev, the energy dependence is rther simple, ll monotonic vritions. Strting t bout 1 ev nd extending up to bout 1 KeV, there re mny shrp resonnces. Above 10 KeV the cross sections return to smooth behvior. In the context of using urnium s the fuel for nucler rector, we cn see the reson for building therml rectors - to tke dvntge of the lrge fission cross sections. Since neutrons from fission re emitted in the MeV rnge, one lso hs the problem of slowing down the neutrons pst the resonnce region where there is high probbility of cpture down to the therml rnge in order to continue the chin rection.. 6
Thus, the optimum design of nucler system, whether it is nucler rector, prticle ccelertor, or nything else, often comes down to mtter of mterils properties, or in this cse the cross sections. When one compres the cross sections of U 35 with those of nother isotope U 38, shown in Fig. 5, severl fetures cn be noted. The bsence of therml fission rection in U38 results in rther low cross section vlues in the therml energy region. In the resonnce region, the chrcteristic interference behvior in elstic scttering is even more pronounced thn tht sketched in Fig. -. In the Mev region, fst fission is seen to be threshold process. U 38 is fertile nucleus in the sense tht when it cptures neutron, it becomes U39 which undergoes two β -decys to rech Pu 39 which is fissile. Fig. -5. Cross sections of urnium isotope 38. At the opposite extreme of light nucleus we cn consider the cross section of the proton, the nucleus of hydrogen tom, shown in Fig. -6. The most notble feture here is the essentilly constnt vlue of the cross section t 0 brns over n extended energy from bout 1 ev to well beyond 10 KeV. Given tht the therml bsorption cross section is 0.3 brns, the entire 0 brns cn be ttributed to potentil scttering. An interesting question is since the proton is the smllest possible nucleus, why should its scttering cross section be so lrge, bout fctor of or more compred to ll the other nuclei. This clerly suggests tht in the cse of neutron scttering by hydrogen, the cross section is not determined only by the size of the nucleus. As we will see lter, the nswer lies in the contribution to the cross section from the coupling of the neutron nd proton spins (ech hs spin 1/). This explntion is fully consistent with our knowledge of nucler interctions s being spin-dependent Fig. -6. Neutron scttering cross section of hydrogen in the low-energy region. In contrst to the hydrogen cross section, the cross section of crbon, shown in Fig. -7, shows fetures which rise from the physicl stte of trget toms. In this cse the trget is crystlline smple. One sees shrp edge in the cross section vrition which rises from Brgg scttering, constructive interference effect. Below the edge, the cross section behves like 1/v nd is temperture-sensitive, higher mgnitude t higher temperture. Beyond bout 0.1 ev the cross section reches constnt t bout 5 7
brns, the sme potentil scttering behvior s mentioned bove. We will come bck to discuss these fetures in more detil lter. Fig. -7. Neutron scttering cross section of polycrystl of nturl crbon in the low-energy region. Nucler Dt Compiltion nd Evlution As the lst prt of this lecture we tke up the importnt topic of how informtion on nucler dt is gthered, nlyzed, nd mde vilble to users on world-wide bsis. Given the importnce of hving ccurte knowledge of cross sections nd other dt in nucler clcultions, it is not surprising tht there is whole industry on nucler dt technology tht hs been long estblished nd continues to be ctive in mintining nd improving the dtbse. The student should bewre tht our discussion is bsed on source mteril dted round the mid 70's; it is quite possible tht significnt chnges hve since tken plce. While detiled informtion my hve chnged over the yers, the role of nucler dt in nucler system nlysis nd design remins s vitl s ever. Compiltion nd evlution of nucler dt re crried out t mny ntionl lbortories nd reserch centers ll over the world. In the U.S. the focl point of this ctivity hs been the Ntionl Neutron Cross Section Center t the Brookhven Ntionl Lbortory. Perhps the most importnt contribution of this Center is librry of evluted nucler dt, known s the Evluted Nucler Dt File (ENDF), which ws developed for the storge nd retrievl of nucler dt needed for neutronic nd photonic clcultions. There re two librries. ENDF/A is collection of useful evluted dt sets. ENDF/B contins the reference dt sets recommended by the Cross Section Evlution Working Group. At ny given time there is only evluted dt set for prticulr mteril. Most users re concerned only with the B version. Evlution of nucler dt mens the ssignment of the most credible vlue fter considertion of ll the pertinent informtion. The evlution is supported by documenttion giving description of how the vlue ws determined nd n estimte of its uncertinty. As of 1975 the informtion contined in ENDF consists of: Resonnce prmeters, cross section tbles, ngulr distributions, energy distributions, double differentil dt in ngle nd energy, scttering lw dt, nd fission prmeters. Four primry centers with responsibilities for collecting nd disseminting nucler dt informtion hve been estblished to serve the world-wide community. The Neutron Cross Section Center, Brookhven Ntionl Lbortory, serves U.S. nd Cnd. The Neutron Dt Compiltion Center, Scly, Frnce, serves Western Europe nd Jpn. The Nucler Dt Center, Obninsk, USSR, serves the Soviet Union. The Nucler Dt Section, IAEA, serves essentilly the rest of the world. 8
For brief look t the contents of ENDF/B librry, we note tht ENDF/B-II contins 3 bsic types of dt stored on 13 mgnetic tpes. 1. Scttering lw dt for 1 modertor mterils. For exmple, S ( α, β ) for series of tempertures, 10 tempertures between 96 o K nd 000 o K in the cse of grphite.. Neutron cross sections for 78 fissile, fertile, structurl, nd other mterils, ech being n element or n isotope -- totl nd ny significnt prtil cross sections, rections producing outgoing neutrons, ngulr nd energy distributions, rdioctive decy chin dt, fission product yield, ν ( E). 3. Photon interction cross sections for 87 elements from Z = 1 to 94 -- photon cross sections, ngulr nd energy distributions of secondry photons. Over the yers number of ctive compiltion groups hve issued librry files: KEDAK from Krlsruhe, Germny, JAERI (Jpn), UKNDL from Englnd, nd vrious librries from U.S. ntionl lbortories such s Ok Ridge, Los Almos, nd Lwrence Livermore. 9