Applied Nuclear Physics (Fall 2004) Lecture 23 (12/3/04) Nuclear Reactions: Energetics and Compound Nucleus

Transcription

1 .101 Appled Nuclear Physcs (Fall 004) Lecture 3 (1/3/04) Nuclear Reactons: Energetcs and Compound Nucleus References: W. E. Meyerhof, Elements of Nuclear Physcs (McGraw-Hll, New York, 1967), Chap 5. Among the many models of nuclear reactons there are two opposng basc models whch we have encountered. These are () the compound nucleus model proposed by Bohr (1936) n whch the ncdent partcle nteracts strongly wth the entre target nucleus, and the decay of the resultng compound nucleus s ndependent of the mode of formaton, and () the ndependent partcle model n whch the ncdent partcle nteracts wth the nucleus through an effectve averaged potental. A well-known example of the former s the lqud drop model, and examples of the latter are a model proposed by Bethe (1940), the nuclear shell model wth spn-orbt couplng (cf. Chap 9), and a model wth a complex potental, known as the optcal model, proposed by Feshbach, Porter and Wesskopf (1949). Each model descrbes well some aspects of what we now know about nuclear structure and reactons, and not so well some of the other aspects. Snce we have already examned the nuclear shell model s some detal, we wll focus n the bref dscusson here on the compound nucleus model. As we wll see, ths approach s well suted for descrbng reactons whch show sngle resonance behavor, a sharp peak n the every varaton of the cross secton. In contrast, the optcal model, whch we wll not dscuss n ths course, s good for gross behavor of the cross secton (n the sense of averagng over an energy nterval). Energetcs Before dscussng the compound nucleus model we frst summarze the energetcs of nuclear reactons. We recall the Q-equaton ntroduced n the study of neutron nteractons (cf. Chap 15) for a general reacton depcted n Fg. 1.1, 1

2 Fg A generc two-body nuclear reacton wth target nucleus at rest. M 3 1/ Q = T T 1 1 M 1 ( M 1 M 3T 1T 3 ) cosθ (1.1) M 4 M 4 M 4 Snce Q = T 3 +T 4 T 1, the reacton can take place only f M 3 and M 4 emerge wth postve knetc energes (all knetc energes are LCS unless specfed otherwse), T 3 + T 4 0, or Q + T 1 0 (1.) We wll see that ths condton, although qute reasonable from an ntutve standpont, s necessary but not suffcent for the reacton to occur. We have prevously emphaszed n the dscusson of neutron nteracton that a fracton of the knetc energy brought n by the ncdent partcle M 1 goes nto settng the center-of-mass nto moton and s therefore not avalable for reacton. To see what s the energy avalable for reacton we can look nto the knetc energes of the reactng partcles n CMCS. Frst, the knetc energy of the center-of-mass, n the case where the target nucleus s at rest, s T = 1 (M 1 + M )v o o (1.3) where the center-of-mass speed s v = [M /(M + M )]v 1, v 1 beng the speed of the o 1 1 ncdent partcle. The knetc energy avalable for reacton s the knetc energy of the

3 ncdent partcle T 1 mnus the knetc energy of the center-of-mass, whch we denote as T, M T = T 1 T o = M 1 + M T1 1 M v o = V M (1.4) The second lne n (1.4) shows that T s also the sum of the knetc energes of partcles 1 and n CMCS (we follow the same notaton of usng captal letters to denote velocty n CMCS). In addton to the knetc energy avalable for reacton, there s also the restmass energy avalable for reacton, as represented by the Q-value. Thus the total energy avalable for reacton s the sum of T and Q. A necessary and suffcent condton for reacton s therefore E aval = Q + T 0 (1.5) We can rewrte (1.5) as T1 M 1 + M Q (1.6) M If Q > 0, (1.6) s always satsfed whch s expected snce the reacton s exothermc. For Q < 0, (1.6) shows that the threshold energy, the mnmum value of the ncdent partcle knetc energy for reacton, s greater than the rest-mass defct. The reason for needng more energy than the rest-mass defct, of course, s that some energy s needed to put nto the center-of-mass. At threshold, Q + T = 0. So M 3 and M 4 both move n LCS wth speed v o (V 3 and V 4 = 0). At ths condton the total knetc energes of the reacton products s 3

4 (T + T 4 ) = 1 (M M 4 )v (1.7) thres o Snce we have M 3 V 3 = M 4 V 4 from momentum conservaton, we can say n general Q + T = M V ( V 3 M ) (1.8) M 4 Wth Q and T 1 gven, we can fnd V 3 from (1.8) but not the drecton of V 3. It turns out that for T 1 just above threshold of an endothermc reacton, an nterestng stuaton exsts where at a certan scatterng angle n LCS one can have two dfferent knetc energes n LCS, whch volates the one-to-one correspondence between scatterng angle and outgong energy. How can ths be? The answer s that the one-to-one correspondence that we have spoken of n the past apples strctly only to the relaton between the knetc energy T 3 and the scatterng angle n CMCS (and not wth the scatterng angle n LCS). Fg. 1. shows how ths specal stuaton, whch corresponds to the double-valued soluton to the Q-equaton, can arse. Fg. 1.. A specal condton where a partcle can be emtted wth two dfferent knetc energes at the same angle (n LCS only). 4

5 Energy-Level Dagrams for Nuclear Reactons We have seen n the prevous chapter how the varous energes nvolved n nuclear decay can be convenently dsplayed n an energy-level dagram. The same argument can be appled to nuclear reactons. Fg. 1.3 shows the energes nvolved n an Fg Energy-level dagram for an endothermc reacton. endothermc reacton. In ths case the reacton can end up n two dfferent states, dependng on whether the product nucleus M 4 s n the ground state or n an excted state (*). T f denotes the knetc energy of the reacton products n CMCS, whch one can wrte as T f = Q + T 1 = M V V M 4 Snce both T and T f can be consdered knetc energes n CMCS, one can say that the knetc energes appearng n energy-level dagram should be n CMCS. (1.9) Compound Nucleus Reactons The concept of compound nucleus model for nuclear reactons s depcted n Fg The dea s that an ncdent partcle reacts wth the target nucleus n two ways, a scatterng that takes place at the surface of the nucleus whch s, properly speakng, not a reacton, and a reacton that takes place after the ncdent partcle has entered nto the nucleus. The former s what we have been studyng as elastc scatterng, t s also known 5

6 Fg Compound nucleus model of nuclear reacton formaton of compound nucleus (CN) and ts subsequent decay are assumed to be decoupled. shape elastc or potental scatterng. Ths part s always present, we wll leave t asde n the present dscusson. The nteracton whch takes place after the partcle has penetrated nto the target nucleus can be consdered an absorpton process, leadng to the formaton of a compound nucleus (ths need not be the only process possble, the others can be drect nteracton, multple collsons, and collectve exctatons). Ths s the part that we wll now consder brefly. In neutron reactons the formaton of compound nucleus (CN) s qute lkely at ncdent energes of ~ Mev. Physcally ths corresponds to a large reflecton coeffcent n the nsde edge of the potental well. Once CN s formed t s assumed that t wll decay n a manner that s ndependent of the mode n whch t was formed (complete loss of memory). Ths s the basc assumpton of the model because t then allows the formaton and decay to be treated as two separate processes. The approxmaton can be expressed by wrtng the nteracton as a two-stage reacton, a + X C * b + Y the astersk ndcatng that the CN s n an excted state. The frst arrow denotes the formaton stage and the second the decay stage. For ths reacton the cross secton σ (a, b ) may be wrtten as 6

7 σ (a,b) = σ C ( T ) P ( E ) (1.10) b whereσ C (T ) s the cross secton for the CN formaton at knetc energy T, whch s the avalable knetc energy for reacton as dscussed above, and P b (E) s the probablty that the CN at energy level E wll decay by emsson of partcle b. It s understood that σ C and P b can be evaluated separately snce the formaton and decay processes are assumed to be decoupled. The energy-level dagram for ths reacton s shown n Fg. 1.5 for an endothermc reacton (Q < 0). Notce that E s the CN exctaton and t s measured Fg Energy-level dagram for the reacton a + X b + Y va CN formaton and decay. relatve to the rest-mass energy of the nucleus (a+x). If ths nucleus should have an excted state (a vrtual level) at E* whch s close to E, then one can have resonance condton. If the ncomng partcle a should have a knetc energy such that the knetc avalable for reacton has the value T *, then the CN exctaton energy matches an excted level of nucleus (a+x), E = E*. Therefore the CN formaton cross secton σ C (T ) wll show a peak n ts varaton wth T, an ndcaton of a resonance reacton. The condton for a reacton resonance s essentally a relaton between the ncomng knetc energy and the rest-mass energes of the reactants. Fg. 1.5 shows that ths relaton can be stated as T = T*, or 7

8 (M + M X )c + T * = M a+ X c + E * (1.11) a Each vrtual level E* has a certan energy wdth, denoted as Γ, whch corresponds to a fnte lfetme of the state (level), τ = h / Γ. The smaller the wdth means the longer the lfetme of the level. The cross secton for CN formaton has to be calculated quantum mechancally [see, for example, Burcham, Nuclear Physcs, p. 53, or for a complete treatment Blatt and Wesskopf, Theoretcal Nuclear Physcs, pp. 398]. One fnds σ (T ) = π D g Γ Γ a C J * ( T T ) + Γ / 4 (1.1) J + 1 where g = ( I + 1 I X + 1) and J = I a + I X + L a. In ths expresson D s the J )( a reduced wavelength (wavelength/π ) of partcle a n CMCS, J s the total angular momentum, the sum of the spns of partcles a and X and the orbtal angular momentum assocated wth partcle a (recall partcle X s statonary), Γ s the energy wdth (partal a wdth) for the ncomng channel a+x, and Γ (wthout any ndex) s the total decay wdth, the sum of all partal wdths. (The dea here s that CN formaton can result from a number of channels, each wth ts own partal wdth. In our case the channel s reacton wth partcle a wth partal wdth Γ.). Gven our relaton (1.11) we can also regard the a CN formaton cross secton to be a functon of the exctaton energy E, n whch case σ C (E) s gven by (1.1) wth ( E E *) replacng the factor ( T T denomnator. * ) n the To complete the cross secton expresson (1.10) we need to specfy the probablty for the decay of the compound nucleus. Ths s a matter that nvolves the exctaton energy E and the decay channel where partcle b s emtted. Treatng ths process lke radoactve decay, we can say 8

9 P b (E) = Γ b (E) / Γ (E) (1.13) where Γ (E) = Γ (E) + Γ (E) + wdth of any other decay channel allowed by the a b energetcs and selectve rules. Typcally one ncludes a radaton partal wdth Γ γ snce gamma emsson s usually an allowed process. Combnng (1.1) and (1.13) we have the cross secton for a resonance reacton. In neutron reacton theory the result s generally known as the Bret-Wgner formula for a sngle resonance. There are two cross sectons of nterest to us, one for neutron absorpton and another for neutron elastc scatterng, σ (n,γ ) = π D g Γ Γ γ n J * ( T T ) + Γ / 4 (1.14) * Γ n ( T T ), n ( T T ) + Γ / 4 + 4π Dg aγ J * J n * ( T T ) + Γ σ ( n ) = 4π a + π D g / 4 (1.15) In σ ( n, n ) the frst term s the potental scatterng contrbuton, what we had prevously called the s-wave part of elastc scatterng, wth a beng the scatterng length. The second term s the compound elastc scatterng contrbuton. The last term represents the nterference between potental scatterng and resonant scatterng. Notce the nterference s destructve at energy below the resonance and constructve above the resonance. In Fg. 1.6 we show schematcally the energy behavor of the absorpton cross secton n the form of a resonance peak. Below the peak the cross secton vares lke 1/v as can be deduced from (1.14) by notng the energy dependence of the varous factors, along wth Γ n ~ T, and Γ γ ~ constant. Notce also the full wdth at half maxmum s governed by the total decay wdth Γ. Fg. 1.7 shows a well-known absorpton peak n Cd whch s wdely used as an absorber of low-energy neutrons. One can see the resonance behavor n both the total cross secton, whch s domnated by absorpton, and the elastc scatterng cross secton. 9

10 Fg Schematc of Bret-Wgner resonance behavor for neutron absorpton. Fg Total and elastc neutron scatterng cross sectons of Cd showng a resonant absorpton peak and a resonant scatterng peak, respectvely. We conclude our bref dscusson of compound nucleus reactons by notng an nterestng feature n the elastc scatterng cross secton assocated wth the nterference effect between potental scatterng and resonance scatterng. Ths s the destructve effect of nterference n the energy regon just below the resonance and the constructve effect just above the resonance. Fg. 1.8 shows ths behavor schematcally, and Fg. 1.9 shows that such effects are ndeed observed. Admttedly ths s generally not so obvous, so the present example s carefully chosen and should not be taken as beng typcal. 10

11 Fg Interference effects n elastc neutron scatterng, below and above the resonance. Fg Expermental scatterng cross secton of Al7 showng the nterference effects between potental and resonance scatterng, and an asymptotcally constant value (potental scatterng) suffcently far away from the resonance. (from Lynn) 11