22.101 Applied Nuclear Physics (Fall 2006)

Lecture 2 3 (12/6/06)

Nuclear Reactions: En ergetics an d Compound Nucleus

References :

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), Chap 5.

A. Foderaro, The Elements of Neutron Interaction Theory (MIT Press, Cambridge, 1971), Chap. 6.

Am ong the m a ny models of nuclear r eactions there are two opposing basic models which we have encountered. Thes e are (i) the com pound nucleus model proposed by Bohr (1936) in which the incident particle interacts strongly w ith the entire target nucleus, and the decay o f the resultin g com pound nucleus is independent of the m ode of for m ation, and (ii) the independent particle m odel in which the incide nt particle interacts with the nucleus through an effective averag ed potential. A well-known exam ple of the for m er is the liquid drop model, and three ex am ples of the latter are a m o del proposed by Bethe (1940), the nuclear shell m ode l with sp in-orbit coupling (cf. Chap 9), and a m o del with a com p lex potential, known as the optical model, proposed by Feshbach, Porter and Weisskopf (1949). Each m odel describes well som e aspects of what we now know about nuclear structure and reactions, and not so well som e of the ot her aspects. Since we have alre ady exa m ined the nuclea r shell model is so m e detail, we will f o cus in the b r ief discussion here on the com pound nucleus m odel, which in som e sense m a y be considered to be in the s a m e class as the liquid d r op m odel. As we will s ee, this appr oach is well suited for describing reactions which show singl e resonance behavior, a sharp peak in the energy va ria tion of the c r oss sec tion. In contr a st, the optical model, which we will no t discuss in this course, is good for gross beha vior of the cross sect ion (in the sense of averag ing o v er an energ y interval).

Energetics

Before discussing the compound nucleus m odel we first summ arize the energetics of nuclear reactions. W e recall the Q-e quation introduced in the study of neutron interactions (cf. Chap 15) for a general reaction depicted in Fig. 21.1,

Fig. 21.1 . A generic two-body nuclear reacti on with target nucleus at rest.

 M 3 

 M 1  2

1 / 2

Q  T 3  1  M

  T  1 

4  

 

4  M 4

 M 1 M 3 T 1 T 3 

cos  (21.1)

Since Q = T 3 +T 4 – T 1 , the reaction can take place only if M 3 and M 4 em erge with positiv e kin e tic energ i es (all kine tic ener gies ar e LCS unless specif i ed o t herwise),

T 3  T 4  0 , or Q  T 1  0 (21.2)

We will see that this con d ition, altho ugh quite reasonable fro m an intuitiv e standpo int, is

necessary but not sufficient for the reaction to occur.

We have previously em phasized in the discussion of neutron intera ction th at a fraction of the kinetic energy brough t in by the incident particle M 1 goes into the m o tion of the center-of-m ass and is therefore not av a ilab l e f o r rea c tio n. To see what is the energy available for reaction we can look in to the kinetic energies of the reacting particles in CMCS. First, the kinetic energy of th e center-of-m ass, in the case where the target nucleus is at rest, is

T  1 ( M

o 2 1

 M 2

) v 2 (21.3)

where the center-of-m a ss speed is v o   M 1 /( M 1  M 2 )  v 1 , v 1 being the speed of the inciden t particle. The k i netic energy availa ble f o r rea c tion is the kinetic energy of the inciden t par t icle T 1 m i nus the kin e tic energy of th e center-of-m ass, which we denote as T i ,

T i  T 1

 T o

 M 2 T M 1  M 2

= 1 M V 2  1 M v 2 (21.4)

2 1 1 2 2 o

The second line in (21.4) shows that T i is also the sum of the kinetic energies of particles 1 and 2 in CMCS (we follow the same notation as bef o re in us ing capital lette rs to d e n o te velocity in CMCS). In addition to the kineti c en ergy availab l e for reactio n, there is also the res t -m ass energy av ailab l e for reaction, as rep r esented by the Q-value. Thus the to tal energy available for reaction is the su m of T i and Q. A necessary and sufficient condition for reaction is therefore

E avail  Q  T i  0 (21.5)

W e can rewrite (21.5) as

T   Q M 1  M 2 (21.6)

1 M

2

If Q > 0, (21.6) is always satisfied, which is expected s i nce th e r eaction is exotherm ic. For Q < 0, (21.6) shows that the threshold energ y , the m i ni mum value of the incident particle k i netic en ergy for reaction, is greater th an the res t -m ass deficit. Th e reason for needing m o re energy than the res t -m ass deficit, o f course, is that energy is needed for the kinetic energy of the center-of-m a ss.

At t h r e s h ol d, Q + T i = 0. So M 3 and M 4 both m o ve in LCS with speed v o (V 3 and V 4 = 0). At this condition the total kinetic energies of the reaction products is

 T 3

 T 4

thres

 1  M

2 3

 M 4

 v 2 (21.7)

Since we have M 3 V 3 = M 4 V 4 from mom e ntum conservatio n, we c a n s a y i n ge ne ra l

1 1  M V  2

Q  T  M V 2  3 3 (21.8)

i 2 3 3

2 M 4

W ith Q and T 1 given, we can find V 3 from (21.8) but not the direction of V 3 . It turns o u t that f o r T 1 just above threshold of an endotherm ic reaction, an in teresting situation exists where at a certain scattering ang l e in LCS one can have two d ifferent k i netic en ergies in LCS. A situation which violates the one-t o-one correspondence betw een scattering angle and outgoing energy. How can this be? Th e ans w er is that the one-to - on e correspondence that we have spoken of in the past applies st ri c t l y onl y to the re lation between the kinetic energy T 3 and the scattering angle in CMCS (and not with the scattering angle in LCS). Fig. 21.2 shows how this spec ial situation, whic h correspon ds to the double-valued solution to the Q-equation, can arise.

v b

v b '

V '

b

V b

v 0

v 0



v a

Figure b y MIT OCW .

Fig. 21.2. A specia l con d ition whe r e a particle ca n be em itted at the sam e angle but with two different kinetic energies, which can occur only in LCS.

4

Energy-Level Diagrams for Nuclear Reactions

We have seen in the previous chapte r how the various en ergies involved in nuclear decay can be conveniently displaye d in an energy-lev e l diag ram . The sam e argum ent applies to nuclear r eactions. Fig. 21.3 shows the en ergies involved in an

Fig. 21.3 . Energy-level diagram for an endotherm ic reaction .

endotherm ic reaction. In this case the reaction can end up in two different states, depending on whether the product nucleus M 4 is in the ground state or in an excited state (*). T f denotes the kinetic energy of the reaction products in CMCS, whi c h one can write as

T f  Q  T i

= 1 M V 2  1 M V 2 (21.9)

2 3 3 2 4 4

Since both T i and T f can be considered kinetic energi es in CMCS, one can say that the kinetic energies appearing in the en ergy-level diagram should be in CMCS.

Compound Nucleus Reactions

The concept of com pound nucleus m odel for nuclear reactions is depicted in Fig.

21.4. The id ea is that an inciden t par t icle reac ts w ith th e ta rget nucleus in two ways, a

scattering th at tak e s place at th e surf ace of the n u cleus which is, properly speaking, not a reaction, an d a reaction that ta kes p l ace after the inciden t particle has en tered in to the nucleus. The for m er is what we have been st udy ing as elastic scatteri ng, it is also known

Fig. 21.4. Com pound nucleus m odel of nuclear reaction – form ation of compound nucleus (CN ) and its subsequent d ecay are assum e d to be decoupled.

as shape e l a s tic o r poten tia l sca tte rin g . This proc ess is a l way s presen t in that it is a llo wed under any circum stances , we will leave it aside f o r a while in the followin g discuss i on .

The inte raction which ta kes plac e af ter the p a rticle has pene tr ated in to the targe t nucle us can be considered an absorption process, l eading to the form ati on of a compound nucleus (this n eed n o t be the on ly process po ssible, the o t hers can b e direc t in tera ction, m u ltip le collisions, a nd colle ctiv e excitations , etc.). Th is is the p a rt th at we will n o w consider briefly.

In neutron reactions the for m ation of com pound nucleus (CN ) is quite lik ely at incident energies of ~ 0.1 – 1 Mev. Physi cally this correspo nds to a larg e reflection coefficient in the ins i de edge of the potential we ll. Once CN is f o rm ed it is assum e d that it will decay in a m a nner that is independent of the m ode in which it was for m ed (com plete lo ss of m e m o r y ). This is the ba sic ass u m p tion of the m odel because one can then treat th e for m ation and decay as two sepa rate process e s. The approx im ation of two processe s be ing indepen d ent of each other is exp r essed by writing the in te raction as a two-stage reaction,

a  X  C *  b  Y

the as ter i sk indicating th at the CN is in an excited state. The first arrow denotes th e for m ation stage and the second the d ecay stag e. For this reacti on th e cross section

 ( a , b ) m a y be written as

 ( a , b )   C ( T i ) P b ( E ) (21.10)

where  C ( T i ) is th e cross se ctio n f o r the CN f o rm ation a t kine tic en e r gy T i , which is the available k i netic energy for reaction as discussed above, and P b (E) is the probability that the CN at en ergy lev e l E will dec a y b y em issi on of partic le b. It is im plied that  C and

P b are to b e evaluated separately s i nce the fo rm ation and decay proce sses are decoup led. The energy-level diagram for this reaction is shown in Fig. 21.5 for an endotherm ic reac tion (Q < 0). Notic e that E is the CN excitation and it is m easured relative to

Fig. 21.5 . Energy-level diagram for the reaction a + X  b + Y via CN form ation and decay.

the rest-m ass energy of the nucleus (a+X). If this nucleus should have an excited state (a virtu a l level) at E* whic h is clos e to E, th en one can have a resonance condition. If the incom i ng particle a should have a kinetic ener gy such that th e kinetic energy available for reaction has the value T i *, then the C N excita tion energy m a tches an exc i ted lev e l of the

nucleus (a+X), E = E*. Therefore the CN for m at ion cross section  C ( T i ) will sh ow a peak in its v a ria tion with T i as an ind i cation of a resonance reaction.

The condition for a reaction resonance is thus a relation between the incom i ng kinetic energy and the rest-m ass energies of the reactan ts. Fig. 21.5 shows that this relation can be stated as Ti = Ti*, or

( M a

 M X

) c 2  T *  M

a  X

c 2  E * (21.11)

Each virtual level E* has a cer tain energy width, denoted as  , which corresponds to a f i nite lif e tim e of the sta t e (leve l ),   ħ /  . The sm aller the width m e ans the long er the lif etim e of the leve l.

The cross section for CN for m ation has to be calculated qu antum m echanically [see, for example, Burcham , Nuclear Physics , p. 532, or for a com p lete treatm e nt Blatt and W e isskopf, Theoretical Nuclear Physics , pp. 398]. One finds

 ( T )   D 2 g

J  T

 a 

 T *  2   2 / 4

(21.12)

where g J   2 I

2 J  1

 1   2 I X

 1  and J  I a  I X  L a . In this expression D is the

reduced wavelength (wavelength/2 ) of partic le a in CMCS, J is the total angular mom e ntum , the sum of t h e spins of particle s a and X and the orbital angular m o m e ntum associa t ed w ith pa rticle a (rec a ll p a rticle X is s t ationary),  a is th e energy width (partial width) for the incom i ng channel a+X, and  (witho u t any index ) is th e to tal decay width, the sum of all pa rtial wid t hs. The ide a here is tha t CN f o rm ation can resu lt f r om a num b er of channels, each with its own partia l width. In our case th e cha nnel is reac tion with par tic le a, and the p a rtial wid t h  a is a m easure of the stren g th of this c h annel.

Given our relation (21.11) we can also regard the CN for m ation cross section to be a

function of the excitatio n energy E, in which cas e  C ( E ) is given by (21.12) with

 E  E *  2 replacing the factor  T  T *  2 in the denom inator.

To com p lete the cros s section exp r es sion (21.10) we need to s p ecify the probability f o r the de cay of the com p ound nucle u s . This is a m a tter that involves th e excitation e n ergy E and the decay ch annel w h er e particle b is em itted. Treating this process like radioactiv e decay, we can say

P b ( E )   b ( E ) /  ( E ) (21.13)

where  ( E )   a ( E )   b ( E ) + wid t h of any oth e r decay channel allowed by the energe tics a nd selec tive rules. Typ i c a lly one includes a radiation partial width   since gamma e m ission is usually an allowed process. Com b ining (21.12) and (21.13) we have the cros s section for a res onance reaction. In neu t ron rea c tion theory this r e sult is generally known as the Breit-W i gne r form ula for a sing le res onance. Th ere are two cross sections of interest to us, one f o r ne utron absorption and anot her for neutron elastic scattering. They are us ually written as

 ( n ,  )   D 2 g

J  T

 n   (21.14)

 T *  2   2 / 4

  2

 T  T * 

J  T  T *  2   2 / 4

J n  T

 T *  2   2 / 4

In  ( n , n ) the firs t term is the potential s catteri ng contribution, what we had previously called the s - wave part of elas tic s cattering, with a being the scattering leng th. The seco nd term in (21.15) is the compound el astic scattering contribution. It is the term responsible for the peak behavior of the cross section. The las t term represents th e interference between potential scattering and resonant sca tte ring. N o tice the interferen ce is destructive at energy b e low the reson a nce and constructive above the resonance. Below

w e w ill see that the s e ar e charac ter i stic sign atur es of the prese n ce of interf erence in a resonance reaction. In F i g. 21.6 we show schem a tically the energy behavior of the absorption cross section in the form of a resonan ce peak. Below the peak the cros s section varies like 1/v as can be deduced from (21.14) by noting the energy dependence

of the various fact ors, along with  n ~ , an d   ~ constan t . N o tice a l so the f u ll

width at half m a xim u m i s governed b y the to tal d ecay width  . Fig. 21.7 shows a well- known absorption peak in Cd which is wide ly used as an absorber of low-energy neutrons. O n e can see the resonance behavior in both the to tal c r oss se ction, w h ich is dom i nated by absorption, and the elastic s catter i n g cross se ction in the ins e t.

Fig. 21.6 . Schem a tic of Breit-W i gne r resonan ce behavior for neutron absorption.

10 4

10 3

10 2

10

1

0.001

0.01

0.1 1 10 100

Neutron Lab. Ener gy T n , ev

F i g u r e b y M I T O C W .

Fig. 21.7. Total and elas tic neu t r on scattering cross sections of Cd showing a resonant absorption peak and a resonant scattering peak, respectively.

W e conclude our brief discussion of com pound nucleus reactions by returning to the feature of constructive and destructive interference between pot ential scattering an d resonance scattering in th e elas tic scattering cross section. Fig. 21.8 shows this behavior schem a tically, and Fig. 21.9 shows that such effects are indeed observed [J. E. Lynn, Th e Theory of Neutron Resonance Reactions (Cla ren don Press, Oxford, 1968]. Adm ittedly this featu r e is not always seen in the data ; the present exam ple is carefully chosen an d should not be taken as being a typical situation.

Fig. 21.8. Interferen ce effects in elastic n e utron scattering, below and above the resonance.

10 2

10

1.0

0.1

0

10 20

E (keV)

30 40 50

Figure b y MIT OCW .

Fig. 21.9 . Experim e ntal scattering cross section of Al27 showing the interference effects between potential and resonan ce scattering, and an asym ptotically constant value (poten tia l sc atte ring ) suf f i ci ently far away from the resonan c e. (from Lynn)