NU EJ\ R R ~ J\ 1'1 NS P A 1'R PHY Prin 'iplcs, alculalion and App licalions of

Low-Energy Reactions

Nil ' Iear pro esses in stars produce the chemical elements for planets and life . Thi s hook shows how similar processes may be reproduced in laboratories using exotic ileall1s, ancl how these results can be analyzed.

B 'g inning with one-channel scattering theory, the book builds up to complex 1\ ':I~' tions within a multi-channel framework. It describes both direct and compound I \':ll'l ions, making the connections to astrophysics. A variety of theories are covered III (il-Iai l, includi ng the adiabatic model and the CDCC method for breakup, eikonal Iliod ' Is ror stripping, R-matrix techniques, and the Hauser-Feshbach theory for \ IlIlipound nucleus reactions.

I'm '1 ic~tI app lications are prominent in this book, confronting theory predictions \\ II It data throughout. The associated reaction program FRESCO is described, dlowillg readers to apply the methods to practical cases. Each chapter ends \\ lilt ' ' rcises so that readers can test their understanding of the materials 111\\ I 'd. upplementary materials at www.cambridge.org/9780521856355 include Ilh' FRI \S('O program, input and output files for the examples given in the book, and

hllll ~ and graphs related to the exercises.

I N.I . TIIOMPSON is a Nuclear Physicist in the Nuclear Theory and Modeling Group II Illl' L;lwrence Livelmore National Laboratory, USA, having been Professor of I'll ' Il' S at the Univers ity of Surrey, UK, until 2006. His research deals with coupled-I 1t,1I111 'Is and few- body models for nuclear structure and reactions , especially I Illh ' 111 i ng ha lo nuclei. He is a Fellow of the Institute of Physics.

I 1I111o. 11 NA M. NUNES is an Associate Professor in the Department of Physics . lIld slronomy, and at the National Superconducting Cyclotron Laboratory, of 1\ 11 \ Itl 'an State U niversity. Her research has focused mainly on direct nuclear II ,1\ IIIlI1S

NUL A rATIONS FOR ASTROPHYSICS

Princ iples, Calculation and Applications of Low-Energy Reactions

IAN J. THOMPSON Lawrence Liverm.ore National Laboratory, USA al1d

Un.iversity of Surrey, UK and

FILOMENA M . NUNES National Superconciucting Cyclotron. LaboratOlY,

Mich.igan StClte Un.iversity, USA

CAMBRIDGE UNIVgH SI 'I'Y I'ltESS

I i\~11 1 1 1 11l11i I IN I VI' I(,~ 11 \ I' I!I 'SS ( 'lI llI il , id ' " N\'w V," I" M,' 11 111I II 'lI' , Mlldl ld, ( ' lI p~ TUWII , SI Il I'l 'lHII V, Sil l I 1'11,,111, 1)1'1 1"

( 'll lll1l , llIg' LJ"lv ' I'si ly PI' 'ss 'J' II ~ Hdlllilul'gl, lI11ilLl illg, 'II 111 bri Ige 132 HRU, U

Published in Ihe nil ' d Slnl 'S or A merica by ambridge ni vcrsily p",;ss, New York

www,cambridge,org In I'ormal ion on Ih is I itle: www.cambridge.org/97 052 1856355

I. J. Thompson and F. M. Nunes 2009

Th is publication is in copyri ght. Subject to statutory exception an Ito the provisions of relevant co llective licensing agreements,

no reproduction of any part may take place WiUlout the written permi ss ion of Cambridge U niversity Press.

First published 2009

Prin tcd in the United Kingdom at the University Press, Cambridge

/I ("O IO/OK record for Ihis publication is available from the British. Library

ISBN 978-0-521-85635-5 hardback

Cll mh, iuge ni vcrsily Press has no responsibility for the persistence or 1I('\'III'lIl'y Ill' UI L~ 1'0 1' ex ternaJ or third-party internet websites referred to

III lili , publ ica tion, and does not guarantee that any content on such W 'hsilCS is, or wi ll remain , accurate or appropriate.

" " /1/( '1' ~'I/II( t' ,I' (l quotations 1,/,//(/\\'/edRements I Nu ' Ici in the Cosmos

1,1 Nuclei I , Primordial nucLeosynthesis I , \ Reactions in light stars 1,,1 Ileavy stars I , Explos ive production mechanisms 1.(, utlook I{vil ,tions of nuclei ) I Kinds of state and reactions 1,2 Time and energy scales ), \ II is ions ) I ross ection S 'nI l 'ring theory I I b lastic cattering from spherical potentials I, ) Multi -channel catteri ng I, I 1 III gral forms 1,1 Idcnti 'al particles

1 ~ I 'ctromagnetic channels I~ (':lV I ion 111 chan isl11S I I Optica l potentials I 1 Si n ,I ' -nucl n binding potentials I. I 'oup ling potcntials I I In ' Iastie couplings I. Pari i ' I ' r ' arrang mcnts

v

page IX XI I

XIII

I 6 9

17 21 26 28 28 31 34 42 48 48 75 91

105 ] 14 129 129 132 137 14 1 ISO

vi

5

I)

10

II

( '(111 11' 111,\'

,.6 Isospin transitioll s 4.7 I holo-nuclcl.lr coullings

onn cling structure with reactions 5. 1 Summary of tructure models

.2 Folded potentials 5.3 Overlap functions 5.4 General matrix elements Solving the equations 6. I Cia. lic cattering 6.2 lass ifications 6.. Multi -channel equations 6.4 Multi -channel bound states 6.5 I -matrix methods () .() oupl d asymptotic wave functions Appro inwI' solutions 1. 1 Pl'W body adiabatic scattering I ... Hikllilal m ' (hods I \ Pi rsl Ol't! 'I' s '111 iclassical approximation I I W I II approxi l11ali on 1IIl'i il lip K I 'I'hl'l' ' hody way cq uations H. ) ( '() lItilllllllll I iscr tized Coupled Channel method H, \ Olll 'I' hI' 'akup measures and methods ' l 'hlt'V hody nuclei 9, 1 J) 'fi nitions of halo and deeply bound states () ._ Thrc -body models for bound states 9.. Three-body continuum 9.4 Reaction with three-body projectiles R-malrix phenomenology 10. 1 R-matrix parameters 10.2 10 .. In.

ing le-channel R matrix oupled-channels R matrix ombining li rect and re onant contributions

'oll1pound-nuclcus av raging 11 . 1 oillpound-nuci us ph nOll1ena 11 ._ Approximations n 'g l 'cting int rfer n 11 .1 11 :IIISl' r JI 'shhii 'h ll1od ' ls II . I ] .l'w l (il'llsit iL's II .. V(' 1'1I1'l' IIllIplillldl'S li nd til' opti 'n l mod ' I

158 161 173 173 186 190 196 200 200 202 204 215 217 226 229 229 237 245 247 254 254 259 270 274 274 276 285 287 296 296 297 305 3 11 314 314 3 17 318 327 329

11111/ '111 \

I ' ,'kllnl' I' 'n 'Iioll rates :llld Il ,twllr'ks I ). I Til 'l' ll llti it V Ta Ji ll J I ), ) R ' :\ tioll 11 'I works I ). , 1\(Juilibri a I I S 'Ilsitiv iti s t nuc l ar data

I \ ( 'llIlIl' 'I ion 10 'x periment I 1. 1 N 'W acc lerators and their methods I I, I ' I Lion I I I I i r' t m asurements

I I , ' ]I 'l' 11'()S 'opy I I, I Trt1ns~ I' spectroscopy I I. I no 'kout spectro copy I I. \ In ' Ias ti spectroscopy I 1, 1 Hr'akul spectroscopy I I ' hal') - xchange spectroscopy

I I 1' l1tlllF dala I I X measures

]:ilting cr ss-section harmonic multipoles Jiil1in ~ optical potentials Multi - hannel fitting S 'ar 'hing

1'1" Iltil A Symbols 1'1 " IIliI II "tting started with FRESCO

\, // , 1IIIhliogmfJlly /11,1, \

VII

,0 . 40 350 . 54

60 64

364 367 373 379 379 390 3 4 3 6 398 40 403 409 410 4J2 414 4 19 433 455 457

II \ 01 II rniny day in Dec mb I' and we were sitting in an office at the Nuclear 1'11\ II ., ( ' ' III 'r in Lisbon deeply involved in a heated di scussion about the openin ) ,01 IIII hoo!.. . Should we fo llow the standard practice, or should we paint th bi 1'" 11111' '/ '1'111 ' 10 our main motivation, after hours we finally agreed.

1111 illIlIWI) fascination for a clear starry sky is timeless. It has been around since 1111 I I, I dllYs of mankind and includes the most diverse cultures. Only in the la t " 111111 , 1111 '1 'ar physics has started to make a very important contribution to our 1111111 I lillldil1 1 or thes I henomena in the sky. And until the present day, many lot i' I (III ., llons conn t d to iluc lear reactions remain to be answered. One of the 1'111111 I al\1pl's li sted among t the eleven most impOltant physics questions for 11111 11111111 is this: ' How and where are the heavy elements produced?' .

'11v ,I!lolh'r book? For decades we have come across colleagues, including I I II 1llIll' lIla li sts, who would like to learn more about reactions. Some have become 11111 III III rllnning reacti on codes, but cannot find a book at the right level to 11,1111 IIII' III 'ory ass ciated with the calculations they are performing. Probab ly 1111 1.t1 ~'1 sl push toward embarking on the adventure of writing this book came after I I I,d Vl'lIl'S of leaching reaction theory to graduate students. The reference nuclear

II ,II 11,,11 I looks ha ve been arou nd for decades, and even though there have been some 1111111 1\ 11'I11l'fforts, nowhere could we find the appropriate level , detail, connection III 1111 PII 'SI' III 'xperimenlal scene, the guiding motivation of astrophysics, and the l "lito IIllllilSisl ' nl with that motivation. So, five years ago, we convinced ourselves 1111 \ ,I Mil" ' Ihin 1 worth doing.

\ III, h I It ' book fo r? T his book is primarily directed to physics graduate students 1111 III 1111 'r'sl in nU '1 ar physics and astrophysics. It should serve as a practical

'1IIIh 1III'xl1 'rim ' ntali sls that need a better understanding of the reaction theorieil I 1II.Ihll' 101' Ih ' various pI' c sses. We hope it an al 0 be a useful reference book

1111 1111 I (lvll s in lh malt ' r.

What is di!'!' ' I' ' lit about th is hook ' II '\)l ll ui Ii S III ' stlilidard dir 'CI I' ' :I ' li on 111 'ory starlin) I'rom th ' two-botly s 'all ' rin ' prob l ' Ill bul , I'ath ' I' than ' pantlin I toward th 'ori 'S that hav ' nOI b n impl m ' nl ' tI , il 1'0 'us s o n those lhat arc in us or are b 'ing d ' ve loped. W have tr ied to PI' s nl a ll derivat io ns, 0 that it is easier for the slud ' nt to 1'01 low. We have a lso tried to make clear the limi ts of app licability of sp 'e ifi ' m de ls, and to how examples that can be directly compared with data.

Il ow is the book organized? The first two chapters were written at an introductory I 'v I, wh re the ,tage of nuclear astrophysics is set and the basic defi nitions are i nl rodu 'cd . Nexllhere are eighty pages of solid scattering theory, which is by far the bi , 1 ' st hurdle astudent will have to overcome. This is the central theory component, 10) ' Ih ' I' with the next two chapters on coupling potentials and structure models. W ' ha y ' provided a chapte r on the most common approximations used in this field, MOl" ad van 'cd chapte rs then cover specific types of reactions. And eventually we Iwi n 'th I' ' ad I' back t astrophysics, introducing the reaction rates into reaction Il l' lwork s in Slilrs and xplos ive environments.

'1'111011 ,1I0ul I h ' hook , as the various reaction mechanism s are discussed, we IlItl vidl' s ]ll'l' ill ' l' 1I1llpl 's o r re levance to astrophysics and connect back to the I IHIP" s ll'n l Sl' ' Iwri os s ,t in our first chapter.

111 .lIldlll tl ll III III ' astrophys ics motivation, we have kept in mind a strong I 1111111 I III III 10 I' X Pl' l i "Ic n!. II ' I' , ca lculations are important, so there is a chapter Ii , ilil 111 ' 11 III 1111111 ,Ii 'ill m ' thods. Data is important, so there is a chapter on I 111' 111111 ' 111 11 1 d 'llIil s, Alld Ih comparison between theory and experiment is IIlIp11l111l1 , IIIlI s Ih ' 'hapl I' on fitting data.

lIollIl' l l'SS ' Ill ia I 'ompon nt of thi s project is the assisted hands-on experience. Til , hook 'OllleS with a reaction code (FRESCO), and for many examples addressed inlll ' book W ' provide the inputs to the reaction code so that the readers can perform 111 ' 'al 'ulati on by themse lves. An appendix for 'Getting started with FRESCO' is also provid '(I.

What is le ft out? AJthough we expanded on the number of pages significantly, il is cI ' ar that thi s book does not cover everything that could be contained in such !llill ' as ' Nu lear Reactions for Astrophysics .' From the start, our decision was to I'll ' li S on d ir t reactions, and leave out the whole area on central collisions and the sP ' ' iI i

Source of quotation

'haptcr I London: Guardian (23 August 2001) ; Chapter 2 Pierre Curie with Alllohiographical Notes, translated by Charlotte and Vernon Kellogg, New York: Mil 'm i Il an ( 1923), p. 167; Chapter 3 Statement of 1963, as quoted in Schrodinger: I ,i/i' (flld 'rtlOlIghl by Walter J . Moore, Cambridge: Cambridge University Press (I'!)!) ). p, I; hapter 4 A Dictionary of Scientific Quotations by Alan Lindsay MIIl' kli , I3ri s tol : Institute of Physics Publishing (1991), p. 35; Chapter 5 'How NIIlll'l Plil'. winncrs Get That Way ' (December 1969) by Mitchell Wilson, WlI l> hi 1I~'loll : The Allantic; Chapter 6 Nature 403, 345 (27 January 2000), said when ... llt lWIl Ill' r'sull s of a large quantum mechanics calculation; Chapter 7 private l illll llllilli ';I li on, Michigan, August 2008; Chapter 8 Brighter Than a Thousand S IIII .I': A p('J'.I'o//o/ History of the Atomic Scientists , by Robert Jungk, translated by .Il1l11 'S Icugh, New York: Harcourt Brace (1958), p. 22; Chapter 9 letter to h ' I' brothcr; C hapter 10 Communications in Pure and Applied Mathematics 13 ( 1959) I ; C hapter 11 Lise Meitner: A Life in Physics, by Ruth Lewin Syme, H ' rk ' Icy and Los Angeles: University of California Press (1997), p. 375; Chapter 12 ' /\ IiI" in phys ics': Evening Lecture at the International Center for Theoretical Phys ics, Tricste, Italy, supplement of the IAEA Bulletin (1968), 24; Chapter 13 I iCliOIl Clt)l of Scientific Quotations by Alan Lindsay Mackay, Bristol: Institute or Physics Publi . hing ( 1991); Chapter 14 Nuclear Principles in Engineering by Taljana Jcv remov ic, New York: Springer (2005), p. 397; Chapter 15 'Physics and Philosophy : T he Revolution in Modern Science,' Lectures delivered at University or SI. Andrews, cotland, Winte r 1955-56.

11111'1 Ill l' li sl or p op le that, in one form or another, made this book possibl . V I I III wi lh ac know ledging a ll those who provided figures for the book: Thomas

11111111111111 , Harry Davi Is, Erich Ormand , Marc Hausmann , Neil Summers, J n \ IIIIIII ~' W' also thank Ruth Syme for he lp with the quotations.

I'll 1IIIIIIlilry v ' rsions or the book were di stributed to a few experts in the summ I' III '01) / ' I 'Ill' COll1ments w got back were impo rtant to correct and improve the jill I IILIII!)I!. W thank Goran Arbanas, Edward Brown, Raquel Crespo, Jutta 1 I Iii I, flllIll" 1 i'trich, h-ristian Forssen, Alexandra Gade, David Howell , Ron 1111111 1111 , 11Ioni( M ro, Petr Navratil , Jorge Pereira, Sofia Quaglioni , Hendrik , I h.II , IHIi' ' as. chi lI e r, Andrew Steiner, Paul Stevenson, Michael Thoennes en, II II 1" ... lni ll , I ' mco Zegers , and Michael Zhukov. We will forever be in debt Lo 1 III IIl lI la, WIH) wenL Lhrough ha lf of the chapte rs with a magnifying glass and

", , I IIHllill ' J1ts h Iped v ry much in bringing the language to the right leve l. I1II hllilk was origimlily to be written by tlu'ee authors . Although in the end , Ana

I 1111 I I II lid 1101 h i 11 V I ved in the actual writing, we would like to thank her for a II 1111 I 1111I1I ~ III S Ill , th ' many di cus ions, and shaping the content to be included.

\II I

I NU'I'ilnli1 ' 'OS Il1 S

There is a coherent pl an in the universe, though I don 't know what it' s a plan for.

Fred Hoyle

III III d 'J" to understand about the composition of stars and how they produce energy, \ I' II' 'd to know about nuclei, and about the reactions which they undergo. This I lI.lpl ' r provides an in troduction to the description of nuclei , and surveys the rang It! ~lT n a rios in which important reactions occur. We begin with the Big Bang, then iii ,' !ISS energy production ycles in stru's, and finish with an outline of some of the 111 111' 'ss 'S by wh ich we thi nk that heavy elements are produced in supernovae and 111111'1 slellar enviroments. The more detailed discussion of nuclear physics begins III ( 'hapter 2, to which the more advanced student is directed.

1.1 Nuclei 1.1.1 Properties of nuclei

I ,II 11 isotope (A, Z), characterized by mass number A and chru'geZ, has in its ground 1111 ,' a rest mass mA ,Z . This total mass is less than the sum of the masses of th

I nll,1 ilucnt protons and neutrons due to the binding energy of the system. Energy is II II ilSl'd when the bound state is formed . The binding energy may be calcu lated by

( 1.1.1 ) .llId is Ih' energy required to break up the nucleus into its A constituent nucleons. 1111 ' IIlImber or neutrons is N = A - Z. A unit atorrUc mass ( I u) has rest energy

II/I I . 1.494 Me Y. I'll ' hilldillR energy per nucleon B/A diclates whether energy must be suppli cI

III will h ' r -I ' ''sed in th ru sion of two nu '1 ' i Lo form Lhei r compos ite. The values or 1/( \,Z )/;\a r 'shown in Pig. 1.1 foralllh ' I()I) 1 li v'disotop s. The largerth en r y 11/11 ' Ill ' 'ds 10 suppl y 10 r I 'as ' a 1111 ' koll , 11l l' Illorl' SLah l is th nu ' Ieus, Th most

Nllclei ill 'Ii (' (J ,\'I/I f1.1' 10 , , , , , 1

10 6Fe

> 1\lr

'I Nlll'il'i ill IIII' CO,I'III II,I

1'0 1" S slill hav il11l orla nl 1'01 's 10 play in nu I 'a rastl'Op ll si ... '1''' ,,' II 'I ' fJ'()/IIO J,: IP fi . r I' is r sp nsible ror Ih oul omb repulsion b ' lw ' ' 11 pmlnns ill 1111 ' I ' i, and the de 'r as in binding for heavy nuclei seen in Fig. 1. 1. 'I'll . w('ok illt ' raclion plays a ro le whenever reactions involve neutrinos; we will see so me examples of this laler in this chapter (Eqs. (1.2.1) and (1.2.4)). The gravitational attraction is not signi ficant inside nuclei, but is responsible for creating galaxies and stars in the first place, and then compressing them to the stage where nuclear reactions begin.

1.1.4 The Coulomb barrier In order that a nuclear reaction takes place, the nuclei involved have to be close 10 each other, but this is hindered by the Coulomb repulsion between the protons, which acts at longer distances compared with the nuclear force of short range. The verall potential energy between two charged nuclei separated by a distance R th 'r rore follows the pattern shown in Fig. 1.3. There is a repulsive Coulomb barrier or Il'ighl VB, and scattering at energies E < VB still exists because of quantum tU1I11 ,lin through the barrier.

'rh ' 'xponential reduction of reaction rates for charged particles reacting at low I' lalive energies will be extremely important in all astrophysical scenarios,

:;-Q)

6 :Iii C

~ 0...

20 ~----~----~------~-----r----------~

10

0

- 10

-20

-30

- 40

-- ... ...

...

... ...

"'- .. 1/R Coulomb tail

..................... "t ... ...... ;';;._ .- .- .- .-.-.-. /

_/

/ /

R1 /

I I

I I

- - Coulomb - - Nuclear only -- Nuclear + Coulomb

- 50 L-____ ~ ____ -L ____ ~ ______ L-____ ~ ____ ~ o 5 10 15

Radius of separation R (1m)

I ,'i ' . I . . Th' Jlu ' l 'ar fi nd ouloillb polential energi s between a prOlon and 40Ca as II run ti on or tit , di stan " R I ' twe' ll th ir C'll t rs, wh ' r I?I is the radius or 110 II. Th ' 'o l11hill ' I pot 'nti ;iI (solid lin ,) has tI m:lx imum h ' i "1 t or VII , ronning til ' C'Oli lOl llh hll l'l'i 'I',

I I NlII/"1

111.1 \ III V' I on'll h ' Ill, lililitill )' I'lI l ' lIlI I'll! ' IlU ' I 'ar 1"" ti )lIS. W' wi ll s" I "Itll ll ) I) thnt r'll ' lion rat 's un: ddillnl hy Ih ' quan til y a, " II 'c.I lh e/"Oss

,11"/1 Ikva ll Nl' cross s' ' li ) II S 0 (I~' ) drop rapidly with de I' as ing 'cnl 1'- 0 1'-1111 I IH' I!, I ~', dll' to th ' ou lornb rcp " sion , we raclori ze out a simile energy 01 , I" Illh IIC(': I 'cord in ) to

1 a(E) = _ e- 27f 17S(E)

E ( 1.1.4)

'" 01, III II' .III (f.l'fl'Ol l /IY.I' i ca l S-Ja tor SeE) which hould vary less strongly with II'I '\' '1'11(' I I I~' "ol11ctri ca l factor i associated with the wavelength of the

III' llllIlli )' p:llti ' l " and the exponenti al factor repre ents the penetrability through tli (lll1lll1l1h haiTi 'I'. It depends on 1'} , the Sommerfeld parameter, defi ned as 'I I ,(,'lUlu) (Eq. ( . 1.71)) where Z,Z2e2 is the product of charges and v lh ' I III' I 1I1l ' lIkll l v ' Ioc ily. In Fig. 1.4 we show, in the upper panel, the cross sec ti on 111 1 til 'v 1'lI pllll" on :l lie to synthesize 7Be. The reaction cross section fall s fT I '111011 I I ~ Iliv 'II 'rgy dccreases, whereas the S-factor, shown in the lower panel, is III III. Illll , ... IHIlt.

() ()

(1963) P. D. Parker, R. W. Kavanagh (1969) K. Nagatani , M. R. Dwarakanath , D. Ashery (1988) M. Hilgemeier et al. A (1969) P. D. Parker

500 En rgy (k V, c.m,)

1000 1500

I I I I I h'Pl' lId ' lI 'l' of'

1.2 Primordial 1I11l'1i:OI

1: n .... p 2: p(n ,y)d 3: d(p, y)3He 4: d(d,n)3He 5: d(d ,p)t 6: t(d ,n) O' 7: t( 0', y) 7Li 8: 3He(n,p)t 9: 3He(d,p)4He

10: 3He(O', y) 78e 11 : 7Li(p ,O')4He 12: 78e(n,p?Li

Nuclei ill 1/11' ( 'n ,11I1I11

n

Pig. 1.6. The dominant reactions in primordial nucleosynthesis, after Kawano [1].

7 minutes. Had these two numbers not properly matched, there would have been no l1ulrons to initiate the whole primordial nucleosynthesis.

By li me I ~ 250 s, the thermal energy E was 0.1 MeV, and all these primordial rL'a 'I ions came to a stop, except for the decays of neutrons, tritons and 7Be. Tit's' last three nuclei were produced in primordial nUcleosynthesis, but are not 11lL'l1lsclves stable, as they decay with lifetimes of 10.3 minutes, 12.3 years and 53 days, f'speclive ly, by weak interactions in what is called f3-decay:

n -+ p + e- + ve ,

t-+ 3He+e- +ve ,

7Be -+ 7Li + e+ + J)e. (1.2A)

Eventually, all the neutrons and radioactive nuclei transmuted into stable nuclei , such that only very small fractions of 7Li, and practically no 6Li remained. As a consequence, the initial composition of the Universe was almost entirely p, d, 3He, (II Ie, - , y particles and neutrinos . This primeval ratio of abundances, listed in Tab l 1.1, can still be observed if we avoid regions where further reactions have lak ' 11 place, uch as in low-metal stars.

V'ry rew nuclei heavier than helium are formed at this stage. One reason for Ihi s is thal there are no stable nuclei with 5 nucleons, nor with 8 nucleons. Th' long'st-lived isotopes with 5 nucleons are SHe and sLi, but these emit neutrons !Inti protons respectively. For element production there are thus bottlenecks at mass lIumb 'rs or 5 and 8 that had to be later bridged by other means.

V' ry lillie further happ ned until the Universe reached time I = 3.8 x 105 y, wh n Ih ' I ' l1lp ' ratur 'Inti l1 ' r'i 'S w I' I w enough (T "-' 4 x IOI K and E "-' OAeV) ror ' I ' 'Irons to r main boUl d 10 nu '1 ' i in aloills . Al Ihill poinl , III atom ic era SIUrl'(1. Arl'r I _ IO() , Sial'S and 'a laxi 's w' r ' rOIIIl'd . ,ivill' way 10 st ' llar

rrrm , II, 1/ ,I'mrs

' I'll hi . 1, 1, 1,I'o l n / liI ' fI/ III/rifll/ ('I 'S )'/Jiw lI /II';II/ ol'(/io / IIIII'/(,(I.I'VIII/I('.I';.I'I I , ( /t :/ill(' r1/J\lI /t l'.Ii'(/( 'I;OIl (~t llllcl;d ',\' ; 10 1/1( ' 1111111/)(' 1' (~/,o l/ IIIIe/eOIl .l'. '/'l/(, 11I1e/eoll 1IIIIIIher d ell .l';ly ;.1' 1111'11 , A, Y, (l "lle/eol/.I' ill 1"01 i SOIOp ' oj mos.I' Ai. NOl /lloliZfllioll i .l' Li i = I.

I ~ l)ll)pl' Nu ' Iidc fracti on Yi Nucleon frac Lion Xi II( 0.75 0.75 ' II d 2.44 x 10- 5 4.88 x 10- 5 ' 11 \' 1.0 x 10 5 3.0 X 10- 5 Ilk 0.062 0.248l "1 I 1.1 x 10- 14 6.6 X 10- 14 ' I j 4,9 x 10- 10 34.3 x 10- 14

III' 1II I ~ 11111I' 1o. i s , liv ' I1tually some star coll apsed, heated up, and completely new Ii (II 11I1l'lI'HI' I' '1 'lions t ok place. This was how many heavier nuclei were

111101111 I d l ' IIl' ~' pro ' ss 'S c ntinued to repeat themselves until the present day.

1.3 Reactions in light stars

III t I II lIlV 1'01'111'(( by gravitati onal attraction, their continued contraction 111111111 I lit ' 'ollslitu'nl gas s an I rai e their temperature. If the star has a mass 1111 I 01 11111111111111101' aboul 0.1 so lar masses (0. 1 Mo ), then the temperature ri ses I' I () I IO() K and the d nsity to p "-' 102 g cm- 3, and nuclear hydrogell

'"' 1/1/1 ' 11111 IHII. Th' rc leas or energy in the resulting nuclear reactions is 111110 II III It) ~ Iop rUl'lh ' r gravitational co ll apse, ruld the star remains in a phas' II 11 dl" 11111 ('qlli lihrium. The compressive grav itational pressure is balanced by It I 11,1 11 I " r as pI' ssur ' or material heated by the nucleru' reactions. Dirrerenl 11111/11 ,,11.11 IIlIISS'S , iv' ris , in thi s phase, to the range of main equenc III 111111 "'I' lllv

I () NII I'/ l' i i ll 1/1(' ( '11.1'111 11,1

10000 ~ ~ ~ 100 ~4?.qltv ~ c S~Q ~ :::J Co ~~I\I '0 C'~ ~ ~

100R 0

~ ~ Sun 'iii 0 S~ ~ c 'E efqf) , :::J 0.01 S /qly -l

0.001

30000 10000 6000 3000 Surface temperature (K)

Fi ' . 1.7. II-R diagram: main sequence stars are represented by the thick curve, wh 'r-as the thin lines represent Stefan's law. Figure courtesy of Jon Whiting.

IY pal'li 'I- (a 4He nucleus) , along with two positrons (2 e+), two neutrinos (2 li) , lIlId ~ ) M Vol' released energy. It does not do this in one step, but via a chain that sl:lIls with

p + p ---+ d + e + + lie (Q = 1.44 MeV) . (1.3.1)

Thi s reaction, involving neutrinos, proceeds by the weak interaction, and has a very low reaction rate. In fact, its rate is so low that it has never been measured directly. The slowness of this initial step is what is responsible for the long lifetime of stars in their hydrogen-burning phase.

oll owing the formation of the deuteron d e H), a subsequent proton capture reaction

d + P ---+ 3He + y (Q = 5.49 MeV) (1.3.2)

may readily occur. The reaction d + d ---+ 4He + y may also occur, but is less likely since protons are much more abundant than deuterons at thi s stage: about 1 deuteron for every 10 18 protons.

A s eoncl proton capture on 3He cannot succeed because 4Li is unbound, but (ll h I' poss ible reactions in vo lving 3He are (in Chain I):

4Hc+2p + y (Q = 12.86 MeV) , (1.3.3) or

111 , I d II ' 1.59 M ' V) . ( I. .4)

/ . I I\ t 'llt 1/11//1/11 /1 ,1' ;'1 ,lIlli ',\' II

1111 11k do 's 11 0 1 lusl Ion ' , hU I produl"~ Iwo (y p

Nlll'iei ill lit I' ( '/1,\'11111,\

I '

el IV 11111 II I (I II 0) I'k I (I ) 0)

80 K v nogll ( I Vaugn (1 970) ... Wiezorek (1977) ... Filippone (1983) ~ Hammache (1988)

-? 60 + Hass (1999) > x Hammache (2000) ~ o Junghans 0

It"Lj -- Fit to Filippone U ~ 40 l en

20

o L-~~~~~~~~-L~-L~-L~-L~~~ o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

p_7Se energy (MeV, em)

Pi ' . 1.9. Measured astrophysical S-factors S (E) for the 7Be(p,y )8B reaction , with Iii ' I I r'sonance at 640 keY. The astrophysical rate is needed at ~ 20 keY, requiring downward 'x trapolations from the experimental energies. The curve is an El direct l'npillr' model combined with an MI resonance in a hybrid R-matrix treatment, lill 'd 10 III Fil ippone data.

Many orthese reactions have been measured in the laboratory, all the way down to ' ncrgies re levant for the stellar environment (E ~ 0.1 MeV). An example is shown in Fig. 1.4 for 4HeeHe,y)7Be: cross sections decreasing exponentially at small 'ncrgie ::u-e shown in the upper half and S-factors, nearly constant with energy, are shown in the lower half. As mentioned before, the cross section for fusions of charg d particles is strongly dependent on energy because of the Coulomb barrier. Most of the energy dependence is given by the e- 2rr 1) / E factor in Eg. (1.1.4), which wi ll be derived in Chapter 7.

or the 7Be(p,y)8B reaction, the S-factor is only constant away from the I ' r sonance at 640 keY that is prominent in Fig. 1.9. Reaction theory will be needed 10 d scribe how these resonances are superimposed upon the smoother non-resonanl background cross ections. In addition, different data sets shown in Fig. 1.9 hav dirh' nt normalization s at low energies, where the measurement is hardest.

I olhr act ions,4 Hee H ,y)7 Beand 7Be(p,y)8B ,are notimportant fortheenergy production r a star. llow vcr Ih yare c nnected to the amount of neutrinos ' 1Ilill'd f'rom Ih' star and ar' very important contribulions 10 th so lar neutrino 'x I" rin1'l1l s I. I.

I , I N t'ltt lillll \ ill ligltl .1'1/11'.1' 1.1

1.3.2 'I'ri"It,-

N I/('/!'i i ll Ih i' 'OSI//OS

whi 'h r leas s an energy = 7,27 M Y. Th n 't I' 'sult is .\( y I ) ', th ' triple-a 1":\ lion. lali stical equ ilibrium will be li scus ed in d ' tai l ill ' hapt ' r 12.

, trong ly connected to the triple-a reaction is the 12 (0' , y) 16 . Th rate of this r a 'lion, re lative to the 3a capture, determines the post-He burning C/O ratio, wh ich, in turn , affects the abundances of heavier elements produced in subsequent ph 'ls s. Al present, the C/O ratio is believed to playa crucial role in the last phases of a massive star. In particular, whether the final remnant following a supernova 'x plos ion is a neutron star or a black hole is affected by this ratio [6]. Because th ' . a reac tion is comparatively well known, the 12C(a, y) 160 reaction is still the mosl important source of uncertainty in the C~O ratio.

1.3.3 eNO cycles In SO Il1 stars there will be small fractions of carbon nuclei, either from the 3a r ';t ,t ion or fro m the remnants of earlier stars that have completed their evolutionary , ' I ' , II' th'se new stars are heavy enough (M> 1.5 M o ), then the internal Il' lllp ' mllil" is hi gh enough (T9 > 0.03) that there is another cycle that bums II I ii 0' 'I) in lo h lium, but proceeds at a faster rate. This is the CNO cycle illustrated III I,' i " I , ll , which uses the initial carbon as a catalyst: it is not consumed, but is I' ' , ' II 'ru t 'd at the end of the cycle, which proceeds as

12 (p,y) 13N (e+, v) 13C (p,y) 14N (p,y) ISO (e+, v) 15N (p,a) 12c.

T he initial Coulomb barrier in the p+12C reaction is higher than in the reactions or the pp chain but, if the temperature is high enough, this chain is faster because il does not involve the very slow p+p fusion reaction (1.3.1). The speed of the

NO cycle is still limited by weak interactions, namely a 10-minute lifetime for the ,B-decay of 13N, and 2 minutes for 150. The total energy released in one CNO

(p,,),) i Fi ' . 1. 11 . Th ' NO cY'1 'also 'O Il V ' rl s rOllr protons inl O on' 0' parli '1" whil Ihe inilial I ' is 1I t.: lIllil yslllial is I'" 'n ' rnl L;d al 111 ' ' nd oj' 111 ' 'y' I '.

10 I

10 2

12C( )' 3C p,,), gl

t

I , I N,'rI, fl, 11/ \ III I, ' If ,1'flll ',I'

'~"'I" ,,~roo .\ .......

. ~". ....

....

.. ..

..

.

+-'--1-'-'--+--+ I ;-.-+1 -

I ()

103

102

:;-Ol

101 ~ .ci E-O

10 U J2 ch

10- 1

10- 2

o

0

N/lclei ill '" l' '(lSII/O ,I'

GS Schrod r GS Luna to 6.18 MeV, Schroder .. to 6.79 MeV, Schroder ... to 6.79 MeV, Luna

,

2 Proton energy (MeV, lab)

(a)

338 17 resonance 1028 keV 1

1I't: I I

1:

I 1:

1:

resonance

1640 keY I resonance

Hebbard (1960) Rolfs & Rodney (1974)

0.5 1 1.5 Energy (MeV)

(b)

2 2.5

3

I'i . 1. 13. (a) The S-raclors for the cross sections of" 14N(p,y) ISO reaclion, to the ground slale, and 6. 18 MeV and 6.79 MeV exciled slale. of" 150. The curves are hybrid R-malrix lils discu, s d in;\pp ndix B. (b) The -f"a cl rf"onhe 15N(p,y)160 r'acli on.

I I " "tll ' \ 1111/ I

"0

1 (e ' + v)

I'i ' . 1.14. The e NO bi-cycle is a breakout from the original eNO cycle, and pmduce 16, 170 and 17F.

1 I

ddilionaJ cycles may also occur when a (p,y) reaction on 170 competes wilh IIH' (p,a) reaction shown in Fig. 1.14. This leads to a new cycle involving 18F an 1 I () (nol shown here) , which after a (p,a) reaction, returns to l5N in Fig. 1.11 and I I'tllual ly regenerates 12C for continued operation of the eNO cycles.

liv 'Illually no new nuclear reactions are able to produce enough energy to sta ll /11111 1\'1' lravitationaJ collapse. In light stars, less than about 8 solar masses, the sta r 1'1 1II IIIaily contracts into a dwarf star. Outer layers of the star are shed off while the I 1111' continues to contract uRder gravity. If the remaining core mass is less than 1.4

111111 Illasses, then it will compress to electron-degenerate matter, forming a white d" .Id ,

1.4 Heavy stars

III IH'avy stars (more than 8 Mo ), all of the above pp, triple-a and CNO cycles 111 \ III in early evolutionary stages. These cycles produce residues of carbon and II ~'I' 1I as before, but now there is sufficient gravitational pressure to compress and III .II III 'se res idues so that further transmutation reactions may occur: the averag ' 1IIIIIIwi ' nergy i sufficient to overcome the Coulomb batTiers between the reactin 1 11111 11'1. In this ca. e, some reaction chains occur which do not eventually cyc le back III 1'(" initiating the production of heavier elements.

1IIIII is section, we will discuss the main processes for heavy element producti n. I11 I 11111 rasllo the production of light elements, here the number of reactions involved I \ 1'1 large and one can no longer enumerate specific reactions. Instead, we wi II 1111 '1111011 Ihe main mechani sms and the a trophysicaJ sites in which these are mosl Iii I I 10 occur. poradically, we will provide spec ific examples.

/.4./ a-bll"";I1~ '1111,, (''Iu ' Ill r '(1 'lions wi lh 0' parli '1 's , 1'01' l'Xil lllPk' , produ ' hcavi r l1ucl i wilh N ,lIld / hOlh 'v 'n, nail) ' I 160, )ON' , -I M~' 111'111 1KS i, wlli '11 is lh dominant r 's idu '

IH Nlll'il'i ill 171(' 11.1'11111.1'

Hydrogen, Helium ---r-.,,--g

Helium, Nitrogen - -1:-..--"':""11-' Iron

Carbon, Oxygen, Neon --t-....,.... ....... f Silicon, Sulfur

Oxygen, Neon, Sodium

Iii " 1.1 5, Layers in a red giant star before explodjng in a supernova. Figure 'ourtcsy or JOIl Whiting.

(or ' ash') of th process. These reactions only occur when the temperature is high l' lHlll ,11 ; Ih ' ou lomb barriers involved are large because of the increased charge PlIltili 'Is %1%2. me neutrons are also produced by (a,n) reactions, giving rise to 1111 ' I ' i wi lh masses LhaL are not multiples of fouL

SliII'S go Ihrough a seq uence of carbon, oxygen, neon burning and so on, as 111 ' 1 ' llIP ' rulur ' in reases with progressively more gravitational contraction. We s ' , from Fi g. 1. 1, however, that less and less nuclear energy is released in these Sll .(; ' ssivc slages, so there is diminishing return of energy in this advanced burning, and lh ' stages pass progressively more quickly. Stars with these reactions, such as I' 'U g ianLs in advanced stages of evolution, are therefore less luminous. A diagram or lhe compo ition of a red giant is given in Pig . 1.15. Eventually, nuclei near 56Pe aI" produced (at the core of the massive star), and then no new nuclear reactions are ab l ' to produce enough energy to stall further gravitational collapse. This results in ;I r ' hound exp los ion where a new sequence of reactions can occur, to be discussed ill S (; tion 1.5.

1.4.2 s-process neutron reactions

I ~ x()lh Tlllic reactions (Q > 0) at temperatures just su ffi c ient to surpass their entrance 'oulomb barriers will never produce e leme nts heav i I' than 56Fe. Even at energies

ahov ' a ll the ul o lllb barri e rs, the general decrease f binding e n in Fig. 1. 1 for la r" A impli es, via th Saha quation , Ih al Ih pr bability f pI' ducing, say, ~()H ph in 'q uilihriul11 with 5(, 1' ., is 'x iI' 111 ' Iy low. Il owev r, many heav ie r ' 1' 111 ' Ills mu sl b ' I rodu ' 'd so nl 'w h '1" , us w' S ' Ih ' 111 in Ih ' Sun , a cording to

111 ' III ' aslll' ,tt so lar ahundan "S or Iii, . 1. 16. Th 'I" lIlu sl 111 ' I' 'ror' 'x isl si l s in

------,,'

, I I '

,.. 10 I "

" 10 2 )..

~ 10 3 10 4 10 5 ~ 10 6 til Sl 10 7

~ 10 8 Li 10 9 00 OJ B (.) 10 10 Be

10 11 10-12

10-13 0 20

I ,

40

. , I I , 1

alar abundances II s process r- process .. p- process

T

60 80 Proton number Z

------1')

Th

u 100

1'1)' I, I (1. Abu ndances of chemical elements in the Sun (percentages): observation IIlId 11l1' (;onlributions from three important nucleosynthesis processes.

1111 ( 'm ilIOS which have sufficiently high temperature to enable the endothermi I, ,I' 1IlliiS prod uc ing heavy elements, and neutrons should primarily drive the 1'"11 I ~,

III I 11illl':tsl to proton captures, neutron captures are not hindered at very I w III I ~"I '~, WC wi ll see late r that neutron cross sections rise at low energies as 1/'11 1111 1\ I.lli v' veloc ity v, and furthermore that the coefficient of V - I increases for I" ,II II I Illl ' le i. A sequence of neutron capture reactions may thus occur at modera l ' I. "Ipl "Ilur S generating nuclei as heavy as uranium. This is what is referred to as " \ 1'1 () , 'ss , the 's' for slow.

III Il'd l iants (Asymptotic Giant Branch stars) during their a-burning tag s, ,1".Illhlll :1I ncutrons may be produced by (a,n) reactions , for examp le on I3C or

'NI 'I'hl'se may be captured by seed nuclei in the iron group, and by progres ive ly III ,1\ 1\ I Ill! '1 ' i, but at a rate much lower than their ,B-decay rates . The mechani m r I' 1111 III .Ivy 'I m nt production in the s-process proceeds with (n,y) on each stabl 1111, hil S (%, N) , producing neutron-rich isotopes N + 1, N +2, ... Thi s g ives nuc le i 111.11 ,Ill' a r'w nucie ns away from stability, until a radioactive species ,B-decays 11\ I II 'l' llOn ' mission to the nex t lemenl (.7.+ I, NfJ - I ) in the Segre chart in an I IIIIII!\' wi lh th sam' mass . This is 'a ll d a hrollc/lillf{ point . Severa l branchin _ 1"IIIIIIIII ' I ' iar ' id nlifi ' dinPi J . 1.1 7. N 'w( n,y) 'll plur s

Fig. 1. 17 . Diagram illustrating the s-process with some key branching points .

1.0

:0-~0.5 ,..

c b

Mohr et al., 1997 o - Beer et al. , 1996

Kappeler et al., 1985

0.0 L-_---'--__ ~_--' __ __L __ ~ _ __' 0.0 0.1 0.2 0.3

Ec.m. (MeV)

Fig. 1. 18. Neutron capture reaction on 48Ca, from [7]. Reprinted with permission from P. Mohr etal., Phys. Rev C 56 (1997) 1154. Copyright (1997) by the American Physica l Society.

sp 'ci s can be produced, with observable abundance ratios. A diagram illustrating a part f the s-process is shown in Fig. 1. 17 . Because the rate of capture reactions is slower than the ,B-decay rates which bring nuclei back towards stability, the s-proc ss proceeds close to the valley of stability in the Segre chart. In Fig. 1.16 w show the contribution of the '-process to the solar abundances. Ob ervations of III ta l-poor stars show that the s-process is not universa l and depends strongly on the 111 tallicily f the star. This inlrod uc s som unc rtainties on lhefi nal abundances till ' to th ori inal 'ompos ilion of Ihe slar.

,--,

p-procoss

40 (-process

20 Observed nuclei Stable nuclei

OD-_---'--__ L-_~ __ ~_~ __ ~_~ __ ~ __ ~ o 20 40 60 80 100 120 140 160 180

Neutron number

h f' 1.19. Predicted paths of the rp- and r-processes, to the left and r ight, 11 ' P , ' lively, of the vall ey of stability. Figure courtesy of Marc Hausmann.

! I

II \ xam ple of an s-process reaction, 48Ca(n,y)49Ca, has been measured by , 'l i d rl'searchers [7] and is shown in Fig. 1.18. We will revisit this capture rate

III II Wl' discuss transfer reactions in Chapter 14 because transfer reactions offer III IIHllI l' '( melhod for extracting this type of astrophysical information.

1.S Explosive production mechanisms 1111 \ I" () "ss alone can on ly explain about half of the observed abundances of th h . 1 \ 1'Il'1I1 ' nts. Thi has led to a search for astrophysical sites in the Cosmos that ar

1(1111)" lim -vary ing, and where the equilibrium probabilities are not app licab le. IId"IIIl ~' an alternate path for the production of heavy elements. The most probab l

JI"' ~ I V' scenario contributing to a large portion of the abundances of heavy .1. 1111 Ilt s is core-coJiapse supernovae. In this environment, high temperatures and I hll )'\ ' IIhundance of neutron enables a progression of capture reactions extending III' Illlllllnium. Thi is ca lled the r-process for rapid, as opposed to the s-proce, s,

1111 II Is slow. The capture reactions, however, do not necessaril y pass through lilt .!.dlk isOlop S, bUl produce many neutron-rich isotopes that are radioactive, IlIld "\('llllIally fJ -decay inlO other more stab le pecies well after the expl sion has 11111 lH'd . Th'l' ar also some nuclei lhal annot be pI' c1uced by eilh r lh s- or lilt I PIO ' ss Ill' 'hani sms, su 'stin' Ih ' ex isl n ' of an additional process which I" "dll( ('s til '!H1U'1 i. Th' isolo!" 'S involVl'd ill III 'r proc ss and Ihe p-process

Nlll'il'i ill Tlt l' CO,I'IIII I,I

J\ llhou ' h nOI 'onlrihulill l significantly to Ill ' OVl' litil pltHli1 ' 1I(lIi 0 1 ' I ' ments, oth r exp l s ive cnvi ronm nts where the proton d ns il y is hi , II liv' ri s' to another III chan ism, the rp (rapid proton) process. While the s-process pro 'ceds along the va lley f stab ility, the rp-process goes along proton-rich paths above and to the left or the va ll ey on the Segre chart, as shown in Fig. 1.19.

1.5.1 r-process neutron reactions

It is widely believed that in supernovae, after core collapse, there are abundant n ' utrons produced, at least for a few seconds, leading to rapid capture sequences that 'x t nd th horizontal isotopic chains to the right, well beyond the first radioactive isotopes (the haded region in Fig. 1.19). Neutrons are progressively captured by (n,y) reactions until the production of extremely neutron-rich isotopes is limited by Ih' increased probability of (y,n) photo-disintegration reactions. This probability in Tcases as the neutrons become less and less bound. The n B- y balance point h Iweeen capture and disintegration defines the position of the r-process path on Ih . cgre chart, and is thought to be between the lines shown in Fig. 1.19. Some or th nuclei that are produced will ,B-decay to heavier chemical elements (as with lit ' ,I'-process), giving a new seed nucleus for another series of neutron captures.

' los d- hell nuclei are usually very stable. These shell closures are believed to b ' ul the places marked by the vertical lines in Fig. 1.19 at N = 82, 126, and 184, :lnd here the r-process capture times become comparable to ,B-decay halflives. The rapid proce s s lows down as ,B-decay wins over neutron capture, and the r-process path move closer to the valley of stability. Nuclei where this happens are known as Iliai linR points in the r-process. Waiting-point nuclei around N = 126 are depicted by Ih open quares in Fig. 1.20.

The nucleosynthesis produces heavier isotopes until eventually, in the actinide r gi n, fis sion becomes more probable. It is possible that super-heavy elements (Z > 114) may also be produced.

The n utron-rich nuclei will eventually experience a 'freeze-out' after the neutron !lu x has passed, and wiJl no longer be replenished but will,B-decay toward stability. Thi s PI' duces an isotopic population progressively drifting from the neutron-rich s id ' IowaI'd the valley of stability. In the Segre chart, this corresponds to a drift diu 'onally up and to the left from the r-process path. In this sense, waiting-point 111I '1 'i are the progenitors fo r the production of table nuclei of similar mass. 'I'll r'lative abundance of each stable i otope is approxjmately proportional to 111 ' lif'e tim r its origi nati ng nuc leus on the r -pr cess path. Peaks in the abundance or slab l nucl i with A = 80, I 0, ancl 192, aris due to the neutron closed shells al N = 50, 82, and 126. In Fi '. 1.1 6 w show th Ol1lrihulion of' Ih r -process to Ih ' solar ahul1(hn . s.

1011 11 11 111 UII III ' II '

1111 1111 11 11 IHI I 111 " ' " 11.' / 11 1 11111

110 11111111 1 IIH lllllI'll

UU,nnll 01\ lJ.UIUIIJI

I'll' 1.20. Waiting-point nuclei (black open squares) : nuclei produced in the I plO css and that live long enough to be important signatures in the observed IIhlll1dance of stable elements [9] . Reprinted with permission from H. Schatz and I II 'crs, Astra. 1. 579 (2002) 625.

~ 1.5 > ~ .0 ~

o ~-L-L~~WL __ L-~~~ 10 100

En (keV) 1000

I

invo lv 'd ill Ih ' r -pro ' 'ss Ilav' nol ' v ' n t " n ohs 'rv 'd . ' I'IH'I ' IOll' , II I PI' s ' nt, Ihe mod ' lin 101' Ih ' 1'-1 1'0 "ss r ' li 'S h av il y o n theory.

Nume rous ,B -decay rates arc important inputs to r -pro ' 'ss n 'lwork m dels. T he r are direcLmethods to measure this , but a lte rnativ lyon ca n deLe rmine thi s information thro ugh charge-exchange reactions. In F ig. 1.22, data fo r s8Ni(t ,3He) is show n as a function of the excitation e nergy of S8Co [10, 11]. The mechanism for these reactio ns will be introduced in Chapter 4, and a discussion of how to extract Ih n eded structure information from such data will be addressed in Chapter 14. This p'lrtic ul ar reaction is also very relevant to the mechanism of the explosions in supernovae.

a) 5S Ni(t,3H~) 1.5 E(t)=115 MeV/nucleon

0.5 8 Iai He)

1.6 Outlook

1.6.1 Implicationsj'or nuclear/,ll 'sics T h nuclides produced in the s-process are aJmost all long-I ived en ugh to be targets in laboratory reaction measurements, and (n,y) cro s sections have been measured for many of these. The rp- and r-process nuclei, by contrast, have much shorter Ii fet i mes, and are more difficult subjects for laboratory measurements. We will see that s me of them have sufficient lifetimes to be produced in radioactive beams, and then used in subsequent secondary reactions to examine their properties . Those with 'v n shorter lifetimes can still be produced as final states in secondary reactions, and some of their properties determined. Reaction theory will be needed to analyze the secondary reactions, and connect those measurements to the reaction rates relevant in as trophysical environments. In the next chapter, we examine nuclear reactions a nd see what properties of nuclei can be measured, and then in later chapters develop s 'a ltering theory so that we have a theoretical framework to describe these nuclear I' 'ac t ions. Eventually, in later chapters, we will close the circle by applying the va l ious I' 'action theories to many of the astrophysical reactions we introduced in thi s 'hapt ' l'.

1.6.2 Nuclear astrophysics: an openjield Althou1h the rest of the book focuses on the description of nuclear reactions, t h 'S' contribute in multiple ways to many open questions in astrophysics. In all til' standard processes producing the elements desClibed in this chapter, there aI" abundance mismatches that need better constraints from nuclear physics, in particular, nuclear structure and nuclear reaction input. One of the greatest challenges has been blending nuclear physics input with astrophysics modeling in a way that meaningful constraints on the parameters can be made. In present-day modeling, there are specific conditions that need to be introduced artificially and arc yet awaiting a better understanding. In some cases, there is uncertainty on the nu leosynthesis path (as in the r-process); in others, there is uncertainty on the endpoint (as in the weak s-process). Perhaps even more exciting are the new 'merging ideas that have not been discussed here. There is the weak s-process introduced earlier to account for the light nuclei , but more recently the Light Iil ' m nt Primary Process (LEPP) was added to the list, to explain the eady galactic udunda n es and, maybe related to it, the neutrino p-process, thought to occur in all 'ore-coll apse supernovae, and a possible site for the production of Sr and other -1' 11 nts beyond Fe in very early stages of galactic evolution . Studie aiming for a h ' tt ' r understanding of the Universe will continue for decades to come. For more d 'lail-d information on astrophysi 's and the topics c vered in lhi s chapler we refer

III tl'x tho() )., s Sll ' II as ' lay tOil II I. I'ldh Ilid 1' lId ll l' 1131, A I'll 'II II I, Pa I ' III . I, Ilimli s 1161 and 8 ' nil 'tl11 7 1.

I{ ' fer '!Ices III L. Kawano, Fermj National Accelerator Laboratory, Report

F RMILAB-Pub-92/04-A ( 1992). I I P. D. Serpico, S. Esposito, F. Iocco, G. Mangano, G. Miele and O. Pisanti , 1. olCosr/'/..

and Astro. Phys. 12 (2004) OlD. . I \ I R. G. H. Robertson, Prog. Part. Nuc!. Phys. 57 (2006) 90. I 'I F. Hoyle, Astro. 1. Supp!., 1 (1954) 121. I I II. O. U. Fynbo et aI., Nature, 433 (2005) 136. 1(11 T. A. Weaver and S. E. Woosley, Phys. Rep. 227 (1993) 65 . 111 P. Mohr et al., Phys. Rev. C 56 (1997) 1154. I KIN. C. Summers and F. M. Nunes, Phys. Rev. C 78 (2008) 011601R; Phys. Rev. C 78

(2008) 069908. III I II. Schatz and T. Beers, Astro. 1. 579 (2002) 626.

11 01 A. L. Cole eta!., Phys. Rev. C74 (2006) 034333. III I R. Zegers, NSCL White Paper 2007. II ) I D. D. Clayton 1984, Principles of Stellar Evolution and Nucleosynthesis, Chicago:

University of Chicago. II \) . E. Rolfs and W. S. Rodney 1988, Cauldrons in the Cosmos, Chicago: University

of Chicago. II 'I D. Arnett 1996, Supernovae and Nucleosynthesis , Princeton: Princeton University

Press . -II I B. E. 1. Pagel 1997, Nucleosynthesis and Chemical Evolution of Galaxies,

ambridge: Cambridge University Press. II () I ,Iliadis 2007, Nuclear Physics of Stars, Weinheim: Wiley. I1 1I J. Bennett, M. Donahue, N. Schneider, M. Voit, The Essential Cosmic Perspective,

an Francisco; Toronto: Addison-Wesley.

2 Reactions of nuel

I was laught that the way of progress was neither swift nor easy. Marie Curie

III ord ' r to understand nuclear reactions, we have first in Section 2.1 to name the :11 nlll J ' Ill 'nt o f nucleons in a nucleus in terms of the quantum-mechanical state or

()

in I v I li ag ram s. ba 'h reson'mcc has a widlll I ', III 'IIS III' 'd ill M 'Y as (in mo t 'ascs) thc full width at half maximum of thc r sonun ., p 'uk . R'sOln nces can inl rf' re with eacb other, and overlap each other at higher ncrgies where the widths increase, and thus give more complicated patterns in experiments. A resonance can I regarded as a composite system that, because of its energy spread, lasts according to the uncertainty principle in proportion to e-t / r for a time duration of r ~ Ii/ r ca ll ed the lifetime. The half-life, after which half the nuclei will have decayed, is 11 /2 = (I n 2) r. When the cOlTesponding decay channels are included in the picture, onc can think of all radioactive nuclei as resonant states.

2.1.2 Kinds of reactions I n I' actions of type B(A,C)D, the nuclei A and B usually start in their ground states. I I' I h 'y remain in their initial states, we have elastic scattering, written as B(A,A)B. Th' el ir ctions of motions of the two nuclei will have changed by the scattering ollMle 0 shown in Fig. 2.2, while the relative kinetic energy of their motion E (d ' fin d on page 35) will remain unchanged.

I I' one or both of the incident nuclei A, B gets changed to an excited state during Ih' rcaction, this is called inelastic scattering. Excited states are often denoted by a + superscript, so B(A,A)B* is the reaction where B finishes in some excited state b, say. Thc re lative kinetic energy after the reaction will be decreased by the amount (/1 of' cn rgy that has gone into exciting the nucleus B, and the Q-value will be Q = - b fo r this endothermic reaction.

I crhaps during the reaction a proton or a neutron is moved from one nucleus to unoth r. Thi s is called a transfer reaction. If the nucleus A can be regarded as a

D

/ lD A~PA ----O~~- 0 B Ps

o ",/ / / / / Reaction region

c / c

Ili '. 2.2. ca lterin an Ic in th SeA, )D I' action showing the incoming and oul 'oin 11l0ll1enl tl. 111 Ih c 'nler-of-mass i'ralll , Ih IwO 0 an 'Ies

.L

br akup reacti ons are g n ra ll y direct I' actions or Ihi s killd . SO Il1 ' times reactions with resonances htlve peaks in the way the cros ' secti ons va ry with energy. These are measured by the full widths at half maximum or these p aks I , and narrower resonances last for longer times in inverse proportion to their width. Finally, there are extremely narrow resonances from unbound compound nucleus states that, by the time they decay, will have lost practically all information about the direction of the inc ident nuclei, and will therefore decay isotropically.

2.2.1 Direct reactions

The rastest reactions only involve very few nucleons on the surface of the nucleus, or only the nucleus as a collective whole. These are called direct reactions, and are more likely to occur at high incident energies because then the reaction is typica ll y finished more quickly and fewer internal collisions are possible. In these rast I' actions, the directions of the final nuclei are much more influenced by the inilial direction , and will typically have large cross sections at small Bern: large r '" ' I ion rat s in the direction of the incoming nuclei (see Fig. 14.3 for example).

Qua ntum mechankally, direct reactions are much more often modeled as a one-.1'11'1' 1/'{//I.I'ilio/l between the initial and final scattering states. Most of the Big Bang I' 'll 'I ions 'a n be well described as one-step reactions, for which, we will see in ( 'Il apt 'I' 14, the d istorted-wave Born approximation (DWBA) will prove very useful. Trolls t' 'r processes such as A(d,p)B stripping reactions are usually modeled by the I WI3A, assuming a direct-reaction mechanism. Of course, production ofthe A + d 'ol1lpound nuc leus system is still possible at lower energies, but the decay of the ' ol11pound nucleus gives isotropic angular distributions, which can be distinguished rrom the forward-peaked (d,p) cross sections, and subtracted if necessary.

ne-step theories may be improved by including two and higher-order steps , as in a perturbation series. If some of the interaction potentials are strong, however, this s ri s may not converge, and coupled-channels methods must be used, as discussed in hapt rs 6 and 14.

2.2.2 Resonance reactions

W' havc se n many resonances in Chapter 1 as peaks in cross sections when plOl1 '

0.8

:0 -; 0.6 .Q t5 Q) (f) (f)

e 0.4 ()

0.2

{

I? (' II (' I ilil/ ,I' (~ 111/1 II' (a) Labor tory frame

A

(b) Ce~r-of-mass frame .-.

---~

S

v ' A 8 '=0

(c) Vector addition

v ' B

Iii '. 2.4. Laboratory (a) and center-of-mass (b) velocities before the collision, wh 'n B is a stationary target. Part (c) shows the way the laboratory velocity v may h ' d ' 'o l11posed into a velocity in the c.m. frame Vi and the velocity of that frame S (IS v Vi + S. The azimuthal angles are out of the page, and are not changed hy Ih . changes of reference frame.

The center-oj-mass coordinate system I r a II the forces acting in the reaction are only between the nuclei and are not 'X I rna lly imposed, then it is most useful to change the reference frame to one in which the center of mass of A and B is at rest. This is called the center-oj-mass (c.m.) frame. When the forces between A and B depend only on R and not at all on S, then th v locily S is constant, and the c.m. frame remains an inertial frame. We denote v' loci Li s in the center-of-mass frame by primes.

W ' may et the origin of the center-of-mass reference frame at the point S, so the I'ralll' is de fined by S' = S' = 0 as shown in Fig. 2.4(b). The unprimed laboratory y ' 10 ' iti s and primed center-of-mass velocities are related in general by

v = Vi + S or Vi = V - S ,

Ill ' v' ' lor add ition illu tratecl in F ig. 2.4(c).

(2.3 .7)

onservalion lawsJor collision.l In a I' a 'li on (A , ) thai I ads lo final nuc l i and ,(; ns rval ion laws limi t the ra n ) o l'th' and . Non-r lati visli cally,

Wl' hav' s 'paral ' II/{/,\',\' , (' 11 1'1:11 1' 111I11I11I/1I/ '/II/II"lllll l-l ' I vn li o ll l-l : 1

1/11\ III/I 111 ( ' 1111),

Q -I- /:':/1 I /o.:/J t:: , Eo, PA + I>B = P + PD,

II

(2 . . 8) (2.3.9)

(2.3. 10)

lI'spectively, where Q (as before) is the internal energy released in the reaction. 'I'll 'se laws apply in both the laboratory and (for primed quantities) in the center-III mass coordinate frames. '

Laboratory and center-oj-mass scattering angles 1,lIhoratory experiments may measure scattering angles, commonly when B is a tnliona.ry target. Most theories , however, predict cross sections as functions o r

11'lller-of-mass angles ecm, which are different from the laboratory angles elab Ill'cause the center-of-mass frame is now moving in the laboratory with constant l'iocily

(2.3.1 J)

\i II 'n 1"8 = 0 as in Fig. 2.4(a). , nsider some outgoing particle C. If it has lanoratory velocity vc, then its

l' locity in the c.m. frame, by Eq. (2.3.7), is

V~ = Vc - S. (2.3.1 2) l ,l'l us measure the angles e of C from the incident beam direction: the direcli on III S. Then the lateral and parallel components of Eq. (2.3. 12), according Lo th ' tll ilngle of Fig. 2.4(c), give

v~ sin ecm = Vc sin elab S + v~ cos ecm = Vc cos elab

R('(/(' /; ril/ .I' r~f' 1111/'1",

wh'r w de fin p = . /'U~ . To de termine p W ' n " d III li S - 111 ' 'ons rvation Laws. If F:. is th re lative ene rgy in the incident channe l, Ih ' n in Ih ' 'x it hannel we have in the c.m. frame

I ,2 I ,2 Q + E = Ee + ED = '2mev e + '2mDV D O ' , = meVe + mDVD (2.3 .15)

Eliminating v~ from these equations, we find Q + E = ~ :~ (me + mD)v'2 ombining thi s with Eqs. (2.3.6) and (2.3.11) we get

to

[mAme E J ~ p=+ -----mBmDQ+E

(2.3.16)

In ' Iustic scattering A= C, B = D and Q = 0, so we have simply p = mA/mB. To find the C. m. ang les in terms of the laboratory angles, Eq. (2.3.14) can be rearranged liS a qu ad ratic in sin ecm .

In Ih ' remainder of this book we will drop the subscript cm and the primes, and liS' () 10 directly refer to the scattering angles in the center-of-mass frame. 2

2.3.2 Relative and center-oj-mass wave Junctions 'I'll ' above transformations from space fixed to center-of-mass coordinates have Ih ' ir counterpart in quantum mechanics. The Schr6dinger equation for the motion or two partic les A and B with total energy Etot and some potential V (rA - rs) that a ' ts b tween them3 is

(2.3.17)

Using the same coordinate transformations that led to Eq. (2.3.3), but without assuming that B is at rest, the kinetic energy terms in this equation may be rewritten as a sum or operators using center-of-mass and relative coordinates. We may also li S' these coord inates for the wave function, yielding

[ h 2 2 n2 2 ] 0

---V8 - -VR + VCR) - Etot ~(S , R) = . 2mAB 2fL

(2.3.18)

1 Nol ' IIlso IIiBI they may somclimcs bc named 'c.m. scattering anglcs', jusl as Ihe ve loci ti es Vi are sometimes l'lI lkd '~ . llI . v Ioci lics'. Howcvcr, Ihc angles, ve locities and momenta ill (nOI 0./) the ccntcr-o f-mass frame are

ill l ~ lIdcd. Purlhcrmor , I)~ und I>;J arc somelimes ea llcd ' rclal ivc momcnla ' , bul Ihis is misleading since they 11 1 ' 1101 r 'Iuli v' III 111 ' olher Pllfl ic ll:, oilly 10 Ihe cenl 'f-Oj' mass j'ran lc. S in ~' I)~ = 1>;/. Ih 'y afC cqual and opposil IIIOl11'nl 1i i ll Ih ' '. 111 . 1'1'11111 '. ....

I W'II\SlIllI ' llIul V( I{ ) (I w ll~ 1I N , . M) Ih ' 101111 'nClgy /:'1111 I ~ 1I 1 ~0 111 ' 10 1111 ~III ' II ' ' II 'I'!\Y 0 /111 inilia l Pl lll k I L'~ lI iw ll" 1\1I ' . (l . \ .. ) IliHI (J,,\ . 1) .

I , / I /II/II/jI/I \ .19

W- 1l0W look for s larahl ' SUIIlIIlIIl 'o, III Ill ' rorm qJ (S , R) =

10 U ('(f(' liO/l.l' (~f'/I/1('/"t

We ' mploy a 4-vecl I' nOlalion wher lhc 1110111 IIIUIII I v' ' 101' is

P = mOy(v)(c, v) = (mc, p) = (EI " p) (2.3.2 1)

with invari ant p 2 = m2c2 - p2 = m6c2 , for a particle of rest mass mo moving in a g iv n frame with velocity v, momentum p, and total energy E = moc2 + T for kin ti c energy T.

When there are several particles, we cannot define a center-of-mass frame using I ~q . (2 .. 1) because the mass coefficients are frame dependent. What we can do is d fine a 'enter-oJ-momentum (COM) frame4 in which the summed 4-momentum oj' all the particles is purely time-lik~, with the' spatial part being zero, L Pi = 0 (Rind ler [2, 30), Goldstein [3 , 7.7)). In this frame, the summed 4-momentum for two parlicles is

Ptot = mA + ms)c, PA + Ps), = mA + ms)c, 0)

ill Ill ' COM j'rame, with invariant 2 ( . )2 2 _ M2 2 Ptot = mA + ms c = c

(2.3.22)

(2.3.23)

I() Ionns l

M 2 2 2 co 111 + mos - mOA 2 Es = y(vco111)Es = c . 2M

(2.3.27)

1IIIIIl' M frame the energies sum to Mc2, so

(2.3 .28)

11 11 'momenta of A and B in the COM frame must be equal and opposite. Their I '111111 magnitude p may be found from El = p 2c2 + m6ic4 for either i = A or B. Th \ d V(' Ilumber for relative motion, which we will need for later wave equations, ea n III ('valuated as

(2.3.29)

\ I will also need the Sommerfeld parameter 17 , which uses the relative velocity "" I between A and B. This is most easily found via y == y(Vrel) as

II

TA y(Vrel) = --2 + 1,

mOAC

1,11 IlIlt: structure constant a = e2lnc.

(2.3.30)

(2.3. I)

III an ex it channel, the variation of the reduced masses of particles C and D wi II 1IIIIIIIIlHlically take into account the Q-value by Eq. (l.l.3). Their 4-momenta wh n 1lIlIl ll'd will yield the same invariant Mc of Eg. (2.3.23). In the COM frame the ir \ IIIOIl1Cnta will again sum to zero, so their energies may be found by equations IlIlIilur lO (2.3.27 , 2.3.28) . The wave number for the outgoing relative motion is j'lWIl by an eq uation of the same form as Eq. (2.3.29). These all reduce to their 11 1111 l" lativistie limi ts for TA mOA c2 , or Vrel c. Photons, with zero reSlml:lSS, 11111 1>1 always be lreated relalivi tica ll y.

Til' above eq uations are sum ienl to d fine lhe energ ies whi h ent I' inlO a 'Ill illl llllll 111 'hani

th s' th read I' is reI' rred to Goldstein 13, 7 .7 1. '!'olal ' Il) ~~ Sl' ' Ii ons, as ratio of /lu xes, ar n l changed by Lorentz transformati ns.

This treatm nt of re lativity is of course far hOI'l of field -lh oretic treatments, 1'0 1' we rather simply use the above energies and wave numbers in the non-I' lativi ti c Schrodinger equation. In this way, the theoretical treatment may be al lea t consistent with the most accurate descriptions of experimental beams and fi nal scattered particles.

2.4 Cross sections.

Th reaction rates of nuclear reactions are described as cross sections, which

1,1 U I'(/cI ;O/l ,I' (!f' ""l'It 'l

S ill ' 'LIs. 2 .. 1.) and (2.3 . 14), we may deriv ' III ' 1\' llIlll ill

(2.4.7)

where we use p from Eq. (2.3.16).

2.4.3 Experimental and theoretical cross sections

The cross section is an effective meeting point between experiments and theory. Most often, experimentalists transform their measurements into the cross sections a (0,

Th asymptoti I' I'm 1'01' th comb ined in ' ill ' nl lind N ' Ilinill!' WilV s is thus

(2.4.12)

The label ' asymptotic' here means that this is the form in the free space outside the range of the interaction potential between the particles.

The scattered angular flux per steradian is}f = R2jr, namely ./f = vf IAI2lf (e,

3 Scattering theory

I am now convinced that theoretical physics is actually philosophy. Max Born

I n order ton nd the cross sections for reactions in terms of the interactions between Ill ' r ' a ling nuc lei, we have to solve the Schrbdinger equation for the quantum-III ' 'hanica l wave function. Scattering theory tells us how to find these wave I'UIl 'Ii oll s ror the positive (scattering) energies that are needed. We start with the s i IIlpl 's l ' af; ofn nite spherical real potentials between two interacting nuclei in S ' '1 ion . . I, and use a partial-wave analysis to derive expressions for the elastic s 'all 'r in ross sections. We then progressively generalize the analysis to allow ror Ion r-ranged Coulomb potentials, and also complex-valued optical potentials. S ,tion 3.2 presents the quantum mechanical methods to handle multiple kinds r r act ion outcomes, each outcome being described by its own set of pa11ial-

wave channels, and Section 3.3 then describes how multi-channel methods may be reformulated usi ng integral expressions instead of sets of coupled differential quat ion . We end the chapter by showing in Section 3.4 how the Pauli principle r ~q uires us to describe sets of identical particles, and by showing in Section 3.5 how Maxwell's equations for an electromagnetic field may, in the one-photon approximation , be comb ined with the Sch.rbdinger equation for the nucleons. Then w an describe photo-nuclear reactions, such as photo-capture and disintegration ina LI n i fo rm fra mework.

3.1 Elastic scattering from spherical potentials

Wh n the potenti a l between two in teracting nuc le i does not depend on the ori nl iation of the vector between them, we say that th potential is spherical. In Ihal cas, the only rea tion that ca n occur is hsti' scattering, which we now pro '''d 10 'alculal' lI s in ' th m Ihod or xpa nsion in pa rli al wav s.

IK

.1. 1 I I /OSl it ' ,1'/ '11 11"1 11,1 I' rill I I/d,, " "II I/ III II'lIl io l ,I'

3. 1.1 Partial- WllIle ,"{'(II/t!d,,}: ,/'1'/1111 "jl"lIt, spherical potential W' slurl ou r deve lopm nl o r s 'a ll 'rill !, 11 1t'()! h linding the last i scalt rin rrol11 I pol ' nlial V(n ) that is spherica ll y SY 1ll1I1 ' lri ' and socan be written as VCR). Fi nite pul ' Illi als will be dealt with first: Ihos' ror whi 'h VCR) = fo r R ~ R'b where R"

III' finite range of the potenti a l. T hi s exc ludes Coulomb potentials, which will 11 ' d 'a ll with later.

W' will examine the solutions at positive energy of the time- independent ,'l' llri)clinger equation with this potential, and show how to find the scattering ililp liludef(e,

, 0

I ... . J ~ I

SC(f llerillg Iii I' III V

Incoming plane wave exp(ikz)

I I I

I

.-

I

\

"

I I

.... - - ....

--,

" R=O \

" ( .... _ .... " .

-- _ ....

"

"

, \

\

I

Beam direction +z

I Outgoing spherical waves exp(ikR)/R .

Fig. 3. 1. A plane wave in the +2 direction incident on a spherical target, giving ri se to pherically outgoing scattering waves.

Hamilto ni an T + V commuting with all components of the angular momentum v c tOI' operator L, which we write as [T + V, L] = O.

The~ angular independences mean that, since the initial wave function is 'y linclri cally symmetric and no potential breaks that symmetry, the final state must

11 ;lv a wave function that is cylindrically symmetric too, as well as its external s 'all ering amplitude. Thus we need only consider wave functions 1jf(R, e) and IIl1lplitu les fee , cp) = fee) that are independent of cp. In quantum mechanical I ' rms, the potential commutes with Lz, so the Lz eigenvalue is conserved during the r 'acti on. Its conserved value of m = 0 implies that the wave function and scattering amplitudes cannot vary with cp.

Ul' proble m is therefore to solve

[T + V - E]1jf(R, e) = o. (3.1.5) Th scattering wave functions that are solutions of this equation must, from 12q. (2.4. 12), match smoothly at large distances onto the asymptotic form

e ikR 1jfasYIll(R, e) = e ikz +f(e)R' (3.1.6)

W will thu s find a scattering amplitude! (e) and hence the differential cross section (j 0) for e lasti c scattering from a spherical potential.

Partial- wave expansions T h wave fun cti on V/(R, e) is now expanded using Legendre polynomia lsPL (cOS e) , in what is called a partial-wa ve expansion. We choose these polynomi als as they are i 'cnf'unc ti o ns of both 'j} and Lz, with igenva lue L(L+ I) and m = 0 respectively.

W saw ahov" Ihat in th ' pr 'S nt 'asc we o nly need so lutio ns with rn = 0, as these

.1. 1 1:IO.l'II! ' ,1'('(1 11/ '1 II , /1 1/1/1 11/II I'I II'II /l llJ l f'II, i({I,I' . I

Ul'lli u ll s arc indcp 'ndcnt or

'ca ltering 1111'1)/ ) 1

Coulomb functions That Ft(17 , p) is regular means FL(1], p=O) = 0, and irregularity means GrJI7, p=O) t= 0. They are related by the Wronskian

) dFL(1],P) F ( )dGL(1],P)_1 GL(1], P dp - L 1], P dp -

or W(G,F) == GF' - G' F = k. (3.1.11)

Note that mathematics texts such as [1] usually define G' as dG/dp, but we denote this by G. Since p is the dimensionless radius p-= kR, we will use the prime for derivatives with respect to R, so G' = kG, etc. The Wronskian is equivalently Gi'- GF=1.

The Coulomb Hankel functions are combinations of F and G,

Coulomb functions for 1] = 0 Thc I} = functions are more directly known in terms of Bessel funcdons:

h(O,p) = ph(p) = (TCp/2)1/2h +1/2(P) G[(O,p) = -PYL(p) = -(TCp/2)1 /2 fL+l/2(P),

(3.1.12)

(3.1.13)

where the irregular spherical Bessel function YL (p) is sometimes written as nL (p) (the Neumann function). The Jv and Yv are the cylindrical Bessel functions. The 1] =

oulomb functions for the first few L values are

Fo(O,p) = sinp, Go(O,p) = cosp; FI(O,p) = (sinp - p cos p)/p, GI(O,P) = (cosp + psinp)/p; F2(0, p) = 3_p2) sin p - 3p cos p)/ p2, G2(0, p) = 3_p2) cos p + 3p sin p)/ p2.

Thcir behaviour near the origin, for p L, is

1 HI FL(O, p) '" (2L+ 1)(2L- 1) ... 3.1 P GrJO,P) "'" (2L- I) . .. 3.1 p - L,

(3.1.14)

(3.1.15)

(3.1.16)

(3.1.17)

(3. J.l8)

3. 1 mosl/I ' ,1'1'11/11 '1111 ,11 /,(JI/I l/dl"l it 'III /JII lelllials

Ilox .1 (Colllinued) II lId their asymptotic behaviour wh 'II II / , is

FdO,p) '" si n(p - LTC /2) GL(O, p) '" cos(p - LTC /2)

Hi: (0, p) rv ei(p-Ln /2) = j=fLeiP. So lit describes an outgoing wave e ip , and Hi: an incoming wave e-iP . ( 'o\llot11b functions for 1) t= 0 are described on page 62, and Whittaker functions on page 135.

Box 3.1 Coulomb functions

(3.1.19) (3.1.20)

. 3

'I'll . spherical nature of the potential is crucial in allowing us to solve for each partial -\ IV function separately; this corresponds to angular momentum being conserved \ h ' n potentials are spherical.

I :quation (3.1.10) is a second-order equation, and so needs two boundary i'iHI(Jilions specified in order to fix a solution. One boundary condition already kl1 0wn is that XL(O) = O. The other is fixed by the large R behavior, so that it lI 'produces the external form of Eq. (3.1.6). Since f (8) is not yet known, the role ill liq. (3 .1.6) is to fix the overall normalization of the XdR). We show below how III II . 'omplish these things.

As u ual in quantum mechanical matching, both the functions and th i r dl'l'ivatives must agree continuously. We therefore match interior and exterior 11111 '1 ions and their derivatives at some matching radius R = a chosen outsicl 1!Iv linite range Rn of the nuclear potential.

Radial solutions for zero potential 1101' N 2: a we have V (R) = 0, so at and outside the matching radius the radial wav ' 11111 ' Ii on must attain their external forms, which we name xIxt(R). The free- fi lei Ii IIfi al-wave equation may be simplified from Eq. (3. 1.10), and rewritten using a "" III 1 of variable from R to the dimensionless radius

p == kR, (3. 1.2 1 ) illh . x~xt(R) satisfy

[ d2 L(L+ I ) ] eX I

dp2 - p2 + I XI. (pjk) = O. ( . 1.22)

I II i,llplk,s 111(11" pl" ' is ' Iy Ih ll l XI,( /( ) O( N) liS N n .

54 'collerillH /111'(11'1'

This quati on 1'0 1' the externa l form xzxt is a sp "ia l ' tt~ ' 1'01' I} = 0 o r the mo re g ne ra l ouLomb wave equation

[ d2 L(L+ 1) 2r] ]

- 2 - 2 - - + 1 XLC I7, p) = 0, dp p P

(3.1.23)

which has solutions defined in Abramowitz and Stegun [1 , ch. 14] and described in 111 0 1' de tail in Box 3.1.2 This second-order equation has two well-known solutions that 'Ire Ii nearl y independent: the regular function F L (r], p) and the irregular function

'- (I] , p). A regular function is so named because it is zero at p = 0, and an ilTc' ul ar function because it is non-zero at p .= O. Any solution XL can be written as XL = bF L + cGL for some constants band c chosen according to the boundary co nditi ons.

Wc may construct H t = GL iFL, which are also two linearly independent so luti ons of Eq . (3 .1.23). By Eq . (3.1.20), these functions are asymptotically propoltional to e ikR. Since the radial momentum operator is p = ~ a~' the linear 'ombin ati ons Ht asymptotically have radial momentum eigenvalues nk, and this In 'ans that Hi describes a radially outgoing wave and HZ an incoming wave.

Thc partia l-wave expansion ofEq. (3.1.9) can be found for any function l{f(R, e). In particu lar, it can be proved [2] that the partial-wave expansion for the incident plan wave is

. 00 1 e

1kz = I)2L+l)jLpL(cose) kRFLCO,kR),

L=O (3.1.24)

us ing ju t the regular Coulomb functions FL(O, kR). Their appearance in this quat.ion i the reason that (2L+1)iL were defined in Eq. (3 .1.9), as we now have

XL = /< L in the plane-wave limit when the potential is zero. In t nns of the Ht = GL iFL, the plane-wave expansion is equivalently

ikz = "(2L+ l ) i LpL(COSe) ~~[HZ(0,kR) - Hi(O, kR)]. ~ kR2 L=O

(3.1.25)

I ,'rom thi s equatio n we see that the incident beam has equal and opposite amplitudes Il l' th ' r'ld ia ll y ingoing wave HZ and the radially outgoing wave Ht This describes a plan wave approaching the target, and leaving it again unchanged.

RadiaL soLutions with. a potential ut s icle the potentia l we know the xterna l form ofEq . (3. 1.6), but no t the scattering

alllp litud sf (e). Atthe ri g in wc know that xLCO) = 0, but notlhe de rivatives xUO). III IhL, se till ll W' n " d j ll st Ih' sp' 'i: II 'lIse ur 'I - O. bill W' IlH V ' h '~ 1II1 " '1" ' wi th II d ' !illitioll or 111 complete ('OIl IUlllil rUII '11011 ,' . liS II" ~ / l l fl l ~ I ' 1I1 wi ll I" 'lIflP 'lIl" illih' II l.:x l S,'l 'l ill ll wh 'lI ('ew llll ll b PIlI'lIlilll s III" illiruduced.

.1. / Woslh' ,\'/ '/11 /1' '/11,1 /11111/ 1/"II ' I{I " tI / III/(' III;(I /,I'

1\ . '

6 S(,(IIII' I'ill l> IIII'()/ ~Y

IJqualion ( . 1.27) impli s thallh full scallerin ' wav . I'uncli on has the external I'orm o r th partial-w'lve sum

llr(R, B) R~?II ~ L(2L+ l)iLPL(cos B)AdHZ (0, kR) - S LHi(O, kR)], (3.l.31) kR L=O

which we now have to match with Eq. (3 .l.6) in order to determinef(e) in terms or the S L.

ubstituting Eq . (3.1.25) in Eq. (3 .l.6) , equating to the right-hand side of Eq. ( . 1.3 1), multiplying by kR and using the asymptotic forms of Eq. (3.1.20) for the liZ functions, we have

L (2 L+I)iLPL(cosB)A L [iLe- ikR - SLi -LeikR] L=O

00 .

= L(2L+l)iLPL(cos e) ~(iLe-ikR - i-LeikR) + kf(e)eikR L=O

(3.1.32)

wh n both R > Rn and kR L. Collecting together the separate terms with eikR

and e- ikl? factors, we find

e"l1 [t. (2L+ l)iL PL( cos e) I ALSLi-L - ~i-L ) + /if (e) ] = e-

ikR [f (2L+l)iLPL(COse) {ALiL - ~iL}]. =0

(3.1.33)

ince the eikR are linearly independent, and the two [ ... J expressions in this qU:ltion are independent of R, they must each be identically zero. Furthermore,

us in the orthogonality (3 .l.8) of the Legendre polynomials, the second { .. . } 'xpr ss ion must also be zero, which implies AL = i/2. Substituting this result int o the first zero [ .. . J expression, we derive

1 00 f(B) = 2ik L(2L+l)PL(cosB)(SL - 1).

L=O (3.1.34)

This important equation (3 .1 .34) constructs the scattering amplitude in terms of the . -matrix e lement . The e lastic differenti al cross section is thus

la aCe) == - = dQ (3 .1 .35)

.1. 1 h'llIs /lt , ,1'1'111/"1 II, I"~ 'II' 1/,1" '1 I,Ii IW // 'lIl llI ls . 7

'I'll ' r'sultin fu ll s 'a ll 'rill' WilVl' 11111('1 Iill iN

V/(R, B) = , I ( I . I I )1"/ ' ,. cos 0) kR XL(R) , L= ()

(3. 1.36)

wll ' r the radial functions have external form for R > Rn in detail as

(3.1.37)

Phase shifts 1':11 'h matrix element SL is equivalently described by a phase shiftth for each parti a l

IV ' by

(3 . 1.38)

I) tak ing complex logarithms as fh = -it In SL. Phase shifts are thus defined up to I( It! iii ve multiples of n. We often add suitable integer multiples neE) of n ,

1 fh(E) = 2i In SL + n(E)n (3 .1 .39)

Itl IIl ake oL(E) continuous functions of energy E for each separate partial wave L, hili no cross section should depend on this addition as e2irrn == 1.

111 terms of the phase shift OL, the scattering amplitude can be written as

1 00 . f(B) = k L(2L+l)PL(cosB)e'OLsinoL, (3. 1040)

L=O

IIIHllhe external form of Eq, (3.1.37) of the wave function is equivalently

(3. 1.41 )

1IIIhis form we can see the reason for the name 'phase shift. ' In the asymptotic region \ lin' both R > Rn and kR L, we may use Eq. (3.1.19) to write Eq . (3.104 1) as

xfxt(R) --+ eiOL[cos OL sin (kR-Ln /2) + sin OL cos (kR-Ln /2)] = eih sin(kR + OL - Ln /2). (3 .1042)

I 'h\' ON ' ill ation are therefore , hifted to small er R when OL is positive, which occurs 1111 11111"1. 'tive potential (at least when th yar weak). The oscillatory patterns shift 1111 11 1' , ' I' R when OL < Of I' repu lsi v ' POI '111 ials. Physicall y, attracti ve potentials pull IIII' os ' illati ons in to it. , ran re, and I' 'pulsiv ' p()l ~ nlial s tend to expel the, call ring II (' illalions .

H Snlll(,l'ill )i, l!tell l 'Y

Th ' ex terna l so luti on ca n be also be wriLLen as

X~X I (R) = /< L (0, kR) + T LJ-/ t (0, kR) , (3 .1.43) wher

T i8L' ~ L = e Sin UL (3.1.44)

is 'a ll ed the partial wave T-matrix element. By Eg. (3.1.43), T L is the coefficient or lIt (0, kR) , an outgoing wave, and is related to the S-matrix element by

S L = 1 + 2iTL . (3.1.45)

Th scaLLering amplitude in terms of the T Lis

1 00 f (e) = k 2 )2L+ l)PL(cos e)T L L=O

(3. 1.46)

'or zero potential, 8t = T L = 0, and xIxt(R) = FL(O, kR) only, the regular partial-wave component of the incident plane wave.

A third form of the external wave function is

(3.1.47)

with KL = tan OL, called the partial-wave K-matrix element. The S-matrix element in t 'rms of th is is

S 1 + iKL L = . 1- iKL

(3 .1.48)

For 'I. 1'0 I otential, KL = 0. The K-matrix element may be directly found from the R-ll1a tri x element RL of the interior solution at the matching radius a as

(3 .1.49)

wh'r the arguments of the FL and GL are omitted for clarity. All of the above s 'at t >ring phase shift , OL and partial-wave elements T L, S L, and KL are independent or 0 , provid d that it is larger than the range Rn of the potential.

FrOI11 Eq. (3. 1.49) we can see the consequences of V (R) being real. In th is case, th tri al fun ction u(R) may be real, and hence also RL by Eq . (3. 1.28), KL by Eg. (. . 1.49), and henc ' Ot wi ll b rea l since tan OL = KL. It is for thes reasons that s 'att 'rin ' from a rea l pot 'nt ial is most o rt n d scribed by a (rea l) phase shift. This 'orr sponds to til ' Illatrix ' 1 '111 'nt 51, = 2iol havin) unit modulus, IS LI = I.

3.1 Il /lI ,l'l /( ' ,\'( '/111"1 111,1/ !/r l ill l /d' I' III " tI/IIIII'"lill l,\'

Tub l . 1. R' !olio/1.\' bel weell III(' 11'1 /1 '1' ,/1111"/ /11 11 ,1' ,,"rllile phase sh.i:fis, K-, T- and -lIlOlri.x elemenls, for an a/'hil/,fl/ :I 1111/ '1/,,/ l \lfl l ' I ' , !Ja/'Iia!-wave indices and the

III'/{ umenls of the Coulomb/uIIClioll,i' f l/ 'f ' ollliI/N I P}/" clarity. The last two lines lisl lite onsequences of zero and rea! pOI ' 1I Iio!.\'.

t Is in g: 0 K T S

x (1< ) = ei8 [F cos 0 + G sino] 1 I 1 _ iK [F + KG] F+ TH+ -[H - - S H +l 2

11 = 0 arctan K T 1 arctan --- In S

1 +iT 2i

K= tan 0 K T .1- S I--I +iT 1 + S

ei8 sin 0 K i = T -(1 - S ) 1 - iK 2 e2i8

1 +iK = 1 + 2iT S 1- iK - 0 0=0 K =O T =O S = I

\I I" al oreal K real 11 + 2iTI = 1 lS I = 1

'I'll ' I' lations between the phase shifts and the K-, T- and S-matrix elements are IIl11marized in Table 3.1.

Angle-integrated cross sections

Prom the cross section aCe) = If(e) 12 given by Eg. (3.1.35), we may int 'rat' II V ' I' the entire sphere to find the angle-integrated cross section3

00

rr " 2 = k 2 L..-(2L+ I)ll - S LI

L=O

4rr ~ . 2 = k2 L..- (2L+ I) Sin OL , (3.1.50)

L=O

II , III , th orth gonality and norma lization q. (3 . 1.8) orth LegendI' polynomia ls. I N,II ' Ihul thi s inl gnll tI cross S 'clioll is SOli I ' l i lll 'S '"Ikd Ih ' 101ft!. TOSS S 'I;on b .,us il is Ih ' lolld 111'1 ' r

"I I') "1111011, (IV r ll ll lll'gi 'S, Ilow 'v~ r, W' " 'S 'I've Ihe I '1'111 ' 10 111 1' III ill ' Illd ' 111 1 lion ' 11Isl; , linlll SIl, I 'S 100, liS wi l l h,' IIsedin slIhs' 'l inll :1 , . 1,

60 .')('Olle rill l:{ Iheory

There ex ists an opti 'al th. orem whi ch re lat s th ' ;\II'k illl . 'nit cI cross section (J' I to the zero-angle scattering amplitude. Using Pc( I ) = I , w . ha v

1 00 f(O) = 2ik I)2L+1)(e2iOI. - I ),

L=O

(3 .1.51)

so

1 00 Imf(O) = k 2:)2L+1) sin2 8L

L=O k

= -(Jel 4n (3.1.52)

Thi s r lation exists because the incident and scattered waves at zero scattering angle havc a fixed relative phase, and interfere in a manner that portrays the total loss of Ilu x 1'1' m the incident wave to the scattered waves.

Scattering using rotated coordinate systems ' Ib rind the scattering from an incident beam in direction k (not necessarily in the 1~ 1. dircction) to direction k ', the Legendre polynomial PL(COSe) = PL(k . k') in 8q. ( . 1.34) may be simply replaced using the addition theorem for spherical harmon ics [2] ,

L 4n L M ~ M~' * Pdcose) = -- YL (k)YL (k) .

2L+1 M= -L (3.1.53)

wh re k is the notation for a unit vector in the k direction. ThiS,; from Eq. (3.1.24), ives the partial-wave expansion of a plane wave in direction k as

e ik .R = 4n "iLYi CR)Yi (k) * ~FL(O,kR). L.- kR (3.1.54) LM

Thus Ih amplitudef(k'; k ) for scattering from direction k to k ' is

f(k '; k) = ~: LYi(k') Yi(k)* (e2i81. -1) LM

(3.1.55)

= 4: LYi (k' )Yi (k)* T L, LM

(3. 1.56)

and the full scattering wave function depends on the inc ident momentum as

Vf(R ; k ) = 4n I>LYf (R )yf (k )* k~?XL(R) . (3.1.57) I~M

.1. 1 1I'IlisIit ' ,1'/ '1111, ' 1 11 ,1 /111111 1/,/t"I/(U//luI I'IIlill/.l'

III I (l lh c.;as 's , Ih v 'lo r k al'! 'I' IIl l' . \' llll ~' (l Ii)l1 illdi ';.\1 ' S th in id nt mom ' nlulIl. Nol ' Ihat a sphcri ca l harmoni ., I'()I' it s Ullit " Ill l' al' 'umcnt in the +z dirc tion, is I'M (z) - 8 fiL+I th I' 1' 1' 2 ' I ~ - MOY 41r ' oano 'I' o rnl O \q . . 1. 4) ls

(3. 1.58)

lil r Ihe pl ane wave in the +z direction.

Coulomb functions for 17 =1= 0 'I'll ' runetions F L(I}, p), GL(I}, p) and Ht(l}, p) are the solutions of Eq. (3. 1.23) for I} i= O. In terms of a 'Coulomb constant'

CL( ) = 2Le- 7t'17/21f(1 +L+il})1 I} (2L+l ) ! (3 .1.59)

Il IH.llhe confluent hypergeometrie function IFL (a; b; z) == M (a, b, z), the regul ar rUll 'Ii on defined in Abramowitz and Stegun [1 , ch. 13] as

where ei ther the upper or lower signs may be taken for the same result. The 1/,'1 (a; b; z) is defined by the series expansion

(3 .1.60)

I FI (a; b; z) = 1 + ~ ~ + a(a+ 1) z2 + a(a+ 1)(a+2) Z3 ... b l! b(b+ 1) 2! b(b+ 1)(b+2) 3! + . (3. 1.6 1)

'I'h ' lWO irregular functions have the corresponding de finitions

Ht(l}, p) = GL(T/, p) ih(l}, p) = ei(~(=F2ip) 1+Li,) V(l +L iI} , 2L+2, =F2ip)

(3 .1 .62) (3. 1.63)

wI! ' I'e U (a, b, z) is the corresponding irregular confluent hypergeometric fUllction Ikli llCd in r I , eh. 131, The e == p - Ln /2 + O'L(I}) - I} In (2p), and

0'[(17) = arg f( I + L + i17 ) (3 .1.64)

I, ca ll ed thc. Coulomb phase sli!ft. The fun ctions may easily be calculated numeri ca lly II . ~ I, also lor complex arguments 141,

Th ' ir behav ior ncar Ihe orig in is Ihus

(3 .1 .65)

(,

Box 3.2 (Continued) noting that

S 'a llu illg Ih ' (NY

(3.1.66)

A tJ"lUJsition from small-p power law behavior to large-p oscillatory behavior occurs outside the classical turning point. This point is where T =21] /p + L(L+1) / p2, namely

Ptp = 1'] J 1']2 + L(L+ 1). In nuclear reactions 1] is usually positive, so with purely Coulomb andcelltrifugal " potentials there is only one turning point. Classically, the turning point is atthe di~tance of closest approach, Rnear ofEq. (3.1.79), and these quantities are related by Pip = kRllcm if the classical impact parameter b is related to the quantum Hlechanical partial wave L according to

k b = J L(L+ 1)R:: L + ~. The asymptotic behaviom of the Coulomb functions outside the turning point (p PIp) is

"iEl and HL (1'],p) "-' e . (3.1.69)

Box 3.2 Coulomb functions for T} i= 0

3.1.2 Coulomb and nuclear potentials In I n ra l, we saw in Chapter 1, nuclei have between them both a short-range nil racli ve nuc lear potential and a long-range Coulomb repulsion. This Coulomb '(lIl1pOnenl has the I /R form shown in Fig. 1.3, and invalidates the theory presented

IIhove ro r finite-range potential s. We develop below a theory for a pure l/R 'o mp nent, and then see how to add to it the effects of an additional finite-range ' 0 1'1' c ti on. We still assume both the Coulomb and nuclear potentials to be spherical.

Pure point-Coulomb scattering If w ' consider only the point-Coulomb potentia] between two particles with charges %1 and 2 2 times the uni t charge e, we have

C l .70)

6. 110 1' scaU ring with re lati ve v ' 10 il / . Wt' d '(ill ' a dimensionl.ess Sornrnelj' lel I)(/I"C/rneter T) , as menti one I beror " hy

(3. I. 71 )

where the energy of relative motion is E = n2k 2 /2 /L. A beam in the direction k is no longer e1k-R when T} i= O. Fortunately, the Schrbdinger equation with the Coulomb potential can be solved exactly, and the solution in terms of hypergeometric ru nc tions is

(3. 1.72)

d ' fi ned using the gamma function r (z) and the confluent hypergeometric fu nctio n 1 F I (a; b; z) of Eq. (3.1.61).

I n partial-wave form, the generalization of the standard +z plane-wave expans io n 1 1. (3. 1.24) is

00

1/Ic(kz, R) = L(2L+ l)iLpLCcos 8)..!...-FL(T}, kR) . - L=O kR

(3.1.73)

We are now using the regular Coulomb function FL(T}, kR) with T} i= O. Detail of I,',. and GL for general T}, and their asymptotic forms for small and large p, are g iven in Box 3.2. In particular, the asymptotic form of FLCT}, kR) is

FL(T}, kR) "" sin(kR - Ln / 2 + O'L(T}) - T} In (2kR) , (3. 1.74) wi lh the Coulomb phase shift O'L(T}) given by Eq. (3.1.64). The logarit.hmi . I 'I'l)l ill thi s expression is needed to accommodate the l / R Coulomb potenti a l. T hus 1/)(' pha 'e shift caused by the Coulomb potential is O'LC T}), once kR T} In (2k!?).

T he outgoing part in 1/Ie(kz, R) is found by matching at large values of R-z 10 R-z--+oo . ei [kR- 1) In 2kR]

1/Ic(kz , R) -+ e l [kz+1) l