Relativistic Coulomb excitation of the giant dipole resonance in nuclei: How to calculate transition probabilities without invoking the Liénard-Wiechert relativistic scalar and vector potentials

PHYSICAL REVIEW C 73, 014907 (2006) Rela i is ic Coulomb exci a ion o he gian dipole esonance in nuclei: How o calcula e ansi ion p obabili ies wi hou in oking he Li´ ena d-Wieche ela i is ic scala and ec o po en ials C. H. Dasso and M. Galla do Depa amen o de F´ ısica A ´ omica, Molecula y Nuclea Facul ad de F´ ısica, Apa ado 1065, E-41080 Se illa, Spain (Recei ed 4 Augus 2005; published 27 Janua y 2006) The conclusions ex ac ed om a ecen s udy o he exci a ion o gian dipole esonances in nuclei a ela i is ic bomba ding ene gies open he way o a u he simpli ica ion o he p oblem. I consis s in he elimina ion o he ela i is ic scala and ec o elec omagne ic po en ials and he amilia nume ical di icul ies associa ed wi h hei p esence in he calcula ion scheme. The inhe en ad an age o a e o mula ion o he p oblem o ela i is ic Coulomb exci a ion o gian dipole esonances along hese lines is discussed. DOI: 10.1103/PhysRe C.73.014907 PACS numbe (s): 21.60.Jz I. INTRODUCTION In a ecen a icle [1] i was shown how i is possible o calcula e accu a e p obabili ies o exci a ion o he gian dipole esonance a he one- and wo-phonon le els (GDR and DGDR, espec i ely) in e y simple e ms. A key conclusion o his e e ence was ha he esul s ob ained ollowing he s anda d o malism o Win he and Alde [2,3] a e— o hose pa ial wa es whe e hey can be conside ed eliable—much less dependen on he speci ic s uc u al de ails p o ided by a mic oscopic desc ip ion o he collec i e mode han p e iously assumed. In ac , o bomba ding ene gies up o a leas 5–10 GeV pe nucleon and o si ua ions ha a e compa ible wi h a unca ion o he model space a he le el o he DGDR i is no compelling o ha e an elabo a e pic u e o he dis ibu ion o cha ges and cu en s wi hin he nucleus. The ele an in o ma ion is con ained in hei lowes momen s, namely he posi ion o he cen e o cha ge and i s eloci y. A a ie y o s uc u al models may sha e hese ea u es and his ealiza ion makes he in es men o a signi ican e o in his di ec ion a he imp ac ical. Fo he conc e e ask o unde s anding he main ea u es exhibi ed by he da a ob ained wi hin p esen expe imen al condi ions he simples , amilia model o an oscilla ion o he p o on densi y agains he neu on densi y does su ice. Le us he e ecall ha he basic pa ame e s ha a e necessa y o implemen such pic u e a e in e ed om he known exci a ion ene gy o he mode and he mass and cha ge o he nucleus in ques ion. Wi h his back ound in mind we p opose in his con ibu ion o go a s ep u he and elimina e om he calcula ion scheme o Re . [1] he p esence o he scala Li´ ena d-Wieche po en ial o a poin like p ojec ile [4] (x,y,z, )=γZ Pe (x−b)2+y2+γ2(z− P )2,(1) and, e en mo e deside ably, i s ec o coun e pa  A= P c. (2) To his end we sugges o cas he p oblem in he e e ence sys em o he p ojec ile, whe e he elec ic and magne ic ields in he whole space and a any ins an o ime a e simply  E( )=ZPe 2 , B( )=0,(3) ha is, he same exp essions ha a e no mally used a non ela i is ic ene gies. The p ice o be paid o such simplici y and he absence o ec o po en ials in he p oblem is ha we mus , o cou se, ans o m he es o ing o ces associa ed wi h he gian dipole esonance om a sys em in which he “sp ing” is a es o ano he in which i mo es wi h a speed ≈c. This, as we shall see, is no ha di icul o do. We would like o s ess, howe e , ha he eason o posing his p oblem is no en i ely “academic.” No ice ha i one manages o inco po a e success ully such ans o ma ion laws in he o malism i should be possible o compu e he exci a ion p obabili y o he mode ia any o he ield whose exp ession in a sys em a es is known. Le us expand on he p e ious s a emen . The elec omag- ne ic po en ials a e speci ically easy o ans o m be ween ela i is ically mo ing sys ems o e e ence because he combina ion (  A, ) o ms a ou - ec o whose change is go e ned by he ou -by- ou Lo en z ans o ma ion ma ix. In ac , he Li´ ena d-Wieche exp essions quo ed abo e a e no mo e han he Lo en z- ans o med e sion o he elec ic and magne ic po en ials gene a ed by a poin cha ge a es . I is well known, howe e , ha he ans o ma ion law o p ac ically any o he in e ac ion is no ha s aigh o wa d o ob ain. I is because o his ecu en di icul y ha i appea s p omising o inco po a e in he o malism, once and o all, he ans o ma ion p ope ies o he in insic o ces ha accoun o he esponse o he gian dipole esonance. Upon a success ul comple ion o his p og am one would be able o es he e ec s o a a ie y o exci a ion couplings ( o ins ance, nuclea ) wi hou ha ing o wo y abou hei p ope ies unde Lo en z ans o ma ions. In his con ibu ion we se ou o explo e he p ac ical implemen a ion o hese ideas. In Sec. II we wo k ou he o mal aspec s o sol ing he p oblem o exci a ion o he GDR as seen om he ame o e e ence o he p ojec ile. The p oblem o he a ge ecoil, al eady ea ed in de ail in Re . [1], is b ie ly e iewed in Sec. III. A compa ison 0556-2813/2006/73(1)/014907(5)/$23.00 014907-1 ©2006 The Ame ican Physical Socie y C. H. DASSO AND M. GALLARDO PHYSICAL REVIEW C 73, 014907 (2006) S S’ yy’ xx’ zz’ eloci y V = –V a ge b p ojec ile z R FIG. 1. (Colo online) Re e ence ames used in he ex . The p ojec ile ame is Sand he a ge ame ( he nucleus whose gian dipole mode is exci ed) is S. The la e mo es owa d he le in he zdi ec ion wi h a eloci y compa able o he speed o ligh . The o igin o he wo sys ems coincide o =0 and he a ge -p ojec ile ela i e coo dina e  Rin he sys em Shas componen s (Rx,R y,R z)= (−b, 0,− ), whe e bis he impac pa ame e . o he esul s ob ained wi h he p ocedu es p esen ed in he p e ious Sec. wi h esul s ex ac ed ollowing he con en ional o malism o Alde and Win he is he subjec o Sec. IV. A summa y o he a icle and some closing ema ks a e le o Sec. V. II. FORMALISM Because in ela i is ic hea y-ion collisions he exci a ion ene gies ha en e in o play (≈10–20 MeV) a e much smalle han he bomba ding ene gies, we can conside he ela i e mo ion be ween p ojec ile and a ge o be uni o m and assume ha he classical ajec o ies a e well app oxima ed by s aigh lines. Wi hou loss o gene ali y we ake ha di ec ion o be along he zaxis. The p oblem we in end o app oach is illus a ed in Fig. 1. He e we see wo sys ems o e e ence, Sand S, which mo e wi h espec o each o he a a e y la ge speed  . The sys em Sis chosen so ha he p ojec ile is always a es in he posi ion (b, 0,0), whe e bplays he ole o he classical impac pa ame e . I is in his ame o e e ence ha we would like o w i e down he equa ions o mo ion ha desc ibe he ex e nally d i en oscilla ion o p o ons agains neu ons in he a ge . We exploi in his sec ion he same assump ion made a he beginning o Re . [1], namely ha he cen e o mass o he p o on and neu on dis ibu ions—i.e., he o iginal “ a ge ”— emains a es a he coo dina es (x,y,z )≡(0,0,0), he o igin o sys em S.We e i y in Sec. III ha , al hough ap io iunjus i ied, he use o his no- ecoil app oxima ion does no a ec he conclusions o he analysis in any signi ican way. The ime scales a e chosen so ha he xaxes in Sand So e lap a = =0. Unde he condi ions speci ied abo e, he coo dina es in Sand Sa e connec ed by he ollowing: x=x, y=y, (4) z=γ(z+ ), =γ + z c2, whe e is posi i e and he ac o γassocia ed wi h he Lo en z ans o ma ion be ween he wo e e ence sys ems is γ=1/1−( /c)2.(5) The se o equa ions (4) can be supplemen ed wi h he one ela ing he spacelike eloci ies in he wo sys ems, namely ˙ x=˙ x γ(1 +˙ z /c2), ˙ y=˙ y γ(1 +˙ z /c2),(6) ˙ z=˙ z+ (1 +˙ z /c2). I is use ul o ecall a couple o o mal ela ionships in ol ing he γ ac o s o he mo ion o pa icles in S and S(impo an a ela i is ic speeds in ei he sys em). We use hem, below, o a i e a ou inal esul s. Calling u2=(˙ x2+˙ y2+˙ z2) and u2=(˙ x2+˙ y2+˙ z2) we in oduce ˜γ=1/1−(u/c)2,(7) ˜γ=1/1−(u/c)2. Wi h hese de ini ions i is possible o show ha he h ee quan i ies γ, ˜γ, and ˜γsa is y γ˜γ=˜γ/(1 +˙ z /c2), γ˜γ=˜γ/(1 −˙ z /c2),(8) γ2=1 (1 +˙ z /c2)(1 −˙ z /c2). We can igno e om now on equa ions in ol ing one o he spa ial o ien a ions because he mo ion is cons ained o ake place on he [x,z](o [x,z ]) plane and y=y=0 h oughou . Keeping his in mind, we w i e he spacelike componen s o he o ce ac ing a he end o he elonga ed sp ing in he sys em S. They a e as ollows: FHO x=−Cx, FHO y=0,(9) FHO z=−Cz. He e we use he same no a ion as in Re . [1]. The alue o he es o ing o ce pa ame e C=D(¯hω)2is de i ed om he exci a ion ene gy ¯hω o he mode and he educed mass D= (ZTNT/AT)m, whe e mis a nucleon mass. S a ing om hese exp essions we cons uc hecomponen s o Minkowski’s ou - ec o o ce in S[8], K 1=−˜γCx, K 2=0,(10) K 3=−˜γCz, K 4=−i c˜γC(x˙ x+z˙ z), 014907-2 RELATIVISTIC COULOMB EXCITATION OF THE . . . PHYSICAL REVIEW C 73, 014907 (2006) which, Lo en z- ans o med in o S, yield K1=−˜γCx, K2=0, K3=−Cγ ˜γγ(z+ )− c2x˙ x γ(1 +˙ z /c2) (11) +γ(z+ )˙ z+ (1 +˙ z /c2), K4=−iCγ ˜γ cγ(z+ )+x˙ x γ(1 +˙ z /c2) +γ(z+ )˙ z+ (1 +˙ z /c2). F om he p e ious exp essions i is easy o iden i y he in insic es o ing o ces ha should be used o cons uc he ela i is ic equa ions o mo ion o he collec i e a iable (x,y,z) in he sys em o coo dina es Sand ollow i s e olu ion wi h espec o he a iable , FHO x=−Cγ(1 +˙ z /c2)x, FHO y=0,(12) FHO z=−Cγ(z+ )+Cγ c2x˙ x. No ice he wo d ela i is ic in he p e ious pa ag aph. I is qui e clea ha ypical eloci ies in he sys em Swill be close o he speed o ligh and he e o e he classical (meaning in his con ex “no quan al”) equa ions we need o sol e a e Eins ein’s se d d D˙ x 1−(˙ x2+˙ z2)/c2=Fx=FHO x+ZPZTe2δ R2b R, d d D˙ z 1−(˙ x2+˙ z2)/c2=Fz=FHO z+ZPZTe2δ R2 R. (13) To de ine he componen s o he o al o ce Fx,F zwe ha e added o he es o ing o ce [Eq. (12)] he (now i ial) con ibu ion om he Coulomb in e ac ion. Wi hin he dipole app oxima ion one can use he alue o he ield a any poin in he neighbo hood o he o igin o Sand we speci ically ake R=√b2+ 2 2. No ice he ac o δ=NT/(NT+ZT) in he exp ession o he elec ic o ce. I appea s (c . Re . [1]) because we se up he e equa ions o mo ion o he collec i e a iable  =(x,y,z) and igno e he o e all accele a ion o he cen e o mass o he a ge . I should no en e in he o mula ion (and in ac i does no ) i one in eg a es sepa a ely o he p o on and neu on componen s, as i is la e done in Sec. III. The exp essions (13) a e no ye cas in a con enien o m o be sol ed by s anda d in eg a ion me hods. Fo his pu pose we need o isola e he second ime de i a i es o x,z. In oducing he auxilia y quan i ies A(˙ z)=1−˙ z2 c2, B(˙ x, ˙ z)=˙ x˙ z c2,(14) C(˙ x)=1−˙ x2 c2, we a i e o a e y compac se o ime-dependen , i s -o de coupled di e en ial equa ions in he a iables x,z, ˙ x, ˙ z o be nume ically p opaga ed om hei ini ial alues, namely dx d =˙ x, dz d =˙ z, (15) d˙ x d =1 D˜γ3 FxC−FzB AC−B2, d˙ z d =1 D˜γ3 FzA−FxB AC−B2. The inal exci a ion ene gy o he dipole mode is calcula ed a e in eg a ion o [Eq. (15)] and—di iding by ¯hω—con e ed in o an a e age numbe o phonons N∞. In u n, his numbe is con e ed in o exci a ion p obabili ies o he g ound s a e and he one- and wo-phonon le els, jus as i was done in Re . [1] o si ua ions whe e N∞1. A compu e code called REVRCE has been w i en o imple- men his p esc ip ion. To illus a e he ou mos simplici y o his app oach we show in Fig. 2 an o e iew o he comple e lis ing o his Fo an p og am. The inse iden i ies he only piece o he p og am whe e he Coulomb in e ac ion appea s [c . Eq. (13)]. We compa e in Sec. IV he esul s ob ained using he REVRCE code wi h s a e-o - he-a calcula ions ollowing he o mula ion o Alde and Win he . III. TARGET RECOIL In Re . [1] i was discussed in de ail he impac o he common p ac ice o calcula ing he exci a ion o he dipole mode in a coo dina e sys em a ached o he a ge . Being his a nucleus wi h a ne posi i e cha ge i is ac ually accele a ed du ing he collision p ocess and he p esence o ine ial o ces should no be igno ed wi hou a p ope in es iga ion. I is no necessa y he e o adap he en i e line o a gumen s o he p esen si ua ion. Ra he , we limi ou sel es o quo e he equa ions o mo ion ha should be sol ed in sys em S o ollow he sepa a e mo ion o he cha ged and neu al componen s o he nuclea densi y. In e ms o he wo independen collec i e a iables  p=(xp,yp,z p)(p o p o ons) and  n=(xn,yn,z n) (n o neu ons) he se o equa ions, now eigh in o al, is 014907-3 C. H. DASSO AND M. GALLARDO PHYSICAL REVIEW C 73, 014907 (2006) FIG. 2. An o e iew o he o an p og am REVRCE ha cal- cula es he exci a ion p obabili ies o he GDR and he DGDR acco ding o he o malism p esen ed in Sec. II. The p og am is comple e and sel -con ained wi h he excep ion o a single call o a s anda d in eg a ion ou ine D02BAF [9]. The boxes a e included o d aw a en ion o he only pa o he p og am whe e he elec o- magne ic in e ac ion en e s in he equa ions o mo ion, assuming he simple o m gene a ed by a cha ge a es in a neighbo hood o he a ge (dipole app oxima ion). as ollows: dxp d =˙ xp, dzp d =˙ zp, (16) d˙ xp d =1 ZTm˜γ3 p Fp xCp−Fp zBp ApCp−Bp2, d˙ zp d =1 ZTm˜γ3 p Fp zAp−Fp xBp ApCp−Bp2. and dxn d =˙ xn, dzn d =˙ zn, (17) d˙ xn d =1 NTm˜γ3 n Fn xCn−Fn zB AnCn−Bn2, d˙ zn d =1 NTm˜γ3 n Fn zAn−Fn xBn AnCn−Bn2. In hese exp essions Ap(˙ zp)=1−˙ zp2 c2, Bp(˙ xp,˙ zp)=˙ xp˙ zp c2, Cp(˙ xp)=1−˙ xp2 c2, (18) An(˙ zn)=1−˙ zn2 c2, Bn(˙ xn,˙ zn)=˙ xn˙ zn c2, Cn(˙ xn)=1−˙ xn2 c2. The o ces ac ing on he cha ged and neu al componen s a e, espec i ely Fp x=−Cγ(1 +˙ zp /c2)(xp−xn)+ZPZTe2 R2b R, Fp z=−Cγ(zp−zn)+Cγ c2(xp−xn)˙ xp+ZPZTe2 R2 R, (19) and Fn x=−Cγ(1 +˙ zn /c2)(xn−xp),(20) Fn z=−Cγ(zn−zp)+Cγ c2(xn−xp)˙ xn. No ice ha he ac o δdoes no —as an icipa ed—scale he Coulomb in e ac ion e m ac ing now only on he cha ged densi y. A compu e p og am called REVRCE2c (also e y simple) has been w i en o implemen his p esc ip ion. We compa e in he ollowing sec ion he esul s ob ained using his code wi h hose o calcula ions pe o med aco ding o he s anda d app oach. IV. COMPARISON WITH THE STANDARD FORMALISM In his b ie sec ion we compa e esul s o he p obabili y o exci a ion o he gian dipole esonance o 40Ca in he eac ion 208Pb+40Ca a he ela i is ic bomba ding ene gies o 500, 1000, and 4000 MeV pe nucleon. On he one hand, we ha e he p edic ions o he codes REVRCE and REVRCE2c acco ding o he p esc ip ions discussed in Sec. II and III, espec i ely. On he o he hand, he esul s o Bayman and Za di ob ained wi h high compu a ional p ecision ollowing he app oach o Alde and Win he [5,6]. Resul s o a la ge numbe o impac pa ame e s a e collec ed in Fig. 3. The lowe limi in he ange o pa ial wa es ep esen ed in he d awing is de e mined—as i was al eady men ioned in Re . [1]—excluding om he se hose alues ha a e incompa ible wi h a “sa e” unca ion o he model Hamil onian a he wo-phonon le el. This is, equi alen ly, he egime o alidi y o pe u ba ion heo y whe e P0≈1 h oughou . On he la ge impac -pa ama e side 014907-4 RELATIVISTIC COULOMB EXCITATION OF THE . . . PHYSICAL REVIEW C 73, 014907 (2006) 0.96 1 1.04 P0 Bayman-Za di p og am REVRCE p og am REVRCE2c 500 MeV/nucleon -6 -5 -4 -3 -2 -1 log10 P1 40 80 -12 -10 -8 -6 -4 -2 log10 P2 40Ca + 208Pb 1000 MeV/nucleon 0 40 80 120 Impac Pa ame e ( m) 4000 MeV/nucleon 100 200 300 FIG. 3. P obabili ies o he exci a ion o he GDR (P1) and he double GDR (P2)in40Ca in he eac ion 208Pb+40Ca a bomba ding ene gies o 500, 1000, and 4000 MeV pe nucleon. The ene gy o he GDR was assumed o be 11.6 MeV. In he i s ow he p obabili y P0 o emaining in he elas ic channel is also shown. The h ee se s o esul s gi e he p edic ions ob ained om he simples classical model (REVRCE), wi h he inclusion o a ge ecoil (REVRCE2c) and in he con en ional app oach (Bayman-Za di). he p ac ical limi is se by he eliabili y o he adi ional cal- cula ions, whose accu acy is, na u ally, much mo e di icul o man ain. We can see ha he di e en me hods yield, in all ci cum- s ances, e y simila esul s. The ag eemen ob ained wi h he codes REVRCE and REVRCE2c should no come as a su p ise, hough, because he alidi y o he no- ecoil app oxima ion had been es ablished in Re . [1] and is no a ec ed by a e o mula ion o he p oblem h ough Lo en z ans o ma ions. V. SUMMARY AND CONCLUSIONS In his manusc ip we se ou o e i y ha an al e na i e app oach o he p oblem o ela i is ic Coulomb exci a ion o gian dipole esonances is possible. Ou aim was he e alua ion o ansi ion p obabili ies a ela i is ic bomba ding ene gies a oiding he in oduc ion in he calcula ion scheme o he Li´ ena d-Wieche po en ials. Such p ojec could ha e no been con empla ed wi hou ha ing a ailable he esul s p e iously ob ained in Re . [1]. The e we lea ned ha —wi hin he ange o impac pa ame e s whe e he ac i e eac ion channels in ol e only he g ound s a e and he i s wo exci ed s a es— he in insic mo ion can be sa is ac o ily modelled by a collec i e oscilla ion o he cha ged and neu al componen s o he o al nuclea densi y agains each o he . The p oblem is hen echnically o mula ed in e ms o a mac oscopic a iable  ha ep esen s he displacemen o cen e s o each dis ibu ion densi y wi h espec o hei equilib ium posi ion. This is, o cou se, one o he oldes isualiza ions o he in insic mo ion associa ed wi h a gian dipole esonance in nuclei. Wha was no ob ious, pe haps, is ha such simple scheme could yield accu a e exci a ion p obabili ies and handle so well he cu en end o push bombading ene gies up in o he ela i is ic egime. Desc ibing he in insic mode in e ms o an ha monic ib a ion i has been ela i ely s aigh o wa d o ecas he solu ion o he p oblem h ough a new se o equa ions o mo ion whe e he elec ic in e ac ion en e s in he simples possible o m indica ed in Eqs. (13) and (19). The p ac ical ad an age o he o mula ion can be app ecia ed in he ema kable simplici y o he calcula ion ool shown in Fig. 2 and he excellen ag eemen s displayed in Fig. 3. No ice, also, ha in he igh -hand sides o Eqs. (13) and (19) one could easily add addi ional e ms o conside he e ec o o he ypes o couplings. In he case o he GDR he analogy wi h a classical h ee-dimensional sp ing can be exploi ed o i s ulles ex en . Le us men ion, howe e , ha he e a e well-es ablished echniques ha employ a simila semiclassical language o handling he exci a ion o collec i e ha monic ib a ions o o he mul ipola i ies in eac ions wi h hea y ions [7] ha can be a sou ce o inspi a ion o u u e de elopmen s. ACKNOWLEDGMENTS Suppo is acknowledged om he Minis y o Educac ion and Science unde p ojec numbe s FIS2005-01105 and FPA2005-04460. [1] C. H. Dasso, M. Galla do, H. M. So ia, and A. Vi u i, Phys. Re . C 70, 044903 (2004). [2] A. Win he and K. Alde , Nucl. Phys. A319, 518 (1979). [3] K. Alde and A. Win he , Elec omagne ic Exci a ion (No h Holland, Ams e dam, 1975). [4] The esp essions co espond o he elec omagne ic po en ials gene a ed by a classical mo ion o he p ojec ile along he zaxis wi h impac pa ame e band a eloci y  P. [5]B.F.BaymanandF.Za di,Phys.Re .C68, 014905 (2003). [6]B.F.BaymanandF.Za di,Phys.Re .C59, 2189 (1999). [7] R. B oglia, C. H. Dasso, and A. Win he , in P o- ceedings o he In e na ional School o Physics “En ico Fe mi” (No h Holland, Ams e dam, 1981), Cou se LXXVII, p. 327. [8] H. Golds ein, Classical Mechanics (Addison-Wesley, Reading, MA, 1950). [9] NAG Fo an Lib a y, The Nume ical Algo i hms G oup L d, Ox o d. 014907-5