Role of thermal fluctuations in the damping of the giant dipole resonance of spherical and deformed nuclei: 90Zr and 164Er

V01ume 191, num6er 3 PHY51C5 LE77ER5 8 11 June 1987 R0LE 0F 7HERMAL FLUC7UA710N5 1N 7HE DAMP1N6 0F 7HE 61AN7 D1P0LE RE50NANCE 0F 5PHER1CAL AND DEF0RMED NUCLE1: 9°2r AND 1~4Er~" M. 6ALLARD0, F.J. LU15 Departament0 de F151ca At6m1ca y Nuc1ear, Facu1tad de F151ca, Un1ver51dad de 5ev111a, Apartad0 1065, E-41080 5ev111a, 5pa1n and R.A. 8R06L1A D1part1ment0 d1F151ca, Un1ver51ta d1M11an0 and 1NFN 5e210ne d1M11an0, V1a Ce10r1a 16, 1-20133 M11an, 1ta1y and N1e15 80hr1n5t1tute, Un1ver51ty 0f C0penha9en, 81e9dam5vej 17, DK-2100 C0penha9en 0, Denmark Rece1ved 21 January 1987; Rev15ed manu5cr1pt rece1ved 25 March 1987 we 5tudy the damp1n9 w1dth 0fthe 91ant d1p01e re50nance (6DR) 6u11t 0n the 9r0und 5tate and 0n exc1ted 5tate5 0f 9°2r, wh1ch 15 5pher1ca1 at 2er0 5p1n and temperature, and 164EE wh1ch 15 def0rmed. Ne1ther 0f the5e nuc1e1 chan9e5 apprec1a61y the1r e4u1116r1um 5hape when exc1ted. N0nethe1e55, f1uctuat10n5 due t0 f1n1te temperature pr0duce a marked 1ncrea5e 1n the w1dth 0fthe 6DR 0f 9°2r, 1eav1n9 e55ent1a11y unchan9ed that 0f ~64Er. 7h1515 6ecau5e 1n the ca5e 0f def0rmed nuc1e1 therma1 f1uctuat10n51mp1y the 5amp11n9 0f60th 1ar9er and 5ma11er def0rmat10n5 ar0und the e4u1116r1um 5hape, wh11e 5pher1ca1 nuc1e1 can 0n1y fee11ar9er d15t0rt10n5. 1n 5p1te 0f the 5ta6111ty 0f the 6DR 0f ~64Er, the an9u1ar d15tr16ut10n 0f the a550c1ated y-ray5 15 5tr0n91y affected 6y the temperature. A recent c0mp11at10n 0f the pr0pert1e5 [ 1 ] 0f the 6DR can 6e f0und 1n ref. [2]. F19. 17 0f th15 reference d15p1ay5 the 065erved w1dth (FWHM) 0f the a550c1ated 5tren9th funct10n5 6a5ed 0n the 9r0und 5tate and 0n exc1ted 5tate5 f0r a num6er 0f nuc1e1, a5 a funct10n 0f the ma55 num6er 1n the ran9e 40~<A~< 160. At E*=0 the data d15p1ay5 a marked 5tructure. 7he m1n1ma (F = 4 MeV) c0rre5p0nd t0 c105ed 5he11 nuc1e1 wh11e the max1ma (F < 8 MeV) are a550c1ated w1th def0rmed nuc1e1. 7h15 5tructure 15 wa5hed 0ut 1n the ca5e 0f 91ant re50nance5 6u11t 0n exc1ted 5tate5. 1n fact, the data 5h0w5 a un1f0rm va1ue ar0und F = 8 MeV, 1mp1y1n9 that the 6DR 0f nuc1e1 that are 5pher1ca1 1n the 9r0und 5tate have 6ec0me c0n51dera61y 6r0ader, wh11e th05e c0rre5p0nd1n9 t0 we11-def0rmed nuc1e1 rema1n appr0x1mate1y unchan9ed. 7h15 w0rk 15 part1a11y 5upp0rted 6y the cA1cY7 (5pa1n) under c0ntract num6er 2868/83 and 6y a 9rant 0f the C0mmem0rt1ve A550c1at10n f0r the Japan W0r1d Exp051t10n. 7he ma1n mechan15m re5p0n5161e f0r the damP1n9 0f 91ant re50nance5 1n med1um and heavy nuc1e1 ar15e5 fr0m the c0up11n9 t0 10w-1y1n9 5urface 05c111at10n5 [ 3 ]. A 5tat1c 4uadrup01e def0rmat10n a5 0ne 11m1t 0f th15 mechan15m (ad1a6at1c 11m1t), pr0duce5 a 6reak1n9 0f the 91ant d1p01e re50nance and thu5 a 6r0aden1n9 0f the a550c1ated d15tr16ut10n [ 4 ]. An9u1ar m0mentum (1~< 50h) 1ead5, a5 a ru1e, t0 the def0rmat10n 0f 5pher1ca1 nuc1e1, w1th0ut e55ent1a11y chan91n9 the 5hape 0f we11-def0rmed nuc1e1 [ 5-7 ]. 1t 15 thu5 tempt1n9 t0 c0nc1ude that th15 15 the rea50n why the 5tren9th funct10n 0f 6DR 065erved 1n the •/-decay 0f c0mp0und nuc1e1 have a c0n5tant w1dth a5 a funct10n 0f ma55 num6er. 1n the pre5ent paper we check th15 c0njecture 6y 5tudy1n9 the ph0t0a650rpt10n cr055 5ect10n f0r tw0 extreme ca5e5 1n the c0mp11at10n 0f ref. [2]: 9°2r, 5pher1ca1 1n the 9r0und 5tate and 164Er, wh1ch 15 we11 def0rmed. We c0mpare the d15tr16ut10n 0fthe d1p01e 5tren9th 6a5ed 0n the 9r0und 5tate and 0n the c0m222 0370-2693/87/$ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n) V01ume 191, num6er 3 PHY51C5 LE77ER5 8 11 June 1987 p0und nuc1eu5 5tate, def1ned 6y the exper1menta1 c0nd1t10n5 9°2r: 7=0,1=0 and 7=1.5MeV, 1=7.5h, 264Er: 7--0,1=0 and 7=1.2 MeV, 1=21 h. (1) 7he ca1cu1at10n5 were carr1ed 0ut a5 1n ref. [8]. A cranked N11550n p0tent1a1, w1th the parameter5 taken fr0m ref. [6] f0r 9°2r and fr0m ref. [9] f0r 164Er, 15 u5ed a5 the 51n91e-part1c1e f1e1d. F0r 9°2r the5e parameter5 91ve a fa1r d1p01e re5p0n5e at E* = 0 where m05t 0f the 5tren9th 15 c0ncentrated 1n 0ne peak a5 exper1menta11y 065erved. 0n the 0ther hand, a11 the 5et5 0f parameter5 we u5ed f0r 164Er y1e1d an an0ma10u5 d1p01e 5tren9th 5p11tt1n9 at 10w ener91e5 ~. 7he 5et 91ven 1n ref. [ 9 ] 15 1n any ca5e the 6e5t 0ne ava11a61e at pre5ent f0r th15 ma55 re910n. Pa1r1n9 15 n0t 1nc1uded 1n the ca1cu1at10n5. At the temperature5 0f 1ntere5t 1t5 effect 15 expected t0 6e 5ma11 [ 10]. F0r n0n-2er0 temperature5, the 0ccupat10n parameter5 0f the 51n91e-part1c1e 1eve15 are 91ven 6y the Ferm1 d15tr16ut10n [ 11 ]. 7he re51dua1 1nteract10n 15 appr0x1mated 6y a 5epara61e d1p01e-d1p01e f0rce. 7he c0up11n9 5tren9th5 are ch05en t0 f1t the exper1menta1 ener9y centr01d5 0f the 6DR at 7=0, 1=0 (x/x5c=0.765, 0.697 f0r ~64Er and 9°2r, where the 5e1f-c0n515tent va1ue5 [4] are x5c=0.102, 0.274 MeV/fm :, re5pect1ve1y). 7he d1p01e cr055 5ect10n 1n the 1a60rat0ry frame cr 15 ca1cu1ated 1n the 11near re5p0n5e the0ry [ 12,13] (cf. a150 ref. [ 8 ] and n0te an err0r 0f a m1551n9 fact0r 1/3 1n e45. (12) and (15)). 7herma1 f1uctuat10n5 0f the nuc1ear 5hape are 1nc1uded 1n the ca1cu1at10n5. 7hu5 the t0ta1 cr055 5ect10n 15 wr1tten a5 [ 8] ( a(1,7) ) ; fe de d7 P(e, 7; 1, 7)a(e, 7; 1, 7) fe de dy P(e, y; 1, 7) where P(e, y; 1, 7) ~exp[ -F(e, 7; 1, 7)/7]. (2) (3) ~J we n0te that the unpertur6ed d1p01e re5p0n5e 1n the 9r0und 5tate need5 t0 6e 5tud1ed m0re carefu11y. A11 the rea115t1c 51n91epan1c1e p0tent1a15 91ve, 1n 9enera1, a 6ad de5cr1pt10n 0f 1t. 7he 4uant1ty F(e, y; 1, 7) 15 the free ener9y f0r a 91ven temperature 7, 5p1n 1 and def0rmat10n parameter5 (e, 2:). N0 hexadeCap01e def0rmat10n5 are c0n51dered. 7he free ener9y 5urface5 F are ca1cu1ated w1th the 5trut1n5ky meth0d [ 7]. F19. 1 5h0w5 the5e 5urface5 f0r the tw0 nuc1e1 under 5tudy and f0r the va1ue5 0f 7 and 1 d15p1ayed 1n (1). 7he e4u1116r1um def0rmat10n5 rema1n 1n 60th ca5e5 e55ent1a11y unchan9ed. C0n5e4uent1y, the 065erved 6r0aden1n9 0f the 6DR 1n 9°2r 15 n0t due t0 a chan9e 1n 5hape, a5 wa5 the ca5e 1n 1°85n [8], 6ut mu5t ar15e fr0m the 5amp11n9 0f f1n1te d15t0rt10n5 under the 1nf1uence 0f therma1 f1uctuat10n5. F0r 5y5tem5 wh1ch d15p1ay n0n-5pher1ca1 e4u1116r1um 5hape5 at 7,1# 0 11ke 164Er, the r01e 0fthe 5hape f1uctuat10n515 n0t 50 1mp0rtant. 7he therma1 en5em61e 1nc1ude5, w1th 51m11ar we19ht5, 60th 5ma11er and 1ar9er def0rmat10n5 than the 0ne a550c1ated w1th e4u1116r1um, the 5ummed effect 6e1n9 5ma11. 7h15 p1cture 15 60rne 0ut 6y the re5u1t5 0f the ca1cu1at10n5 d15p1ayed 1n f19. 2.7here 15 a 519n1f1cant 1ncrea5e 1n the w1dth 0f the 6DR 0f 9°2r w1th exc1tat10n ener9y. 0n the 0ther hand, the d1p01e 5tren9th 0f ~64Er d15p1ay5 tw0 peak5 and thu5 a 512ea61e w1dth at 7= 1= 0. 7h15 15 6ecau5e 164Er ha5 a 5ta61e def0rmat10n (e=0.26, y=0 °) 1n the 9r0und 5tate. At 7= 1.2 MeV and 1=21 h, 60th peak5 6ec0me 50mewhat6r0ader due t0 the avera91n9 effect 0f therma1 f1uctuat10n5. 7he t0ta1 w1dth 15, h0wever, 11tt1e affected 6e1n9 5t111 c0ntr011ed 6y the 5eparat10n 6etween the peak5 and ref1ect1n9 the m05t pr06a61e 5tat1c def0rmat10n (e = 0.19, y = 0 ° ). Ca1cu1at10n5 0f the 6DR u51n9 a pure harm0n1c 05c111at0r p0tent1a1 a5 the 51n91e-part1c1e ham11t0n1an [14] are 5h0wn 1n f19. 3.7he a550c1ated re5u1t5 d0 n0t d15p1ay Landau damP1n9. C0n5e4uent1y, the effect5 0f r0tat10n and temperature appear c1earer 1n th15 51mp1er m0de1. 70 06ta1n a fa1r de5cr1pt10n 0f the 6DR 1n the 9r0und 5tate, the re5u1t5 0f the ca1cu1at10n5 were f01ded 6y a 10rent21an funct10n w1th an emp1r1ca1 ener9y dependent w1dth ~2 [ 15 ] :2 8ecau5e 0fthe 5tr0n9 Landau damp1n9 pre5ent 1n the ca1cu1at10n5 w1th the N11550n p0tent1a1, the f01d1n9 pr0cedure w111 n0t 1mpr0ve the re5u1t5. 7heref0re we have n0t perf0rmed 1t 1n that ca5e. 223 V01ume 191, num6er 3 PHY51C5 LE77ER5 8 11 June 1987 60* ra 970~r0 MeV 0~k~hk ~ / 2(111(4 •,00 C 161~E r 7:0M v 0y) /•60* /50* /40* •1-30* 20* 10* "••-•" 0* 1" / 60* 04~ /50* /60• 0.4~ /50~ -10 ---~00 Y F19. 1.5trut1n5ky free ener9y 5urface5 a5 funct10n 0fthe def0rmat10n parameter5 f0r 9°2r at 7=1=0 (a) and at 7= 1.5 MeV, 1=7.5 h (6) and L64Er at 7=1=0 (c) and 7= 1.2 MeV, 1= 21 h (d). 7he e4u1p0tent1a111ne5 are repre5ented 1n 5tep5 0f0.5 MeV away fr0m the m1n1mum wh1ch 15 den0ted 6y a fu11 d0t. 6 10 06 02 06 02 902r 6 5 75 25 5 15 25 E (MeV) F19. 2. Ph0t0a650rpt10n cr055 5ect10n 1n the 1a60rat0ry 5y5tem f0r the 6DR 0f 9°2r and 164Er at the c0nd1t10n5 91ven 1n (1). A cranked N11550n p0tent1a1 15 u5ed a5 the 51n91e-part1c1e ham11t0n1an. f0r 7# 0 the t0ta1 cr055 5ect10n 15 ca1cu1ated acc0rd1n9 t0 e0. (2). F=F0(E/Ec) 1"9 • (4) 7he 4uant1ty F015 ch05en t0 f1t the exper1menta1 data at 7=1=0, wh11e Ec 15 the ener9y centr01d 0f the d1p01e 5tren9th d15tr16ut10n. 7he5e parameter5 are kept f1xed thr0u9h0ut the ca1cu1at10n5.7h15 15 50mewhat e4u1va1ent t0 a55um1n9 that the effect 0f the c0up11n9 0f 2p-2h 5tate5 t0 the 91ant v16rat10n d0e5 n0t chan9e ne1ther w1th temperature [ 16 ] n0r w1th 5p1n. 1n f195. 3a, 36 and 3d, 3e we c0mpare the 6DR 6u11t 0n the e4u1116r1um def0rmat10n at 2er0 and n0n2er0 exc1tat10n ener91e5.7he therma1 5hape f1uctuat10n5 are 1nc1uded 1n the re5u1t5 rep0rted 1n f195. 3c and 3f f0r 9°2r and 164Er, re5pect1ve1y. A re5ume 0f the d1fferent w1dth5 15 c011ected 1n ta61e 1. 51m11ar c0mment5 t0 th05e made 1n c0nnect10n 224 V01ume 191, num6er 3 PHY51C5 LE77ER5 8 11 June 1987 0.4 0,2 0 604 0.2 F: 0 ° ~ 5 15 25 5 15 25 5 15 25 E(MeV) a2 02 01 -01 -0.2 ..... 91 16/~ Er 7 =1.2 MeV, 1 =21f~ w1th0ut f1uct.---- w1th .... 5 1L0 1•5 2J0 25 E (Me•C) F19. 3. Ph0t0a650rpt10n cr055 5ect10n f0r the 6DR 1n the 1a60rat0ry 5y5tem ca1cu1ated w1th the harm0n1c 05c111at0r m0de1. (a) and (d) 5h0w the 6DR 6a5ed 1n the 9.5.0f9°2r and 164Er. 1n (6) and (e) the 6DR 15 6u11t 0n the e4u1116r1um def0rmat10n a550c1ated w1th the exc1ted 5tate5 7= 1.5 MeV, 1=7.5 h f0r 9°2r and 7= 1.2 MeV, 1=21 h f0r J64Er. 7he effect 0fthe 5hape f1uctuat10n5 can 6e n0ted 1n (c) and (f) wh1ch 5h0w the avera9e cr055 5ect10n (e4. (2)) f0r the ca5e5 91ven 6ef0re. (9) repre5ent5 the an9u1ar d15tr16ut10n f0r J64Er at 1= 21 h, 7= 1.2 MeV a550c1ated t0 the e4u1116r1um def0rmat10n (th1n 11ne) and the 0ne ca1cu1ated c0n51der1n9 5hape f1uctuat10n5 (th1ck 11ne). (A5 1t 15 very we11 kn0wn, the c0eff1c1ent a2 15 re1ated w1th the an9u1ar d15tr16ut10n, W(0), 0fthe 7-ray5 ar151n9 fr0m d1p01e decay re1at1ve t0 the 6eam ax15, 6y W( 0) 0c 1 + a2P2( c05( 0) ). ) 7a61e 1 Damp1n9 w1dth (FWHM) 0f a var1ety 0f 91ant d1p01e re50nance5. 1n c01umn5 4 and 5 the rat105 6etween the w1dth F 0f the 6DR re5p0n5e funct10n 6a5ed 0n a 5tate w1th temperature 7 and an9u1ar m0mentum 1 and 0n the 9r0und 5tate are rep0rted (cf. f19. 3). 7he ca1cu1at10n5 f0r the ca5e 0f m85n were carr1ed 0ut f01d1n9 the a550c1ate d1p01e re5p0n5e funct10n taken fr0m ref. [ 8 ] w1th a 10rent21an funct10n 0f w1dth P = 5 MeV wh1ch 15 a55umed t0 51mu1ate the c0up11n9 t0 2p-2h 5tate5. 1n the 1a5t tw0 c01umn5 the w1dth F(0) 0f the 6DR re5p0n5e funct10n 6a5ed 0n the 9r0und 5tate 15 5h0wn. 7he ca1cu1at10n5 f0r 9°2r and 2°85n are 06ta1ned e11m1nat1n9 the effect due t0 Landau damp1n9 0f the re5u1t 0f re1". [ 3]. where the c0up11n9 0f the 91ant v16rat10n t0 2p-2h 5tate5 c0nta1n1n9 a 5urface v16rat10n 15 c0n51dered. 7he w1dth 0f the 6DR 0f ~64Er re5u1t5 fr0m the c0nv01ut10n 0f the 5tren9th funct10n re5u1t1n9 fr0m the ca1cu1at10n 0f the v16rat10n 1n the def0rmed p0tent1a1 ( ¢ = 0.2 6 ) and a 10rent21an w1dth F = F 0 ( E/Ec ) ( cf. the text f0r m0re deta115 ). Nuc1eu5 7 (Mev) 1 (h) F(E*)/F(0) F(0)(MeV) exper1ment the0ry exper1ment the0ry 90 2r 1.5 7.5 2 1.4 4 4.6 1°85n 2 60 2.1 1.9 5 5 J64Er 1.2 21 1 0.9 8.2 7 225 v01ume 191, num6er 3 PHY51c5 LE77ER5 8 11 June 1987 w1th f19. 2 a150 app1y here. 70 6e added 15 that even 1f the a650rpt10n cr055 5ect10n5 f0r t64Er when therma1 f1uctuat10n5 are n0t 1nc1uded (f19. 3e) and when they are 1nc1uded (f19. 3f) are very 51m11ar, the ),- decay an9u1ar d15tr16ut10n 5h0w5 a pr0f0und chan9e 1n the 1a5t ca5e. 7h15 15 apparent 1n f19. 39 where the c0eff1c1ent a2 15 p10tted. 1t 15 ca1cu1ated fr0m the expre5510n a2 = 1 ( 3a10/tr1 ) (5) where t7 15 the t0ta1 d1p01e a650rpt10n cr055 5ect10n and t7~0 the 0ne a550c1ated w1th the d1p01e tran51t10n5 carrY1n9 n0 an9u1ar m0mentum w1th re5pect t0 the ax15 0f r0tat10n (a55um1n9 t0ta1 5p1n 1>> 1h). 7he therma1 avera9ed re5u1t (th1ck 11ne 0f f19, 39), ca1cu1ated fr0m the c0rre5p0nd1n9 d1p01e cr055 5ect10n, d15p1ay5 0vera11 a9reement w1th the exper1menta1 data [17]. 1t 1mp11e5 a d1m1nut10n 0f the an150tr0py w1th re5pect t0 the ca5e w1th n0 f1uctuat10n5 (th1n 11ne), a1th0u9h a pr01ate 5hape 0f the nuc1eu5 can 5t1116e 5u99e5ted. 1t 1511ke1y that the 5tudy 0f the •/-ray an9u1ar d15tr16ut10n5 15 the 5pec1f1c t001 t0 1earn a60ut the 5urface therma1 f1uctuat10n5 0f the c0mp0und nuc1eu5 [ 18 ]. We c0nc1ude that the marked var1at10n5 0f the w1dth 0f 6DR rec0rded a5 a funct10n 0f temperature and an9u1ar m0mentum are due t0 chan9e5 1n def0rmat10n and/0r t0 therma1 5hape f1uctuat10n5. 7he effect 0f the C0r10115 f0rce 15 n0t f0und t0 6e 1mp0rtant. Fr0m the kn0w1ed9e 0f the free ener9y 5urface5 a5 a funct10n 0f/and 71t 5eem5 thu5 p055161e t0 1nfer the a550c1ated ev01ut10n 0f the ph0t0a650rpt10n cr055 5ect10n. Reference5 [ 1 ] J.E.Draper et a1., Phy5. Rev. Lett. 49 (1982) 434; J. 8arrete and A.M. 5and0rf1, C0mm. Nuc1. Part. Phy5. 12 (1983) 57; 8. Haa5 et a1., Phy5. Lett. 8 120 (1983) 79; W. Henner1c1 et a1., Nuc1. Phy5. A 396 (1983) 329; A.M. 5and0rf1 et a1., Phy5. Lett. 8 130 (1983) 19; L. La22ar1n1 et a1., Phy5. Rev. Lett. 53 (1984) 1045; J.J. 6aardh0je et a1., Phy5. Lett. 8 139 (1984) 273; J.J. 6aardh~je et a1., Phy5. Rev. Lett. 53 (1984) 148; D. 5chwa1m et a1., Fr0nt1er51n nuc1ear dynam1c5, ed5. R.A. 8r0911a and C.H. Da550 (P1enum, New Y0rk, 1985) p. 33. 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