Damping of hot giant dipole resonance due to complex configuration mixing

Transcription

1 3 December 998 Physics Letters B Damping of hot giant dipole resonance due to complex configuration mixing Nguyen Dinh Dang a,b,,, Kosai Tanabe a, Akito Arima b,c a Department of Physics, Saitama uniõersity, 55 Shimo-Okubo, Urawa, Saitama, Japan b Cyclotron Laboratory, RIKEN, - Hirosawa, Wako, Saitama , Japan c Ministry of Education, Chiyoda-ku, Tokyo, Japan Received 5 June 998; revised 0 November 998 Editor: W. Haxton Abstract An approach is presented to study the width of the giant dipole resonance Ž GDR. at non-zero temperature T, which includes all forward-going processes up to two-phonon ones. Calculations are performed in 0 Sn and 08 Pb. An overall agreement between theory and experiment is found for the GDR width and shape. The total width of the GDR due to coupling of the GDR phonon to all ph, pp and hh configurations increases sharply as T increases up to T;3 MeV and saturates at T; 4 6 MeV. The uantal width GQ due to coupling to ph configurations decreases with increasing T. Itis almost independent of T if the contribution of two-phonon processes at T/ 0 is omitted. 998 Elsevier Science B.V. All rights reserved. PACS:.0.Pc; 4.0.Pa; 4.30.Cz Keywords: Single-particle levels; Thermal and statistical models; Giant resonances The giant dipole resonance Ž GDR. built on compound nuclear states Ž hot GDR., has been the subject of a considerable number of experimental and theoretical studies during the last fifteen years ŽSee wx for the reviews.. The width of the GDR built on the ground state Ž g.s. GDR. is composed mainly of Landau splitting, the escape width G, and the spreading width G x. The Landau splitting represents the distribution of the GDR over a number of harmonic oscillators Ž phonons. with freuencies around the GDR energy. It can be well described within the random-phase approximation Ž RPA. and amounts to only a small fraction of the total width. The escape width G arises from particle emission. It can be studied via coupling to the continuum and x is also small few hundreds kev. The main part of the total GDR width comes from the spreading width G Electronic address: dang@rikaxp.riken.go.jp On leave of absence from the Institute of Nuclear Science and Techniue, VAEC, Hanoi Vietnam r98r$ - see front matter 998 Elsevier Science B.V. All rights reserved. PII: S

2 ( ) N. Dinh Dang et al.rphysics Letters B due to coupling to complicated configurations such as ph and even more complex ones. The extension of the microscopic descriptions of the width of the g.s. GDR such as those in w,3x to non-zero temperature Ž T/ 0. has shown that all the three components of the GDR width depend weakly on temperature w4 7 x. This is in contradictory to the experimental systematic of the width of hot GDR, which increases rapidly at low excitation energies E ) Ž or temperature T.. At higher excitation energies the observed width increases slowly and even ) saturates at E G 30 MeV in tin isotopes w8 4 x. The increase of the width at T/ 0 can be described by the thermal fluctuations of nuclear shapes w5 7 x. They were also included in the recent adiabatic coupling model Ž ACM. w8 x, whose results agree well with the data from the inelastic a-scattering experiments in w4x for the GDR width in 0 Sn and 08 Pb within MeV -TF 3 MeV. The width of the GDR may depend noticeably on the angular momentum J if the latter reaches a rather high value JG 35 " at T,.5.8 MeV in a lighter 06 nucleus Sn w9 x. Recently we have shown that the coupling of the collective dipole vibration Ž GDR phonon. to the incoherent particle-particle Ž pp. and hole-hole Ž hh. configurations appearing at T/ 0 Ž the thermal damping., which is de-facto taking shape fluctuations into account, is decisively important for an adeuate description of the width s increase and its saturation w0 x. It has also been concluded that the uantal damping due to coupling to only ph configurations decreases slowly as T increases. The application of this approach in a systematic study of the hot GDR in 90 Zr, 0 Sn, and 08 Pb has shown a reasonable agreement with the experimental data w0 x. The higher-order graphs such as phmphonon-, p pmphonon-, hhmphonon- orrand two-phonon ones have not been included explicitly in w0 x, but rather effectively in the parameters of the model. The aim of the present letter is to study the contribution of these higher-order processes to the width of the hot GDR. For this purpose a complete set of approximate euations will be derived including all the forward-going processes up to two-phonon ones at T/ 0 and applied in numerical calculations in 0 Sn and 08 Pb. As the present paper is a further development of the approach in w0 x, the latter will be freuently uoted as I hereafter. We consider the same model Hamiltonian as in I for the description of the coupling of collective oscillations Ž phonons. to the field of ph, pp and hh pairs. This Hamiltonian is composed of three terms: Hs Ea a v Q Q F Ž. a a Q Q. Ž. Ý Ý Ý s s s ss s s s ss Ž. The first term in the r.h.s. of E. is the field of independent single particles, where as and as are creation and destruction operators of a particle or hole state with energy EssesyeF with es being the single-particle energy and ef - the Fermi surface s energy. We will simply call the energy Es as the single-particle energy. The second term stands for the phonon field, where Q and Q are the creation and destruction operators of a phonon with energy v. The last term describes the coupling between the first two terms. The indices s and s denote particle Ž p, E ) 0. or hole Ž h, E - 0., while the index is reserved for the phonon state sl,i4 p h with multipolarity l Ž the projection m of l in the phonon index is omitted in the writing for simplicity.. The sums in the last two terms in the r.h.s of E. Ž. are carried over lg. In order to include all the propagators up to two-phonon ones let us introduce the double-time Green functions in the standard notation w3 x, which describe the following processes: a. The propagation of a free phonon: G Ž. ²² Ž.::. Ž tyt s Q t ;Q t,b The transition between a nucleon pair and a phonon: G tyt. ; ss ; s²² a Ž. t a Ž. Ž t ;Q t.::,c. s s The transition between a nucleon pair m phonon configuration and a phonon: y, G Ž. ²² Ž. Ž. Ž. Ž t y t s a t a t Q t ;Q t.::, and d. ss ; s s The transition between two- and one-phonon configura- yy, tions: G Žtyt. s²² Q Ž t. Q Ž t.;q Žt.::. The effect of single-particle damping on the GDR width has been found in I to be rather small up to high T. Therefore we will not consider it here again. The, Ž. ²² Ž. Ž.::, backward-going processes G Ž. ²² Ž. Ž. Ž. Ž t y t s Q t ;Q t, G t y t s a t a t Q t ;Q t.:: ; ss ; s s,, and, G Žty t. s²² Q Ž. t Q Ž. t ;Q Žt.:: will be omitted as their effects on the damping of the GDR are expected to be negligible. Applying now the standard method of the euation of motion for the double-time Green function w3 x, we obtain a set of coupled euations for an hierarchy of Green functions. Employing the decoupling approximation discussed previously in I, we can close this set to the functions Ž. a Ž. d. Making then

3 ( ) N. Dinh Dang et al.rphysics Letters B the Fourier transformation to the energy plane E, we obtain a set of four euations for the Fourier transforms of the Green functions Ž. a Ž. d in the form: Ž Eyv G Ž E. y F Ẋ G Ž E. s, Ž. Ý ; ss ss ; ss p Ý Ž. y, Ž. y, Ž EyE E G E y F G E yf G s s. ss ; s s ss ; s s s s ;Ž E. s0, Ž 3. s Ý yy, Ž. y, Ž. y, Eyv yv G E y F G E F G ; ss ss ; ss ss ; E s0. 4 ss Ý Ý y, EyE E yv G Ž E. y yn n F Ž. G Ž E. Ž n n. F Ž. G Ž E. s s ss ; s s s ss ; s ss ss ; s s Ý yž n yn F Ž. G yy, s s. s s ; Ž E. s0, Ž 5. The first three euations in this set are exact, while the last E. Ž. 5 is approximated due to the decoupling scheme mentioned above. The latter leads to the single-particle occupation number ns and phonon occupation number n, whose explicit expressions have been derived in I. Eliminating G Ž E. from Es. Ž. and Ž. ss ; 3, we y, obtain one exact euation, which relates G E to G Ž E., in the form ; ss, Ž. Ž. y, Ž. y, F F G E yf G ss ss ss ; s s s s ;Ž E. Eyv G; Ž E. y s. Ž 6. EyE E p Ý ss s s yy, y, Eliminating now G E by expressing it in terms of G Ž E. using E. Ž 4. ; and inserting the result into E. Ž. y, 5, we come to the an approximate euation, which relates functions G Ž E. to G Ž E. ss,. Making again the decoupling for all the Green functions under the sums in this euation, which truncates the chain at the second wž Ž.. x Ž. y, order O F of the interaction strength F, we can express G Ž E. in terms of G Ž E. as ss ss ss ;, Ý y, EyE E yv G Ž E. s M Ž E. G Ž E.. Ž 7. s s ss ; ss ; y, Inserting G Ž E. from the approximate E. Ž. 7 into the exact E. Ž. ss ; 6, we end up with the final euation for the propagation of a single phonon Ž s. in the following form G,Ž E. s. Ž 8. p EyvyPŽ E. Ž. Ž. The polarization operator P E and the vertex function M E in Es. 7 and 8 are ss Ý Ž. Ž. Ž. F F M E F M ss s s ss s s s sž E. P Ž E. s y, Ž 9. EyE E EyE E yv EyE E yv ss s s s s s s s ½ Ž yn n.ž n yn. Ž n n.ž n yn s s s s s s. Ž. Ž. Ž. Ž. Ý ss ss ss ss ss s EyEs Es EyEs E s Ž. Ž. Ž. Ž. F F F ss s s ss F ss s s s Ý 5 Eyv yv Eyv yv M E s F F y F F s n Ž n yn. d. Ž 0.

4 4 ( ) N. Dinh Dang et al.rphysics Letters B The damping width GGDR of the hot GDR located at energy vsvgdr is defined via the imaginary part of the analytical continuation of the polarization operator P Ž E. into complex energy plane E s v" i as G sg Ž v. < simp < Ž v"i. <. Ž. GDR vsvgd R vsv GDR The polarization operator P Ž E. includes all ss and ss mphonon processes ŽŽs,s. sž p,h., Žp, p. and Žh,h.. and two-phonon ones at the same second order in the interaction strength. In the limit of high T it is easy to verify that the vertex function M Ž y E in Es. 9 and 0 decreases as OT. ss with increasing T y because of the factor T in front of two-phonon terms on the r.h.s. of E. Ž 0.. Neglecting these two-phonon terms would lead to a constant width at high temperature because the first two terms on the r.h.s. of E. Ž 0. are independent of T under the assumption that the temperature dependence of the interaction, of the phonon energy, and of the difference E s y Es is negligible. The decrease of the uantal width GQ of the GDR as OT Ž y. at high T has also been estimated analytically in the SRPA in w4 x. Although we proceeded the calculations up to TF 6 MeV, the high T-limit indicates the decreasing trend of GQ with increasing T as will be seen below The calculations of the GDR width G from E. Ž. GDR have been carried out in Sn and Pb within the interval 0FTF 6 MeV. The schematic model in I is employed, according to which the g.s. GDR is generated by a single collective and structureless phonon width energy v closed to the energy EGDR of the g.s. GDR. The realistic single-particle energies, calculated in the Woods-Saxon potentials at Ts0 for 0 Sn and 08 Pb, were used in calculations. The calculations in w8x have shown that the major contribution of the shape fluctuations in the increase of the GDR width at T/ 0 comes from the uadrupole shape fluctuations. Therefore, we retain here only dipole and uadrupole phonons in the two-phonon configuration mixing for simplicity. Conseuently, from the sums on the r.h.s. of Es. Ž. 9 and Ž 0. there remain only one dipole phonon with s, which corresponds to the GDR ls and one uadrupole phonon with, which corresponds to the energy of Ž. Ž. state ls. The parameters of the model have been selected as follows. We first set the ratio rsfi rfi Ž. is, and choose v in E. 0 to be close to E. The values of v in E. 0 and of F are then selected in such a way that the solution v of the euation for the pole of the Green function in E. Ž. 8 vyv yp Ž v. s0 is eual to the GDR energy vse while the total width G Ž v. from E. Ž. GDR GDR Ž. reproduces the empirical width of the GDR at Ts 0. The value of F is chosen so that v is stable while T is varied. For 0 Sn the set of parameters has been found as: v s7.0 MeV, F Ž. s 6.6=0 y3 MeV, F Ž. s.845=0 y MeV, and rs 8.603=0 y. For 08 Pb the values v s3.8 MeV, F Ž. s.7=0 y3 MeV, F Ž. s 9.95=0 y MeV, and rs 8.8=0 y have been selected. These parameters are kept unchanged with changing T in the present paper. This ensures that all thermal effects come from the microscopic configuration mixings, but not from varying parameters. In fact, as has been shown, e.g. in wx 5, the first uadrupole state within the RPA has a considerable fragmentation with increasing T. This offers some possibility to finely tune the parameters of the model, which we will not consider in this stage. The inclusion of higher multipolarities such as octupole phonons in 08 Pb, etc. is also expected to improve the results. However it would certainly make the calculations more complicate. The calculations have used a value eual to 0.5 MeV for the smearing parameter in E. Ž.. The results have been checked to be stable against varying within the interval 0. MeVF F.0 MeV The total widths G, calculated in Sn and Pb as a function of T, are shown in Fig. Ž solid. GDR. An overall agreement is found between theory and recent data from the inelastic a-scattering w4x and heavy-ion 0 fusion experiments w8,0, x. In both nuclei the region of width s saturation is at around T;4 6 MeV. In Sn 08 the saturated value of the width is about MeV in agreement with the data from w8 3 x. In Pb it is around 0.5 MeV. As has been demonstrated in I, the total width GGDR is composed of the uantal width GQ due to coupling of the GDR phonon to ph configurations and the thermal width GT due to coupling to pp and hh configurations at T/ 0. The main conclusion of I is that the behavior of the total width at high temperatures is mostly driven by the thermal width GT since the uantal width GQ decreases as temperature increases. In order to see whether this conclusion still holds within the present more refined approximation we have also switched

5 ( ) N. Dinh Dang et al.rphysics Letters B Fig.. GDR width as a function of temperature for Sn Ž. a and Pb Ž. b. Solid: total width G GDR; dashed: uantal width G Q; dash-dotted: uantal width G when the contribution of the two-phonon graphs at T / 0 is omitted. Diamonds: data of w4 x Q ; suares and triangles: data by Enders et al. and Hofmann et al. w x, respectively; cross and asterisk: data from w8x and w0 x, respectively. A level ) density parameter ; Ar8 has been used to translate E to T in fusion data of w8,0x off the coupling to pp and hh configurations in the sums on the r.h.s. of Es. Ž. 9 and Ž 0.. The results obtained are shown by the dashed curves in Fig.. They restore perfectly the results for uantal width GQ in I, which show a clear decrease as T increases. Switching off the two-phonon terms at T/ 0 from E. Ž 0. in calculating the width GQ has resulted in a uantal width, which is practically independent of T as shown by the dash-dotted curves in Fig. in agreement with the conclusion within the NFT at T/ 0 w6,5 x. These results show that the NFT indeed includes the graphs, which are most important for an adeuate description of the uantal width GQ at Ts 0, namely the ph and phmphonon ones. However, in calculating G Q, the NFT neglects entirely the coupling to two-phonon configurations which enter in the same second order of the Ž interaction strength F. ss. These two-phonon processes at T/ 0 indeed lead to the decrease in the uantal width with increasing T as has been mentioned above. Within the NFT, where the phonon operator has a ph nature, the exclusion of two-phonon terms was necessary to avoid double counting. We also notice that the NFT solved an RPA euation at T/ 0 with the sum carried over mainly n yn as has been shown by E. Ž 4. of wx h p 6. The self-energy graphs within the NFT at T s 0 described just the damping with ph-vibration doorways as has been shown by Fig. 6 and Es. Ž B8. and Ž B9. of wx 3. The deformation of Fermi surface at T/ 0 within NFT leaded to some ground-state correlation graph and single-particle renormalization, whose effect on the GDR damping are known to be small ŽFigs. 5, 6, and Es. Ž. Ž 4. of wx. 6. Shown in Fig. are the GDR shapes calculated within our model ŽŽ a. and Ž b.. using the GDR strength GDR y w Ž. function S v s p g v r v y v g v x and those obtained within the ACM w8x ŽŽ c. and Ž d... The areas of experimental divided spectra as well as the results from the ACM have been taken from w6 x. The experimental spectra at low Ž MeV. and high Ž 0 0 MeV. excitation energies are put in a one-to-one

6 6 ( ) N. Dinh Dang et al.rphysics Letters B Fig.. GDR shapes in Sn at low Ž MeV. and high Ž 0 0 MeV. excitation energies. In each panel the vertical bars denote the area of experimental divided spectra. Results of the present work are plotted in Ž. a and Ž. b, where dashed curves show the shapes calculated without an additional parameter Dg Ž see the text. while solid curves denote those obtained by adding Dg to g Ž v. ŽDgs 0.5 MeV in Ž a. and MeV in Ž.. b. Results of ACM using a strength parameter Ss Ž dashed. and 0.8 Ž solid. are plotted in Ž. c and Ž. d. correspondence with the data of w4x at T s.4 MeV and 3. MeV, respectively. The calculated shape Ždashed curves in Figs. Ž a. and Ž b.. is slightly narrower than the experimental one. Among the possible reasons there are the effects of mixing with more complex configurations, other multipolarities, angular momentum w9 x, evaporation width w7x at high T, and coupling to continuum, which are not considered in the present scheme. An effective way to take into account these contributions is to add a smearing parameter Dg to the damping g Ž v. to recover the value of the total width at high T obtained in I. By doing so the a better agreement between theory and experiment is achieved Žsolid curves in Figs. Ž a. and Ž b... It is seen in both cases that our approach offers a reasonable agreement with the observed GDR shape in both regions of low and high excitation energies while the calculated strength function in the ACM does not follow a Lorentzian-like shape at high E ). Reducing the value of the GDR energy weighted sum rule value by 0% helped the ACM to improve the agreement with data at high E ) but worsened it at low excitation E ). In summary we have developed an approach to study the width of the GDR as a function of temperature, which includes all the forward-going processes up to two-phonon ones in the second-order of the interaction strength. The present paper show that: Ž. the total width GGDR of the hot GDR arises mainly from the coupling of the GDR vibration to all ph, pp and hh configurations. It increases sharply as temperature increases up to T;3 MeV. At higher temperatures the width increase is slowed down to reach a saturated value of around 0 08 MeV in Sn and around 0.5 MeV in Pb at T;4 6 MeV; Ž. the uantal width GQ of the GDR due to coupling of GDR vibration to only ph configurations decreases as T increases. Neglecting the two-phonon processes in the expansion to higher-order propagators results in a uantal width, which is almost independent of T; Ž. 3 the calculated GDR shape in our model agrees reasonably well with the data. The numerical calculations are limited within a schematic case whose parameters were selected to reproduce the empirical values of the GDR width and its energy at Ts 0. More refined microscopic studies in this direction are highly

7 ( ) N. Dinh Dang et al.rphysics Letters B desirable since the actual complexity of the wave functions can reveal the detail relationship between the eigenstates of the model Hamiltonian and the representation basis, and, therefore, may shed more light on the physics of the problem. Other ingredients such as the temperature dependence of the single-particle energies, the contribution of the evaporation width, the effects of high values of the angular momentum have been also left out in the present study. While there have been some indications on that these effects may not be crucial within the temperature region under consideration andror in nuclei with mass number A G 0 w,4,9,6 x, they certainly deserve thorough studies in the future. Acknowledgements N.D.D. acknowledges the financial support of the Japanese Society for the Promotion of Science. References wx K. Snover, Annu. Rev. Nucl. Part. Sci. 36 Ž ; J.J. Gaardhøje, Annu. Rev. Nucl. Part. Sci. 4 Ž wx V.G. Soloviev, Theory of Atomic Nuclei Quasiparticles and Phonons, IOP, Bristol, 99. wx 3 G.F. Bertsch, P.F. Bortignon, R.A. Broglia, Rev. Mod. Phys. 55 Ž wx 4 H. Sagawa, G.F. Bertsch, Phys. Lett. B 46 Ž wx 5 N. Dinh Dang, J. Phys. G Ž 985. L5. wx 6 P.F. Bortignon et al., Nucl. Phys. A 460 Ž wx 7 N. Dinh Dang, Nucl. Phys. A 504 Ž wx 8 J.J. Gaardhøje et al., Phys. Rev. Lett. 53 Ž ; 56 Ž ; 59 Ž wx 9 D.R. Chakrabarty et al., Phys. Rev. C 36 Ž w0x A. Bracco et al., Phys. Rev. Lett. 6 Ž ; 74 Ž wx G. Enders et al., Phys. Rev. Lett. 69 Ž ; H.J. Hofmann et al., Nucl. Phys. A 57 Ž wx J.H. Le Faou et al., Phys. Rev. Lett. 7 Ž w3x D. Perroutsakou et al., Nucl. Phys. A 600 Ž w4x E. Ramakrishnan et al., Phys. Rev. Lett. 76 Ž ; T. Baumann et al., Nucl. Phys. A 635 Ž w5x M. Gallardo et al., Nucl. Phys. A 443 Ž w6x N. Dinh Dang, J. Phys. G 6 Ž w7x Y. Alhassid, B. Bush, S. Levit, Phys. Rev. Lett. 6 Ž ; Y. Alhassid, B. Bush, Nucl. Phys. A 509 Ž ; A 53 Ž 99., 39; B. Bush, Y. Alhassid, Nucl. Phys. A 53 Ž w8x W.E. Ormand, P.F. Bortignon, R.A. Broglia, Phys. Rev. Lett. 77 Ž ; W.E. Ormand et al., Nucl. Phys. A 64 Ž w9x M. Matiuzzi et al., Nucl. Phys. A 6 Ž w0x N. Dinh Dang, A. Arima, Phys. Rev. Lett. 80 Ž ; Nucl. Phys. A 636 Ž wx N. Dinh Dang, F. Sakata, Phys. Rev. C 57 Ž ; C 55 Ž wx N. Dinh Dang, Phys. Rep. 64 Ž w3x D.N. Zubarev, Sov. Phys. Uspekhi 3 Ž w4x K. Tanabe, Nucl. Phys. A 569 Ž c. w5x P. Donati et al., Phys. Lett. B 383 Ž w6x G. Gervais, M. Thoennessen, W.E. Ormand, Phys. Rev. C 58 Ž R. w7x Ph. Chomaz, Phys. Lett. B 347 Ž 995..