Brack, Matthias

Meyer, J.

Quentin, P.

English

Brack, M.

HAL-IN2P3

A sum rule description of giant resonances at finite temperature

University of Regensburg Publication Server

Volume 133B, number 5 PHYSICS LETTERS 22 December 1983 A SUM RULE DESCRIPTION OF GIANT RESONANCES AT FINITE TEMPERATURE J. MEYER Institut de Physique Nucl&re (et IN2P3), Universit~ Lyon-l, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne, France P. QUENTIN Laboratoire de Physique Thdorique, Universit~ de Bordeaux-I, Domaine du Haut-Vigneau, 33 170 Gradignan, France and M. BRACK Institut far Theoretische Physik, Universitiit Regenburg, 8400 Regensburg, Fed. Rep. Germany Received 28 September 1983 A generalization f the sum rule approach to collective motion at finite temperature is presented. The m 1 and m-1 sum rules for the isovector dipole and the isoscalar monopole lectric modes have been evaluated with the modified SkM force for the 2°spb nucleus. The variation of the resulting iant resonance energies with temperature is discussed. Recent experimental developments in the domain of light projectile induced reactions (e.g. P3' [ 1] ) as well as heavy ion reactions [2] make it important to quantitatively describe the behaviour of nuclear giant resonances when increasing the excitation energy. As a natural extension of the zero excitation energy case one may describe relevant strength functions in the statistical approximation through finite temperature RPA calculations. This has been performed for the di- pole mode in the schematic force model and recently with more realistic forces but within a non fully self- consistent approach [3]. The technical difficulties as- sociated with such RPA calculations provide an incen- tive to develop a tractable yet sufficiently accurate method to grasp the bulk of the phenomenon under study. In the zero temperature case the sum rule approach has been shown to be well suited to that purpose (see e.g. ref. [4] ). Moreover it has appeared that a semi- classical determination of some sum rules, such as the inverse energy weighted sum rule, yielded very good 0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) approximations to quantal (i.e. Hartree-Fock)  results [5,6]. The finite temperature generalisation f such an approach implies an extension of some basic results to the non zero excitation energy case. It is the aim of this letter to discuss uch a generalisation a d to provide some illustrations of the methods in the par- ticular cases of isovector dipole and isoscalar mono- pole electric modes. For a statistical mixture defined by its density matrix D = ~Pn In) (hi, (1) n whose dynamics is governed by a hamiltonian H with eigenvectors In) and eigenvalues En, the kth order mo- ment of the strength function associated to a given operator Q may be defined as mk(a)= ~ I(nlOlP)12Pn(Ep - En) k . (2) n,p¢n For all moments which can be expressed in the pure 279 Volume 133B, number 5 PHYSICS LETTERS 22 December 1983 state case as expectation values of operators Ok involv- ing a sum of products of powers of H and Q one still has in the non zero temperature case mk(Q) = tr(DOk). (3) This is the case for positive odd k values. One finds for instance ~ [(n tQ [p)12 Pn(Ep - E n) n,p =51tr{D[Q , [H, Q] ] } , (4) In the pure case the m_ 1 moment is known to be pro- portional to the static polarisability a associated to the operator Q. For a statistical mixture (1) corresponding to a canonical (or grand canonical) thermodynamical equilibrium, i.e. with en = z -1  exp( - /3En) ,  (5) it can be shown [5] that the static polarisability a* 1 is given by [(nlQlp)l 2 °t= 2 C Pnn*p -~p -- En The first term of the rhs in eq. (6) is equal to 2m_ 1- The two other terms are vanishing at zero temperature, leading thus back to the well-known result mentioned above. This is still the case however, even at finite tem- perature, whenever the operator Q has no diagonal matrix element between eigenstates of H, i.e. if Q breaks a symmetry of such states. These two extra terms are also vanishing if the set of states In) can be considered as resulting from a harmonic oscillator in the "coordinate variable" Q - Z n Pn (n [Q[n), due to the constancy of(n IQ In) in such a case. In so far as the RPA provides a reasonable stimate of the collective motion under study and the operator Q a satisfactory ansatz for the collective operator, it is expected that even when Q does not break any symmetry of the eigenstates of H, one has approximately i m_ 1 ~ ~0t. (7)  *t Of course the polarisability must be understood in the fin- ite temperature case as resulting from a constraint with re- spect to Q on the variational solutions corresponding tothe Helmholtz free energy (H)-TS and not to the energy (H>. We will discuss quantitatively below the validity of eq. (7) on a typical example. In what follows we will evaluate for the 208pb nu- cleus the temperature dependence of the m 1 and m_ 1 sum rules for the isovector dipole and isoscalar mono- pole electric modes. Calculations will be performed in the independent particle (Hartree-Fock) approxima- tion using the modified SkM force which has recently been shown [7] to provide an excellent description of both static and low energy dynamical properties. As for the operators Q we take with usual notation D = ~ "= Oiz i - -~  0 i z i , (8)  for the dipole mode (0 i being the third isospin compo- nent) and A M = ~ r 2 , (9) i=l  for the monopole mode. The m 1 sum rule is computed as indicated in eq. (4). One finds in the Hartree-Fock approximation h 2 ml(D) - 2m NZ + ¼ (t 1 + t2 ) f Pn (r)pp(r) d3r , (10a) 2h 2 m 1 (M) = ~ f [Pn(r) + pp(r)] r2d3r .  (10b) In this approximation, the evaluation of m 1 moments necessitates only the knowledge of the one-body re- duced diagonal (i.e. local) densities Pn(r) and Op (r). Their variational determination (minimizing the Helmholtz free energy) has been performed with the energy density method using the extended Thomas- Fermi kinetic energy density functional r[p] (up to fourth order in h) with four-parameter "modified Fermi" density profiles [8] pq(r) = PqO{1 + exp[(r -Rq)/~q]}~q , q = n, p. (11) The results of this semi-classical pproximation have been shown to very well reproduce on the average those of full Hartree-Fock calculations [7,8]. At finite temper- ature, a suitable modification of the extended Thomas- Fermi method has also been shown recently [9] to yield a good reproduction of microscopic Hartree- 280 Volume 133B, number 5 PHYSICS LETTERS 22 December 1983 Table 1 Parameters of the variational semi-classical density parameters a functions of the temperature T (in MeV) with the notation of eq. (11). The Pq0 (q = P, n) are reported in units of fm -3, the aq in fm while the "rq are dimensionless. The Helmholtz free energy F (in MeV) and the energy TS (in MeV), where S is the entropy, are also reported. T PpO OnO C~p an "rp "rn F TS 0 0.0622 0.0911 0.532 0.661 1.42 1 .57  -1608.9 0.0 1 0.0621 0.0909 0.537 0.663 1.42 1 .53  -1628.5 38.5 2 0.0615 0.0899 0.542 0.662 1.37 1 .42  -1685.2 149.5 3 0.0605 0.0882 0.544 0.648 1.27 1.22 -1778.0 333.5 4 0.0591 0.0857 0.543 0.627 1.11 0.96 -1908.4 602.0 Fock results [10], in particular at temperatures above T ~ 2.5 3 MeV where the single-particle (shell) effects are washed out. The parameters of the densities eq. (11) are displayed in table 1 for temperatures varying from 0 up to 4 MeV. At a temperature higher than ~3 - 4 MeV the grand canonical approximation cannot be considered to be valid any longer due to the importance of the particle emission phenomenon asquantitatively checked in recent calculations [11]. For the dipole mode one observes a very small decrease o fml (D  ) with an increasing temperature (~2% from 0 to 3 MeV). The variation of (r 2), and thence of m 1 (M), is already known [I0] to be also very slow (~5% from 0 to 3 MeV). To evaluate the m_ 1 moments we have also com- puted the corresponding polarisabilities a in a semi- classical approximation. Indeed a fully quantal deter- mination of a from constrained Hartree-Fock calcula- tions is technically feasible a priori but is plagued by serious accuracy problems (e.g. at zero temperature the a value so calculated for the dipole mode has been found in ref. [ 12] to be at variance with a direct evalu- ation from the RPA strength function by ~15% where- as the Thouless theorem ensures their equivalence). It is our opinion that the numerical errors on a in con- strained Hartree-Fock calculations are at least of the same order of possible shell effects on m_ 1- For technical details concerning the evaluation of m_ 1 moments the reader is referred to refs. [13,14] where the zero tem- perature valuation of such moments i discussed in detail. In the dipole case (n ID In) is vanishing for sym- metry reasons and thus the m_ 1 moment is exactly equal to the polarizability divided by 2. In the mono- pole case this is a priori no longer the case. However. the extra contribution to m_ 1 has been found to be negligible in the following approximate approach. The relevant one-dimensional Bohr hamiltonian has been evaluated in the scaling approximation (see, e.g. ref. [ 14] ) and diagonalized in a large harmonic oscillator basis. The resulting matrix elements (n [MIn)have then to be calculated. From the spectrum En, the probabili- ties Pn have been computed leading for all tempera- tures under consideration tovery small terms t3[~n Pn((nlMIn)2)- ( ~n Pn(n[Mln))2 3. Resulting m_ 1 moments for both the dipole and the monopoie modes (in the latter case the scaling approxi- mation was used) are reported in table 2. As a suggestive way of condensing the results for m 1 and m_ l ,  theE 1 energies E1 = (ml/m_l) 1/2 (12) are displayed in fig. 1. These energies for the 208pb nucleus provide at zero temperature good estimates Table 2 Inverse nergy weighted sum rules m_ 1 for the dipole (D) and monopole (M) modes as functions of the temperature T (in MeV). Results are reported in units of fin 2 MeV -1 for D and fm 4 MeV-1 for M modes. For the breathing mode, the com- K~al(in MeV) in the scaling approximation is also pressibility reported. T m_l(D) m I(M) K~ fl 0 7.11 2875 138.5 1 7.17 2930 136.8 2 7.31 3137 130.5 3 7.50 3491 121.5 4 7.96 4173 108.0 281 Volume 133B, number 5 PHYSICS LETTERS 22 December 1983 E l  (MeV) 14 12 10 ! I I I D M I I I I 1 2 3 4 T(MeV) Fig. 1. Energies E 1 (in MeV) as functions of the temperature T (in MeV) for the dipole (D) and monopole (M) modes. of the location of the giant resonance for both dipole [5] and monopole [14] modes. When increasing the temperature these energies are found to decrease. This result is qualitatively consistent with the sche- matic model results of ref. [3] (in our case, however, the variation is less marked). It is also in agreement with the conclusions of ref. [2] even though the pos- sible occurrence of such giant resonances atrather high spin may lead to a deformation of the resonance structure due to various broadening effects as sketched in the phenomenological approach of ref. [15]. In our calculations, when the temperature increases, the vari- ation o fE  1 results from at least three effects: (i) the vanishing of the particle-hole residual interaction as noted in ref. [3], (ii) a decrease of the stiffness param- eters, (iii) a possible side effect on m 1 from the in- crease of resonance widths with increasing tempera- ture. In the monopole case the above variation (i) is not effective since the global effect of the particle- hole residual interaction on the giant resonance energy at zero temperature is rather small and the variation of the stiffness parameter (i.e. the incompressibility) seems to be dominating. One of us (M.B.) is grateful to the Physics Institutes of the Universities at Lyon and Bordeaux for the warm hospitality extended to him during his visits. The two others acknowledge the use of the semi-clas- sical ETF codes, worked out in a collaboration of M. Brack with C. Guet and H.-B. H]kansson, prior to publication. R efegences [ 1 ] D.H. Dowell, G. Feldman, K.A. Snover, A.M. Sandorfi and M.T. Collins, Phys. Rev. Lett. 50 (1983) 1191. [2] J.O. Newton et al., Phys. Lett. 46 (1981) 1383; J.E. Draper et al., Phys. Rev. Lett. 49 (1982) 434. [3] D. Vautherin and N. Vinh Mau, Phys. Lett. 120B (1983) 261; and private communication. [4] (). Bohigas, A.M. Lane and J. MartoreU, Phys. Rep. 51 (1979) 267. [5 ] J. Meyer, P. Quentin and M. Brack, invited paper 7th Meeting on Nuclear physics (Aussois, 1983), report LYCEN 8302, p. C.13.1. [6] H. Krivine, C. Schmit and J. Treiner, Phys. Lett. l12B (1982) 281. [7] J. Bartel, P. Quentin, M. Brack, C. Guet and H.-B. H~kansson, Nucl. Phys. A386 (1982) 79. [8] M. Brack, C. Guet and H.-B. I-l~kansson, to be published; see also M. Brack, in: M~thodes math~matiques d  la physique nucl~aire,eds. M. Demeur, B. Giraud and P. Quentin (Universit~ Libre de Bruxelles, 1982) p. 251. [9] J. Barrel, M. Brack, C. Guet and H.-B. H°akansson, to be published. [10] M. Brack and P. Quentin, Phys. Lett. 52B (1974) 159; Phys. Scr. A10 (1974) 163; Nucl. Phys. A361 (1981) 35. [ 11 ] P. Bonche and D. Vautherin, private communication. [12] O. Bohigas, Nguyen Van Gian and D. Vautherin, Phys. Lett. 102B (1981) 105. [13] J. Meyer, P. Quentin and M. Brack, to be published. [14] M. Brack and W. Stocker, Nucl. Phys. A388 (1982) 230; Nucl. Phys. A, to be published. [15] M.E. Faber, J.L. Egido and P. Ring, Phys. Lett. 127B (1983) 5. 282 

A sum rule description of giant resonances at finite temperature

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