Boson expansion methods applied to a two-level model in the study of multiple giant resonances

arXiv:nucl-th/9811083v1 24 Nov 1998 Boson expansion methods applied to a two-level model in the study of multiple giant resonances C. Volpe Groupe de Physique Th´eorique, Institut de Physique Nucl´eaire, F-91406 Orsay Cedex, France Ph. Chomaz GANIL, B.P. 5027, F-14076 Caen Cedex 5, France M.V. Andr´es Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, Apdo 1065, 41080 Sevilla, Spain F. Catara Dipartimento di Fisica dell’Universit`a and INFN, Sezione di Catania, I-95129 Catania, Italy E.G. Lanza Dipartimento di Fisica dell’Universit`a and INFN, Sezione di Catania, I-95129 Catania, Italy and Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, Apdo 1065, 41080 Sevilla, Spain Abstract We apply boson expansion methods to an extended Lipkin-Meshkov-Glick model including anharmonicities in analogy with previous microscopic calculations. We study the effects of different approximations present in these calculations, among which the truncation of the hamiltonian and of the space, in connection with the study of the properties of two-phonon and three-phonon states. By comparing the approximate results on the spectrum with the exact ones we conclude that the approximations made in the microscopic calculations on two-phonon states are well justified. We find also that a good agreement with the exact results for the three-phonon state is obtained by using a bosonic hamiltonian truncated at the fourth order. This result makes us confident that such approximation can be used in realistic calculations, thus allowing a theoretical study of triple excitations of giant resonances. 1 1 Introduction Collective excitations have been known for many years in nuclear physics [1], both in the low-lying spectra and in the Giant Resonance (GR) region. A basic microscopic theory for such collective modes is the Random Phase Approximation (RPA) [2, 3] which can be seen as the lowest order in a boson expansion such that the hamiltonian can be put in the form of a sum of hamiltonians of harmonic oscillators, each one corresponding to a collective mode (phonon). Therefore, RPA predicts the existence of one-phonon, two-phonon,...etc states with a harmonic spectrum. In addition to the well known low-lying two-phonon states, recently heavy ion inelastic scattering experiments at intermediate and relativistic energies and double charge-exchange reactions have shown the existence of states in the high excitation energy region which can be described as a GR built on top of another GR [4, 5]. The study of the properties of these states allows to test our comprehension of the GR’s as small amplitude vibrations and therefore the harmonic picture. The systematics on the energies and widths is in qualitative agreement with the harmonic approximation, namely the energy of a doubly excited GR is nearly twice that of the single GR and its width is between √2 and 2 times larger. However, the inelastic cross sections, when calculated within the harmonic picture, are almost always smaller than the measured ones. In particular, the data concerning Coulomb excitation show a discrepancy ranging from 30% up to a factor 4, according to the nucleus studied. In order to understand the origin of this discrepancy, corrections to the harmonic approximation have been proposed [6] by including anharmonicities in the internal hamiltonian and non-linearities in the external field. As shown in [6], small anharmonicities in the excitation spectrum of the target nucleus can lead to a large enhancement of the Coulomb excitation cross section. The model used there was an oversimplified one, namely the target was described as an anharmonic linear oscillator. The parameters of the cubic and the quartic terms in its hamiltonian were fixed so that the energy of the second excited state was shifted down by ≈2MeV with respect to twice the energy of the first excited state. In [7] a 3-dimensional extension of this model was considered and similar effects coming from anharmonicities were found. We remark that, in the case of Coulomb excitation of 136Xe in the reaction 136Xe + 208Pb at E/A = 700 MeV, the experimentally observed peak, interpreted as the Double Giant Dipole Resonance (DGDR), is shifted by ≈ −2 MeV from that expected in the harmonic case [8]. A more refined study was performed in [9], where anharmonicities and non-linearities were included by starting from RPA and extending it by means of boson mapping techniques [3, 10]. Such microscopic approach was applied in [9] to realistic cases. In the case of 208Pb + 208Pb at E/A = 641 MeV, for example, it was found that the inclusion of anharmonicities and non-linearities gives rise to an enhancement of 30% of the cross section in the region of the DGDR, bringing the theoretical results closer to the experimental ones. Important contributions coming from other two-phonon states in the same energy region were also found. The studies reported in the above quoted papers have of course some limitations. The model used in [6] is clearly very schematic and has, among others, the drawback that it is based on a purely bosonic description so that Pauli blocking effects are not 2 included. The approach presented in [9] is certainly much more realistic and the effects of the Pauli exclusion principle are taken into account to some extent. However, for computational reasons, only one- and two-phonon states were considered there. The purpose of this paper is to present an analysis based on an extension of the Lipkin-Meshkov-Glick (LMG) model [11] including a residual interaction between the phonons, in analogy with the microscopic approach used in ref.[9]. This model has several advantages with respect to the others. Because of its group structure, this model is exactly solvable. Besides, it directly takes into account the Pauli principle. In this context, we discuss several approximations, corresponding to the cases considered in [6, 9] and, by direct comparison with the exact results, we test how severe are the limitations of these approximations. We focus on one hand on two-phonons states, studied in ref.[9] and on the other hand on the properties of three-phonon states. In fact, in order to have a more stringent test on the validity of the harmonic picture, experimental and theoretical studies of triple excitation of GR’s should be envisaged. From the theoretical point of view, the same approach as the one in [9] can be used, but this would imply huge calculations. In order to make them feasible in the context of boson expansion methods, one may consider an approximation in which the same fourth order hamiltonian used in [9] is diagonalized in a space containing up to 3- phonon states. We present a test on how well this approximation works, in the context of the extended LMG hamiltonian. We will see that the exact results are very close to the approximate ones in all the cases studied for both the second and the third excited states. The paper is organized as follows. In section 2 the LMG model is shortly reviewed and our extension introduced. In section 3 the boson mapping of this hamiltonian is presented. In section 4 the approximate results on the energy of the first, second and third excited states, obtained by truncating at different levels the mapping, are compared among themselves and with the exact calculations. 2 The model 2.1 The Lipkin-Meshkov-Glick model In the original Lipkin-Meshkov-Glick (LMG) model [11] a finite number Ω of particles can be arranged in two levels separated by εin energy. Ω different quantum states are available in each level. Each particle is therefore identified by two quantum numbers. The first σindicates if the particle is in the upper (σ= +) or in the lower (σ=−) energy level. The second quantum number sis related to the Ω different available quantum states in each level. We will consider systems whose hamiltonian can be written as function of the operators K+,K−,K0: K+= Ω X s=1 a† +,sa−,s (1) 3 K−= (K+)†= Ω X s=1 a† −,sa+,s K0=1 2 Ω X s=1 (a† +,sa+,s −a† −,sa−,s) where a† σ,s (and the hermitian conjugate aσ,s ) creates (annihilates) a particle in the quantum state (σ, s). Since the operators K+,K−,K0satisfy the SU(2) commutation relations : [K+, K−] = 2K0[K0, K±] = ±K±(2) they are often called the quasi-spin operators. According to (1), there can not be two particles in the same quantum state, that is the Pauli exclusion principle is included in the model. A system of Ω particles has 2Ωstates. However, since the hamiltonian is a function of the generators of SU(2) and K2is a Casimir operator of such algebra, the space can be separated into subspaces, each corresponding to an eigenvalue K. The state with all particles in the lower level, σ=−, is an eigenvector of K2and K0, belonging to their maximum and minimum eigenvalue, respectively. This state, with the ones generated by applying K+to it, are the elements of a subspace of dimension Ω + 1, that we will denote by |Ω/2, miwith m∈[−Ω/2,Ω/2]. Following (1) the original hamiltonian of LMG [11] can be written as : HLMG =ε¯ K0+V1K+K−+V2(K+K++K−K−) (3) where ¯ K0=K0+ Ω/2 (4) The ε¯ K0term can be viewed as corresponding to the Hartree-Fock part. The lowest eigenstate of K0corresponds then to the uncorrelated HF ground state |Ω/2,−Ω/2i. In order to stress the analogy with the microscopic calculations in [9], we will often denote by h(hole) a (−, s) state, i.e. a single particle state which is occupied in |Ω/2,−Ω/2i, and by p(particle) a (+, s) single particle state. Therefore, by looking at eq.(1), we can say that the residual interaction in (3) only contains particle-hole terms a† pa† p′ahah′and a† pa† hap′ah′terms, that is those usually included in RPA. If V1=V2= 0 then HLMG =ε¯ K0. The corresponding eigenvalues are nε with n∈[0,Ω] .From now on we will denote these states by |n >. The energy eigenvalues are equidistant and form a harmonic spectrum truncated at Ω + 1 levels. If V1or V2are different from zero, then the energy spectrum is not harmonic any more and only if V26= 0 the energy eigenvectors correspond to a mixing of |nistates. In fact, V1K+K−is diagonal in the |nispace. This term of ph −p′h′type does not mix states with different particle-hole numbers. On the contrary, the pp′−hh′type term, V2K2 ++K2 −mixes states with nand n±2 particle-holes. To conclude this part, we would like to recall that the LMG model is exactly solvable using group techniques and that it includes the Pauli exclusion principle. 4 2.2 An extension of the LMG model The LMG model in its original form already includes some anharmonicities, essentially those related to the fact that the Pauli principle is treated exactly. However, parts of the residual interaction are neglected, namely the pp′−p′′p′′′,pp′−p′′h,hh′−h′′p, hh′−h′′h′′′ ones, which were considered in the microscopic calculations [9] where the anharmonicities arising from them were also studied. In order to simulate them, we propose an extension of the LMG hamiltonian which is still quadratic in the K+,K− and K0operators : H=HLMG + ∆V=HLMG +V3(K+¯ K0+¯ K0K−) + V4(¯ K0−1) ¯ K0(5) The K+¯ K0term and its hermitian conjugate introduce a coupling between |niand |n±1iwhereas the last term shifts the energies of the |nistates, except those with n= 0 and 1. Therefore, the eigenstates |φα>of the hamiltonian (5) are superpositions of the |n > states |φα>=X n Xα n|n > (6) In section 4 we will compare the exact eigenvalues of the hamiltonian (5) with those corresponding to the bosonic hamiltonian obtained by mapping the fermionic one up to the fourth order. 3 The boson hamiltonian We apply boson expansion methods to the fermionic hamiltonian (5) and truncated the so obtained boson hamiltonian to second and fourth order. In ref.s[12, 13] a similar study was done for the original LMG hamiltonian (3). In the present case, the quadratic hamiltonian corresponds to RPA while the quartic can be directly compared with what was done in [9], where some further approximations were performed, as discussed below. Let’s take the normal ordered Holstein-Primakoff boson expansion of the SU(2) generators [12, 14, 15] : (K+)b=P∞ i=0 ai(b†)i+1bi (K−)b=P∞ i=0 ai(b†)ibi+1 (¯ K0)b=b†b (7) where ai= i X m=0 (−1)i−m m!(i−m)!(Ω −m)1/2(8) In the case of SU(2), the Hage-Hassan-Lambert mapping used in ref.[9] is equivalent to the Holstein-Primakoff mapping (7,8) [16] with the shifted operator ( ¯ K0)b. By this shift one eliminates terms linear in b†and bin the hamiltonian which corresponds to a redefinition of the mean field. Let us then consider the bosonic mapping of our extended hamiltonian (5) in the particular case V2=V1/2 which is the value used in the numerical applications 5 we present in the next section. By including all the terms of the expansion (7) which contribute up to the second order in the b†, b operators, we get the quadratic hamiltonian H(2) b=ε(1 + δ1)b†b+δ1ε 21−1 Ω1/2b†2+b2(9) where δ1= ΩV1/ε (10) is related to the strength of the particle-hole residual interaction. The hamiltonian H(2) bcan be written in diagonal form by introducing the Bogoliubov transformation [3] : b†=XQ†+Y Q b=XQ +Y Q†(11) and imposing H20 = 0 (12) with the condition X2−Y2= 1 (13) which guarantees the correct commutation relation hQ, Q†i= 1 (14) Thus one gets the RPA hamiltonian H(2) b=H11Q†Q(15) where H11 =ε(1 + δ1)(X2+Y2) + 2XY δ1ε1−1 Ω1/2 (16) The conditions (12,13) determine the Xand Y’s amplitudes as solutions of the set of equations :    X2−Y2= 1 ε(1 + δ1)XY +δ1ε 21−1 Ω1/2(X2+Y2) = 0 (17) Of course the terms in V3and V4do not appear at the quadratic order. In eq.(15), the H00 which corresponds to a shift of the eigenenergies was omitted. The violations to the Pauli exclusion principle introduced by truncating the boson expansion at the lowest order as well as a contribution coming from the residual interaction of the pp′−p′′p′′′,pp′−p′′h,hh′−h′′p,hh′−h′′h′′′ type can be partly introduced by going one step further, i.e. including all terms up to the fourth order, as it was done in ref.[9]. At this level, the terms in V3and V4will therefore enter. Their presence, together with the corrections for the Pauli principle, will give rise to an anharmonic bosonic hamiltonian. By mapping the hamiltonian of eq.(5) up to fourth order we get: 6 H(4) b=ε(1 + δ1)b†b+δ1ε 21−1 Ω1/2b†2+b2 +V3√Ω1−1 Ω1/2b†2b+b†b2+ (V4−V1)b†2b2 +δ1ε 21−1 Ω1/21−2 Ω1/2−1b†3b+b†b3 (18) which we rewrite in terms of the Q†and Qoperators as : H(4) b=H11Q†Q+H30(Q†3+Q3) + H21(Q†2Q+Q†Q2)+ H31(Q†3Q+Q†Q3) + H22Q†2Q2+H40(Q†4+Q4)(19) whose coefficients are given in the appendix (eqs.21-27), together with the new equations for Xand Y(20) coming again from the conditions (12,13). In eq.(19) the H00 was omitted as well as a term, linear in the Q†and Qoperators, which would introduce a redefinition of the mean field, in analogy with what was done in [9]. The quartic hamiltonian (19) corresponds to that used in [9] where, however, some approximations were introduced in order to make feasible the calculations in the realistic cases considered there. First, only one- and two-phonon states were considered. Therefore, the terms in H30,H31 and H40 were not effective. Second, the Xand Y amplitudes appearing in the fourth order hamiltonian were not recalculated but taken equal to those obtained at the second order, i.e. the RPA ones. In order to get an indication on how good these approximations on the space, on the hamiltonian and on the Xand Yamplitudes are, in the next section we will study them within the present schematic model. 4 Results and discussion First of all we have to fix the parameters entering in the hamiltonian (5). For the single particle energy we use the parametrization ε= 41/A1/3MeV. We take V2= 0.5V1 and the strength V1is fixed so that the first excited state lies at an energy around 80/A1/3MeV corresponding to the systematics of GDR in nuclei. This criterion gives V1= 1.2 MeV. We have studied the behaviour of the energies of the three lowest states, E1,E2and E3, as a function of V4for two values of V3, namely V3= 0 MeV (Fig.1) and V3= 0.25 MeV (Fig.2). The latter value for V3gives <2|∆V|1>≈1 MeV in analogy with the microscopic calculations [9]. Note that the sign of V3is irrelevant. As far as the sign of V4is concerned, we show results only for negative values, which give a downward shift of E2and E3with respect to 2E1and 3E1, respectively, i.e. the harmonic (RPA) values. In the figures we show the results obtained for Ω = 8. To compare the spectrum of the exact hamiltonian and its different boson expansions we study the energy differences (En−En−1) between the lower excited states. In RPA they have the common value 15.11 MeV for the considered parameters, independent of n, V3and V4. In the figures this value is represented by a dashed line. This should be compared with the exact results for the fermionic hamiltonian eq.(5), shown as solid lines. We see that, even for V3=V4= 0, the RPA results deviate from the exact 7 Figure 1: Energy differences as functions of V4for a fixed value of V3= 0 MeV. The ones corresponding to the fermionic hamiltonian (5) are plotted as solid lines. The others correspond to different approximations on the bosonic expansion of the hamiltonian (see text). ones. Namely, the first three differences obtained in the exact calculations are equal to 15.62 MeV, 14.38 MeV and 13.03 MeV, respectively. The agreement with the exact energies can be improved by using the quartic hamiltonian, eq.(19). When the bosonic hamiltonian is diagonalized in the space containing up to two-phonon states we get the results presented in Fig. 1 and 2 by dotted lines. We remark that now the first excited state is very close to the exact one, while for the second one there is a discrepancy of not more than 250 KeV. In the more realistic microscopic calculations presented in [9] a further approximation was introduced. Instead of reobtaining the Xand Yamplitudes by solving the equations analogous to eqs (20), the RPA amplitudes were used. In the present model, this corresponds to use in eq. (19) the solutions of eqs (17) rather than those of (20). The results obtained are indistinguishable from the dotted lines in Fig. 1 and 2. This supports the validity of the procedure used in [9]. 8 Figure 2: As in fig. 1, but for V3= 0.25 MeV. In order to test how much the truncation of the space affects the results, let us now enlarge the space up to three-phonon states. Of course, in this space, the quartic term H40 in (19) does not play any role. In the figures, we show the results corresponding to the complete quartic hamiltonian of eq.(19), with the Xand Yamplitudes solutions of eq.s (20) (dot-dashed lines). Comparing the energies of the first two states we see that the agreement with the exact ones is now almost perfect. In the enlarged space we can also study the third excited state which, in the harmonic limit, would correspond to a three-phonon state. In this case the approximate results differ from the exact ones by 1 MeV, which can be considered as a good approximation since the excitation energy of this state is about 40 MeV. We would like to point out that the results obtained in the same model space but corresponding to the quartic hamiltonian of eq (19) without the H31 and H30 term and with the use of the RPA Xand Yamplitudes, solution of eqs (17), (represented in figure 1 and 2 like solid circles) are practically coincident with the dot-dashed line. The use of an analogous approximation in realistic microscopic calculations would make them much simpler. In summary, we have seen that quite good results can be obtained for both the second 9