Anharmonic vibrations in nuclei

Nuclear Physics A 729 (2003) 699–712 www.elsevier.com/locate/npe

Anharmonic vibrations in nuclei M. Fallot a,∗,1 , Ph. Chomaz b , M.V. Andrés c , F. Catara d , E.G. Lanza d , J.A. Scarpaci a a Institut de Physique Nucléaire, IN2P3-CNRS, F-91406 Orsay Cedex, France b GANIL, B.P. 5027, F-14076 Caen Cedex 5, France c Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla,

Apdo 1065, E-41080 Sevilla, Spain d Dipartimento di Fisica ed Astronomia, Universitá di Catania and INFN-Catania,

Via S. Sofia 64, I-95100 Catania, Italy Received 18 February 2003; received in revised form 24 September 2003; accepted 1 October 2003

Abstract We show that the non-linearities of large amplitude motions in atomic nuclei induce giant quadrupole and monopole vibrations. As a consequence, the main source of anharmonicity is the coupling with configurations including one of these two giant resonances on top of any state. Twophonon energies are often lowered by one or two MeV because of the large matrix elements with such three phonon configurations. These effects are studied in two nuclei, 40 Ca and 208 Pb.  2003 Elsevier B.V. All rights reserved. PACS: 21.60.Ev; 21.10.Re; 21.60.Jz; 24.30.Cz Keywords: Anharmonic vibrations; Giant resonances; Two-phonon states; Three-phonon states

1. Introduction Many-body fermionic systems possess collective vibrational states which are well described as bosonic modes (phonons). The existence in atomic nuclei of such states, both low-lying and Giant Resonances (GR) is now well established up to the second quantum [1,2]. However, their properties such as energy and excitation probability are still open questions. From the experimental point of view the strong excitation cross * Corresponding author.

E-mail address: [email protected] (M. Fallot). 1 Present address: Subatech, 4 rue Alfred Kastler BP 20722, F-44307 Nantes cedex 3, France.

0375-9474/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2003.10.001

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M. Fallot et al. / Nuclear Physics A 729 (2003) 699–712

section of two phonon states calls for the presence of large anharmonicities but up to now, all the theoretical estimates were pointing to weak deviations from a harmonic spectrum. For a recent study of the mass dependence of the energy shift of the double giant dipole resonance, see Ref. [3]. Vibrational modes are fairly well described by the Random Phase Approximation (RPA). In RPA, the fermionic Hamiltonian is mapped onto a sum of independent harmonic oscillators corresponding to each collective mode. Thus the RPA states are pure one-phonon or multiphonon states and their energy is the sum of the energies of the single phonons. Anharmonicities arise when further terms from the fermionic Hamiltonian are considered in the boson expansion. The eigenstates of the new bosonic Hamiltonian are no more pure multiphonons but superpositions of them, with different numbers of phonons. To our knowledge, only the mixing of one- and two-phonon states has been considered in microscopic calculations so far, with two exceptions. In Ref. [4] the coupling to some specific three-phonon configurations has been included as a mechanism generating the damping width of the Double Giant Dipole Resonance. In Ref. [5] the fragmentation of the doubly excited low lying octupole states in 208Pb has been studied by allowing the coupling to one- and three-phonon configurations with a low energy cut-off introduced to reduce the diagonalization space. For this reason monopole, (GMR) and quadrupole (GQR) contributions which, as we will see, play an important role, were neglected. The present study has been motivated also by the results obtained in an extended, exactly solvable, two level Lipkin–Meshkov–Glick (LMG) model [6]. There we have applied boson expansion methods to the extended LMG Hamiltonian and truncated the boson Hamiltonian to second and fourth order. The truncations of the space and of the Hamiltonian have been studied by comparing the results with the exact ones. This analysis has shown that a quartic Hamiltonian diagonalized in an enlarged space including up to three-phonon states produced results which are very close to the exact ones. In this paper, following Ref. [6] we diagonalize a microscopic quartic Hamiltonian in the space of two- and three-phonon states and we show that a correct description of the states whose main component is a two-phonon configuration requires the inclusion of oneand three-phonon ones. We stress the essential role played by the breathing mode in the nuclear anharmonicity as an important novelty of the present analysis since volume modes are usually not considered in damping or coupling mechanisms. Moreover, we will show that the very collective GQR plays also an important role.

2. The model The starting point of our calculation is a mapping of the fermion particle-hole operators † ap† ah into boson operators Bph as, for example, the one proposed in Ref. [7] † ap† ah → Bph + (1 −

ap† ap →

h

√ † 2) Bp h Bp† h Bph + · · · , p h

† Bph Bp h , ah ah† →

p

† Bph Bph ,

(1) (2)


701

where a † (a) creates (annihilates) one nucleon in an occupied (h) or unoccupied (p) single particle state. The second term on the right-hand side of Eq. (1) is a correction that takes care of the Pauli principle. Then we construct a boson image of the Hamiltonian, truncated at the fourth order in the B † and B operators. HB = H10B † + H11 B † B + H20 B † B † + H21 B † B † B + H22B † B † BB + H31 B † B † B † B + h.c.,

(3)

where we have dropped the indices. The term H10 vanishes in a Hartree–Fock basis. The quartic terms come from the Pauli √ principle corrections. The last term in the bosonic Hamiltonian is of the order of 1/ Ω with respect to that in H21 , where Ω is the number of active single particles states [6] and it couples states differing by two bosons. Therefore, it can be neglected. Introducing the Bogoliubov transformation for bosons: † ν ν Xph (4) Bph − Yph Bph Q†ν ≡ p,h

and imposing that the quadratic part of the boson Hamiltonian in the new operators is diagonal, we obtain the usual random phase approximation (RPA) equations for the X and Y amplitudes. By inverting Eq. (4) and its adjoint, HB can be expressed in terms of the collective Q† and Q operators HB = H11Q† Q + H21 Q† Q† Q + h.c. + H22 Q† Q† QQ + H30 Q† Q† Q† + h.c. + H31 Q† Q† Q† Q + h.c. + H40 Q† Q† Q† Q† + h.c. + · · · (5)

with Hνν 11 = Eν δνν . The H matrices are expressed in terms of the X and Y of transformation (4). The leading terms of the parts of HB in the first line (H11 , H21 , H22 ) do not contain any Y amplitude. On the contrary, in the other parts of HB at least one Y appears. Therefore, since the ratios Y/X are small in closed shell nuclei, one can neglect these terms (i.e., H30 , H31 , H40 ) and retain only those of the first line. In Ref. [6] a similar analysis was done in the two level Lipkin model. It was found that this approximation is well justified and one gets good results in the larger space for the eigenstates whose main component is a two-phonon configuration. Here, we will compare the spectra of 40 Ca and 208 Pb obtained by the diagonalization in the spaces containing up to two-phonon states and up to three-phonon states, respectively.

3. Calculations and results All calculations have been performed by using the SGII Skyrme interaction [8]. We include all natural parity RPA collective one-phonon states with angular momentum J 3 which exhaust at least 5% of the EWSR and all two- and three-phonon configurations built with them, without any energy cut-off, with both natural and unnatural parity. The Hamiltonian Eq. (5) is diagonalized in the space spanned by such states.

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3.1. Results for 40 Ca In Table 1 we show the one-phonon states taken into account. They are the same as used in Ref. [9] where only one- and two-phonon states were considered. In the present paper we extend the previous calculations and diagonalize the Hamiltonian in the space of the states with up to three phonons. In Table 2 we show some results of the diagonalization for a selected set of states. The energies obtained in the space up to two phonons are, of course, equal to those found in Ref. [9] and are reported here just for comparison. The conclusion of this diagonalization limited to the two-phonon states was that the anharmonicities were of the order of a few hundred keV. Let us now study the more complete calculation, including the three-phonon states. As a general comment, one can say that the shift induced by the coupling to threephonon states is fairly large, being in almost all the cases more than 1 MeV, and always downward. This can be understood in second order perturbation which, as can be seen from the table, gives a good estimate for the energies in most cases. In second order perturbation the correction to the energy is given by | ϕj |V |ϕi |2 , (6) Ei = ϕi |V |ϕi + Ei0 − Ej0 j =i where |ϕi is the considered unperturbed state, |ϕj all the other states and E 0 the corresponding unperturbed energies. Since the diagonal, first order, contribution is small in most cases, the sign of the shift Ei is that of the denominator in the second order term. Therefore, if |ϕi is a two-phonon state, the contributions from three-phonon configurations are negative in most cases since most of the three phonon states lye above the two phonon ones. Moreover, whenever a GMR is added on top of any state, the H21 terms (see Eq. 5) are large, of the order of 1 to 2 MeV in 40 Ca. The specific values can be found in Table 1. This strong coupling of all collective vibrations with the breathing mode comes from the fact that in a small nucleus such as 40 Ca any large amplitude motion affects the central density. Therefore, surface modes cannot be decoupled from a density variation in the whole volume as clearly seen in recent TDHF simulations [10]. If the state is a two-phonon one, then the matrix elements coupling it to the state obtained by exciting a breathing mode on top of it are about 3 MeV (up to 5.5 MeV) when the less (more) collective component of the 40 Ca GMR is considered. Even larger matrix elements are obtained, when the states connected by H21 involve several GMR. In that case a Bose enhancement factor appears and all Clebsch–Gordan coefficients entering in the calculation are equal to one. Thus √ the matrix element between the double and the triple GMR located at 18.25 MeV, M1 , is 6 times larger than between the single and the double M1 . That gives a matrix element of −5.22 MeV, and a contribution of −1.49 MeV to the second order energy correction of the double M1 state. An even larger value comes out in the case of the double GMR located at 22.47 MeV, M2 , and the triple M2 , namely, a matrix element of −9.69 MeV giving a −4.18 MeV contribution to the energy shift of the double M2 . This is due to the fact that M2 is more collective than M1 in 40 Ca. Something similar, but less strong, happens also for the matrix elements connecting some state with that built by adding one GQR phonon. We quote two examples. The lowlying component of the giant dipole resonance |D1 has a matrix element of the residual


703

Table 1 RPA one-phonon basis for the nucleus 40 Ca. For each state, spin, parity, isospin, energy and percentage of the EWSR are reported. In the following column, VM1 stands for the matrix element ν|V |ν ⊗ M1 , where ν is the one phonon in the 1st column. The same for VM2 and VQ1 in the next two columns. In the last two columns we report the energies of the phonons after the inclusion of two and three-phonon states, respectively Phonons

JπT

Eharm (MeV)

EWSR (%)

VM1 (MeV)

VM2 (MeV)

V Q1 (MeV)

E2ph (MeV)

E3ph (MeV)

M1 M2

0+ 0 0+ 0

18.25 22.47

30 54

−2.13 −2.03

−2.36 −3.96

– –

18.36 22.00

18.30 21.78

D1 D2

1− 1 1− 1

17.78 22.03

56 10

−1.38 −1.48

−2.12 −2.16

−1.25 +0.73

17.35 21.64

17.29 21.59

Q1 Q2

2+ 0 2+ 1

16.91 29.59

85 26

−1.36 −1.70

−2.49 −2.85

−0.36 −0.00

16.51 29.09

16.44 29.00

3− O1 O2

3− 0 3− 0 3− 0

4.94 9.71 31.33

14 5 25

−1.74 −1.42 −1.69

−2.60 −2.28 −2.72

−0.07 −0.43 −0.31

4.47 9.33 30.80

4.40 9.28 30.89

Table 2 Results for 40 Ca. In the first column, the states are labeled by their main component in the eigenvector and their unperturbed energy (in parentheses). In the second column, the amplitude of the main component c0 of the state in the basis including up to 3 phonon states is reported. Then for each total angular momentum J, we show the results of the calculation in the basis up to 2 phonon states, the present results for the basis extended to 3 phonon states, the corresponding first order perturbation theory energy, and the second order one. The last two columns contain the 2nd main component in the eigenstates and the corresponding amplitude c1 . The sub-index in the two-phonon configurations denotes J. All energies are given in MeV Jπ

2ph

3ph

1st order

2nd order

2nd main component

c1

3− ⊗ 3− −0.91 (9.88) −0.96

0+ 2+

10.96 10.63

9.27 8.89

12.12 10.66

9.20 8.75

0.21 −0.21

−0.96

4+

M1 (3− )22 ⊗ M2

9.85

8.10

9.86

7.96

−0.96

6+

10.88

9.12

10.88

8.99

(3− )26 ⊗ M2

−0.21

D1 ⊗ D1 −0.92

0+

35.27

33.71

35.25

33.59

−0.22

(35.56)

−0.96

2+

35.10

33.66

35.06

33.59

(3− )20 ⊗ M2 (D1 )22 ⊗ M2

D1 ⊗ Q1 (34.69)

0.95 0.96 −0.96

1− 2− 3−

34.83 34.56 34.67

33.35 33.22 33.13

34.72 34.56 34.67

33.24 33.16 33.02

(M2 ⊗ D1 )1 ⊗ Q1 (M2 ⊗ D1 )1 ⊗ Q1 (M2 ⊗ D1 )1 ⊗ Q1

0.19 0.19 −0.19

Q1 ⊗ Q1 −0.87 (33.82) 0.84 0.90

0+ 2+ 4+

33.88 33.82 34.02

32.47 32.47 32.61

33.83 33.82 34.02

32.27 32.26 32.44

(Q1 ⊗ 3− )3 ⊗ O1 (Q1 ⊗ 3− )5 ⊗ O1 (Q1 ⊗ 3− )5 ⊗ O1

0.32 −0.38 −0.32

M2 ⊗ D1 −0.89 (40.25)

1−

40.26

38.14

40.05

37.65

(M2 )20 ⊗ D1

0.26

M2 ⊗ Q1 −0.73 (39.38)

2+

39.62

37.34

39.35

36.80

(O1 )22 ⊗ M1

0.40

M2 ⊗ M2 (44.94)

0+

45.60

42.76

44.87

41.18

(O1 )20 ⊗ M2

−0.55

Main component

c0

0.67

(3− )24 ⊗ M2

−0.21

−0.17

704


interaction with the states |D1 ⊗ M1 , |D1 ⊗ M2 and |D1 ⊗ Q1 equal to −1.38 MeV, −2.12 MeV and −1.25 MeV, respectively. Another example, with total J = 1, is given by the matrix elements between |D1 ⊗ Q1 and |(M1 ⊗ D1 )1 ⊗ Q1 , |(M2 ⊗ D1 )1 ⊗ Q1 and |(Q1 )22 ⊗ D1 equal to −2.74 MeV, −4.61 MeV and −1.41 MeV, respectively. These findings clearly indicate that large amplitude motions are strongly coupled both to surface and volume oscillations, the latter being more important in 40 Ca. It is worthwhile stressing that such large corrections to the energy of two-phonon states are obtained despite the quite large absolute values of the energy difference between the coupled states. Therefore, introducing an energy cut-off in the three-phonon states included in the calculation may lead to erroneous results. Let us consider, for example, the case of the 0+ member of the multiplet of double low-lying octupole states. At first order perturbation, it is shifted up by 2.24 MeV. The second order correction coming from the single GMR states is −0.93 MeV. These two contributions, leading to a total shift of +1.31 MeV, dominate the effects of the coupling with one- and two-phonon states as confirmed by the result of the diagonalization in this subspace. When three-phonon states are included, one gets a further shift down of 1.86 MeV coming from the configuration including a GMR on top of the two octupoles. This contribution is absent in Ref. [5] because the energy cut-off introduced there in order to reduce the number of three-phonon configurations was too low. The same happens for the other members of the multiplet as well as for the double D1 or D2 , the double Q1 and the D1 or D2 ⊗ Q1 states. 3.2. Results for 208 Pb The results for 208 Pb are shown in Tables 3 and 4. The same general remarks already made for 40 Ca also apply in this case. The most relevant difference is that the role played by the GMR and the GQR in 40 Ca is now inverted, the latter being dominant in 208 Pb. This reduced importance of the GMR may come from the fact that in large nuclei the surface vibrations can occur without changing the volume. Concluding about the energy of the two-phonon states one can see that the inclusion of the three phonon configurations induces an anharmonicity of more than 1 MeV in 40 Ca but only of a few hundred keV in 208 Pb. This is related to the fact that collectivity is more pronounced in 208 Pb than in 40 Ca. Because of the location at high energy of the three phonon states, the observed shift is systematically downward. It is important to stress that the considered residual interaction only couples states with a number of phonon varying at maximum by one unit. Therefore, the energy variation of the two-phonon spectrum induced by inclusion of four and more phonon states would be small since it corresponds to a third order perturbation involving two large energy differences in the denominator. If we now analyze the splitting of the two-phonon multiplets we can see that it remains small for giant resonances (about a few hundred keV) while it may go up to 1 MeV for low lying states in 40 Ca. Comparing the splitting and the ordering of the states obtained in first order perturbation and in the full calculation we can see that they remain almost unchanged. Therefore, the diagonal matrix elements of the residual interaction are responsible for this splitting and ordering.


705

Table 3 Same as Table 1 for the nucleus 208 Pb Phonons

JπT

E (MeV)

EWSR (%)

VM1 (MeV)

VM2 (MeV)

V Q1 (MeV)

E2ph (MeV)

E3ph (MeV)

M1 M2 D1 D2 2+ Q1 Q2 3− O

0+ 0 0+ 0 1− 1 1− 1 2+ 0 2+ 0 2+ 1 3− 0 3− 0

13.61 15.02 12.43 16.66 5.54 11.60 21.81 3.46 21.30

61 28 63 17 15 76 45 21 37

−1.87 −1.32 −0.79 0.00 −0.11 −0.64 −0.86 −1.13 −0.99

−0.92 −1.16 −0.59 0.00 0.07 −0.48 −0.63 −0.62 −0.74

– – −0.68 −0.64 −1.18 −0.74 −0.55 −0.90 −0.42

13.42 14.78 12.30 16.61 5.18 11.59 21.69 3.21 21.19

13.48 14.76 12.30 16.60 5.14 11.55 21.68 3.19 21.20

Table 4 Same as Table 2 for the 208 Pb nucleus Main component

c0

Jπ

2ph

3ph

1st order

2nd order

2nd main component

c1

0+

7.88

6.96

8.06

6.90

2+

(3− )20 ⊗ 2+

−0.17

6.57 6.55

7.33 7.16

6.52 6.51

3− ⊗ 3−

−0.95

(6.93)

−0.92 −0.98

4+

7.31 7.16

0.97

6+

7.43

6.63

7.44

6.56

−0.94

1−

9.20

8.26

9.21

8.02

0.97

2−

9.12

8.54

9.12

8.50

0.96

3−

9.17

8.70

9.12

8.56

0.96

4−

9.07

8.61

9.07

8.45

3− ⊗ 2+ (9.01)

2+ − (3 )24 ⊗ M1 (3− )26 ⊗ M1

−0.28 −0.15 0.15

(2+ )22 ⊗ 3−

−0.23

(3− )32

−0.17

(2+ )22 ⊗ 3− (3− )3

0.17 0.19

−0.96

5−

9.06

8.33

9.06

8.16

2+ ⊗ 2+

0.92

0+

11.23

9.88

11.24

9.46

(2+ )22 ⊗ 3− (2+ )3

(11.09)

−0.94

2+

11.27

10.78

11.12

10.61

0.94

4+

(3− )22 ⊗ 2+

11.25

10.39

11.25

10.13

(2+ )3

0.24

D1 ⊗ D1 (24.87)

0.97 0.96

0+ 2+

24.91 24.68

24.42 24.29

24.90 24.68

24.40 24.27

(D1 )20 ⊗ M1 3− ⊗ O

0.11 0.19

D1 ⊗ Q1

−0.96

1−

24.07

23.73

24.02

23.71

(3− )2 ⊗ D1

0.17

0.98

2−

23.97

23.82

23.97

23.80

−0.16

0.96

3−

(3− )22 ⊗ D1

24.03

23.74

24.03

23.71

−0.94

0+

23.20

22.92

23.20

22.86

−0.24

0.95

2+

(3− )22 ⊗ Q1

23.23

23.17

23.18

23.14

−0.95

4+

23.26

23.10

23.26

23.07

−0.22

M1 ⊗ D1 (26.05)

−0.94

1−

(3− )22 ⊗ Q1

26.05

25.35

26.02

25.28

(M1 )20 ⊗ D1

−0.20

M1 ⊗ Q1 (25.21)

−0.92

2+

25.25

24.77

25.22

24.66

(3− )22 ⊗ M1

−0.20

M1 ⊗ M1 (27.22)

0.74

0+

27.52

26.23

27.28

25.95

(2+ )20 ⊗ M2

0.54

(24.03) Q1 ⊗ Q1 (23.20)

(2+ )22 ⊗ D1

(3− )22 ⊗ Q1

−0.18 0.24

−0.31

0.18 0.22

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3.3. Mixing induced by the residual interaction In Tables 2 and 4 the mixing coefficients of the two main components in each state are presented. First we can see that there is always one component that remains very large, explaining the success of the perturbative approach. The important point is that in general we observe large mixing coefficients, namely, about 0.2 to 0.4 or more in 40 Ca and 0.15 to 0.3 in 208 Pb. This may have very important consequences in the excitation process as we will investigate in a forthcoming work. It is worthwhile mentioning that, in some cases, a three-phonon component appears with a large amplitude in the wavefunction of a (mainly) two-phonon state, despite the fact that the residual interaction does not couple directly these configurations to each other. In a few cases, indeed, this is the second main component as can be seen for 40 Ca in Table 2 (the |(D1 )20 and |M2 ⊗ Q1 states) and for 208 Pb in Table 4 (the |(M1 )2 state). This happens because the diagonal matrix elements of the Hamiltonian in the two-phonon and three-phonon configurations are close and the matrix elements coupling the latters with other configurations are large. A similar situation has been found in 208Pb for two onephonon (mainly) states which have a three-phonon configuration as second most important component, even though our Hamiltonian does not couple directly states whose numbers of phonons differ by more than one. This is the case for the state whose main component is |M1 , with amplitude c0 = −0.79, and for which the second most important component is |(3− )22 ⊗ 2+ with c1 = 0.55. Both components have large matrix elements with the two-phonon state |(3− )20 . How this mixing of the monopole resonance may affect the monopole response, and so the usual conclusion about the compressibility, is now under study. The other case is the single high energy octupole resonance |O which is strongly mixed with the states |(2+ ⊗ 3− )J ⊗ Q1 . The energy of all these states are, however, shifted by less than 100 keV. This is coherent because the strong mixing is coming from a quasi degeneracy of the unperturbed states. 3.4. Discussion Until now we have been discussing the properties of the eigenstates |α of the Hamiltonian and their corresponding eigenvalues Eα . Special emphasis has been put on the |α states such that their main component is either a single RPA phonon or a double RPA phonon. We have seen that enlarging the diagonalization space to include up to three phonon configurations has not modified the energy of the |α states whose main component is a single phonon, while it induces a shift of more than 1 MeV (a few hundred keV) in the energy of the |α states of 40 Ca (208Pb) whose main component is a twophonon configuration. We may study how much this energy shift is a property of some specific states or is directly linked to a change in the distribution of the unperturbed collective configurations, |ϕi , each one of them being now fragmented over many energy eigenstates, |α . Let us look at the energy centroid, E¯ i , of the unperturbed configuration, |ϕi . It is given by Eα | α|ϕi |2 = ϕi |H |ϕi = Ei0 + ϕi |H22 |ϕi . (7) E¯ i = α


707

Therefore, the first order perturbation approximation gives the energy centroid, after diagonalization, for each unperturbed configuration. These values are reported in the sixth column of Table 2 for 40 Ca and Table 4 for 208 Pb. When the diagonalization space includes just up to two RPA phonons, the energy centroid, E¯ i , is often close to the energy Eα ( 2pho) of the state whose main component is the unperturbed configuration |ϕi , as can be seen comparing the fourth and sixth columns of Table 2 for 40 Ca and table 4 for 208 Pb. At the same time, the energy centroid is close to the harmonic energy, the shifts being of the order of a few hundred keV with the exception of the double low-lying octupole for which it is larger. These results are in agreement with the findings of [3], where a systematic study was done for different nuclei and it was found that the shift of the centroid of the double dipole giant resonance scales as A−1 and is entirely due to the terms of the residual interaction which correct for the Pauli principle violations in RPA, i.e., the H22 term in our case. The diagonal matrix element ϕi |H |ϕi is independent of the coupling of the two-phonon configuration with the one- and three-phonon ones. Consequently, the energy centroids of the two-phonon configurations are not affected by the enlargement of the diagonalization space which, on the contrary, strongly affects the energies Eα , as can be seen comparing the fourth and fifth columns of Table 2 and Table 4 for 40 Ca and 208 Pb. In summary, if the main component of the energy eigenstate |α is a two-phonon RPA state then we usually have that the eigenvalue Eα is significantly lower than the harmonic value. In contrast, the energy centroid of the unperturbed configuration remains close to the harmonic value. We will discuss the relevance of these two magnitudes. The second central moment of the energy distribution of the unperturbed configuration, |ϕi , can be written down as (Eα − E¯ i )2 | α|ϕi |2 = ϕi |H 2 |ϕi − ϕi |H |ϕi 2 = | ϕj |H |ϕi |2 . (8) α

j ( =i)

Therefore, it can only get contributions from the non-diagonal matrix elements of the Hamiltonian in the unperturbed basis. Note that H22 does not change the number of RPA phonons while H21 and H12 connect states whose phonon number differ by one. Thus the second central moment of the energy of an n-phonon configuration “saturates” if the diagonalization space includes up to (n + 1)-phonons. In the third and fourth columns of Table 5 for 40 Ca (Table 6 for 208 Pb) we can find σ , the square root of the second central moment of the energy distribution, for several unperturbed configurations for the two diagonalization spaces considered. Although the energy centroid, i.e., the first moment of the distribution, is not modified, the second central moment, σ 2 , increases considerably. However, with our Hamiltonian, for the twophonon configurations, an extension of the basis to 4 or more phonon states will not further modify σ . In Figs. 1(a) and (b), the energy distribution corresponding to a double dipole coupled to zero angular momentum for 40 Ca is presented in the two diagonalization spaces. The arrow indicates the position of the energy centroid. When the space contains up to 2-phonon states, there is one eigenstate having an overlap very close to one. Therefore, its energy and E¯ i are essentially the same. All the other eigenstates have a probability of being in a (D1 ⊗ D1 )J =0 configuration several orders of magnitude smaller. Consequently, σ is small, about 0.6 MeV. When we allow up to 3-phonon states in the basis, more states

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Table 5 Results for 40 Ca. In the first column, the unperturbed configurations are labeled. Then for each total angular momentum J , we show the square root of the second central moment of the energy distribution for the calculation in the basis up to 2 phonon states and in the basis extended to 3 phonon states. All energies are given in MeV Unperturbed RPA configuration

Jπ

σ 2ph

σ 3ph

3− ⊗ 3−

0+ 2+ 4+ 6+ 0+ 2+ 1− 2− 3− 0+ 2+ 4+ 1− 2+ 0+

3.30 0.58 0.28 0.36 0.59 0.75 1.38 0.03 0.16 0.86 0.43 0.10 2.12 2.49 3.98

7.13 6.34 6.30 6.32 5.85 5.61 5.96 5.54 5.89 5.98 5.89 5.92 8.71 9.15 11.99

Unperturbed RPA configuration

Jπ

σ 2ph

σ 3ph

3− ⊗ 3−

0+ 2+ 4+ 6+ 1− 2− 3− 4− 5− 0+ 2+ 4+ 0+ 2+ 1− 2− 3− 0+ 2+ 4+ 1− 2+ 0+

1.06 0.82 0.11 0.18 0.19 0.09 0.58 0.04 0.07 0.35 0.96 0.21 0.30 0.24 0.70 0.06 0.05 0.18 0.77 0.13 0.79 0.65 1.87

3.65 3.37 2.98 3.31 3.06 2.53 2.29 2.14 2.73 3.44 2.20 2.68 2.57 2.36 2.59 2.05 2.42 2.74 2.30 2.42 3.72 3.56 5.50

D1 ⊗ D1 D1 ⊗ Q1

Q1 ⊗ Q1

M2 ⊗ D1 M2 ⊗ Q1 M2 ⊗ M2

Table 6 Same as Table 5 for the 208 Pb nucleus

3− ⊗ 2+

2+ ⊗ 2+

D1 ⊗ D1 D1 ⊗ Q1

Q1 ⊗ Q1

M1 ⊗ D1 M1 ⊗ Q1 M1 ⊗ M1


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Fig. 1. Energy distribution associated to the unperturbed |(D1 )2J =0 RPA state on the eigenstates for 40 Ca when the space includes up to two-phonon states (a) and with the inclusion of up to three-phonon states (b). The arrow indicates the position of the energy centroid.

can couple to J = 0, mostly at higher energies giving rise to a much larger σ value, namely 6 MeV (see Fig. 1(b)). Some of these new eigenstates carry probabilities of being in (D1 ⊗ D1 )J =0 of a few percent. Actually for the eigenstate at Eα1 = 58.24 MeV we have | (D1 ⊗ D1 )J =0 |α1 |2 = 4.1% and 1.9% for the one at Eα2 = 53.39 MeV. These probabilities are small, but they come from states lying 20 to 25 MeV away from the main contribution. Therefore, they affect the energy centroid. Actually nearly all the difference between the centroid and the energy of the eigenstate whose main component is (D1 ⊗ D1 )J =0 comes from the two eigenstates just mentioned above. Without them the centroid would be located 1.4 MeV lower. Now it is clear that these two energy regions differing by 20 MeV will be experimentally distinguishable. Therefore, besides the overall energy centroid, it is meaningful to look at the centroids associated with the eigenstates lying in separate energy regions. We would like to emphasize that the set of eigenstates with energies in the region of the unperturbed configuration, have a partial energy centroid practically given by the energy of the eigenstate with the biggest overlap with such configuration. We have checked that these findings are common to all the twophonon configurations of Tables 5 and 6, and are related to their strong coupling with the three-phonon configurations obtained by building a giant monopole on the top of them.

4. Further discussion The couplings of giant resonance states with non-collective 2p–2h configurations give rise to the spreading width. In the review papers [11,12] about this problem, it is shown

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Table 7 Components of the matrix element M1 ⊗ M1 |V |M1 for the nucleus 40 Ca. In each row the values of the components, as indicated in the first column, are given for calculations done with different interactions. In the second column are reported the results for the complete SGII interaction. In the next columns are the results obtained with the SGII interaction where only the parameters shown (t and x) are different from zero. In the last column all the ti are different from zero while all the xi are zero, i = 0, 1, 2, 3. The last two lines refer to the case of a single p–h proton configuration, namely 3d3/2 , 1d3/2 t0 , x0

t3 , x3

t1 , x1 ; t2 , x2

xi = 0

ppph hhhp hppp phhh

−0.2225 −0.0717 −0.0096 −1.8280

0.7013 −0.1835 0.0443 −5.2100

−0.8025 0.1324 −0.0479 3.6471

−0.1217 −0.0207 −0.0059 −0.2601

−0.2405 −0.0592 −0.0110 −1.6621

ppph phhh

−0.1219 −0.4171

0.0769 −1.1790

−0.1282 0.6965

−0.0707 0.0654

−0.1241 −0.4482

Components

SGII

that the total spreading width Γ ↓ of GR is reduced with respect to the predictions of the naive model where Γ ↓ is assumed to be the incoherent sum of the particle and hole widths. This reduction is due to cancellations between different amplitudes. In particular, an exact cancellation is found for an infinite one-component Fermi liquid which, for l = 0, is a consequence of particle number conservation [12]. For nuclei, that are finite and consist of a four-component liquid, the cancellations are reduced. The monopole mode should be the most favorable one for the cancellations to be effective. Even in that case, however, very special conditions are required [13] and they have been explicitly shown to be amply not satisfied in RPA calculations for 208 Pb (see Table 6 of Ref. [13]). In fact, the very strict conditions (separable p–h interaction, degenerate p–h gaps and equal radial wavefunctions for particles and holes) required for an exact cancellation are hardly met in realistic calculations. In order to investigate whether something similar happens for the matrix elements entering in our study of anharmonicities, we have analyzed the matrix element M1 ⊗M1 |V |M1 for 40 Ca in terms of its particle hole components. Cancellations may originate from properties of the residual interaction, from the structure of the wavefunctions of the collective phonon under consideration and from the interplay between these two aspects. As for the residual interaction, one has to consider on one hand that it is the sum of various terms. On the other hand, in the matrix element of V (α, β, γ , δ) between a one-phonon and a two-phonon states there are four contributions: V (p, p, p, h), V (p, h, h, h), V (h, h, h, p) and V (h, p, p, p) where we denote by p a particle state and by h a hole state, respectively. The single particle wavefunctions are those obtained from a Hartree–Fock calculation as superpositions of 16 harmonic oscillator wavefunctions. We have done several calculations by switching on and off various terms of the interaction. From Table 7 we see that the terms in t0 and t3 dominate and partially cancel each other. It is also interesting to remark that the contributions calculated with all xi = 0 and all ti = 0 are very close to those with the total SGII interaction. We see that in all cases the ppph and phhh contributions are larger than the other ones. This is due to the fact that in the latters at least one Y amplitude is involved. The difference in magnitude between the two largest contributions is a consequence of the different behavior of the radial wavefunctions of particle and holes. Both, the particle and


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the hole wavefunctions are normalized to unity, the formers having a much larger radial extension. Therefore, the integrand of the ppph part of the matrix element we are analyzing is almost everywhere smaller than the phhh part. The large values of the contributions to the matrix elements are due to the coherent sum of the various components of the wavefunction of the M1 phonon. Indeed, we have repeated the same calculations considering only the main component of the wavefunction, namely the (3d3/2, 1d3/2) p–h proton configuration, with the X amplitude put equal to one and found values smaller by a factor 2 to 4 for the different contributions (last two lines of Table 7). This point should be kept in mind in comparing our results with those for the spreading width [13]. In fact, in that case one looks at matrix elements of the interaction between a collective RPA state (ν) and a pure 2p–2h configuration ν|v|2p2h . The matrix elements entering in our calculations of anharmonicities couple one-phonon states with two-phonon states. Therefore, in our case a sum weighted with the X- and Y -amplitudes of those states is involved.

5. Summary Summarizing, the spectrum of two-phonon states is strongly modified by their coupling to the three-phonon ones. All of the states appear mixed with those obtained by the excitation of a GMR and GQR on top of them. This is due to the fact that most of the matrix elements of H21 coupling a phonon with the same phonon plus a GMR or a GQR are large. Moreover, because of the Bose enhancement factors, the effect of H21 between the two and three phonon states is even larger. It is also to be noted that many of the important three-phonon states are higher in energy than the two-phonon ones. Therefore, they induce a systematic shift down of the two phonon states as the sum of several quite large negative contributions. This unexpected finding can be understood as a modification of the central density in large amplitude motion leading to an excitation of the breathing mode. The case of the GQR seems to be related to the extreme collectivity of this state leading to a strong quadrupole response to the quadrupole component of the non-linearities of the mean-field. Of course, our findings imply that in order to get a correct three-phonon spectrum one should further enlarge the space up to four-phonons. This is a formidable task which is beyond the scopes of the present paper. We also want to stress that, because of the perturbative nature of the observed phenomenon, the possible introduction of four-phonon states should not modify the above conclusions about two phonon states. This statement is corroborated by the fact that the energies of the one-phonon states are nearly unaffected by the extension of the basis from two- to three-phonon states.

Acknowledgements This work has been partially supported by the Spanish DGICyT under contracts BFM 2002-03315 and FPA 2002-04181-C04-04, by the Italian MIUR under contract PRIM 2001-2003 Fisica teorica del nucleo e dei sistemi a molti corpi, by the Spanish–Italian

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agreement between the CICyT and the INFN and by the Spanish–French agreement between the CICyT and the IN2P3.

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