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Spreading effects on the isovector dipole strength distribution in 208Pb
Volume 149B, number 6
PHYSICS LETTERS
27 December 1984
SPREADING EFFECTS ON THE ISOVECTOR DIPOLE STRENGTH DISTRIBUTION IN 208pb Shizuko ADACHI and NGUYEN VAN GIAI Division de Physique Thdorique 1, lnstitut de Physique Nucldaire, 91406 Orsay Cedex, France Received 5 July 1984 Revised manuscript received 13 September 1984
There is seemingly a difficulty with all Hartree-Fock RPA calculations to describe correctly the distribution of dipole strength in heavy nuclei above the giant dipole region. We investigate in a consistent framework the effects of damping into more complex configurations. We show that in 2°~Pb, this damping leads to a smooth distribution at higher energies while it gives a satisfactory description of the spreading width of the giant dipole resonance.
Numerous calculations have established that the bulk properties of electric giant resonances can generally be understood in the framework of the random phase approximation (RPA). When it is applied consistently, i.e. with a residual interaction derived from the same interaction which generates the single-particle HartreeFock (HF) spectrum, this framework relates quantitatively resonance energies and strengths to nuclear properties such as the compression modulus (monopole modes) or the nucleon effective mass (quadrupole modes). To refine further the theory and obtain some understanding of the spreading widths, one then needs to go beyond the particle-hole ( p - h ) RPA space and include the coupling to 2 p - 2 h and more complex configurations. The general success of the HF RPA approach is, however, overshadowed by a persistent difficulty appearing in the case of the giant dipole resonance (GDR) in heavy nuclei. If one calculates for instance the GDR in 2°spb, the main concentration of strength in the 13.5 MeV region comes out more or less correctly, but a secondary peak appears some 4 - 5 MeV above whereas the data show no such structure in this region. This was already found in the early calculation of Bertsch and Tsai [1 ], and the same type of results is obtained with supposedly better adjusted interactions [ 2 - 5 ] . One possible reason could be that the above mentioned i Laboratoire associd au CNRS. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
interactions do not possess the correct neutron-proton symmetry properties, and hence one would have a handle to adjust this additional degree of freedom in phenomenological interactions. Indeed, it was shown in refs. [4,6] that the unwanted structure could be removed by varying the ratio of surface to volume symmetry energies of Skyrme-type forces without strongly affecting other nuclear bulk properties. However, it is likely that spreading effects beyond RPA could be of importance in redistributing the dipole strength. In this work, we show that the spreading of the RPA states provides an appropriate mechanism for smoothing out the unphysical structure and for describing correctly the total width of the GDR. We consider the GDR in 2°8pb, for which several RPA calculations exist [ 1 - 5 ] . Here, we use the Skyrme force SGII [7] throughout the calculations. We start with a standard RPA diagonalization in a restricted configuration space. The single-particle spectrum is given by a HF calculation. The RPA configuration space is limited by keeping two major shells below the Fermi level, and those unoccupied orbitals which either are bound or clearly correspond to a single partitle resonance. This choice gives two complete major shells above the Fermi level for the protons, whereas for the neutrons it excludes the 4p and 3f orbitals from the second major shell above the neutron Fermi le vel. All wave functions are normalized within a box of radius R = 12 fm. This procedure allows us to work 447
Volume 149B, number 6
PHYSICS LETTERS
27 December 1984
operator [8]:
25 3~r:1 208pb
Y'ph,p 'h ,(60)
=
_ ~
(3)
15 10
[
i
010 11
,,I 12
13
14
15
I
16
17
18
Ex [MeV] Fig. 1. RPA dipole strength distribution in 2°aPb. in a large enough space while it leaves the results fairly insensitive to the choice of the radius R. The RPA dipole strength distribution is shown in fig. 1. It exhibits the same general features as the resuits obtained in larger configuration spaces or with the inclusion of the continuum, namely a large fraction of strength (80%) distributed among several states between 11 and 14 MeV, and a rather strong state (15% of the total strength) at 17 MeV. The latter corresponds to the unwanted structure mentioned above. Its largest p - h compolnents are: 2g9/2-1 h~/2 (p), 2 f7/2-1 g9~2(P), 2hll/2-1i~3/2(n ) and 2g9/2-1hit/2(n). Next, we evaluate the changes in the dipole strength distribution brought about by the coupling of RPA states to 2 p - 2 h states. In the absence of coupling, the RPA Green function G(co) obeys the familiar equation:
B*
A*+~
'i1
0
where the matrices A , B are defined in the p - h representation as: Aph,p, h, = ~ppi~hh,(e p -- ell) + (ph-ll Vreslp'h'-l>, Bph,p, h, = <(ph -1 )(p'h '-1 )[Vres [0).
(2)
In eq. (2), ep (ell) is the HF single particle (hole) energy, Vres is the residual p - h interaction. When the coupling between l p - l h and 2 p - 2 h configurations is included, G(¢o) still satisfies an equation similar to (1) in whichA is replaced b y A + Z(co) where ~(co) is a frequency-dependent contribution to the p - h mass 448
In eq. (3), E N and IN> are the eigenvalues and eigenstates of the hamiltonian diagonalized in the 2 p - 2 h subspace, I'av is an energy averaging parameter which can simulate the damping of 2 p - 2 h states into more complex configurations and particle escape effects. The calculations presented below have been done with Pay = 1 MeV, and we have checked that other choices of I'av between 1 MeV and 2 MeV leave the results essentially unchanged. The set of equations (1), (2) and (3) is equivalent to the so-called second RPA [9]. In practice, however, diagonalizing the hamiltonian in a meaningful 2 p - 2 h subspace is a far too formidable task, and we make the following approximations: (i) The residual interaction is neglected in the 2 p - 2 h subspace, i.e. IN> and E N in eq. (3) are approximated by uncorrelated 2 p - 2 h states and energies. This approximation may miss some effects due to the collective character of some of the states IN>. In other microscopic studies of the spreading of giant resonances, these states IN> have been chosen either as two-phonon states [10], or l p - l h plus one-phonon states [11]. Nevertheless, our approximation has the advantage that it does not introduce any Pauli principle violation or double-counting problem. (ii) Only the imaginary part of Y~(co)is included in eq. (1). In ref. [12], the effect of the real part of ~(co)on the damping of giant resonances has been discussed. However, Re Y~(co)calculated from eq. (3) depends somewhat on the 2 p - 2 h space if one uses a Skyrme-type force or any other force of zero-range type. In principle, Skyrme interactions are effective forces valid only in some limited configuration subspace, but it is not clear how to limit the sum on the rhs of eq. (3). Our approximation amounts to neglecting the energy shifts in the peaks of the strength distribution caused by the 2 p - 2 h states. This should not strongly affect, however, the spreading of these peaks which is the purpose of the present study. Approximations (i) and (ii) have been used in ref. [1 3] where a good description of the spreading widths of charge-exchange monopole resonances was obtained. With these approximations, it is easy to calculate ~(co).
Volume 149B, number 6
PHYSICS LETTERS
In eq. (3), the antisymmetric part of the p - h interaction derived from the Skyrme force is employed [14] since the uncorrelated 2 p - 2 h states are antisymmetrized. Having obtained the Green function G(co), the dipole strength distribution S(co) is readily computed from the linear response function: S(~)
=
Im<'OID+G(~)DI'O),
7r-1
jrr=1 -
~E 16
%
~4
%
i'2
1500
208pb~ SG I[
1000
(4)
where D is the isovector dipole operator and IO) is the correlated ground state. Fig. 2 shows the strength distribution S(co) when the spreading effects are taken into account. We have included 2 p - 2 h states with unperturbed energies up to 22 MeV. As far as we neglect the real part of Z(co), this 2 p - 2 h subspace is large enough to give a numerical convergence and the calculated S(co) does not change if one increases the cut-off energy of 2 p - 2 h states. After choosing 23 1p - 1h components among 47 components, we calculate the matrix elements between these 23 components and 2 p - 2 h states. For each of the five strong RPA states one can see in fig. 1, a sum of squared RPA forward amplitudes of these 23 components is at least 0.96, and these 1 p - l h components are enough for giving the spreading of the peaks in the energy region shown in fig. 2. The dipole strength function in fig. 2 shows a large bump in the energy region from 11 MeV to 15 MeV, where four strong RPA states are located in fig. 1. On the other hand, the RPA state at 17 MeV is practically washed out and we cannot identify a bump at all in this energy region. This result shows that the discrepancy between the experimental result and the l p - l h RPA strength function can be resolved by including the damping of l p - l h states.
~'20
27 December 1984
I'7 E x [MeV]
Fig. 2. Calculated dipole strength distribution including spreading effects.
500
0
10
12
14
16
18
20 22
~-2p2h[M~V] Fig. 3. Number of 2 p - 2 h configurations with jzr = 1-, as a function of excitation energy.
The disappearance of the 17 MeV peak can be attributed mainly to the rapidly increasing level density of 2 p - 2 h states with increasing energy. The magnitude of the spreading width of 1p - l h states, in general, depends on the level density of surrounding 2 p - 2 h states. In fig. 3 we show the number of 2 p - 2 h states withJ 7r = 1- as a function of the unperturbed energy of 2 p - 2 h states. The number of 2 p - 2 h states begins to increase around 12 MeV and it grows up steadily with the energy except for the energy region around 17 MeV where a dip appears caused by a shell structure effect. We can see in fig. 3 that the number of 2 p - 2 h states around 17 MeV is about twice as high as that around 12.5 MeV. The summed strength in the energy region shown in fig. 2 is 57.7 e2fm 2 , while the summed energy weighted strength is 777.8 e2fm 2 MeV. This value is 78% of the energy weighted sum rule (EWSR) value obtained by the 1p - l h RPA calculation. By including the damping of l p - l h states, the dipole strength function has a strong tail above 18 MeV, and this part must contain a sizable fraction of the EWSR. In summary, we have evaluated the isovector dipole strength function in 208pb within the framework of the self-consistent HF RPA using the interaction SGII. The imaginary part of the coupling between l p - l h states and uncorrelated 2 p - 2 h states is included in the RPA response function. Because of the rapidly increasing level density of 2 p - 2 h states with increasing energy, the unphysical peak in the calculated strength func449
Volume 149B, number 6
PHYSICS LETTERS
tion within the l p - l h RPA is shown to disappear while the GDR emerges as a broad, structured resonance.
References [1] G.F. Bertsch and S.F. Tsai, Phys. Rep. 18C (1975) 126. [2] K.F. Liu and Nguyen van Giai, Phys. Lett. 65B (1976) 23. [3] K.F. Liu and G.E. Brown, Nucl. Phys. A265 (1976) 385. [4] O. Bohigas, Suppl. Prog. Theor. Phys. 74-75 (1983) 380. [5] J. Decharg~ and L. Sips, Nucl. Phys. A407 (1983) 1. [6] J. Treiner, Proc. HESANS 83, J. Phys. (Paris) C4 (1984) 265. [7] Nguyen van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379.
450
27 December 1984
[8] S. Yoshida, Suppl. Prog. Theor. Phys. 74-75 (1983) 142. [9] A.M. Lane, Nuclear theory (Benjamin, New York, 1964); C. Yannouleas, M. Dworzeeka and J.J. Griffin, Nucl. Phys. A397 (1983) 239. [10] V.G. Soloviev, Ch. Stoyanov and V.V. Voronov, Nucl. Phys. A399 (1983) 141. [I 1 ] P.F. Bortignon and R.A. Broglia, Nucl. Phys. A371 (1981) 405; J. Wambach, V.K. Mishra and C.-H. Li, Nuel. Phys. A380 (1982) 285. [12] B. Schwesinger and J. Wambach, Phys. Lett. 134B (1984) 29; preprint (1983). [13] S. Adachi and N. Auerbach, Phys. Lett. 131B (1983) 11. [14] S. Adachi and S. Yoshida, Phys. Lett. 81B (1979) 98.
PHYSICS LETTERS
27 December 1984
SPREADING EFFECTS ON THE ISOVECTOR DIPOLE STRENGTH DISTRIBUTION IN 208pb Shizuko ADACHI and NGUYEN VAN GIAI Division de Physique Thdorique 1, lnstitut de Physique Nucldaire, 91406 Orsay Cedex, France Received 5 July 1984 Revised manuscript received 13 September 1984
There is seemingly a difficulty with all Hartree-Fock RPA calculations to describe correctly the distribution of dipole strength in heavy nuclei above the giant dipole region. We investigate in a consistent framework the effects of damping into more complex configurations. We show that in 2°~Pb, this damping leads to a smooth distribution at higher energies while it gives a satisfactory description of the spreading width of the giant dipole resonance.
Numerous calculations have established that the bulk properties of electric giant resonances can generally be understood in the framework of the random phase approximation (RPA). When it is applied consistently, i.e. with a residual interaction derived from the same interaction which generates the single-particle HartreeFock (HF) spectrum, this framework relates quantitatively resonance energies and strengths to nuclear properties such as the compression modulus (monopole modes) or the nucleon effective mass (quadrupole modes). To refine further the theory and obtain some understanding of the spreading widths, one then needs to go beyond the particle-hole ( p - h ) RPA space and include the coupling to 2 p - 2 h and more complex configurations. The general success of the HF RPA approach is, however, overshadowed by a persistent difficulty appearing in the case of the giant dipole resonance (GDR) in heavy nuclei. If one calculates for instance the GDR in 2°spb, the main concentration of strength in the 13.5 MeV region comes out more or less correctly, but a secondary peak appears some 4 - 5 MeV above whereas the data show no such structure in this region. This was already found in the early calculation of Bertsch and Tsai [1 ], and the same type of results is obtained with supposedly better adjusted interactions [ 2 - 5 ] . One possible reason could be that the above mentioned i Laboratoire associd au CNRS. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
interactions do not possess the correct neutron-proton symmetry properties, and hence one would have a handle to adjust this additional degree of freedom in phenomenological interactions. Indeed, it was shown in refs. [4,6] that the unwanted structure could be removed by varying the ratio of surface to volume symmetry energies of Skyrme-type forces without strongly affecting other nuclear bulk properties. However, it is likely that spreading effects beyond RPA could be of importance in redistributing the dipole strength. In this work, we show that the spreading of the RPA states provides an appropriate mechanism for smoothing out the unphysical structure and for describing correctly the total width of the GDR. We consider the GDR in 2°8pb, for which several RPA calculations exist [ 1 - 5 ] . Here, we use the Skyrme force SGII [7] throughout the calculations. We start with a standard RPA diagonalization in a restricted configuration space. The single-particle spectrum is given by a HF calculation. The RPA configuration space is limited by keeping two major shells below the Fermi level, and those unoccupied orbitals which either are bound or clearly correspond to a single partitle resonance. This choice gives two complete major shells above the Fermi level for the protons, whereas for the neutrons it excludes the 4p and 3f orbitals from the second major shell above the neutron Fermi le vel. All wave functions are normalized within a box of radius R = 12 fm. This procedure allows us to work 447
Volume 149B, number 6
PHYSICS LETTERS
27 December 1984
operator [8]:
25 3~r:1 208pb
Y'ph,p 'h ,(60)
=
_ ~
(3)
15 10
[
i
010 11
,,I 12
13
14
15
I
16
17
18
Ex [MeV] Fig. 1. RPA dipole strength distribution in 2°aPb. in a large enough space while it leaves the results fairly insensitive to the choice of the radius R. The RPA dipole strength distribution is shown in fig. 1. It exhibits the same general features as the resuits obtained in larger configuration spaces or with the inclusion of the continuum, namely a large fraction of strength (80%) distributed among several states between 11 and 14 MeV, and a rather strong state (15% of the total strength) at 17 MeV. The latter corresponds to the unwanted structure mentioned above. Its largest p - h compolnents are: 2g9/2-1 h~/2 (p), 2 f7/2-1 g9~2(P), 2hll/2-1i~3/2(n ) and 2g9/2-1hit/2(n). Next, we evaluate the changes in the dipole strength distribution brought about by the coupling of RPA states to 2 p - 2 h states. In the absence of coupling, the RPA Green function G(co) obeys the familiar equation:
B*
A*+~
'i1
0
where the matrices A , B are defined in the p - h representation as: Aph,p, h, = ~ppi~hh,(e p -- ell) + (ph-ll Vreslp'h'-l>, Bph,p, h, = <(ph -1 )(p'h '-1 )[Vres [0).
(2)
In eq. (2), ep (ell) is the HF single particle (hole) energy, Vres is the residual p - h interaction. When the coupling between l p - l h and 2 p - 2 h configurations is included, G(¢o) still satisfies an equation similar to (1) in whichA is replaced b y A + Z(co) where ~(co) is a frequency-dependent contribution to the p - h mass 448
In eq. (3), E N and IN> are the eigenvalues and eigenstates of the hamiltonian diagonalized in the 2 p - 2 h subspace, I'av is an energy averaging parameter which can simulate the damping of 2 p - 2 h states into more complex configurations and particle escape effects. The calculations presented below have been done with Pay = 1 MeV, and we have checked that other choices of I'av between 1 MeV and 2 MeV leave the results essentially unchanged. The set of equations (1), (2) and (3) is equivalent to the so-called second RPA [9]. In practice, however, diagonalizing the hamiltonian in a meaningful 2 p - 2 h subspace is a far too formidable task, and we make the following approximations: (i) The residual interaction is neglected in the 2 p - 2 h subspace, i.e. IN> and E N in eq. (3) are approximated by uncorrelated 2 p - 2 h states and energies. This approximation may miss some effects due to the collective character of some of the states IN>. In other microscopic studies of the spreading of giant resonances, these states IN> have been chosen either as two-phonon states [10], or l p - l h plus one-phonon states [11]. Nevertheless, our approximation has the advantage that it does not introduce any Pauli principle violation or double-counting problem. (ii) Only the imaginary part of Y~(co)is included in eq. (1). In ref. [12], the effect of the real part of ~(co)on the damping of giant resonances has been discussed. However, Re Y~(co)calculated from eq. (3) depends somewhat on the 2 p - 2 h space if one uses a Skyrme-type force or any other force of zero-range type. In principle, Skyrme interactions are effective forces valid only in some limited configuration subspace, but it is not clear how to limit the sum on the rhs of eq. (3). Our approximation amounts to neglecting the energy shifts in the peaks of the strength distribution caused by the 2 p - 2 h states. This should not strongly affect, however, the spreading of these peaks which is the purpose of the present study. Approximations (i) and (ii) have been used in ref. [1 3] where a good description of the spreading widths of charge-exchange monopole resonances was obtained. With these approximations, it is easy to calculate ~(co).
Volume 149B, number 6
PHYSICS LETTERS
In eq. (3), the antisymmetric part of the p - h interaction derived from the Skyrme force is employed [14] since the uncorrelated 2 p - 2 h states are antisymmetrized. Having obtained the Green function G(co), the dipole strength distribution S(co) is readily computed from the linear response function: S(~)
=
Im<'OID+G(~)DI'O),
7r-1
jrr=1 -
~E 16
%
~4
%
i'2
1500
208pb~ SG I[
1000
(4)
where D is the isovector dipole operator and IO) is the correlated ground state. Fig. 2 shows the strength distribution S(co) when the spreading effects are taken into account. We have included 2 p - 2 h states with unperturbed energies up to 22 MeV. As far as we neglect the real part of Z(co), this 2 p - 2 h subspace is large enough to give a numerical convergence and the calculated S(co) does not change if one increases the cut-off energy of 2 p - 2 h states. After choosing 23 1p - 1h components among 47 components, we calculate the matrix elements between these 23 components and 2 p - 2 h states. For each of the five strong RPA states one can see in fig. 1, a sum of squared RPA forward amplitudes of these 23 components is at least 0.96, and these 1 p - l h components are enough for giving the spreading of the peaks in the energy region shown in fig. 2. The dipole strength function in fig. 2 shows a large bump in the energy region from 11 MeV to 15 MeV, where four strong RPA states are located in fig. 1. On the other hand, the RPA state at 17 MeV is practically washed out and we cannot identify a bump at all in this energy region. This result shows that the discrepancy between the experimental result and the l p - l h RPA strength function can be resolved by including the damping of l p - l h states.
~'20
27 December 1984
I'7 E x [MeV]
Fig. 2. Calculated dipole strength distribution including spreading effects.
500
0
10
12
14
16
18
20 22
~-2p2h[M~V] Fig. 3. Number of 2 p - 2 h configurations with jzr = 1-, as a function of excitation energy.
The disappearance of the 17 MeV peak can be attributed mainly to the rapidly increasing level density of 2 p - 2 h states with increasing energy. The magnitude of the spreading width of 1p - l h states, in general, depends on the level density of surrounding 2 p - 2 h states. In fig. 3 we show the number of 2 p - 2 h states withJ 7r = 1- as a function of the unperturbed energy of 2 p - 2 h states. The number of 2 p - 2 h states begins to increase around 12 MeV and it grows up steadily with the energy except for the energy region around 17 MeV where a dip appears caused by a shell structure effect. We can see in fig. 3 that the number of 2 p - 2 h states around 17 MeV is about twice as high as that around 12.5 MeV. The summed strength in the energy region shown in fig. 2 is 57.7 e2fm 2 , while the summed energy weighted strength is 777.8 e2fm 2 MeV. This value is 78% of the energy weighted sum rule (EWSR) value obtained by the 1p - l h RPA calculation. By including the damping of l p - l h states, the dipole strength function has a strong tail above 18 MeV, and this part must contain a sizable fraction of the EWSR. In summary, we have evaluated the isovector dipole strength function in 208pb within the framework of the self-consistent HF RPA using the interaction SGII. The imaginary part of the coupling between l p - l h states and uncorrelated 2 p - 2 h states is included in the RPA response function. Because of the rapidly increasing level density of 2 p - 2 h states with increasing energy, the unphysical peak in the calculated strength func449
Volume 149B, number 6
PHYSICS LETTERS
tion within the l p - l h RPA is shown to disappear while the GDR emerges as a broad, structured resonance.
References [1] G.F. Bertsch and S.F. Tsai, Phys. Rep. 18C (1975) 126. [2] K.F. Liu and Nguyen van Giai, Phys. Lett. 65B (1976) 23. [3] K.F. Liu and G.E. Brown, Nucl. Phys. A265 (1976) 385. [4] O. Bohigas, Suppl. Prog. Theor. Phys. 74-75 (1983) 380. [5] J. Decharg~ and L. Sips, Nucl. Phys. A407 (1983) 1. [6] J. Treiner, Proc. HESANS 83, J. Phys. (Paris) C4 (1984) 265. [7] Nguyen van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379.
450
27 December 1984
[8] S. Yoshida, Suppl. Prog. Theor. Phys. 74-75 (1983) 142. [9] A.M. Lane, Nuclear theory (Benjamin, New York, 1964); C. Yannouleas, M. Dworzeeka and J.J. Griffin, Nucl. Phys. A397 (1983) 239. [10] V.G. Soloviev, Ch. Stoyanov and V.V. Voronov, Nucl. Phys. A399 (1983) 141. [I 1 ] P.F. Bortignon and R.A. Broglia, Nucl. Phys. A371 (1981) 405; J. Wambach, V.K. Mishra and C.-H. Li, Nuel. Phys. A380 (1982) 285. [12] B. Schwesinger and J. Wambach, Phys. Lett. 134B (1984) 29; preprint (1983). [13] S. Adachi and N. Auerbach, Phys. Lett. 131B (1983) 11. [14] S. Adachi and S. Yoshida, Phys. Lett. 81B (1979) 98.