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Monopole resonances and Jastrow correlations
Volume l18B, number 1, 2, 3
PHYSICS LETTERS
2 December 1982
MONOPOLE RESONANCES AND JASTROW CORRELATIONS ~r
J.S. DEHESA, R. GUARDIOLA Departamento de Ffsica Nuclear, Universidad de Granada, Spain and
A. POLLS and J. ROS Departamento de Ffsica Te6rica, Universidad de Granada, Spain
Received 7 July 1982 Revised manuscript received 9 August 1982
The effect of short range correlations on isoscalar monopole resonances in 4He, 160 and 4°Ca is analyzed by using a correlated generator coordinate method. We observe an important increase of the excitation energies of the first resonances with respect to the calculation without src, but small effects on the saturation of the EWSR.
The aim of this work is to study the effect of short. range correlations (arc) on the isoscalar monopole resonances and compression moduli of light doublyclosed shell nuclei. The possible influence of src on these properties has already been pointed out [1 ~2] but their effect has not yet been analysed in a consistent way, mainly because the evaluation of the ma. trix elements between correlated nuclear states was carried out by means of unreliable techniques. In this letter we present the energies of the T = 0 EO states of 4He, 160 and 40Ca nuclei determined in the framework of a correlated generator coordinate method. At the same time the effect of src on the compression moduli and the percentage of the EWSR exhausted by these states is also considered. We describe the nuclear ground state (gs) and monopole resonances in the space spanned by Jastrow-type wave functions with state-independent correlation factor: I(x;a,b) = H ~/I*SM(t~)), i
(1)
where #SM(a) represents a pure harmonic oscillator Work supported by Comisi6n Asesora de Investigaci6n Cienti'fica y T6cnica, Spain. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
shell model state with parameter ,~ = (me.o/h) 1/2 and ~i is the Jastrow correlation factor which we take of the form: f(r) = 1 + a e x p ( - b r 2 ) ,
(2)
in which a and b characterize the depth and range, respectively, of the correlation. The B1 Brink-Boeker interaction has been used. Extensive variational studies [3] of the gs energy with this interaction have shown that the correlation (2) is mainly determined by the short-range part of the potential and is fairly independent of the particular nucleus considered. This suggests, in a first estimate, to take the parameters a and b from the minimization of the gs energy and to keep them fixed in (1). Then we take c~in (1) as the only generator coordinate, and the corresponding Hill-Wheeler [4] * 1 equation, when properly discretized, leads to a generalized eigenvalue problem. As such, it is characterized by the hamfltonian H~a and overlap Oa# matrices. Neither of these matrices is diagonal in the correlated basis (1) but the off-diagonal elements involving oscillator parameters a ~/~ can be expressed in terms ,1 See Wong [4] for a review of theGCM. 13
Volume 118B, number 1, 2, 3
PHYSICS LETTERS
of diagonal matrix elements corresponding to oscillator parameter 7 2 = (or2 + ~2)/2. This important simplification is achieved by using the following property [5] of harmonic oscillator (HO) shell model wave functions q(ot; r 1 ... r a ) : (~/7)N@(~,; r I ... rA) X exp (½(72 - c~2) ~ r 2 )
(3)
,
with
N =
~
oc.shells
C~nl(2n+l-½),
i.e. N(4He) = 6, N ( 1 6 0 ) = 36 and N(4°Ca) = 120. Notice that the elimination of non-diagonal matrix elements in favour o f diagonal and simpler ones indicated by eqs. (4) below is still possible in the presence of correlations as far as these are not m o m e n t u m dependent. With all these observations, the secular equation of our problem is written as follows
(Ha# - EOao) c~ = O, where Oao = (a; a, b I/3;a, b) = Ca#(~/; a, biT; a, b ) , = (ct; a, b IHI/~; a, b) = Ca# ( H ~ -- (Pi2/8m) (ix2 -/32)2(7; a, b l ~ r 213'; a, Ca ~ = [20q3/(a2 + 32)]N
b>),
(4)
7 = [1 (a 2 + 132)] 1/2.
Therefore we are left with the evaluation of correlated expectation values. To this end we use the
2 December 1982
F a c t o r - A v i l e s - H a r t o g - T o l h o e k cluster expansion [6] to third order. This has been recently shown to be a safe procedure [3,7]. The center of mass corrections, although not expli. citly shown in the above equations, are simply incorporated by replacing the exponent N in Co¢ by (N 3/2) and subtracting (3h2/4m) [2a2/32/(~ 2 +/~2)]Oat ~ from the hamiltonian matrix. The results of our work are presented in table 1. The energy of the gs and of the first two isoscalar monopole resonances (relative to the gs) are shown, as is also the percentage of saturation of the EWSR. The values of all these quantities are given both in the case without (HO rows) and with (src rows) shortrange correlations. The spurious center of mass motion has been properly removed in all the quantities shown in table 1 and the translationaUy invariant form of the EWSR has been used. Coulomb energy, on the contrary, has , a t been included. The uncorrelated case was already known from other work [8]. The most remarkable result is the increase of about 35% in the excitation energies of the resonances produced by the inclusion of src. Secondly, we observe a small change in the portion of the classical EWSR exhausted by these resonances. This change goes always in the direction of locating the strength in the first resonance. In 40Ca we l~md 101% of the EWSR exhausted by the first resonance: this, apart from other numerical uncertainties, is due to the fact that the correlated matrix elements are calculated in third order of the FAHT cluster expansion. We think this supports our view that the calculation presented here is consistent and the third order reliable. Otherwise the numerical result for the percentages of EWSR could have been completely unsensical.
Table 1 Properties of monopole resonances computed with (src) and without (HO) short-range correlations. Nucleus
Eg s (MeV)
AE 1 (MeV)
AE 2 (MeV)
%EWSR (1)
%EWSR (2)
4He
HO src
-29.3 -37.4
26.6 33.1
36.0 46.1
59 64
23 21
160
HO src
-106.5 -163.3
22.9 30.9
42.7 60.9
96 99
3 1
4°Ca
HO sre
-323.2 -478.1
20.3 27.0
39.9 53.0
99 101
1 1
14
Volume 118B, number 1, 2, 3
PHYSICS LETTERS
Table 2 Compression modulus of 4He, 160, 4oCa and nuclear matter. Nucleus
K (MeV) Src
4He
t60 4oCa nuclear matter
rio SrC
73
84
86 100 190
88 135 359
Another interesting result of table 1 is the approximate harmonicity of the spectrum: i.e., the spacing of the energies is fairly constant (except for 4He) both with and without src. Furthermore, this occurs not only for the states shown in table 1 but also for other higher states. The last result makes it meaningful to talk of a compression modulus [9] for a nucleus of mass numberA which we deffme as
K A = r2(d2/dr 2) [E(r)/A] , where r is the length parameter of the oscillator and
E(r) is the gs energy calculated with state (1) taking a= 1/r. The value of this quantity for the different nuclei considered are shown in table 2 where for comparison the incompressibility of nuclear matter Knm is also given. Here Knm is defined by
Knm = k 2( d2 / dk 2) [E(k )[A lk:kF , and the values given are obtained with the same interaction used here for the Finite nuclei. The correlated expectation values are calculated by means of the Fermi-hypemetted-chain theory in the approximation FHNC/O where the elementary diagrams are neglected [10]. In this case the correlation factor has been determined by the method of Pandharipande and Bethe [11]. It is seen from table 2 that src produce an appreciable increase of the compression moduli. This was intuitively expected because the equilibrium radius in the presence of Jastrow correlations is noticeably smaller that the equilibrium radius for the uncorrelated nucleus. This means an increase of the nuclear density and correspondingly of the compression modulus. Finally, in the presence of src,KA increases
2 December 1982
with the mass number A, as is also the case without src.
In conclusion, the relevant role of src in the determination of the energies of the isoscalar electric monopole states and compression moduli of light doubly-closed shell nuclei has been stressed. Of course we are aware of the fact that the magnitude of the effect of src might depend also on the precise form of the interaction used, but we expect that the tendency here found will survive. Finally, we would like to point out that, in contrast to previous approaches, our treatment has the merit of a greater coherence since, once the correlation function and the two-body force are chosen, the values of the quantities here evaluated emerge naturally from the calculation. To end up we should mention that it has no sense to compare our results with the experimentally determined properties of the monopole resonances. Actually, the B 1 Brink-Boeker force is an effective interaction designed to be used without src. However, that interaction serves as an adequate prototype for the study of src effects because of the small wound volume corresponding to the optimal correlation.
References [1] M.K. Kirson, Nucl. Phys. A257 (1976) 58; A301 (1978) 93. [2] J.R. Henderson, Nucl. Phys. A323 (1979) 109. [3] R. Guardiola, Nucl. Phys. A328 (1979) 490, in: Lecture Notesin Physics, Vol. 142 (Springer, Berlin, 1981) p. 389; R. Guardiola, A. Faessler, H. M~lther and A. Polls, Nucl. Phys. A371 (1981) 79. [4] D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1102; C.W. Wong,Phys. Rep. 15 (1975) 283. [5] A.E.L. Dieperink and T. de Forest, Phys. Rev. C10 (1970) 543. [6] P. Westhaus and J.W. Clark, J. Math. Phys. 9 (1968) 131. [7] R. Guatdiola and A. Polls, Nucl. Phys. A342 (1980) 385; R. Guardiola, A. Polls and J. Ross, Nuovo Cimento 20A (1980) 419. [8] E. Caurier, B. Bourotte-Bilwes and Y. Abgrall, Phys. Lett. 44B (1973) 411; J. Navarro, Phys. Lett. 59B (1975) 13. [9] J.P. Blaizot, D. Gogny and B. Grammaticos, Nucl. Phys. A265 (1976) 315. [10] S. Fantoni and S. Rosati, Nuovo Cimento 25A (1975) 593. [11] V.R. Pandharipande and H.A. Bethe, Phys. Rev. C7 (1973) 1312.
15
PHYSICS LETTERS
2 December 1982
MONOPOLE RESONANCES AND JASTROW CORRELATIONS ~r
J.S. DEHESA, R. GUARDIOLA Departamento de Ffsica Nuclear, Universidad de Granada, Spain and
A. POLLS and J. ROS Departamento de Ffsica Te6rica, Universidad de Granada, Spain
Received 7 July 1982 Revised manuscript received 9 August 1982
The effect of short range correlations on isoscalar monopole resonances in 4He, 160 and 4°Ca is analyzed by using a correlated generator coordinate method. We observe an important increase of the excitation energies of the first resonances with respect to the calculation without src, but small effects on the saturation of the EWSR.
The aim of this work is to study the effect of short. range correlations (arc) on the isoscalar monopole resonances and compression moduli of light doublyclosed shell nuclei. The possible influence of src on these properties has already been pointed out [1 ~2] but their effect has not yet been analysed in a consistent way, mainly because the evaluation of the ma. trix elements between correlated nuclear states was carried out by means of unreliable techniques. In this letter we present the energies of the T = 0 EO states of 4He, 160 and 40Ca nuclei determined in the framework of a correlated generator coordinate method. At the same time the effect of src on the compression moduli and the percentage of the EWSR exhausted by these states is also considered. We describe the nuclear ground state (gs) and monopole resonances in the space spanned by Jastrow-type wave functions with state-independent correlation factor: I(x;a,b) = H ~/I*SM(t~)), i
(1)
where #SM(a) represents a pure harmonic oscillator Work supported by Comisi6n Asesora de Investigaci6n Cienti'fica y T6cnica, Spain. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
shell model state with parameter ,~ = (me.o/h) 1/2 and ~i is the Jastrow correlation factor which we take of the form: f(r) = 1 + a e x p ( - b r 2 ) ,
(2)
in which a and b characterize the depth and range, respectively, of the correlation. The B1 Brink-Boeker interaction has been used. Extensive variational studies [3] of the gs energy with this interaction have shown that the correlation (2) is mainly determined by the short-range part of the potential and is fairly independent of the particular nucleus considered. This suggests, in a first estimate, to take the parameters a and b from the minimization of the gs energy and to keep them fixed in (1). Then we take c~in (1) as the only generator coordinate, and the corresponding Hill-Wheeler [4] * 1 equation, when properly discretized, leads to a generalized eigenvalue problem. As such, it is characterized by the hamfltonian H~a and overlap Oa# matrices. Neither of these matrices is diagonal in the correlated basis (1) but the off-diagonal elements involving oscillator parameters a ~/~ can be expressed in terms ,1 See Wong [4] for a review of theGCM. 13
Volume 118B, number 1, 2, 3
PHYSICS LETTERS
of diagonal matrix elements corresponding to oscillator parameter 7 2 = (or2 + ~2)/2. This important simplification is achieved by using the following property [5] of harmonic oscillator (HO) shell model wave functions q(ot; r 1 ... r a ) : (~/7)N@(~,; r I ... rA) X exp (½(72 - c~2) ~ r 2 )
(3)
,
with
N =
~
oc.shells
C~nl(2n+l-½),
i.e. N(4He) = 6, N ( 1 6 0 ) = 36 and N(4°Ca) = 120. Notice that the elimination of non-diagonal matrix elements in favour o f diagonal and simpler ones indicated by eqs. (4) below is still possible in the presence of correlations as far as these are not m o m e n t u m dependent. With all these observations, the secular equation of our problem is written as follows
(Ha# - EOao) c~ = O, where Oao = (a; a, b I/3;a, b) = Ca#(~/; a, biT; a, b ) , = (ct; a, b IHI/~; a, b) = Ca# ( H ~ -- (Pi2/8m) (ix2 -/32)2(7; a, b l ~ r 213'; a, Ca ~ = [20q3/(a2 + 32)]N
b>),
(4)
7 = [1 (a 2 + 132)] 1/2.
Therefore we are left with the evaluation of correlated expectation values. To this end we use the
2 December 1982
F a c t o r - A v i l e s - H a r t o g - T o l h o e k cluster expansion [6] to third order. This has been recently shown to be a safe procedure [3,7]. The center of mass corrections, although not expli. citly shown in the above equations, are simply incorporated by replacing the exponent N in Co¢ by (N 3/2) and subtracting (3h2/4m) [2a2/32/(~ 2 +/~2)]Oat ~ from the hamiltonian matrix. The results of our work are presented in table 1. The energy of the gs and of the first two isoscalar monopole resonances (relative to the gs) are shown, as is also the percentage of saturation of the EWSR. The values of all these quantities are given both in the case without (HO rows) and with (src rows) shortrange correlations. The spurious center of mass motion has been properly removed in all the quantities shown in table 1 and the translationaUy invariant form of the EWSR has been used. Coulomb energy, on the contrary, has , a t been included. The uncorrelated case was already known from other work [8]. The most remarkable result is the increase of about 35% in the excitation energies of the resonances produced by the inclusion of src. Secondly, we observe a small change in the portion of the classical EWSR exhausted by these resonances. This change goes always in the direction of locating the strength in the first resonance. In 40Ca we l~md 101% of the EWSR exhausted by the first resonance: this, apart from other numerical uncertainties, is due to the fact that the correlated matrix elements are calculated in third order of the FAHT cluster expansion. We think this supports our view that the calculation presented here is consistent and the third order reliable. Otherwise the numerical result for the percentages of EWSR could have been completely unsensical.
Table 1 Properties of monopole resonances computed with (src) and without (HO) short-range correlations. Nucleus
Eg s (MeV)
AE 1 (MeV)
AE 2 (MeV)
%EWSR (1)
%EWSR (2)
4He
HO src
-29.3 -37.4
26.6 33.1
36.0 46.1
59 64
23 21
160
HO src
-106.5 -163.3
22.9 30.9
42.7 60.9
96 99
3 1
4°Ca
HO sre
-323.2 -478.1
20.3 27.0
39.9 53.0
99 101
1 1
14
Volume 118B, number 1, 2, 3
PHYSICS LETTERS
Table 2 Compression modulus of 4He, 160, 4oCa and nuclear matter. Nucleus
K (MeV) Src
4He
t60 4oCa nuclear matter
rio SrC
73
84
86 100 190
88 135 359
Another interesting result of table 1 is the approximate harmonicity of the spectrum: i.e., the spacing of the energies is fairly constant (except for 4He) both with and without src. Furthermore, this occurs not only for the states shown in table 1 but also for other higher states. The last result makes it meaningful to talk of a compression modulus [9] for a nucleus of mass numberA which we deffme as
K A = r2(d2/dr 2) [E(r)/A] , where r is the length parameter of the oscillator and
E(r) is the gs energy calculated with state (1) taking a= 1/r. The value of this quantity for the different nuclei considered are shown in table 2 where for comparison the incompressibility of nuclear matter Knm is also given. Here Knm is defined by
Knm = k 2( d2 / dk 2) [E(k )[A lk:kF , and the values given are obtained with the same interaction used here for the Finite nuclei. The correlated expectation values are calculated by means of the Fermi-hypemetted-chain theory in the approximation FHNC/O where the elementary diagrams are neglected [10]. In this case the correlation factor has been determined by the method of Pandharipande and Bethe [11]. It is seen from table 2 that src produce an appreciable increase of the compression moduli. This was intuitively expected because the equilibrium radius in the presence of Jastrow correlations is noticeably smaller that the equilibrium radius for the uncorrelated nucleus. This means an increase of the nuclear density and correspondingly of the compression modulus. Finally, in the presence of src,KA increases
2 December 1982
with the mass number A, as is also the case without src.
In conclusion, the relevant role of src in the determination of the energies of the isoscalar electric monopole states and compression moduli of light doubly-closed shell nuclei has been stressed. Of course we are aware of the fact that the magnitude of the effect of src might depend also on the precise form of the interaction used, but we expect that the tendency here found will survive. Finally, we would like to point out that, in contrast to previous approaches, our treatment has the merit of a greater coherence since, once the correlation function and the two-body force are chosen, the values of the quantities here evaluated emerge naturally from the calculation. To end up we should mention that it has no sense to compare our results with the experimentally determined properties of the monopole resonances. Actually, the B 1 Brink-Boeker force is an effective interaction designed to be used without src. However, that interaction serves as an adequate prototype for the study of src effects because of the small wound volume corresponding to the optimal correlation.
References [1] M.K. Kirson, Nucl. Phys. A257 (1976) 58; A301 (1978) 93. [2] J.R. Henderson, Nucl. Phys. A323 (1979) 109. [3] R. Guardiola, Nucl. Phys. A328 (1979) 490, in: Lecture Notesin Physics, Vol. 142 (Springer, Berlin, 1981) p. 389; R. Guardiola, A. Faessler, H. M~lther and A. Polls, Nucl. Phys. A371 (1981) 79. [4] D.L. Hill and J.A. Wheeler, Phys. Rev. 89 (1953) 1102; C.W. Wong,Phys. Rep. 15 (1975) 283. [5] A.E.L. Dieperink and T. de Forest, Phys. Rev. C10 (1970) 543. [6] P. Westhaus and J.W. Clark, J. Math. Phys. 9 (1968) 131. [7] R. Guatdiola and A. Polls, Nucl. Phys. A342 (1980) 385; R. Guardiola, A. Polls and J. Ross, Nuovo Cimento 20A (1980) 419. [8] E. Caurier, B. Bourotte-Bilwes and Y. Abgrall, Phys. Lett. 44B (1973) 411; J. Navarro, Phys. Lett. 59B (1975) 13. [9] J.P. Blaizot, D. Gogny and B. Grammaticos, Nucl. Phys. A265 (1976) 315. [10] S. Fantoni and S. Rosati, Nuovo Cimento 25A (1975) 593. [11] V.R. Pandharipande and H.A. Bethe, Phys. Rev. C7 (1973) 1312.
15