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Two- and three-body correlations in the photonuclear dipole sum
Volume 51B, number 4
PHYSICS LETTERS
19 August 1974
TWO- AND THREE-BODY CORRELATIONS IN THE PHOTONUCLEAR DIPOLE SUM M. FINK Gymnasium Bismarckstrasse, Hattingen, Germany M. GARI, H. HEBACH Ruhr-Universit~t Bochum, lnstitut f'dr Theoretische Physik, Germany and Max-Planck-Institut f't2r Chemie, Abteilung Kernphysik, Mainz, Germany J.G. ZABOLITZKY* Ruhr-Universit~t Bochum, lnstitut f'dr Theoretische Physik, Germany Received 1 June 1974 The enhancement factor K in the photonuclear dipole sum rule is calculated using Brueckner-Hartree-Fock (BHF) wave-functions, i.e. accurate solutions of the coupled Bethe-Goldstone- and Hartree-Fock equations with approximate inclusion of the three-body Bethe-Faddeev amplitudes. The effect of these two- and three-body correlations is discussed in dependence of various nucleon-nucleon potentials. Results are presented for 4 He and 160.
Recently there have been renewed efforts [1,2] in calculating the photonuclear electric dipole sum 27r2e2h NZ (1 + r). t~(EL)(E) d E = m ~ - c
(1)
A
o The interesting quantity K, the so-called enhancement factor, is given by A M
= ~ h-~-.
(2)
D z is the electric dipole operator .4
Dz=~l
z ~%z,
(3)
ct=l
V is the nucleon-nucleon interaction and Iff) is the exact ground state wave function belonging to this same potential. Experiments [3] on the total photonuclear absorption cross section have shown surprisingly large values for r, namely K ~ 0.6 for light nuclei like Be and C and r ~ 1.0 for heavier nuclei like O, A1, Si and Ca. Calculations of K [1,2] have been done by the use of nuclear wave functions including two-body * Supported by the Deutsche Forschungsgemeinschaft.
320
short range nucleon-nucleon correlations. While calculations with short range correlations computed with the reference spectrum method [1 ] are able to explain large values of K, the use of solutions of the combined Hartree-Fock and Bethe-Goldstone equations [2] give values of K generally around 0.6. In the present paper we improve our calculations (i) by a detailed investigation of the influence of different nucleon-nucleon potentials, (ii) by taking into account two-body correlations and the most important three-body correlations resulting from the Bethe-Faddeev equations. We also discuss the connection to ref. [1] by investigating the effect of the reference spectrum method in solving the Bethe-Goldstone equation. Results for r are presented for 4He and 160 and are discussed in connection with the binding energy and charge radius obtained by the same method. The wave functions used in this work have been calculated by use of the exp(S)-method [4] : A
[~b)= eXP(n~=l Sn) l¢)
(4)
here 14>denotes a Slater determinant of single particle wave functions. Sn denotes the n particle correlation function. The exp(S)-method allows the derivation of a set of coupled equations for the amplitudes S n in
Volume 51B, number 4
PHYSICS LETTERS
a hierarchical order, (i) generalized Hartree-Fock (HF) equations, (ii) generalized Bethe-Goldstone (BG) equations, (iii) three-body Bethe-Faddeev (BF) equations, (iv) four-body BF equations and so on. In order to solve these equations numerically one has to truncate this set. Truncation after the BG equations yields the Brueckner-Hartree-Fock (BHF) method [5]. One has to perform a HF calculation, where the effective interaction is to be calculated via the BG equation and
RSC
1.0
various self-consistency schemes have to be considered Going beyond this method, it has also been possible [6] to truncate after the three-body BF equations so that we are left with three types o f equations: the HF equations, the BG equation which determines the effective interaction, and the BF equations which determine the particle potential to be used with the BG equation. In addition, further self-consistency requirements are imposed. The solution o f this coupled
~.o
REFS
08
19 August 1974
RSC
08 FBHF
06
-
0.4
OZ
02
O2
0
0 ~
02
-02
FBHF
a)
REFS
c)
0.,~
- 0%
0
08
1.6
2.4
1.0
r[fm]
0
4.0
08
1.6
2.~
~0
SSC
0.8
r[fm]
410
SSC
0.8 FBHF
06 OZ
Oz
02
0.2
0
0
-0.2
~~ 2
REFS
b)
-0,~ 0
08
1.6
2.4
r [ f m ] ~.0
J I
d)
FBHF
-0.4 / 0
0.8
1.6
2.4
r [ f r n ] 40
Fig. 1. a) Relative two-particle wave functions for the RSC potential. These functions are not those actually employed in our calculations, but for the sake of comparison harmonic oscillator orbitals were used. The function starting linearly at the origin is the uneorrelated one. The 3Dl state function is given negative values, the 3S 1 state positive ones. The solution of the Bethe-Goldstone equation is compared to the reference spectrum approximation (REFS). b) Same as a), for the SSC-C potential, c) Same as a), the effect of three-body correlations is shown, i.e. the FBHF type wave function is compared to the solution of the BG equation. d) Same as a), for the SSC-C potential. 321
PHYSICS LETTERS
Volume 51B, number 4
set of equations yields the Faddeev-Brueckner-HartreeFock (FBHF) wave function. In ref. [1 ] different approximations in the calculation of the wave functions were used. While the BHF calculation results in single-particle functions (the HF functions) as well as in correlated two-particle wave functions (solution of the BG equation with double self-consistency) these authors used oscillator functions for the s.p. wave functions and employed the reference spectrum approximation in the calculation of the two-body correlation functions. In figs. la and lb we show the solution of the BG equation as well as the one resulting from the use of the reference spectrum approximation. We see that because of the strong overshoot larger resuits for r are to be expected when we use the reference spectrum method. It has been shown [6, 8] that to a good approximation the solution of the three-body BF equation may be represented by a spatially constant, attractive particle potential of about - 8 MeV per particle inserted into the BG equation. The calculations presented in this paper were done in this approximate way only in order to save computer time. In figs. 1c and 1d we show the relative two-body wave functions including two-body correlations only (BHF) and the one with the additional three-body effects (FBHF) calculated in this way. The use of a BHF type wave function is equivalent to the incorporation of the diagrams shown in fig. 2. At first sight these diagrams equal those ofref. [1] but this is not the case because of the different meaning of the lines (HF orbitals instead of harmonic oscillator
(a)
(b)
(c)
(d)
Fig. 2. Contributions from two-particle correlations. ~ denotes the operator, -..,,.,,denotes the solution of the BG equation.
322
19 August 1974
(c)
Fig. 3. Contributions from three-particle correlations. Meaning of the tl~ee-particle box see fig. 5.
orbitals) and G-matrices (solutions of the BG equation instead of reference spectrum approximation). Note that our method does not only treat the Pauli principle correctly but also avoids a separation of intermediate particle states into low lying and higher excited ones. If FBHF wave functions are employed, additional diagrams as shown in fig. 3 are taken into account. This is due to the additional correlations incorporated into the wave function. As the threeparticle correlation function S 3 itself is not calculated, but only its effect onto the two-particle Correlation function S 2 (see fig. 3), the diagram of fig. 4 is not included. This one, however, would be proportional to IIS3 n2 which is a very small quantity (IlS3 l[< 0.1 [kS2 II). The three-particle box introduced in fig. 3 includes all three-particle ladder diagrams and some other higher order diagrams. The lowest order is shown in fig. 5, which contributes about one half of the threebody correlation part (exchange diagrams are not shown). In tables 1 and 2 we present the results of our calculation for the enhancement factor K. For the sake of comparison we also list the binding energy BE/A
Fig. 4. Three-body correlation contribution which is not included.
Volume 51B, number 4
__
PHYSICS LETTERS
~
"Jr- . . .
Fig. 5. Lowest order of the three-particle box (exchange diagrams omitted). calculated w i t h the very same wave functions. Also the calculated charge radius is given. For 4He r varies b e t w e e n 0.5 and 0.6. The differences o f r for various soft and hard core potentials are not exciting. The results are in good agreement w i t h the e x p e r i m e n t a l
19 August 1974
values ( n o t yet very well determined), l~or 160 K varies b e t w e e n 0.58 and 0.74. Also here the d e p e n d e n c e on the n u c l e o n - n u c l e o n potentials is n o t dramatic. The largest value K = 0.74 obtained for the Yale potential is smaller than the experimental value o f r = 1.0. In tables 1 and 2 the effect o f t h r e e - b o d y correlations (already included in K) A = K -- r 2 is listed separately. We see that the three-body correlations give an e n h a n c e m e n t o f 5 - 1 2 % . The differences b e t w e e n hard and soft core potentials are remarkable. In order to compare our results w i t h the larger values o f r obtained f r o m the S t o n y Brook group [1]
Table 1 Enhancement factor r, norm of the defect wave function II$2 II, binding energy and charge radius for 4 He. The results are given with the inclusion of the three-body effects. The contribution A = K- K2 of the three-body correlation is listed separately. Potentials are Reid soft core (RSC), super soft core (SSC) in three versions, Gogny-Pires-de Tourreil (GPT), Eikemeier-Hackenbroieh (EH), Brueckner-Gammel-Thaler (BGT), Yale and Hamada-Johnston (HJ).
soft core
hard core
Potential
IIS211
BE/A [MeV]
r c [fm]
r
A = r - K2
RSC SSC-A SSC-B SSC-C GPT EH
0.129 0.077 0.069 0.073 0.043 0.102
-5.16 -5.87 -5.99 -4.85 -6.66 -5.52
1.73 1.70 1.69 1.75 1.70 1.69
0.58 0.53 0.53 0.49 0.60 0.56
0.04 0.03 0.03 0.03 0.02 0.03
BGT YALE HJ
0.123 0.232 0.191
-5.34 -4.29 -4.33
1.74 1.89 1.85
0.51 0.61 0.59
0.03 0.07 0.05
1.63
0.33
-
- 7 . 0 7
-
-
1.71
0.57 t71
experimental
-
Table 2 Same as table 1 for 160.
soft core
hard core
experimental
Potential
IIS211
BE/A [MeV]
r c [fm]
K
£x = r - r 2
RSC SSC-A SSC-B SSC-C GPT EH
0.166 0.108 0.098 0.100 0.070 0.139
-5.17 -6.38 -6.71 -5.14 -7.51 -5.46
2.68 2.58 2.57 2.66 2.61 2.61
0.67 0.62 0.61 0.58 0.67 0.66
0.05 0.03 0.03 0.03 0.03 0.04
BGT YALE HJ
0.157 0.284 0.238
-5.79 -3.48 -3.64
2.62 2.97 2.90
0.60 0.74 0.72
0.04 0.10 0.08
2.65
1.00
-
- 7 . 9 6
-
+
2.73
0.07 [3]
-
323
Volume 51B, number 4
PHYSICS LETTERS
Table 3 Results of the enhancement factor r are given including only two-body correlations (K = r2). Here we show the relative importance of the tensor correlations. KS denotes the scalar part, KT denotes the tensor contribution. Potential
RSC SSC-A SSC-B SSC-C GPT EH BGT YALE HJ
19 August 1974
Table 4 Norm IIS211of the defect wave function, binding energy, charge radius and enhancement factor K (K = K2) in BHF method and reference spectrum approximation (REFS). The data given are for 160, calculated by the use of RSC potential.
~S
4 He KT
K2
KS
rT
160 K2
Method
IIS211
BE[A [MeV]
r c [fro]
K2
0.22 0.28 0.28 0.25 0.45 0.27 0.20 0.22 0.22
0.31 0.22 0.22 0.21 0.13 0.26 0.28 0.32 0.32
0.53 0.50 0.50 0.46 0.58 0.53 0.48 0.54 0.54
0.25 0.32 0.33 0.29 0.49 0.31 0.22 0.25 0.26
0.37 0.27 0.25 0.26 0.15 0.31 0.34 0.39 0.38
0.62 0.59 0.58 0.55 0.64 0.62 0.56 0.64 0.64
BHF REFS
0.137 0.277
-4.30 -7.80
2.71 2.57
0.62 0.93
body theory. Further discussions will be meaningful when the experiments have reached their final status.
References we list in table 3 the contributions o f the scalar and tensor correlations separately (although the separation cannot be done very clean from our program). Our results confirm the strong importance of tensor correlations, generally o f about 50% of the total K. In table 4 we compare our m e t h o d with the reference spectrum approximation. We see that the reference spectrum method leads to a r which is 50% larger than the one obtained with the correct treatment o f the Pauli projection operator. This is due to the strong overshoot obtained in the R E F S m e t h o d (compare figs. l a and lb). In conclusion we remark that although our calculations o f r for 4He (and 12C [2] ) are in good agreement with the experiments, the additional three-body correlations included in this paper are not able to explain a value ofK = 1.0 for 160. It seems to us that this discrepancy is not due to a shortcoming o f many-
324
[1] W.T. Weng, T.T.S. Kuo and G.E. Brown, Phys. Letters 46B (1973) 329. [2] M. Fink, M. Gari and H. Hebach, Phys. Letters 49B (1974) 20. [3] J. Ahrens et al., Proc. Intern. Conf. on Nuclear structure studies, Sendai, Japan, 1972. [4] F. Coester, Nucl. Phys. 7 (1958) 421; F. Coester, H. K~mmel, Nucl. Phys. 17 (1960) 477; H. Kfimmel, Nucl. Phys. A176 (1971) 205; H. K~mmel and K.H. L~hrmann, Nucl. Phys. A191 (1972) 525; K.H. Lu'hrmann and H. K~mmel, Nucl. Phys. A194 (1972) 255. [5] H. Kfimmel, J.G. Zabolitzky, Phys. Rev. C7 (1973) 547; J.G. Zabolitzky, Nucl. Phys., in print. [6] J.G. Zabolitzky, Phys. Letters 47B (1973) 487; J.G. Zabolitzky, Nucl. Phys., in print. [7] W.E. Meyerhofand S. Fiarman, Proc. Asilomar Conf. (1973) 385. [8] W. Schelongowski and J.G. Zabolitzky, to be published.
PHYSICS LETTERS
19 August 1974
TWO- AND THREE-BODY CORRELATIONS IN THE PHOTONUCLEAR DIPOLE SUM M. FINK Gymnasium Bismarckstrasse, Hattingen, Germany M. GARI, H. HEBACH Ruhr-Universit~t Bochum, lnstitut f'dr Theoretische Physik, Germany and Max-Planck-Institut f't2r Chemie, Abteilung Kernphysik, Mainz, Germany J.G. ZABOLITZKY* Ruhr-Universit~t Bochum, lnstitut f'dr Theoretische Physik, Germany Received 1 June 1974 The enhancement factor K in the photonuclear dipole sum rule is calculated using Brueckner-Hartree-Fock (BHF) wave-functions, i.e. accurate solutions of the coupled Bethe-Goldstone- and Hartree-Fock equations with approximate inclusion of the three-body Bethe-Faddeev amplitudes. The effect of these two- and three-body correlations is discussed in dependence of various nucleon-nucleon potentials. Results are presented for 4 He and 160.
Recently there have been renewed efforts [1,2] in calculating the photonuclear electric dipole sum 27r2e2h NZ (1 + r). t~(EL)(E) d E = m ~ - c
(1)
A
o The interesting quantity K, the so-called enhancement factor, is given by A M
= ~ h-~-
(2)
D z is the electric dipole operator .4
Dz=~l
z ~%z,
(3)
ct=l
V is the nucleon-nucleon interaction and Iff) is the exact ground state wave function belonging to this same potential. Experiments [3] on the total photonuclear absorption cross section have shown surprisingly large values for r, namely K ~ 0.6 for light nuclei like Be and C and r ~ 1.0 for heavier nuclei like O, A1, Si and Ca. Calculations of K [1,2] have been done by the use of nuclear wave functions including two-body * Supported by the Deutsche Forschungsgemeinschaft.
320
short range nucleon-nucleon correlations. While calculations with short range correlations computed with the reference spectrum method [1 ] are able to explain large values of K, the use of solutions of the combined Hartree-Fock and Bethe-Goldstone equations [2] give values of K generally around 0.6. In the present paper we improve our calculations (i) by a detailed investigation of the influence of different nucleon-nucleon potentials, (ii) by taking into account two-body correlations and the most important three-body correlations resulting from the Bethe-Faddeev equations. We also discuss the connection to ref. [1] by investigating the effect of the reference spectrum method in solving the Bethe-Goldstone equation. Results for r are presented for 4He and 160 and are discussed in connection with the binding energy and charge radius obtained by the same method. The wave functions used in this work have been calculated by use of the exp(S)-method [4] : A
[~b)= eXP(n~=l Sn) l¢)
(4)
here 14>denotes a Slater determinant of single particle wave functions. Sn denotes the n particle correlation function. The exp(S)-method allows the derivation of a set of coupled equations for the amplitudes S n in
Volume 51B, number 4
PHYSICS LETTERS
a hierarchical order, (i) generalized Hartree-Fock (HF) equations, (ii) generalized Bethe-Goldstone (BG) equations, (iii) three-body Bethe-Faddeev (BF) equations, (iv) four-body BF equations and so on. In order to solve these equations numerically one has to truncate this set. Truncation after the BG equations yields the Brueckner-Hartree-Fock (BHF) method [5]. One has to perform a HF calculation, where the effective interaction is to be calculated via the BG equation and
RSC
1.0
various self-consistency schemes have to be considered Going beyond this method, it has also been possible [6] to truncate after the three-body BF equations so that we are left with three types o f equations: the HF equations, the BG equation which determines the effective interaction, and the BF equations which determine the particle potential to be used with the BG equation. In addition, further self-consistency requirements are imposed. The solution o f this coupled
~.o
REFS
08
19 August 1974
RSC
08 FBHF
06
-
0.4
OZ
02
O2
0
0 ~
02
-02
FBHF
a)
REFS
c)
0.,~
- 0%
0
08
1.6
2.4
1.0
r[fm]
0
4.0
08
1.6
2.~
~0
SSC
0.8
r[fm]
410
SSC
0.8 FBHF
06 OZ
Oz
02
0.2
0
0
-0.2
~~ 2
REFS
b)
-0,~ 0
08
1.6
2.4
r [ f m ] ~.0
J I
d)
FBHF
-0.4 / 0
0.8
1.6
2.4
r [ f r n ] 40
Fig. 1. a) Relative two-particle wave functions for the RSC potential. These functions are not those actually employed in our calculations, but for the sake of comparison harmonic oscillator orbitals were used. The function starting linearly at the origin is the uneorrelated one. The 3Dl state function is given negative values, the 3S 1 state positive ones. The solution of the Bethe-Goldstone equation is compared to the reference spectrum approximation (REFS). b) Same as a), for the SSC-C potential, c) Same as a), the effect of three-body correlations is shown, i.e. the FBHF type wave function is compared to the solution of the BG equation. d) Same as a), for the SSC-C potential. 321
PHYSICS LETTERS
Volume 51B, number 4
set of equations yields the Faddeev-Brueckner-HartreeFock (FBHF) wave function. In ref. [1 ] different approximations in the calculation of the wave functions were used. While the BHF calculation results in single-particle functions (the HF functions) as well as in correlated two-particle wave functions (solution of the BG equation with double self-consistency) these authors used oscillator functions for the s.p. wave functions and employed the reference spectrum approximation in the calculation of the two-body correlation functions. In figs. la and lb we show the solution of the BG equation as well as the one resulting from the use of the reference spectrum approximation. We see that because of the strong overshoot larger resuits for r are to be expected when we use the reference spectrum method. It has been shown [6, 8] that to a good approximation the solution of the three-body BF equation may be represented by a spatially constant, attractive particle potential of about - 8 MeV per particle inserted into the BG equation. The calculations presented in this paper were done in this approximate way only in order to save computer time. In figs. 1c and 1d we show the relative two-body wave functions including two-body correlations only (BHF) and the one with the additional three-body effects (FBHF) calculated in this way. The use of a BHF type wave function is equivalent to the incorporation of the diagrams shown in fig. 2. At first sight these diagrams equal those ofref. [1] but this is not the case because of the different meaning of the lines (HF orbitals instead of harmonic oscillator
(a)
(b)
(c)
(d)
Fig. 2. Contributions from two-particle correlations. ~ denotes the operator, -..,,.,,denotes the solution of the BG equation.
322
19 August 1974
(c)
Fig. 3. Contributions from three-particle correlations. Meaning of the tl~ee-particle box see fig. 5.
orbitals) and G-matrices (solutions of the BG equation instead of reference spectrum approximation). Note that our method does not only treat the Pauli principle correctly but also avoids a separation of intermediate particle states into low lying and higher excited ones. If FBHF wave functions are employed, additional diagrams as shown in fig. 3 are taken into account. This is due to the additional correlations incorporated into the wave function. As the threeparticle correlation function S 3 itself is not calculated, but only its effect onto the two-particle Correlation function S 2 (see fig. 3), the diagram of fig. 4 is not included. This one, however, would be proportional to IIS3 n2 which is a very small quantity (IlS3 l[< 0.1 [kS2 II). The three-particle box introduced in fig. 3 includes all three-particle ladder diagrams and some other higher order diagrams. The lowest order is shown in fig. 5, which contributes about one half of the threebody correlation part (exchange diagrams are not shown). In tables 1 and 2 we present the results of our calculation for the enhancement factor K. For the sake of comparison we also list the binding energy BE/A
Fig. 4. Three-body correlation contribution which is not included.
Volume 51B, number 4
__
PHYSICS LETTERS
~
"Jr- . . .
Fig. 5. Lowest order of the three-particle box (exchange diagrams omitted). calculated w i t h the very same wave functions. Also the calculated charge radius is given. For 4He r varies b e t w e e n 0.5 and 0.6. The differences o f r for various soft and hard core potentials are not exciting. The results are in good agreement w i t h the e x p e r i m e n t a l
19 August 1974
values ( n o t yet very well determined), l~or 160 K varies b e t w e e n 0.58 and 0.74. Also here the d e p e n d e n c e on the n u c l e o n - n u c l e o n potentials is n o t dramatic. The largest value K = 0.74 obtained for the Yale potential is smaller than the experimental value o f r = 1.0. In tables 1 and 2 the effect o f t h r e e - b o d y correlations (already included in K) A = K -- r 2 is listed separately. We see that the three-body correlations give an e n h a n c e m e n t o f 5 - 1 2 % . The differences b e t w e e n hard and soft core potentials are remarkable. In order to compare our results w i t h the larger values o f r obtained f r o m the S t o n y Brook group [1]
Table 1 Enhancement factor r, norm of the defect wave function II$2 II, binding energy and charge radius for 4 He. The results are given with the inclusion of the three-body effects. The contribution A = K- K2 of the three-body correlation is listed separately. Potentials are Reid soft core (RSC), super soft core (SSC) in three versions, Gogny-Pires-de Tourreil (GPT), Eikemeier-Hackenbroieh (EH), Brueckner-Gammel-Thaler (BGT), Yale and Hamada-Johnston (HJ).
soft core
hard core
Potential
IIS211
BE/A [MeV]
r c [fm]
r
A = r - K2
RSC SSC-A SSC-B SSC-C GPT EH
0.129 0.077 0.069 0.073 0.043 0.102
-5.16 -5.87 -5.99 -4.85 -6.66 -5.52
1.73 1.70 1.69 1.75 1.70 1.69
0.58 0.53 0.53 0.49 0.60 0.56
0.04 0.03 0.03 0.03 0.02 0.03
BGT YALE HJ
0.123 0.232 0.191
-5.34 -4.29 -4.33
1.74 1.89 1.85
0.51 0.61 0.59
0.03 0.07 0.05
1.63
0.33
-
- 7 . 0 7
-
-
1.71
0.57 t71
experimental
-
Table 2 Same as table 1 for 160.
soft core
hard core
experimental
Potential
IIS211
BE/A [MeV]
r c [fm]
K
£x = r - r 2
RSC SSC-A SSC-B SSC-C GPT EH
0.166 0.108 0.098 0.100 0.070 0.139
-5.17 -6.38 -6.71 -5.14 -7.51 -5.46
2.68 2.58 2.57 2.66 2.61 2.61
0.67 0.62 0.61 0.58 0.67 0.66
0.05 0.03 0.03 0.03 0.03 0.04
BGT YALE HJ
0.157 0.284 0.238
-5.79 -3.48 -3.64
2.62 2.97 2.90
0.60 0.74 0.72
0.04 0.10 0.08
2.65
1.00
-
- 7 . 9 6
-
+
2.73
0.07 [3]
-
323
Volume 51B, number 4
PHYSICS LETTERS
Table 3 Results of the enhancement factor r are given including only two-body correlations (K = r2). Here we show the relative importance of the tensor correlations. KS denotes the scalar part, KT denotes the tensor contribution. Potential
RSC SSC-A SSC-B SSC-C GPT EH BGT YALE HJ
19 August 1974
Table 4 Norm IIS211of the defect wave function, binding energy, charge radius and enhancement factor K (K = K2) in BHF method and reference spectrum approximation (REFS). The data given are for 160, calculated by the use of RSC potential.
~S
4 He KT
K2
KS
rT
160 K2
Method
IIS211
BE[A [MeV]
r c [fro]
K2
0.22 0.28 0.28 0.25 0.45 0.27 0.20 0.22 0.22
0.31 0.22 0.22 0.21 0.13 0.26 0.28 0.32 0.32
0.53 0.50 0.50 0.46 0.58 0.53 0.48 0.54 0.54
0.25 0.32 0.33 0.29 0.49 0.31 0.22 0.25 0.26
0.37 0.27 0.25 0.26 0.15 0.31 0.34 0.39 0.38
0.62 0.59 0.58 0.55 0.64 0.62 0.56 0.64 0.64
BHF REFS
0.137 0.277
-4.30 -7.80
2.71 2.57
0.62 0.93
body theory. Further discussions will be meaningful when the experiments have reached their final status.
References we list in table 3 the contributions o f the scalar and tensor correlations separately (although the separation cannot be done very clean from our program). Our results confirm the strong importance of tensor correlations, generally o f about 50% of the total K. In table 4 we compare our m e t h o d with the reference spectrum approximation. We see that the reference spectrum method leads to a r which is 50% larger than the one obtained with the correct treatment o f the Pauli projection operator. This is due to the strong overshoot obtained in the R E F S m e t h o d (compare figs. l a and lb). In conclusion we remark that although our calculations o f r for 4He (and 12C [2] ) are in good agreement with the experiments, the additional three-body correlations included in this paper are not able to explain a value ofK = 1.0 for 160. It seems to us that this discrepancy is not due to a shortcoming o f many-
324
[1] W.T. Weng, T.T.S. Kuo and G.E. Brown, Phys. Letters 46B (1973) 329. [2] M. Fink, M. Gari and H. Hebach, Phys. Letters 49B (1974) 20. [3] J. Ahrens et al., Proc. Intern. Conf. on Nuclear structure studies, Sendai, Japan, 1972. [4] F. Coester, Nucl. Phys. 7 (1958) 421; F. Coester, H. K~mmel, Nucl. Phys. 17 (1960) 477; H. Kfimmel, Nucl. Phys. A176 (1971) 205; H. K~mmel and K.H. L~hrmann, Nucl. Phys. A191 (1972) 525; K.H. Lu'hrmann and H. K~mmel, Nucl. Phys. A194 (1972) 255. [5] H. Kfimmel, J.G. Zabolitzky, Phys. Rev. C7 (1973) 547; J.G. Zabolitzky, Nucl. Phys., in print. [6] J.G. Zabolitzky, Phys. Letters 47B (1973) 487; J.G. Zabolitzky, Nucl. Phys., in print. [7] W.E. Meyerhofand S. Fiarman, Proc. Asilomar Conf. (1973) 385. [8] W. Schelongowski and J.G. Zabolitzky, to be published.