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Core polarization via core diagonalization in oxygen
Volume 51B, number 3
PHYSICS LETTERS
5 August 1974
C O R E P O L A R I Z A T I O N V I A C O R E D I A G O N A L I Z A T I O N IN O X Y G E N P. GOODE* Department of Physics, Rutgers University, New Brunswick, New Jersey 08903, USA and
M.W. KIRSON Department of Nuclear Physics, tCeizmann Institute**, Rehovot, Israel and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA
Received 17 May 1974 A comparison is made between inclusion of high order perturbation theory terms in the effective interaction by diagonalization of core phonons and by infinite partial summations. The final results of the two methods are found to be qualitatively similar.
Several attempts have recently been made to calculate the effect on the nuclear effective interaction of selective inclusion of specific processes to all orders in perturbation theory. Most work has concentrated on the core polarization contribution [1 ], in which two valence nucleons interact indirectly by virtual excitation of the closed-shell core (see fig. 1). The aim has been to improve the treatment of the core excitation by including high order terms in perturbation theory, while retaining the basic structure of the lowest order process. The three main avenues of approach have been infinite partial summation of interacting particle-hole pairs [2] (to be referred to as the "summation method"); diagonalization of the bare interaction, generally a Brueckner reaction matrix G, in the space of two-particle (2p) and three-particle, one-hole (3plh) states [3] ("large space method"); and preliminary computation of the core excited states by large-space diagonalizafions of G, prior to the inclusion of these states in the core polarization diagram [4] ("core * Work supported in part by the National Science Foundation. ** Permanent address.
Fig; 1. Lowest order core polarization.
a.)
b.)
c.)
Fig. 2. Schematic representation of core polarization involving summation of selected higher order diagrams. (a) Summation method (hatched circles represent sums over exchanged particle-hole pairs). (b) Large space method. (c) Core-phonon method. phonon method"). The three methods which are indicated schematically in fig. 2 overlap only partially, each including specific higher order terms neglected by the others. An attempt is made here to relate the first and third of these approaches, with the aim of understanding the similarities and differences between the results generated, and a hybrid calculation is presented which partly merges the two methods. One significant effect omitted by the core-phonon method is renormalization of the vertex coupling the valence particles to the excited core. In the core-pho-, non method, this vertex is always a single G-matrix interaction, while both the other methods include processes in which the vertex contains higher order corrections, and these are known to be important [5, 2]. Thanks to the way in which excitation of the core 221
Volume 51B, number 3
PHYSICS LETTERS
and coupling of the valence particles are separated in the core-phonon method, it is easy to build a hybrid calculation in which the coupling vertex is taken from the summation method and the core excitation from the core-phonon method. The results of such a calculation are presented here and compared with those of the pure summation method. As usual, the discussion centers on the mass-18 nuclei. The original core-phonon method [4] involved diagonalizing G in the space of 0p-0h, l p - l h and 2p-2h harmonic-oscillator states up to 2h 6o excitation energy in 160, and using the resulting excited states of the core in computing the core-polarization contribution to the effective interaction. However, it has become clear that the inclusion of the 0p-0h state in the diagonalization is incorrect. It leads to a large depression of the ground state, through the coupling to many 2p-2h states, and hence to increased excitation energies of the core excited states and decreased core polarization. But since the mass-18 calculations are referred to the experimental ground-state binding energy of 160 (and use experimental mass-17 single-particle energies), this effect has already been included, and its introduction in the core-phonon method involves serious double counting. In diagram terms, the use of 0p-0h in the diagonalization implies allowing the 0p-0h core state in intermediate states of the valence-nucleon interaction, and this produces unlinked diagrams, which have already been summed to obtain the core binding energy and the single-particle energies of the valence particles. The calculations have thus been repeated using only a 1p-1 h + 2p-2h space for the core diagonalization, including all oscillator states of this character with positive parity and excitation energy 2 h w = 28 MeV. (Note that only the J~rT = 0+0 core states are directly affected by this change, though all states appear to shift because their excitation energy was previously referred to the over-depressed ground state.) Experience has proved that the most important core excited states for the core polarization process are the 0÷0 and 2+0 multipoles. In the present calculation, the lowest such multipoles occur at excitation energies of 0.79 MeV and 0.22 MeV, respectively, but are both 96% 2p-2h, hence contributing very little to core polarization. (As is clear from fig. 2(c), the core excited states are coupled to the valence particles only through their lp-lh components.) The lowest such 222
5 August 1974
Table 1 Shifts (in keV) in lowest eigenvalues of the effective interaction upon including core polarization by (a) the vertex-renormalized core-phonon method; (b) the summation method [2]. JlrT
(a)
(b)
JlrT (a)
(b)
0÷1
-206 + 65 +207 -162 + 85 +192 +335 +145 +687
-209 - 52 +212 -122 + 40 +162 +185 +103 +555
1÷0
+ 664 +151 +229 +518 + 95 -222 - 74 +290 -107
1÷1 2+1 3÷1 4+1
2÷0 3÷0 4÷0 5+0
+ 672 +341 +187 +558 +240 -376 - 20 +345 -102
multipoles with significant lp-lh strength occur at 4.92 MeV and 6.06 MeV, respectively. These should be compared with the 10.39 MeV and 15.25 MeV excitation energies of the lowest corresponding multipoles in a pure l p - l h diagonalization (known as TDA). Since all other multipoles stay near 2hw, in both lp-lh + 2p-2h and TDA, they contribute to core polarization essentially as they would in lowest order. They also change very little from one calculation to the next, giving essentially the same contribution to core polarization in lowest order, TDA and the corephonon method. Only the collective 0+0 and 2+0 multipoles drop significantly in energy, leading to enhanced core polarization in TDA, and even more enhancement in the core-phonon method. In a typical effective interaction matrix element the contributions of these multipoles increase by 30% between TDA and the core-phonon approach. In terms of matrix diagonalization, it is easy to see why the core method produces larger enhancements. When the space for diagonalization is increased, the low-lying lp-lh states spread their strength over neighboring 2p-2h configuration, and the lowest of them are depressed in energy. However, it is not so easy to find a diagram explanation for this effect. The corephonon technique includes diagrams like fig. 3(a), generally called "self-screening" and known to reduce collective enhancements [6, 2], and fig. 3(b), a singleparticle-energy correction affecting both particle and hole energies in intermediate states. In an effort to understand the influence of various diagrams, their monopole averages [7], defined as
Volume 51B, number 3
o.)
PHYSICS LETTERS
b.)
Fig. 3. Some higher-orderdiagrams automatically included in the core-phonon method.
F,j,T(2J+I)(2T+I ) VjT/ZJ, T(2J+I)(2T+I ) for a given matrix element V(abab) of a diagram, were computed. For orientation, the (2Sl/2) 2 diagonal matrix element of lowest order core polarization (fig. 1) has a monopole average of 100 keV (repulsive), while the corresponding TDA matrix element has a monopole average of - 4 0 0 keV (attractive). Terms like fig. 3(a) have a monopole average of - 4 0 keV, terms like fig. 3(b), 30 keV, showing a tendency to cancel and no strong enhancement. A cursory examination of the diagrams of next higher order included by the core-phonon method reveals a similar tendency of the monopole averages to cancel. This investigation leaves unclear the diagram source of the 2p-2h enhancement. It may arise from the all-orders summation of many small contributions. (In contrast, 70% of the TDA monopole enhancement occurs in the lowest two orders.) The numerical results of the core-phonon method are similar to those of one of the intermediate steps in the summation method, where TDA phonons are generalized to RPA phonons (interacting particle-hole pairs go backwards, as well as forwards), but are screened by the process of fig. 3(a), while vertex renormalization of the valence-particle-core coupling is ignored 12]. This produces some small enhancement over TDA which differs in degree only from the corephonon results though each contains classes of diagrams omitted from the other. A simplified version of the summation method was used to sum the whole set of diagrams common to both methods, and produced rather different results, though a detailed study revealed the following strange feature. For all noncollective multipoles, the core-phonon method and the simplified summation method produced near
5 August 1974
identical results. However, for the collective 0+0 and 2+0 multipoles, the core-phonon method produced numbers almost exactly 50% larger than those from the simplified summation method (some typical examples are - 1.299 versus -0.849, -0.107 versus -0.070, -0.853 versus -0.557, and 0.538 versus 0.351 MeV!). It is not understood why the additional diagrams included by the core-phonon method cancel for non-collective multipoles, while producing a uniform 50% enhancement of all collective multipoles checked. As a final step, a hybrid calculation was performed, using the core exbited states from the core-phonon method and the valence-particle-to-core coupling vertex from the summation method (self-consistent blackbox vertices). In accordance with previous experience [2], the black-box vertices shuffled the coupling strength of the core multipoles, damping the collective multipoles and enhancing certain repulsive non-collective multipoles, the net result being a greatly reduced core polarization. The vertex-renormalized core-phonon method agrees remarkably well with the summation method, as shown in the table, though the similarity between individual core polarization matrix elements is not nearly as striking, except that they are quite generally small compared to G. In sum, the enhancement of the core-phonon method over TDA is understandable in general terms, though its source in terms of diagrams is not clear. However, the source must lie in diagrams not included in the summation method, since the diagram intersection of the two methods does not contain the enhancement. When coupled to the valence particles by renormalized vertices, the core-phonon method's core excited states produce results similar to those of the summation method. The large space method [3] differs from the other two in abandoning their simple valence-core-valence structure. It allows repeated interactions between the valence particles and the excited core but omits part of the summation method's vertex renormalization and the self-screening and other 2p-2h core effects. However, it also produces small net core polarization. Though quantitative differences between the three methods do not permit the core polarization to be precisely determined, all three methods do find it to be generally quite small.
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Volume 51B, number 3
PHYSICS LETTERS
References [1] G.F. Bertsch, Nucl. Phys. 74 (1965) 234; T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966) 40. [2] M.W. Kirson, Ann. Phys. (N.Y.) 66 (1971) 624; 82 (1974) 345. [3] N. Lo Iudice, D.J. Rowe and S.S.M. Wong, Nucl. Phys. A219 (1974) 171.
224
5 August 1974
[4] P. Goode, Nucl. Phys. A172 (1971) 66. [5] B.R. Barrett and M.W. Kirson, Nucl. Phys. A148 (1970) 145, and erratum A196 (1972) 638. [6] J. Blomqvist and T.T.S. Kuo, Phys. Lett. B29 (1969) 544; E. Osnes, ToT.S. Kuo and C.S. Warke, Nucl. Phys. A168 (1971) 190. [7] P. Goode and D.S. Koltun, Phys. Lett. 39B (1972) 159.
PHYSICS LETTERS
5 August 1974
C O R E P O L A R I Z A T I O N V I A C O R E D I A G O N A L I Z A T I O N IN O X Y G E N P. GOODE* Department of Physics, Rutgers University, New Brunswick, New Jersey 08903, USA and
M.W. KIRSON Department of Nuclear Physics, tCeizmann Institute**, Rehovot, Israel and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA
Received 17 May 1974 A comparison is made between inclusion of high order perturbation theory terms in the effective interaction by diagonalization of core phonons and by infinite partial summations. The final results of the two methods are found to be qualitatively similar.
Several attempts have recently been made to calculate the effect on the nuclear effective interaction of selective inclusion of specific processes to all orders in perturbation theory. Most work has concentrated on the core polarization contribution [1 ], in which two valence nucleons interact indirectly by virtual excitation of the closed-shell core (see fig. 1). The aim has been to improve the treatment of the core excitation by including high order terms in perturbation theory, while retaining the basic structure of the lowest order process. The three main avenues of approach have been infinite partial summation of interacting particle-hole pairs [2] (to be referred to as the "summation method"); diagonalization of the bare interaction, generally a Brueckner reaction matrix G, in the space of two-particle (2p) and three-particle, one-hole (3plh) states [3] ("large space method"); and preliminary computation of the core excited states by large-space diagonalizafions of G, prior to the inclusion of these states in the core polarization diagram [4] ("core * Work supported in part by the National Science Foundation. ** Permanent address.
Fig; 1. Lowest order core polarization.
a.)
b.)
c.)
Fig. 2. Schematic representation of core polarization involving summation of selected higher order diagrams. (a) Summation method (hatched circles represent sums over exchanged particle-hole pairs). (b) Large space method. (c) Core-phonon method. phonon method"). The three methods which are indicated schematically in fig. 2 overlap only partially, each including specific higher order terms neglected by the others. An attempt is made here to relate the first and third of these approaches, with the aim of understanding the similarities and differences between the results generated, and a hybrid calculation is presented which partly merges the two methods. One significant effect omitted by the core-phonon method is renormalization of the vertex coupling the valence particles to the excited core. In the core-pho-, non method, this vertex is always a single G-matrix interaction, while both the other methods include processes in which the vertex contains higher order corrections, and these are known to be important [5, 2]. Thanks to the way in which excitation of the core 221
Volume 51B, number 3
PHYSICS LETTERS
and coupling of the valence particles are separated in the core-phonon method, it is easy to build a hybrid calculation in which the coupling vertex is taken from the summation method and the core excitation from the core-phonon method. The results of such a calculation are presented here and compared with those of the pure summation method. As usual, the discussion centers on the mass-18 nuclei. The original core-phonon method [4] involved diagonalizing G in the space of 0p-0h, l p - l h and 2p-2h harmonic-oscillator states up to 2h 6o excitation energy in 160, and using the resulting excited states of the core in computing the core-polarization contribution to the effective interaction. However, it has become clear that the inclusion of the 0p-0h state in the diagonalization is incorrect. It leads to a large depression of the ground state, through the coupling to many 2p-2h states, and hence to increased excitation energies of the core excited states and decreased core polarization. But since the mass-18 calculations are referred to the experimental ground-state binding energy of 160 (and use experimental mass-17 single-particle energies), this effect has already been included, and its introduction in the core-phonon method involves serious double counting. In diagram terms, the use of 0p-0h in the diagonalization implies allowing the 0p-0h core state in intermediate states of the valence-nucleon interaction, and this produces unlinked diagrams, which have already been summed to obtain the core binding energy and the single-particle energies of the valence particles. The calculations have thus been repeated using only a 1p-1 h + 2p-2h space for the core diagonalization, including all oscillator states of this character with positive parity and excitation energy 2 h w = 28 MeV. (Note that only the J~rT = 0+0 core states are directly affected by this change, though all states appear to shift because their excitation energy was previously referred to the over-depressed ground state.) Experience has proved that the most important core excited states for the core polarization process are the 0÷0 and 2+0 multipoles. In the present calculation, the lowest such multipoles occur at excitation energies of 0.79 MeV and 0.22 MeV, respectively, but are both 96% 2p-2h, hence contributing very little to core polarization. (As is clear from fig. 2(c), the core excited states are coupled to the valence particles only through their lp-lh components.) The lowest such 222
5 August 1974
Table 1 Shifts (in keV) in lowest eigenvalues of the effective interaction upon including core polarization by (a) the vertex-renormalized core-phonon method; (b) the summation method [2]. JlrT
(a)
(b)
JlrT (a)
(b)
0÷1
-206 + 65 +207 -162 + 85 +192 +335 +145 +687
-209 - 52 +212 -122 + 40 +162 +185 +103 +555
1÷0
+ 664 +151 +229 +518 + 95 -222 - 74 +290 -107
1÷1 2+1 3÷1 4+1
2÷0 3÷0 4÷0 5+0
+ 672 +341 +187 +558 +240 -376 - 20 +345 -102
multipoles with significant lp-lh strength occur at 4.92 MeV and 6.06 MeV, respectively. These should be compared with the 10.39 MeV and 15.25 MeV excitation energies of the lowest corresponding multipoles in a pure l p - l h diagonalization (known as TDA). Since all other multipoles stay near 2hw, in both lp-lh + 2p-2h and TDA, they contribute to core polarization essentially as they would in lowest order. They also change very little from one calculation to the next, giving essentially the same contribution to core polarization in lowest order, TDA and the corephonon method. Only the collective 0+0 and 2+0 multipoles drop significantly in energy, leading to enhanced core polarization in TDA, and even more enhancement in the core-phonon method. In a typical effective interaction matrix element the contributions of these multipoles increase by 30% between TDA and the core-phonon approach. In terms of matrix diagonalization, it is easy to see why the core method produces larger enhancements. When the space for diagonalization is increased, the low-lying lp-lh states spread their strength over neighboring 2p-2h configuration, and the lowest of them are depressed in energy. However, it is not so easy to find a diagram explanation for this effect. The corephonon technique includes diagrams like fig. 3(a), generally called "self-screening" and known to reduce collective enhancements [6, 2], and fig. 3(b), a singleparticle-energy correction affecting both particle and hole energies in intermediate states. In an effort to understand the influence of various diagrams, their monopole averages [7], defined as
Volume 51B, number 3
o.)
PHYSICS LETTERS
b.)
Fig. 3. Some higher-orderdiagrams automatically included in the core-phonon method.
F,j,T(2J+I)(2T+I ) VjT/ZJ, T(2J+I)(2T+I ) for a given matrix element V(abab) of a diagram, were computed. For orientation, the (2Sl/2) 2 diagonal matrix element of lowest order core polarization (fig. 1) has a monopole average of 100 keV (repulsive), while the corresponding TDA matrix element has a monopole average of - 4 0 0 keV (attractive). Terms like fig. 3(a) have a monopole average of - 4 0 keV, terms like fig. 3(b), 30 keV, showing a tendency to cancel and no strong enhancement. A cursory examination of the diagrams of next higher order included by the core-phonon method reveals a similar tendency of the monopole averages to cancel. This investigation leaves unclear the diagram source of the 2p-2h enhancement. It may arise from the all-orders summation of many small contributions. (In contrast, 70% of the TDA monopole enhancement occurs in the lowest two orders.) The numerical results of the core-phonon method are similar to those of one of the intermediate steps in the summation method, where TDA phonons are generalized to RPA phonons (interacting particle-hole pairs go backwards, as well as forwards), but are screened by the process of fig. 3(a), while vertex renormalization of the valence-particle-core coupling is ignored 12]. This produces some small enhancement over TDA which differs in degree only from the corephonon results though each contains classes of diagrams omitted from the other. A simplified version of the summation method was used to sum the whole set of diagrams common to both methods, and produced rather different results, though a detailed study revealed the following strange feature. For all noncollective multipoles, the core-phonon method and the simplified summation method produced near
5 August 1974
identical results. However, for the collective 0+0 and 2+0 multipoles, the core-phonon method produced numbers almost exactly 50% larger than those from the simplified summation method (some typical examples are - 1.299 versus -0.849, -0.107 versus -0.070, -0.853 versus -0.557, and 0.538 versus 0.351 MeV!). It is not understood why the additional diagrams included by the core-phonon method cancel for non-collective multipoles, while producing a uniform 50% enhancement of all collective multipoles checked. As a final step, a hybrid calculation was performed, using the core exbited states from the core-phonon method and the valence-particle-to-core coupling vertex from the summation method (self-consistent blackbox vertices). In accordance with previous experience [2], the black-box vertices shuffled the coupling strength of the core multipoles, damping the collective multipoles and enhancing certain repulsive non-collective multipoles, the net result being a greatly reduced core polarization. The vertex-renormalized core-phonon method agrees remarkably well with the summation method, as shown in the table, though the similarity between individual core polarization matrix elements is not nearly as striking, except that they are quite generally small compared to G. In sum, the enhancement of the core-phonon method over TDA is understandable in general terms, though its source in terms of diagrams is not clear. However, the source must lie in diagrams not included in the summation method, since the diagram intersection of the two methods does not contain the enhancement. When coupled to the valence particles by renormalized vertices, the core-phonon method's core excited states produce results similar to those of the summation method. The large space method [3] differs from the other two in abandoning their simple valence-core-valence structure. It allows repeated interactions between the valence particles and the excited core but omits part of the summation method's vertex renormalization and the self-screening and other 2p-2h core effects. However, it also produces small net core polarization. Though quantitative differences between the three methods do not permit the core polarization to be precisely determined, all three methods do find it to be generally quite small.
223
Volume 51B, number 3
PHYSICS LETTERS
References [1] G.F. Bertsch, Nucl. Phys. 74 (1965) 234; T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966) 40. [2] M.W. Kirson, Ann. Phys. (N.Y.) 66 (1971) 624; 82 (1974) 345. [3] N. Lo Iudice, D.J. Rowe and S.S.M. Wong, Nucl. Phys. A219 (1974) 171.
224
5 August 1974
[4] P. Goode, Nucl. Phys. A172 (1971) 66. [5] B.R. Barrett and M.W. Kirson, Nucl. Phys. A148 (1970) 145, and erratum A196 (1972) 638. [6] J. Blomqvist and T.T.S. Kuo, Phys. Lett. B29 (1969) 544; E. Osnes, ToT.S. Kuo and C.S. Warke, Nucl. Phys. A168 (1971) 190. [7] P. Goode and D.S. Koltun, Phys. Lett. 39B (1972) 159.