- Email: [email protected]
Relativistic theory of pairing in symmetric nuclear matter
Progress in Particle and Nuclear Physics PERGAMON
Progress in Particle and Nuclear Physics 46 (2001) 175-176 http://www.elsevier.nl/locate/npe
Relativistic Theory of Pairing in Symmetric Nuclear Matter M. SERRA, A. R U M M E L and P. R I N G Physik-Department, TechnisheUniversitdtMfinchen, 85748 Garching, Germany Abstract The 1S0 pairing gap at the Fermi surface for symmetric nuclear matter at zero temperature is calculated in the framework of a fully consistent relativistic model. A relativistic bate interaction is used to investigate the pairing properties and the resulting gap is compared with a non-relativistic calculation based on a phenomenological density dependent force. Good agreement between the relativistic and the non-relativistic solution of the gap equation is found.
Although it is well known for decades that pairing correlations are essential for astrophysical [2] and nuclear structure [1] calculations, a number of questions is still open on the subject. According to the fact that in principle the effective force in the pairing channel should be the particle-particle K-matrix, we present a solution of the relativistic gap equation using as force in the pp-channel a relativistic version of the Bonn potential [3], a realistic bare nucleon-nucleon interaction. As shown in Ref.[6], the relativistic equation for the pairing gap reduces to the non-relativistic BCS-equation, therefore it reads A(p)
1 f~
vpp(p, k)
A(k)
k2dk
(1)
in which e(k) is now the eigenvalue of the Dirac hamiltonian h, and A the chemical potential. As we investigate pairing in infinite nuclear matter, we may consider, up to a good approximation, only the a and the w fields in the ph-channel. This leads to the following expressions for e(k) and A e(k)
=
V + v ~ + M .2
(2)
A =
V+V/--k2F+M*2
(3)
where the vector field and the effective mass are given by V = g,,,w and M* = M + g,,a respectively. As already mentioned, as force in the pairing channel v~(p, k) we use a relativistic version of the Bonn potential [3], which originates mainly from the exchanges of the a- and w mesons [4]. The resulting gap at the Fermi surface A F is shown in Fig. 1 as function of the Fermi momentum kF for the Bonn-B potential and compared with the corresponding quantity obtained in a non-relativistic calculation [7] based on the Gogny force D1 [5]. We find excellent agreement between the two solutions up to a k f of roughly one fifth of the nuclear matter density, i.e. k f = 0.8 fin-1, where pairing correlations are maximal with m F ,~, 2.8 MeV. This is in agreement with the usual statement that pairing is a surface phenomenon. At larger densities the relativistic solution drops to zero faster than the non-relativistic one and this is due to the fact that at large kF the repulsive contribution of w becomes stronger than the attractive contribution of a. Whether nuclear matter is superfluid at saturation, kF ---- 1.35 fm -1, is hard to decide as it seems to depend critically on the details of the interaction: for the Bonn potential there is no pairing at this density, whereas for the Gogny force a small gap of roughly 0.5 MeV is left. Fig.2 shows the contributions to the gap parameter at the Fermi surface of the different one-meson exchanges which define the Bonn potential. As for the potential, we notice that the gap AF results from the difference between two large contributions: the large and positive contribution from the attractive a-exchange and the negative contribution from the repulsive w-exchange. Nevertheless we point out 0146-6410/01/$ - see front matter © 2001 Elsevier ScienceBV. All rights reserved PII: S0146-6410(01)0012I-1
176
M. Serra et al. / Prog. Part. Nucl. Phys. 46 (2001) 175-176 3.0
2.0
/
/
//
\',,,
\",,,
0.0
1.0
0.0 0.0
0.5
(~")
1.0
1.5
-7.5 0.0
0!5
(fro")
11o
Figure 1: The gap parameter at the Fermi surface AF as a function of the density represented by the Fermi momentum kF for the relativistic Bonn-B potential and the Gogny force D1. Figure 2: Contributions of the different meson-exchange potentials to the gap parameter at the Fermi surface A F as a function of the density represented by the Fermi momentum k f . The thick full line corresponds to the total gap as shown in Fig.l, while the thin lines refer to the different meson exchange contributions. The total gap mainly results form the cancellation between the large positive term of the a-meson and the large negative term of the w-meson. It can be noticed that the r-exchange potential gives a non-vanishing contribution of the order of a few percent of a and w.
that, although small, the contribution of the ~- and p-meson exchanges in the pairing channel does not vanish. In this investigation we have shown that pairing properties of infinite nuclear matter at the Fermi surface calculated using a relativistic bare interaction are in good agreement with phenomenologically adjusted Gogny's results, which are supposed to describe correctly pairing correlations in finite nuclei. Although further terms of the K-matrix should be taken into account for a fully microscopic description of nuclear superfluidity, the fact that the application of a bare nucleon-nucleon interaction in the BCS-equation leads to very reasonable pairing correlations at the Fermi surface gives us confidence to believe that the renormalization effects of higher order terms in the K-matrix can be neglected in the 1S0 pairing channel. To confirm this statement the effect of the polarization diagram into the pairing potential should be investigated.
References [1] [2] [3] [4] [5] [6] [7]
A. Bohr, B.R. Mottelson, and D. Pines, Phys. Rev. 110 (1958) 936 T. Takatsuka, Prog. Theor. Phys. 48 (1972) 1517 R. Machleidt, Adv. NucL Phys. 19 (1989) 189 M. Serra, A. Pummel, and P. Ring, Submitted for Publication J. Decharge and D. Gogny, Phys. Rev. C 21 (1980) 1568 H. Kuchaxek and P. Ring, Z. Phys. A 339 (1991) 23 H. Kucharek, P. Ring, P. Schuck, R. Bengtsson, and M. Girod, Phys. Lett. B 216 (1989) 249
Progress in Particle and Nuclear Physics 46 (2001) 175-176 http://www.elsevier.nl/locate/npe
Relativistic Theory of Pairing in Symmetric Nuclear Matter M. SERRA, A. R U M M E L and P. R I N G Physik-Department, TechnisheUniversitdtMfinchen, 85748 Garching, Germany Abstract The 1S0 pairing gap at the Fermi surface for symmetric nuclear matter at zero temperature is calculated in the framework of a fully consistent relativistic model. A relativistic bate interaction is used to investigate the pairing properties and the resulting gap is compared with a non-relativistic calculation based on a phenomenological density dependent force. Good agreement between the relativistic and the non-relativistic solution of the gap equation is found.
Although it is well known for decades that pairing correlations are essential for astrophysical [2] and nuclear structure [1] calculations, a number of questions is still open on the subject. According to the fact that in principle the effective force in the pairing channel should be the particle-particle K-matrix, we present a solution of the relativistic gap equation using as force in the pp-channel a relativistic version of the Bonn potential [3], a realistic bare nucleon-nucleon interaction. As shown in Ref.[6], the relativistic equation for the pairing gap reduces to the non-relativistic BCS-equation, therefore it reads A(p)
1 f~
vpp(p, k)
A(k)
k2dk
(1)
in which e(k) is now the eigenvalue of the Dirac hamiltonian h, and A the chemical potential. As we investigate pairing in infinite nuclear matter, we may consider, up to a good approximation, only the a and the w fields in the ph-channel. This leads to the following expressions for e(k) and A e(k)
=
V + v ~ + M .2
(2)
A =
V+V/--k2F+M*2
(3)
where the vector field and the effective mass are given by V = g,,,w and M* = M + g,,a respectively. As already mentioned, as force in the pairing channel v~(p, k) we use a relativistic version of the Bonn potential [3], which originates mainly from the exchanges of the a- and w mesons [4]. The resulting gap at the Fermi surface A F is shown in Fig. 1 as function of the Fermi momentum kF for the Bonn-B potential and compared with the corresponding quantity obtained in a non-relativistic calculation [7] based on the Gogny force D1 [5]. We find excellent agreement between the two solutions up to a k f of roughly one fifth of the nuclear matter density, i.e. k f = 0.8 fin-1, where pairing correlations are maximal with m F ,~, 2.8 MeV. This is in agreement with the usual statement that pairing is a surface phenomenon. At larger densities the relativistic solution drops to zero faster than the non-relativistic one and this is due to the fact that at large kF the repulsive contribution of w becomes stronger than the attractive contribution of a. Whether nuclear matter is superfluid at saturation, kF ---- 1.35 fm -1, is hard to decide as it seems to depend critically on the details of the interaction: for the Bonn potential there is no pairing at this density, whereas for the Gogny force a small gap of roughly 0.5 MeV is left. Fig.2 shows the contributions to the gap parameter at the Fermi surface of the different one-meson exchanges which define the Bonn potential. As for the potential, we notice that the gap AF results from the difference between two large contributions: the large and positive contribution from the attractive a-exchange and the negative contribution from the repulsive w-exchange. Nevertheless we point out 0146-6410/01/$ - see front matter © 2001 Elsevier ScienceBV. All rights reserved PII: S0146-6410(01)0012I-1
176
M. Serra et al. / Prog. Part. Nucl. Phys. 46 (2001) 175-176 3.0
2.0
/
/
//
\',,,
\",,,
0.0
1.0
0.0 0.0
0.5
(~")
1.0
1.5
-7.5 0.0
0!5
(fro")
11o
Figure 1: The gap parameter at the Fermi surface AF as a function of the density represented by the Fermi momentum kF for the relativistic Bonn-B potential and the Gogny force D1. Figure 2: Contributions of the different meson-exchange potentials to the gap parameter at the Fermi surface A F as a function of the density represented by the Fermi momentum k f . The thick full line corresponds to the total gap as shown in Fig.l, while the thin lines refer to the different meson exchange contributions. The total gap mainly results form the cancellation between the large positive term of the a-meson and the large negative term of the w-meson. It can be noticed that the r-exchange potential gives a non-vanishing contribution of the order of a few percent of a and w.
that, although small, the contribution of the ~- and p-meson exchanges in the pairing channel does not vanish. In this investigation we have shown that pairing properties of infinite nuclear matter at the Fermi surface calculated using a relativistic bare interaction are in good agreement with phenomenologically adjusted Gogny's results, which are supposed to describe correctly pairing correlations in finite nuclei. Although further terms of the K-matrix should be taken into account for a fully microscopic description of nuclear superfluidity, the fact that the application of a bare nucleon-nucleon interaction in the BCS-equation leads to very reasonable pairing correlations at the Fermi surface gives us confidence to believe that the renormalization effects of higher order terms in the K-matrix can be neglected in the 1S0 pairing channel. To confirm this statement the effect of the polarization diagram into the pairing potential should be investigated.
References [1] [2] [3] [4] [5] [6] [7]
A. Bohr, B.R. Mottelson, and D. Pines, Phys. Rev. 110 (1958) 936 T. Takatsuka, Prog. Theor. Phys. 48 (1972) 1517 R. Machleidt, Adv. NucL Phys. 19 (1989) 189 M. Serra, A. Pummel, and P. Ring, Submitted for Publication J. Decharge and D. Gogny, Phys. Rev. C 21 (1980) 1568 H. Kuchaxek and P. Ring, Z. Phys. A 339 (1991) 23 H. Kucharek, P. Ring, P. Schuck, R. Bengtsson, and M. Girod, Phys. Lett. B 216 (1989) 249