Bonn potential

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

Since we will perform only an exploratory calculation with respect to the density modification, instead of highly sophisticated parameterization of the interaction such as the PARIS and BONN potential we will use a simple, separable Yamaguchi interaction. 

Guided by history, we review the major developments concerning realistic nucleon-nucleon (NN) potentials since the pioneering work by Kuo and Brown on the effective nuclear interaction. Our main emphasis is on the physics underlying various models for the NN interaction developed over the past quarter-century. We comment briefly on how to test the quantitative nature of nuclear potentials properly. A correct calculation (performed by independent researchers) of the Z /datum for the fit of the world NN data yields 5.1,3.7, and 1.9 for the Nijmegen, Paris, and Bonn potential, respectively. Finally, we also discuss in detail the relevance of the on- and off-shell properties of NN potentials for microscopic nuclear structure calculations. 

However, if (properly) the pp version of the Bonn potential (see appendix) is confronted with the pp data, a fVdatum of 1.9 is obtained (cf. Table 1). 

Now, from the way pp and np phase shift analyses are conducted, it is clear that a potential that fits the np phase shifts well, will automatically fit the pp phase shifts well, qfier the small but important corrections for Coulomb and charge-dependence have been applied (similarly to what is done in the phase shift analysis). A good example for this is the Bonn potential that has originally been fit to np [15] and has a f /datum of 1.9 for the world np data (cf. Table 1). Now, when the Coulomb force is included and the S scattering length is adjusted to its pp value, then the world pp data are reproduced with a f /datum of also 1.9 (instead of 641 [31], which is a meaningless number). 

To evaluate the OBE amplitudes we use in this work the Bonn potential as it is defined by the meson parameters in Table A.2 of Ref. [7]. For further discussion of this topic, see e.g. Machleidt s contribution in these proceedings. There are three sets of meson parameters which then define three potentials, referred to as the Bonn A, B and C potentials. These potentials differ in the strength of the tensor force, which is reflected in the probability of the D-state of the deuteron. The significance of the tensor force for both nuclear matter and finite nuclei will be discussed in Section 2.3. The coupling constants, cutoffs and masses of the various mesons of Table A.2 of [7] are redisplayed in Table 1. These meson parameters are obtained through a solution of the scattering equation for... 

The reason we include in our discussion, is the fact that this nucleus with T =, the JT = 10 two-body matrix elements, discussed above in connection with the increased binding provided by a potential with a weak tensor force, come into play. For which has isospin T = 2, these matrix elements do not occur. It is therefore instructive to compare the spectra obtained with all three Bonn potentials for these two nuclei. 

Fig. 15. The low-lying Ca spectra relative to Ca. The labels A, B, and C refer to the various versions of the Bonn potential discussed in the text. See text for further explanation. All energies in MeV.

In this work we have reviewed several perturbative many-body techniques which are appropriate to reproduce an effective interaction in low-energy nuclear physics. Moreover, to calculate the reaction matrix G, three recent versions of the Bonn potentials defined in Ref. [7], have been used. We list below our main conclusions. 

For the two-body operators we consider two models first only pion-exchange processes are retained and evaluated in the soft-pion limit, and second pion-exchange processes are evaluated at finite pion energy and momentum, the hard-pion limit, and augmented by heavy-meson-exchange processes (HPPHM). In the latter case, all the heavy-meson coupling constants and form factors are taken from the Bonn potential, OBEPR [13]. The explicit form of the two-body operators are given in Eq. (43) of Ref. [9]. 

Bonn potential. Machleidt et al. (1987) proposed a comprehensive field-theoretical meson-exchange model for nucleon-nucleon interaction. The potential contains many components (one-Tt, two-Tt, p, T, 03, etc. meson exchange). 

Dependence of the SE on NN interaction. Engvik et al. [8] have performed lowest-order BHF calculations in SNM and PNM for all modern potentials (CD-Bonn, Argonne vl8, Reid93, Nijmegen I and II), which fit the Nijmegen NN scattering database with high accuracy. They concluded that... 

Johansson, T. B., McCormick, K., Neij, L. and Turkenburg, W. (2004). The potentials of renewable energy thematic background paper. International Conference for Renewable Energies. Bonn (January 2004). 

The first informal communication of the Commission s decision on Ireland s NAP was received in mid-June 2004 when many of the relevant officials were chairing EU expert groups at the meeting of the subsidiary bodies to the Conference of the Parkes (COP) in Bonn. DG Environment were proposing that the Commission reject the NAP on two grounds first due to the potential for ex-post adjustment to certain allocations and second due to an over-allocation resulting for unsubstantiated policies on the purchase of Kyoto credits by the Irish government. 

B.P. Kreft University of Bonn, Germany Oral MnClj and potential detection of hepatic tumors in rats NS... 

Since the model potential approach yields valence orbitals which have the same nodal structure as the all-electron orbitals, it is possible to combine the approach with an explicit treatment of relativistic effects in the valence shell, e.g., in the framework of the DKH no-pair Hamiltonian [118,119]. Corresponding ab initio model potential parameters are available on the internet under http //www.thch.uni-bonn.de/tc/TCB.download.html. 

Martin, J.M.L., Sundermann, A. Correlation consistent valence basis sets for use with the Stuttgart-Dresden-Bonn relativistic effective core potentials The atoms Ga-Kr and In-Xe, J. Chem. Phys. 2001,114,3408. 

Fig. 2. Phase-shifts of NN scattering for the (a) Pi,(b) and(c) partial wave. Predictions are shown for the Bryan-Scott (B-S) potential of 1969 [12] (long dashes), Nijmegen potential [13] (short dashes), Paris potential [14] (dotted), and the Bonn full model [15] (solid line). The solid squares represent the energy-independent phase shift analysis by Arndt et al. [16].

Fig. 3. (a) Neutron-proton spin correlation parameter Cnn at 181 MeV. Predictions by the Nijmegen potential [13] (long dashes), the Paris potential [ 14] (dotted), and the Bonn full model [ 15] (solid line) are compared with the data (solid squares) from Indiana [18], The ( /datum for the fit of these data is 54.4 for Nijmegen, 3.22 for Paris, and 1.78 for Bonn. The experimental error bars include only systematics and statistics there is also a scale error of 8%. In the calculations of the all three error have been taken-into account [25]. (b) Same as (a), but at 220 MeV with the data from TRIUMF [19]. The z /datum for the fit of these data is 121.0 for the Nijmegen, 16.1 for the Paris, and 0.49 for the Bonn B potential [6]. In addition to the experimental error shown, there is a scale uncertainty of 5.5%. In the calculation of the all errors were taken into account [25]. 

Fig. 7. Phase shifts of some peripheral partial waves as predicted by a field-theoretic model for the 2n exchange (solid line, BONN [15]) and by dispersion theory (dotted line labeled P 73 [21]). Both calculations also include OPE and one-to-exchange. The dotted line labeled P 80 is the fit by the parametrized Paris potential [14]. Octagons represent the phase shift analysis by Arndt et al. [23] and triangles the one by Bugg and coworkers [24].

Fig. 8. itp contributions versus phenomenology in (a) Dj.ib) and(c) Dj. The curves labeled 2it Paris and 27t Bonn represent the predictions by the Paris and Bonn model, respectively, when only the contributions from n, 2it, and cu are taken into account. Adding the phenomenological short-range potential yields the dotted Paris curve (parametrized Paris potential [14]). Adding the up contributions (Fig. 9) yieids the solid 2it + np Bonn curve (Bonn full model [15]). 

In Table 1, we give a summary and an overview of the theoretical input of some meson-theoretic NN models discussed in the previous sections. Moreover, this table also lists the ( /datum (as calculated by independent researchers [25]) for the fit of the relevant world NN data, which is 5.12, 3.71, and 1.90 for the Nijmegen [13], Paris [14], and Bonn [15] potential. The compact presentation, typically for a table, makes it easy to grasp one important point the more seriously and consistently meson theory is pursued, the better the results. This table and its trend towards the more comprehensive meson models is the best proof for the validity of meson theory in the low-energy nuclear regime. 

In Fig. 11, we give an overview of the fit of phase shifts by some modern meson-theoretic potentials. The solid line represents the prediction by the Bonn full model [15] while the dashed line is the Paris [14] prediction. The Bonn full model is an energy-dependent potential. This energy-dependence is inconvenient in nuclear structure applications. Therefore, a representation of the model in terms of relativistic, energy-independent Feynman amplitudes has been developed, using the relativistic, three-dimensional Blankenbecler-Sugar method [30]. This representation has become known as the Bonn B potential [6] . The dotted line in Fig. 11 shows the phase-shift... 

To give an example when the np versions of the Argonne [8] and Bonn [15] potentials are (improperly) confronted with the pp data, a of 824 and 641, respectively, is obtained [31]. 

In a more recent series of x calculations [32], the Argonne and the Bonn np potentials are again (improperly) confronted with the pp data. For the pp data in the energy range 2-350 MeV a ( /datum of 7.1 and 13, respectively, is obtained (while for the range 0-350 MeV, the corresponding numbers are 824 and 641, as mentioned above). However, to cut out the range 0-2 MeV is... 

MeV ( ,ab = 2q /M). The abscissa, k, is the variable over which the integration in Eq. (5) is performed. It is seen that, particularly for large off-shell momenta, the Bonn B potential is smaller than the Paris potential. However, notice also that at the on-shell point (q = k, solid dot in Fig. 12) both potentials are identical (both potentials predict the same Ei parameter). Thus, the Bonn B potential has a weaker off-shell tensor force than the Paris potential. Since the Bonn B and the Paris potential predict almost identical phase shifts, the Born term (central force) in the Si state will be more attractive for the Bonn B potential than for the Paris potential. 

Fig. 12. Magnitude of the half off-shell potential < S, F(q,fe)pD, > is held fixed at 153 MeV. The solid curve is the Bonn B potential [6] and the dashed curve the Paris potential [14]. The solid dot denotes the on-shell point (fe = q).

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