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The nuclear shell model is one of the most successful microscopic nuclear structure models[1]. In the shell model, the quantum many-body problem becomes diagonalizing a Hamiltonian matrix in a space spanned by multi-nucleon configuration basis states, with the assumption that valence protons and neutrons occupy several single-particle states. A shell-model effective Hamiltonian includes single-particle energies and two-body interactions. Many efforts have been devoted to description of low-lying states of atomic nuclei using the shell model with realistic nucleon-nucleon interaction, based on various renormalization and core polarization correction approaches[2-4]. Compared with experimental data, shell-model calculations with these realistic interactions deteriorate as the number of valence nucleons increases. Thus empirical corrections or fitting of an effective interaction are necessary[5-9]. Here we mention two typical types of interactions: the effective two-body-matrix-element (ETBME) Hamiltonian and the phenomenological P+Q interaction (i.e. the pairing plus quadrupole-quadrupole interaction).
An ETBME Hamiltonian is specified by a number of parameters, i.e., single-particle energies and two-body matrix elements (TBMEs), in a given single-particle space. A few linear combinations of these parameters are derived from the least square fit, while the remaining linear combinations are fixed by a realistic interaction. This approach is successfully applied to obtain a few famous ETBME Hamiltonians, such as the USD, USDA, and USDB interactions in the
$ sd $ shell[5-6], the GXPF1 interaction in the$ pf $ shell[7], and the JUN45 interaction in the$ 1p_{3/2} 0f_{5/2} 1p_{1/2} 0g_{9/2} $ shell[8]. The advantage of the ETBME Hamiltonians is that interpolations and predictions for low-lying states are accurate. The drawbacks are that in higher shell regions there are too many TBMEs to evaluate and the configuration space is too gigantic to handle with the present computational ability, and it is not easy to figure out the physical meaning from the parameters.The advantage of the P+Q interaction is that it has much less independent parameters and provides us with transparent pictures in physics. Due to its simplicity, the P+Q interaction has been extensively used to study low-lying states in nuclear structure models[10-22]. It has been demonstrated to be a reasonable approximation of the effective interaction for low-lying collective states, but unfortunately the fitting parameters vary widely for different nuclei in the same single-particle space.
In this work, we study low-lying states of nuclei in the
$ sd $ and$ pf $ shells using the shell model with the phenomenological P+Q interaction. We shall show that by considering the “correct” monopole Hamiltonian, low-lying states of even-even nuclei can be well described by a unified set of parameters of the P+Q interaction. We shall also show that the pairing interaction is not independent of the monopole interaction; by removing the monopole component from the pairing interaction, the deviation of calculated binding energies from experimental data shall be reduced. This paper is organized as follows. In Sec. 2 we introduce the P+Q interaction and the parameters used in our shell-model calculation. In Sec. 3 we study low-lying states and binding energies, and extract the symmetry energy and Wigner energy coefficients. In Sec. 4 we summarize our results. -
The P+Q interaction is given as follows.
The first term
is the single-particle energy, and the second term is the monopole interaction[23], which is written as
where
$ V_{JT}(j_{1}j_{2}j_{1}j_{2}) $ is the TBME. The single-particle energy and monopole interaction represent the spherical single-particle mean field in the framework of the shell model, and play a dominant role in describing binding energies of atomic nuclei[1, 23-24]. The third term is the quadrupole-quadrupole interactionwhere
$ Q $ is the quadrupole operator, and$ \kappa $ is a parameter. The quadrupole-quadrupole interaction generates rotational motions and quadrupole deformations in a nucleus. The fourth and fifth terms are the isovector monopole and quadrupole pairing interactions, respectively, i.e.,where
$ {A^{(JT)}_{m\tau}}^{\dagger} $ is a creation operator of a collective nucleon pair with spin$ J $ and isospin$ T $ , and$ G_{J} $ is a parameter. In this work we also investigate the isoscalar$ J = 1 $ and$ J = 2j_{\max} $ pairing interactions, i.e.,The pairing interactions defined in Eqs. (6)~(9) are not independent of the monopole interaction. Thus we remove the monopole component from
$ V_{P0} $ ,$ V_{P2} $ ,$ V_{P1} $ , and$ V_{P{\max}} $ , and the modified pairing interactions are denoted by$ V^{\prime}_{P0} $ ,$ V^{\prime}_{P2} $ ,$ V^{\prime}_{P1} $ , and$ V^{\prime}_{P{\max}} $ , respectively. Meanwhile, the monopole-removal P+Q interaction should be written as -
We study yrast states of even-even nuclei in the
$ sd $ shell and the$ pf $ shell, using the shell model with the P+Q interaction. The NuShellX shell-model code is used[25]. In order to obtain reasonable spherical mean field, the single-particle energy,$ H_0 $ , and the monopole interaction,$ V_{\rm{m}} $ , are directly taken from ETBME Hamiltonians; here we adopt the USDB interaction[6] for the$ sd $ shell and the GXPF1 interaction[7] for the$ pf $ shell.The parameters
$ G_{J} $ and$ \kappa $ for the$ sd $ shell are optimized by fitting the excitation energy data of yrast even-$ J $ states for even-even nuclei 18-24O, 20-28Ne, 24-30Mg, 28-34Si, 32-36S, 36,38Ar, and those for the$ pf $ shell are optimized by fitting the data for 42-48Ca, 44-50Ti, 48-52Cr. The detailed procedure is as follows. We define the weighted root-mean-square deviation of the excited energies between the calculations and experiments byHere we take
$ w_i \!=\! 1.5,\; 1.3,\; 1.2,\; 1.1,\; 0.9,\; 0.5 $ for the$ 2^+,\; 4^+,\; 6^+,\; 8^+,\; 10^+,\; 12^+ $ states, respectively. Our goal is to find the set of$ G_{J} $ and$ \kappa $ for which$ \sigma^2 $ reaches a minimum value. Here we make use of the conjugate gradient method[26-27], an iterative algorithm for unconstrained optimization problems, which we describe it as follows.(1) We define the parameter vector
$ \vec p \!=\! \{ p_{1}, p_{2},\ldots, p_{n} \} $ where$ n $ is the number of parameters (e.g.,$ \{ G_0, G_2 , \kappa \} $ for$ H_{\rm{ P+Q}} $ ), and$ \sigma^2 $ is the function of$ p $ . We denote the initial guess for$ p $ by$ p_0 $ .(2) We compute the negative gradient of
$ \sigma^2 $ at$ p_0 $ , denoted by$ -{ g}_1 $ , and use it as our initial search direction$ { d}_1 $ . The selection of the search direction will change in further iterations.(3) Going from
$ p_0 $ in the direction of$ { d}_1 $ , we search for the solution$ p_1 \!=\! p_0 + x_1{ d}_1 $ for which$ \sigma^2 $ reaches the local minimum. The solution$ p_1 $ is assumed to be the initial vector of the next iteration.We repeat steps 2 and 3 until convergence, except that the search direction in the
$ i $ th iteration$ { d}_i\! =\! -{ g}_i + {\vec d}_{i-1} |{ g}_i|^2 / |{ g}_{i-1}|^2 $ .Table 1 presents the parameters
$ G_J $ and$ \kappa $ of$ H_{\rm{ P+Q}} $ ,$ H^{\prime}_{\rm{ P+Q}} $ and$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1} $ for the$ sd $ and$ pf $ shells. The values of$ G_0 $ ,$ G_2 $ and$ \kappa $ are a little different from those given in Refs. [28-29] ($ G_0, G_2, \kappa \approx -0.40, -0.10, -0.66 $ MeV for the$ sd $ shell, and$ -0.48, -0.18, -0.15 $ MeV for the$ pf $ shell). We calculate a few TBMEs with$ J \!=\! 0 $ and$ T \!=\! 1 $ of our P+Q interactions. For example, we obtain$ V_{01}(d_{3/2}d_{3/2}d_{3/2} d_{3/2}) \!=\! -1.57 $ MeV,$ V_{01}(d_{5/2}d_{5/2}d_{5/2}d_{5/2}) \!=\! -2.21 $ MeV,$ V_{01}(f_{7/2}f_{7/2}f_{7/2}f_{7/2}) \!=\! -2.26 $ MeV for$ H^{\prime}_{\rm{ P+Q}} $ ; these values are reasonably close to those given in the USDB and GXPF1 interactions ($ -1.90, -2.56, -2.24 $ MeV).Interaction $G_{0}$ $G_{2}$ $G_{1}$ $\kappa$ sd shell $H_{\rm{ P+Q}}$ –0.182 –0.056 –0.513 $H^{\prime}_{\rm{ P+Q}}$ –0.194 –0.118 –0.512 $H^{\prime}_{\rm{ P+Q}}+V^{\prime}_{P1}$ –0.197 –0.114 –0.049 –0.511 pf shell $H_{\rm{ P+Q}}$ –0.477 –0.108 –0.113 $H^{\prime}_{\rm{ P+Q}}$ –0.461 –0.116 –0.117 $H^{\prime}_{\rm{ P+Q}}+V^{\prime}_{P1}$ –0.451 –0.109 –0.322 –0.107 Table 1. The parameters (in the unit of MeV) of the pairing interactions,
$G_J$ , and the quadrupole-quadrupole interaction,$\kappa$ , determined by fitting the excitation energy data in the$sd$ and$pf$ shells. All the parameters given in the table should be multiplied by$(\frac{A}{18})^{-0.3}$ in the$sd$ shell and by$(\frac{A}{42})^{-\frac{1}{3}}$ in the$pf$ shell. -
Let us begin with the discussion of the low-lying spectrum. The yrast states of the even-even nuclei are well described by our shell-model calculation with the P+Q interaction. We exemplify this with the cases of 20O, 20Ne, 22Ne, 24Mg, and 28Si in the
$ sd $ shell (see in Fig. 1). 20O is a typical semimagic nucleus with a spherical shape, and 20Ne, 22Ne, 24Mg, 28Si are deformed nuclei with a ground rotational band. The excitation energies obtained by$ H_{\rm{ P+Q}} $ ,$ H^{\prime}_{\rm{ P+Q}} $ , and$ H^{\prime}_{\rm{ P+Q}} +$ $V^{\prime}_{P1} $ are all in good agreement with the data or the USDB results. The root-mean-square deviation of the excitation energies between the$ H^{\prime}_{\rm{ P+Q}} $ results and the experimental data is$ 0.51 $ MeV, and that for the USDB interaction is$ 0.46 $ MeV. The monopole component in the pairing interactions and the isoscalar$ J \!=\! 1 $ pairing interaction are not crucial in reproducing the low-lying spectra. The strength of the isoscalar$ J \!=\! 1 $ pairing interaction,$ G_1 $ , is small.Figure 1. (color online) Excitation energies of the yrast states of 20O, 20Ne, 22Ne, 24Mg, and 28Si obtained from experimental data and by shell-model calculations with different interactions in the
$sd$ shell.Similar results are found for the
$ pf $ -shell nuclei. Fig. 2 compares for the yrast states of 44Ca, 44Ti, 46Ti, 48Cr, and 52Fe the experimental data and the shell-model results. The excitation energies obtained by$ H_{\rm{ P+Q}} + V_{P1} $ ,$ H^{\prime}_{\rm{ P+Q}} $ , and$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1} $ are all in good agreement with the GXPF1 results. The root-mean-square deviation of the excitation energies between the$ H^{\prime}_{\rm{ P+Q}} $ results and the experimental data is$ 0.37 $ MeV and that for the GXPF1 interaction is$ 0.57 $ MeV (we exclude the$ 10^+ $ and$ 12^+ $ states of 44Ti, due to the large discrepancy, in the calculation of the root-mean-square deviation). It is worth mentioning that the data of 52Fe are not used in the parameter fitting, and our prediction for 52Fe is good. The above result indicates the monopole component in the pairing interactions and the isoscalar$ J \!= \!1 $ pairing interaction do not play an important role here. In Table 1 we find the optimized parameter$ G_1 $ of$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1} $ in the$ pf $ shell is not small, this can be explained as follows: the excitation energies are not sensitive to the$ G_1 $ -parameter variation, and the value can be regarded as a result of overfitting.Figure 2. (color online) Same as Fig. 1 except for 44Ca, 44Ti, 46Ti, 48Cr, and 52Fe in the
$pf$ shell.We also investigate
$ V^{\prime}_{P{\max}} $ , namely the isoscalar spin-aligned$ J $ =7 pairing interaction, in the$ pf $ shell calculation. The result of Ref. [30] shows that the quadrupole-quadrupole interaction is correlated with the$ V_{P{\max}} $ interaction, and the wave functions generated by the pure quadrupole-quadrupole interaction and by the strong isoscalar spin-aligned pairing interaction are almost identical. We obtain a similar result: By incorporating the isoscalar spin-aligned pairing interaction in the Hamiltonian,$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P {\max}} $ predicts prominently stronger quadrupole collectivity in 44Ti, especially for higher-spin states (see in Fig. 2). The isoscalar spin-aligned pairing interaction is not important. -
Now let us focus our attention on the binding energy. We predict the binding energy of even-even nuclei using the shell model with the P+Q interactions,
$ H_{\rm{ P+Q}} $ ,$ H^{\prime}_{\rm{ P+Q}} $ , and$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1} $ . We compare the binding energy between the P+Q results and the ETBME (i.e., USDB and GXPF1) results, and calculate the difference between them, namely$ \Delta B\! =\! B_{\rm{ P+Q}} - B_{\rm ETBME} $ . Fig. 3 presents$ \Delta B $ for even-even nuclei 18-24O, 20-28Ne, 24-30Mg, 28-34Si, 32-36S, 36,38Ar, 42-48Ca, 44-50Ti, 48-52Cr, and 52Fe. One sees a large discrepancy between the$ H_{\rm{ P+Q}} $ and ETBME results. We extract the monopole component,$ V_{T}(j_{1}j_{2}) $ , from the pairing interactions in$ H_{\rm{ P+Q}} $ by using Eq. (4), and find that all of them are attractive. This explains the increasing$ \Delta B $ as the number of valence nucleons. The change of relative energies between the single-particle orbits caused by the monopole component is not drastic, thus it does not have a significant effect on excitation energies.Figure 3. (color online) The difference between the binding energy calculated by the P+Q interactions (
$H_{\rm{ P+Q}}$ ,$H^{\prime}_{\rm{ P+Q}}$ ,$H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1}$ ) and that by the ETBME Hamiltonian (i.e., USDB and GXPF1).The agreement between the
$ H^{\prime}_{\rm{ P+Q}} $ and ETBME results is much better. Yet the binding energy obtained by$ H^{\prime}_{\rm{ P+Q}} $ is$ \sim 2 $ MeV larger than that by the ETBME. It is worth mentioning that the parameters of the P+Q interaction are evaluated by fitting the excitation energy data; the results can be improved if further considering the binding energy data in the fitting procedure. One might suppose that the deviation for the nuclei with$ A \!=\! 18 $ and 42, which have 2 valence nucleons above the doubly magic cores, is attributed to single-particle energy difference between the P+Q and ETBME Hamiltonians. Indeed, in the P+Q Hamiltonian the quadrupole-quadrupole interaction contributes an additional single-particle energy term, due to the fact that the scalar product of the quadrupole operators is not a pure two-body interaction. According to out calculation, the additional single-particle energies in$ H^{\prime}_{\rm{ P+Q}} $ range from$ -2.0 $ to$ -0.4 $ MeV. Considering other multipole interactions (e.g., the$ \sigma \tau \cdot \sigma \tau $ interaction) might compensate for the difference.By calculating the double difference of binding energies between neighboring four nuclei, one obtains the empirical proton-neutron interaction between the last two protons and the last two neutrons in a nucleus[31-35], i.e.,
We calculate
$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ by using the binding energies from the AME2016 data table[36] and the shell-model calculation with the P+Q and the ETBME interactions. The results are presented in Fig. 4.Figure 4. (color online) The empirical proton-neutron interaction
$\delta V_{2{\rm{p}} - 2{\rm{n}}}$ calculated by using the binding energies from the AME2016 data and the shell-model calculation.For
$ N \neq Z $ nuclei,$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ obtained by the P+Q interactions are in good agreement with the data or the ETBME results. In the Weizsäcker nuclear mass formula, the empirical proton-neutron interaction of$ N \neq Z $ nuclei is explained by the symmetry energy[37]. In Ref. [38] the empirical proton-neutron interaction is used to constrain symmetry energy coefficients. In this work we constrain the coefficient of the$ I^2 $ volume and surface terms,$ c^{(\rm V)}_{2} $ and$ c^{(\rm S)}_{2} $ , using the same method as in Ref. [38]. The results are presented in Table 2. We find$ c^{(\rm V)}_{2} $ and$ c^{(\rm S)}_{2} $ obtained by$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1} $ are in good agreement with those constrained from experiment or the ETBME results, close to the values$ c^{(\rm V)}_{2} = 32.1 $ MeV and$ c^{(\rm S)}_{2} = 58.91 $ MeV suggested in Ref. [38].Coefficient Expt. $H_{\rm{ P+Q}}$ $H^{\prime}_{\rm{ P+Q}}$ $H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1}$ ETBME $c^{(\rm V)}_{2}$ 34.98 30.44 29.45 34.33 32.82 $c^{(\rm S)}_{2}$ 60.51 43.30 40.36 54.13 54.50 $a_W$ 44.45 32.52 31.21 31.18 44.01 Table 2. The symmetry energy and the Wigner energy coefficients extracted by using the empirical proton-neuron interaction
$\delta V_{2{\rm{p}} - 2{\rm{n}}}$ (in the unit of MeV).In Fig. 4 one sees
$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ for$ N = Z $ nuclei is much stronger than that for their$ N \neq Z $ neighbors. This significant enhancement is interpreted as a consequence of the Wigner energy[32-33, 39-40]. For even-even nuclei the Wigner energy is simply defined byA typical value of the Wigner energy coefficient
$ a_W\! =\! 42.7 $ MeV[41]. The Wigner energy has attracted much attention in nuclear physics society. The microscopic mechanism has been studied in terms of the SU(4) spin-isospin supermultiplet theory[39-40], the seniority scheme (or the SU(2) theory)[42], the monopole and isovector pairing interactions[43-45], etc.In the previous work Fu et al.[24] decomposed the USDB interaction into the monopole, isovector pairing, and qudrupole-quadrupole interactions, and extracted
$ a_W $ for nuclei in the$ sd $ shell by the local mass relation, in which the binding energies were calculated using the shell model. It turned out that the value of$ a_W $ obtained with those interactions are smaller than that extracted from the experimental binding energy data. Similarly, in this work we extract$ a_W $ by using binding energies obtained with our P+Q interactions. In Fig. 4 and Table 2 one sees that$ H_{\rm{ P+Q}} $ ,$ H^{\prime}_{\rm{ P+Q}} $ , and$ H^{\prime}_{\rm{ P+Q}} + V^{\prime}_{P1} $ predict smaller$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ and$ a_W $ than the experimental data and the ETBME results. This indicates that residual two-body interactions beyond the P+Q interaction are indispensable to the interpretation of the Wigner energy (e.g., the$ \sigma \tau \cdot \sigma \tau $ interaction might be important in reproducing the isospin dependence of the binding energy[22]). -
In this paper we study yrast states of even-even nuclei in the
$ sd $ shell and the$ pf $ shell, using the shell model with the phenomenological P+Q interaction, in which the single-particle energy and the monopole interaction are taken from those in the USDB or GXPF1 interaction, and the parameters of the pairing interactions and the quadrupole-quadrupole interaction (namely,$ G_J $ and$ \kappa $ ) are optimized by fitting excitation energy data. We find that the excitation energies of the spherical nuclei and the deformed nuclei in the$ sd $ (or$ pf $ ) shell are all well reproduced by using the original P+Q interaction,$ H_{\rm{ P+Q}} $ , with a unified set of parameters. However, for binding energies we see increasing discrepancy as the number of valence nucleons increases. By removing the monopole component from the pairing interactions, the prediction of binding energies is much improved. We have also shown that the isoscalar$ J \!=\! 1 $ pairing interaction and the isoscalar spin-aligned pairing interaction are not important in our calculation.We extract the empirical proton-neutron interaction,
$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ , by using the calculated binding energies, and then constrain the symmetry energy coefficients,$ c^{(\rm V)}_{2} $ and$ c^{(\rm S)}_{2} $ , and the Wigner energy coefficient,$ a_W $ . The$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ of$ N \neq Z $ nuclei and the symmetry energy coefficients obtained by the P+Q interaction agree well with those obtained from experiment, while the$ \delta V_{2{\rm{p}} - 2{\rm{n}}} $ of$ N \!=\! Z $ nuclei and the Wigner energy coefficient are underestimated.Considering other multipole interactions (e.g., the hexadecapole-hexadecapole interaction, the
$ \sigma \tau \cdot \sigma \tau $ interaction)[1, 22-23] are reasonably expected to further improve the validity of the P+Q interaction.
Shell Model Study of Even-even sd and pf Shell Nuclei With the Pairing Plus Quadrupole-quadrupole Interaction
doi: 10.11804/NuclPhysRev.37.2019CNPC10
- Received Date: 2019-12-14
- Rev Recd Date: 2020-03-19
- Available Online: 2020-09-30
- Publish Date: 2020-09-20
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Key words:
- shell model /
- P+Q interaction /
- unified parameter /
- monopole component
Abstract: In this paper we study low-lying states of even-even nuclei in the sd and pf shells in the framework of the shell model with the phenomenological pairing plus quadrupole-quadrupole (P+Q) interaction. By adopting the single-particle energy and the monopole interaction from the USDB and GXPF1 interactions, the low-lying spectra of spherical nuclei and deformed nuclei are successfully reproduced by a unified set of parameters. We obtain a reasonably good result for binding energies by removing the monopole component from the pairing interactions. The isoscalar pairing interaction does not play an important role in the states. The monopole interaction provides contributions to the empirical proton-neutron interaction, the symmetry energy, and the Wigner energy.
Citation: | Yeqiu HE, Guanjian FU. Shell Model Study of Even-even sd and pf Shell Nuclei With the Pairing Plus Quadrupole-quadrupole Interaction[J]. Nuclear Physics Review, 2020, 37(3): 509-515. doi: 10.11804/NuclPhysRev.37.2019CNPC10 |