Deformed Shell Model Applications to Weak Interaction Processes

Abstract

The deformed shell model (DSM), based on Hartree–Fock intrinsic states with angular momentum projection and band mixing, has been found to be quite successful in describing many spectroscopic properties of nuclei in the A = 60–100 region. More importantly, DSM has been used recently with good success in calculating nuclear structure factors, which are needed for a variety of weak interaction processes. In this article, in addition to giving an overview of this, we discuss the applications of DSM to obtain cross-sections for coherent and incoherent neutrino nucleus scattering on ^96,98,100Mo targets and also for obtaining two neutrino double beta decay nuclear transition matrix elements for ¹⁰⁰Mo.

1. Introduction

Over the last many years, the deformed shell model (DSM), based on Hartree–Fock intrinsic states with angular momentum projection and band mixing, has been established to be a good model to describe the properties of medium-mass nuclei—see the book [1] for details of DSM and for a summary of some of the applications of the model. DSM calculations are performed in the spirit of a spherical shell model, where one takes a model space and a suitable effective interaction (single-particle orbitals, single-particle energies, and a two-body effective interaction). This procedure has been found to be quite successful in the mass region A = 60–100 in describing spectroscopic properties like band crossing, shape coexistence, electro-magnetic transition probabilities, $T=0$ and $T=1$ bands in N = Z odd–odd nuclei with isospin projection, and so on [1]. Going beyond spectroscopy, DSM has also been employed quite successfully for obtaining nuclear structure factors, which are needed for a variety of weak processes. These include (i) both two-neutrino and neutrinoless double beta decay, including positron double beta decay (all the examples studied so far are reviewed in [1]); (ii) $\mu \to e$ conversion in the field of the nucleus in ⁷²Ge [2]; (iii) event rates for WIMP nucleus scattering off ²³Na, ⁴⁰Ar, ^71,73Ge, ⁷⁵As, ¹²⁷I, ¹³³Cs, and ¹³³Xe nuclei [3,4,5,6]; (iv) coherent and incoherent event rates for the neutrino-nucleus scattering from ²⁸Si, ^64,70Zn, ^70,71,73,76Ge, ⁷⁵As, ¹²⁷I, ¹³³Cs, and ¹³³Xe [4,5,7,8]. In the present paper, results are presented for neutrino-nucleus scattering cross-sections and two-neutrino double beta decay nuclear matrix elements obtained using DSM and involving Mo isotopes. Now, we give a preview.

In Section 2, we present some of the important steps in the formulation of DSM. In Section 3, we discuss its application to neutrino-nucleus scattering in ^96,98,100Mo isotopes. In Section 4, the application of the model to $2\nu$ double beta decay in ¹⁰⁰Mo is presented. Finally, Section 5 gives conclusions.

2. Deformed Shell Model

The details of this model have been described in many of our earlier publications (see, for example, ref. [1]). In this model, for a given nucleus, starting with a model space consisting of a given set of single-particle (sp) orbitals and effective two-body Hamiltonian (TBME + spe), the lowest-energy intrinsic states are obtained by solving the axially symmetric Hartree–Fock (HF) single-particle equation self-consistently. Excited intrinsic configurations are obtained by making particle–hole excitations over the lowest intrinsic state. These intrinsic states ${\chi }_K\left(\eta \right)$ do not have definite angular momenta. Hence, states of good angular momentum projected from an intrinsic state ${\chi }_K\left(\eta \right)$ can be written in the form

$${\psi }_{MK}^J\left(\eta \right)=\frac{2J+1}{8{\pi }^2\sqrt{N_{JK}}}\int d\Omega D_{MK}^{J^{*}}(\Omega )R(\Omega )|{\chi }_K\left(\eta \right)\rangle$$

(1)

where $N_{JK}$ is the normalization constant, given by

$$N_{JK}=\frac{2J+1}{2}{\int }_0^{\pi }d\beta sin\beta d_{KK}^J\left(\beta \right)\langle {\chi }_K\left(\eta \right)|e^{-i\beta J_y}|{\chi }_K\left(\eta \right)\rangle$$

(2)

In Equation (1), $\Omega$ represents the Euler angles ($\alpha$, $\beta$, $\gamma$) and $R(\Omega )$, which is equal to exp($-i\alpha J_z$) exp($-i\beta J_y$) exp( $-i\gamma J_z$), represents the general rotation operator.

The good angular momentum states projected from different intrinsic states are not, in general, orthogonal to each other. Hence, they are orthonormalized and then band-mixing calculations are performed. The resulting eigenfunctions are of the form

$$|{\Phi }_M^J\left(\eta \right)\rangle =\sum _{K,\alpha }{S}_{K\eta }^J\left(\alpha \right)|{\psi }_{MK}^J\left(\alpha \right)\rangle .$$

(3)

3. Cross-Sections of Neutral-Current Neutrino Scattering on ^96,98,100Mo Isotopes

Some years ago, the neutral-current neutrino-nucleus channel was detected for the first time in the COHERENT experiment [9], more than four decades after its theoretical prediction by Freedman [10]. The measurement of coherent elastic event rates of neutrino scattering on a sodium-doped CsI detector may help in answering some key questions related to neutrino properties and provide an understanding of some theories beyond the standard model of the electroweak interactions [11]. Several promising nuclear detectors are on their way to being employed in designed and ongoing experiments [12,13,14,15,16,17,18,19,20,21,22]. Following all these, very recently, cross-sections for neutrino scatterings off ^98,100Mo were calculated by us, using DSM for nuclear structure factors [23]. In addition to briefly describing these results, here, we also report the first results for ⁹⁶Mo.

The details of the relevant formulation that provides the neutral-current $\nu$-nucleus scattering differential cross-sections have been discussed earlier in [24,25,26]. Applying a multipole analysis on the weak Hadronic current [25,26], the neutrino-nucleus scattering cross-section becomes

$$\frac{d^2{\sigma }_{i\to f}}{d\Omega d\omega }(\varphi ,\theta ,\omega ,{ϵ}_i)=\delta (E_f-E_i-\omega )\frac{2G^2{ϵ}_f^2{cos}^2(\theta /2)}{\pi (2J_i+1)}\left[C_V+C_A\mp C_{VA}\right]$$

(4)

The $\delta$-function on the right-hand side of the above equation ensures the energy conservation, so that the excitation energy of the nucleus $\omega$ is given by

$$\omega =E_f-E_i={ϵ}_i-{ϵ}_f$$

(5)

$E_i$ and $E_f$ represent the energy of the initial and final states of the studied nucleus, and ${ϵ}_i$ and ${ϵ}_f$ are the incoming and outgoing energies of the neutrino. In Equation (4), the (−) sign corresponds to the scattering of the neutrinos and the (+) to the scattering of the antineutrinos.

The terms $C_V$ and $C_A$ include a summation over the contributions coming from the polar-vector and axial-vector multipole operators. Similarly, the interference term $C_{VA}$ in Equation (4) contains the product of transverse polar-vector and transverse axial-vector matrix elements. All these involve square of the four-momentum transfer $q_{\mu }^2$ and the scattering angle $\theta$. Note that the square of the four-momentum transfer $q_{\mu }^2$ and the magnitude of the three-momentum transfer q $\equiv \mid \mathbf{q}\mid$ can be written in terms of the kinematic parameters, i.e., laboratory scattering angle $\theta$ and neutrino energies ${ϵ}_i={ϵ}_{\nu }$ and ${ϵ}_f=({ϵ}_i-\omega )$, as

$$\begin{gathered} \begin{array}{c}{q}_{\mu }^2\equiv q_{\mu }{q}^{\mu }=-4{ϵ}_i({ϵ}_i-\omega ){sin}^2(\theta /2) \\ q\equiv \mid \mathbf{q}\mid ={\left[{\omega }^2+4{ϵ}_i({ϵ}_i-\omega ){sin}^2(\theta /2)\right]}^{1/2}\hfill \end{array} \end{gathered}$$

(6)

The reduced matrix elements that enter in the formulas for $C_{V\left(A\right)}$ and $C_{VA}$ are evaluated using DSM band-mixed wave function ${\Phi }_M^J\left(\eta \right)$, defined in Equation (3). Full details of the various operators and the method employed for the calculation of various reduced matrix elements are given in [23].

In the calculations for ^96,98,100Mo isotopes, we have adopted the effective interaction $GWBXG$ with the ⁶⁶Ni as the closed core. The details regarding the construction of the effective interaction have been discussed in refs. [27,28]. The active proton orbits are $0f_{5/2}$, $1p_{3/2}$$1p_{1/2}$, and $0g_{9/2}$ with single-particle energies −5.322, −6.144, −3.941, and −1.250 MeV. For the neutrons, the active orbits are $1p_{1/2}$, $0g_{9/2}$, $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, and $2s_{1/2}$. The single-particle energies for the first five orbits are taken to be −0.696, −2.597, 5.159, 1.830, and 4.261 MeV, respectively. The $2s_{1/2}$ orbit produces low-lying large deformed solutions, even though molybdenum isotopes are known to be weakly deformed. Hence, the effect of this orbit is eliminated by taking the corresponding neutron single-particle energy at a high value.

Using the above effective interaction and the single-particle energies, we first carried out an axially symmetric HF calculation by solving the HF equation self-consistently. The detailed spectroscopic results obtained using DSM for ⁹⁸Mo and ¹⁰⁰Mo are given in [23]. The observed low-lying energy levels for both the nuclei are well-reproduced in our calculation, as are the lowest few $B\left(E2\right)$ values. Beyond these two isotopes, here, we report briefly the results obtained for ⁹⁶Mo. For ⁹⁶Mo, the axially symmetric HF calculation is carried out using the above-described model space and effective interaction. It is found that the lowest prolate and oblate HF solutions are almost degenerate, with the oblate solution being lower by ∼0.25 MeV. We obtained excited HF configurations by making particle–hole excitations over these lowest HF configurations. For positive parity, we consider 33 intrinsic configurations (17 oblate and 16 prolate) and, for negative parity, 24 intrinsic configurations (12 oblate and 12 prolate). As described before, good angular momentum states are projected from each of these intrinsic states. These states are not orthogonal to each other. Hence, they are orthonormalized and a band-mixing calculation is performed. States having similar structure and, thus, similar electromagnetic properties are classified in one band. For ⁹⁶Mo, not many collective bands have been observed. In Figure 1, we compare the calculated ground band with the experimentally observed band and we see that the agreement is reasonable. Further details regarding other spectroscopic properties of ⁹⁶Mo will be reported elsewhere.

One of our goals is to calculate the incoherent differential cross-sections for each $J^{\pi }$ state of Mo isotopes with A = 92–100. Here, we report some results for ^96,98,100Mo isotopes and they are obtained by evaluating the transition matrix elements, defining $C_V$, $C_A$ and $C_{VA}$, and connecting the ground $0^{+}$ state to various excited $J^{\pi }$ states through the use of the DSM wave function ${\Phi }_M^J\left(\eta \right)$ defined in Equation (3). The incoming neutrino energy is assumed to be ${ϵ}_{\nu }=15$ MeV. We use a quenching factor of 0.35. Results for ^98,100Mo are given in our recent publication [23] and they are summarized in Figure 2. In addition, recently, we have also considered neutrino scattering off ⁹⁶Mo and details of this study will be presented in a separate publication. Here, we summarize, in Table 1, the calculated ratio of the coherent to the total (coherent + incoherent) cross-section for the three Mo isotopes. These lie, as seen from Table 1, between 85% and 91%. We note that our cross-sections correspond only to the Standard Model (SM) $\nu$-nucleus interactions. It is worth remarking that, in other BSM electroweak processes where the coherent channel is also possible, e.g., the muon-to-electron conversion in nuclei (see e.g., refs. [29,30]), the portion of the coherent branching ratio into the total one represents also about 85–90 % of the total rate.

Figure 1. The ground band observed for ⁹⁶Mo is compared with the DSM predicted values. The experimental data are taken from [31].

Figure 1. The ground band observed for ⁹⁶Mo is compared with the DSM predicted values. The experimental data are taken from [31].

4. $\mathbf{2}\mathbf{\nu }$ Double Beta Decay of ¹⁰⁰Mo

The two-neutrino double beta decay ($2\nu DBD$) is a second-order process of weak interaction. This decay has been observed in more than 10 nuclei and is consistent with the standard model. On the other hand, neutrinoless double beta decay violates lepton number conservation and has not been observed. The successful description of the half-lives (or nuclear matrix elements) for $2\nu \beta \beta$ decay provides a stringent test of the goodness of the nuclear model used, and, with confidence, we can use it to predict the neutrinoless double beta decay matrix elements. A number of nuclear models has been used to study $2\nu DBD$, namely, spherical shell model [28,32,33,34], QRPA and its variants [35], interacting boson model [36], deformed shell model [1], and so on. Recently, Patel et al. [34] have employed large-scale shell model calculations to study the two-neutrino double beta decay in ⁸²Se, ⁹⁴Zr, ¹⁰⁸Cd, ¹²⁴Sn, ¹²⁸Te, ¹³⁰Te, ¹³⁶Xe, and ¹⁵⁰Nd using different effective interactions. Deformed shell model calculations have been performed to study the double beta decay for different nuclei in the A = 60–90 region and the results have been summarized in ref. [1]. The nucleus ¹⁰⁰Mo is a famous double beta decay emitter and has been employed in various experiments like NEMO [37] and MOON [38]. Different variants of QRPA have been used to study $\beta \beta$ decay for this nucleus [39]. But there are very few shell model calculations for this nucleus because of the large matrices to be diagonalized. Recently, Coraggio et al [33] have performed shell model calculation for this nucleus using effective $2\nu \beta \beta$ operators. Here, we discuss the application of DSM for two-neutrino double beta decay of ¹⁰⁰Mo ($0\nu$ $\beta \beta$ decay will be considered elsewhere).

Half-life for the $0_i^{+}\to 0_f^{+}$$2\nu$DBD is given by

$${\left[T_{1/2}^{2\nu }\right]}^{-1}=G_{2\nu }{g}_A^4{\left|M_{2\nu }\right|}^2$$

(7)

$g_A$ corresponds to the axial-vector coupling constant. The kinematical factor $G_{2\nu }$ is independent of nuclear structure and its value $G_{2\nu }=310.6\times {10}^{-20}$ yr⁻¹ [40]. On the other hand, the nuclear transition matrix elements (NTME) $M_{2\nu }$ are nuclear-model-dependent and they are given by

$${\displaystyle M_{2\nu }=\sum _m\frac{〈0_f^{+}\mid \mid \sigma {\tau }^{+}\mid \mid 1_m^{+}〉〈1_m^{+}\mid \mid \sigma {\tau }^{+}\mid \mid 0_i^{+}〉}{\left[E_m-(E_i+E_f)/2\right]/m_e}}$$

(8)

where $\left|0_i^{+}\right.〉$, $\left|0_f^{+}\right.〉$, and $\left|1_m^{+}\right.〉$ are the initial, final, and virtual intermediate states, respectively, and $E_m$ are the energies of the intermediate nucleus ¹⁰⁰Tc. Similarly, $E_i$ and $E_f$ are the ground-state energies of the parent and daughter nuclei. We have, from ref. [41], the atomic masses of ¹⁰⁰Mo, ¹⁰⁰Tc, and ¹⁰⁰Ru as $-86.193$, $-86.0215$, and $-89.227$ MeV, respectively (for nuclear mass, we need to subtract the mass of the electrons from the atomic mass). Experimentally, the lowest $1^{+}$ state in ¹⁰⁰Tc is the ground state. We finally obtain $[E_m-(E_i+E_f)/2]=[1.689+E_{1^{+}}]$ MeV. DSM is used to calculate the energies $E_{1^{+}}$ of the $1^{+}$ states of ¹⁰⁰Tc and the corresponding wave functions. Similarly, we obtained ground-state (with $0^{+}$) wave functions of ¹⁰⁰Mo and the daughter nucleus¹⁰⁰Ru. With these, the reduced matrix elements in Equation (8) are obtained. We have included 29 $1^{+}$ states (up to 5.7 MeV) in the calculation. The calculated half life is compared with experiments in Table 1 and the calculated half life is about 20 times larger compared to the experimental value. Employing the renormalized GT operator or, equivalently, using an effective $g_A$ value, as mentioned in Table 2, will improve the calculated half life, just as in the shell model [33].

5. Conclusions

We have described the application of DSM to evaluate incoherent and coherent cross-sections for ^96,98,100Mo. As a test of DSM, we have also reported the half life for two neutrino double beta decays of ¹⁰⁰Mo and compared them with experimental values. The recently observed neutral-current coherent elastic neutrino-nucleus scattering data, as well as those expected to be measured at the designed CE$\nu$NS experiments with other nuclear detectors, offer the possibility to constrain the parameter space of several models, progressing beyond the standard model, which assumes vector and scalar mediators. Our present predictions on the ^96,98,100Mo may be of help towards this direction since, as is known, these isotopes are prominent detection media. They have been employed previously in appreciably sensitive detectors (for the detection of neutrinoless double beta decay and dark matter) in the MOON and NEMO experiments.

Finally, it may be useful to perform DSM calculations for nuclear structure factors with an improved effective interactions and also by including many more deformed intrinsic states. This will be attempted in the future.