Gamow-Teller and double-beta decays of heavy nuclei within an effective theory

Gamow-Teller and double-beta decays of heavy nuclei within an effective theory

Abstract

We study decays within an effective theory that treats nuclei as a spherical collective core with an even number of neutrons and protons that can couple to an additional neutron and/or proton. First we explore Gamow-Teller decays of parent odd-odd nuclei into low-lying ground-, one-, and two-phonon states of the daughter even-even system. The low-energy constants of the effective theory are adjusted to data on decays to ground states or Gamow-Teller strengths. The corresponding theoretical uncertainty is estimated based on the power counting of the effective theory. For a variety of medium-mass and heavy isotopes the theoretical matrix elements are in good agreement with experiment within the theoretical uncertainties. We then study the two-neutrino double- decay into ground and excited states. The results are remarkably consistent with experiment within theoretical uncertainties, without the necessity to adjust any low-energy constants.

I Introduction

Atomic nuclei are sensitive to fundamental interactions beyond the strong force that binds them. Excited states typically decay due to electromagnetic interactions emitting rays, while unstable nuclear ground states decay via weak interactions emitting or capturing electrons, neutrinos, or their antiparticles. Nuclei are also used as laboratories due to their sensitivity to new-physics interactions beyond the Standard Model Severijns and Naviliat-Cuncic (2011); Baudis (2012); Engel et al. (2013).

The weak interaction is closely connected to ground-state decays. Almost every unstable isotope lighter than Pb decays either via decay or electron capture. The associated half-lives can range from milliseconds to billions of years, with the corresponding nuclear transition matrix elements varying by three or more orders of magnitude. This wide range makes theoretical predictions of decay and electron capture particularly challenging tests of nuclear-structure calculations. Reliable predictions are also especially important for astrophysics because -decay half-lives of experimentally inaccessible very neutron-rich nuclei set the scale of the rapid-neutron capture, or -process, which is responsible for the nucleosynthesis of heavy elements Langanke and Martínez-Pinedo (2003); Arnould et al. (2007); Arcones and Thielemann (2013).

Ab initio calculations of decays are still limited to few-nucleon systems (see, e.g., Refs. Gazit et al. (2009); Baroni et al. (2016); Klos et al. (2016)) or focus on selected lighter isotopes Maris et al. (2011); Ekström et al. (2014). For medium-mass and heavy nuclei theoretical studies typically use the quasiparticle random-phase approximation (QRPA) (see, e.g., Refs. Nikšić et al. (2005); Marketin et al. (2007); Niu et al. (2013a, b); Fang et al. (2013); Mustonen et al. (2014); Shafer et al. (2016)), sometimes combined with more macroscopic calculations Möller et al. (2003), and when possible the nuclear shell model (see, e.g., Refs. Caurier et al. (2005); Suzuki et al. (2012); Zhi et al. (2013); Kumar et al. (2016)). The agreement of these many-body calculations with experimental data demands fitting part of the nuclear interactions and/or the effective operators, such as the isoscalar pairing for the QRPA or a renormalization (“quenching”) factor for the transition operator in shell model calculations Brown and Wildenthal (1988); Martínez-Pinedo et al. (1996). At present, the predictions made by different methods for non-measured decays disagree by factors of a few units. Furthermore, in these phenomenological calculations it is difficult to provide well founded estimates of the associated theoretical uncertainties.

Second-order processes in the weak interaction, double- () decays, have been observed in otherwise stable nuclei. They exhibit the longest half-lives measured to date, exceeding years Barabash (2015). This decay mode is via two-neutrino () decay, to distinguish it from an even more rare type of decay without neutrino emission, neutrinoless () decay, which is so far unobserved. The latter process is not allowed by the Standard Model since it violates lepton number conservation, and can only occur if neutrinos are their own antiparticles. decay is the object of very intense experimental searches Agostini et al. (2017); Gando et al. (2016); Alfonso et al. (2015); Albert et al. (2014) with the goal to elucidate the nature of neutrinos. The calculation of matrix elements for decays is specially subtle Caurier et al. (2012); Vergados et al. (2012); Suhonen and Civitarese (2012); Engel and Menéndez (2017). It faces similar challenges to those of decays, with the additional difficulty that decays are suppressed. As with decays, calculations by different many-body approaches of unknown nuclear matrix elements vary by factors of a few units. The prediction of reliable matrix elements with theoretical uncertainties is specially pressing for the interpretation and planning of present and future decay experiments.

The goal of this work is to use an effective theory (ET) framework to study the , electron capture, and decays of medium-mass and heavy nuclei. Calculations within an ET can provide transition matrix elements with quantified theoretical uncertainties, and are therefore a good complement to existing many-body -decay studies. We focus on Gamow-Teller (GT) transitions for which experimental data is available, leaving the study of decay for future work. The ET used here is valid for transitions involving spherical systems, because nuclei are treated as a spherical collective core coupled to a few nucleons.

Over the last decades ETs have been applied to describe the low-energy properties of nuclei. Effective theories exploit the separation of scales between the low-energy physics governing the processes of interest, which are treated explicitly, and the high-energy physics whose effects are integrated out and encoded into low-energy constants (LECs) that must be fit to experimental data. The ET is formulated in terms of the low-energy degrees of freedom (DOF) and their interactions, consistent with the symmetries of the underlying theory. The ET offers a systematic order-by-order expansion based on a power counting given by the ratio of the low-energy over the breakdown scale, at which the ET is no longer valid. This also allows for the quantification of the theoretical uncertainties at a given order in the ET Dobaczewski et al. (2014); Furnstahl et al. (2015a, b); Wesolowski et al. (2016).

Effective field theories have been very successful in describing two- and three-nucleon forces in terms of nucleon and pion fields van Kolck (1999); Bedaque and van Kolck (2002); Epelbaum et al. (2009); Machleidt and Entem (2011); Hammer et al. (2013). In combination with powerful ab initio methods, chiral interactions have been employed to calculate low-energy properties of light and medium-mass isotopes (see, e.g., recent reviews Barrett et al. (2013); Hagen et al. (2014); Hebeler et al. (2015); Navrátil et al. (2016); Hergert et al. (2016)). Complementary, a different set of ETs has been proposed to describe heavy nuclei in terms of collective DOF Papenbrock (2011); Zhang and Papenbrock (2013); Papenbrock and Weidenmüller (2014); Coello Pérez and Papenbrock (2015a); Papenbrock and Weidenmüller (2015); Coello Pérez and Papenbrock (2015b, 2016). In particular, the ET developed in Refs. Coello Pérez and Papenbrock (2015b, 2016) describes the low-energy properties of spherical even-even and odd-mass nuclei in terms of effective single-particle DOF coupled to a collective spherical core. Using this approach, the low-energy spectra and electromagnetic properties of even-even and odd-mass nuclei (the latter with ground states) were described consistently, in good agreement with experiment Coello Pérez and Papenbrock (2016). The electromagnetic strength between low-lying states, including magnetic dipole transitions in odd-mass systems, was predicted successfully. Since for decays the relevant physics is expected to be like that of magnetic dipole transitions, the ET framework offers a promising approach to describe , electron capture, and decays in nuclei.

This paper is organized as follows. Section II serves as a short summary of key elements from the theory of , electron capture and decays. Section III begins with a brief introduction to the ET of spherical even-even nuclei followed by an extension of the theory to account for the low-lying states of odd-odd nuclei. Next we discuss the GT transition operator that enters decays as well as methods to fix the associated LECs. In Sec. IV we present our ET results for decay matrix elements of spherical odd-odd nuclei with ground states into final states of even-even nuclei corresponding to different collective ET excitations. We compare the ET predictions with experimental data, including the estimated theoretical uncertainties. In Sec. V we test the ability of the ET to consistently describe and decays, without the necessity to adjust additional LECs. We conclude with a brief summary and outlook in Sec. VI.

Ii Types of decays

ii.1 Single- decay and electron capture

The most common weak interaction process is decay. Here either one of the neutrons in a nucleus decays into a proton ( decay), or one of the protons decays into a neutron ( decay). The total number of nucleons remains the same. For the electric charge, lepton number, and angular momentum to be conserved, it is required that an electron () or positron () are emitted along with an electron antineutrino () or neutrino ():

(1)
(2)

Alternatively, proton-rich nuclei can undergo a third process, an electron capture (EC), which involves the capture of an electron by a proton, yielding a neutron. Again, conservation of energy, angular momentum, and lepton number requires an electron neutrino to be emitted:

(3)

At lowest order in the weak interaction, it is possible to distinguish two types of dominant, so-called allowed, transitions. Gamow Teller and Fermi (F) decays differ in the spin dependence of the associated one-body operator. The GT and F operators are defined as

(4)
(5)

where denotes the spin, () is the isospin raising (lowering) operator, and a sum is performed over all nucleons in the nucleus.

In this work, we focus on allowed GT transitions, whose decay rates are related to the reduced matrix elements of the GT operator in Eq. (4) between the corresponding initial () and final () nuclear states:

(6)

The decay’s half-life is given by

(7)

Here the phase-space factor contains all the information on the lepton kinematics,  s is the -decay constant, is the axial-vector coupling, and denotes the total angular momentum of the initial state. The quantity is known as the -value and is directly comparable to the transition nuclear matrix element.

ii.2 decays

In the decay, two neutrons of the parent even-even nucleus decay into two protons. Two electrons and antineutrinos are emitted as well due to charge and lepton-number conservation:

(8)

Such decays have been detected for several nuclei, including few cases of transitions into excited states Barabash (2015). At present the similar but kinematically less favored and the double-electron-capture (ECEC) decays have not been observed experimentally.

In principle, both GT and F operators enter the description of decays. Nevertheless, in decays to low-lying states of the final daughter nucleus only the GT part is relevant. The F operator does not connect states with different isospin quantum numbers, and its strength is almost completely exhausted by the isobaric analogue state, which lies at an excitation energy of tens of MeVs. The decay rate for decay is then Engel and Menéndez (2017)

(9)

where is a phase-space factor, and the nuclear matrix element is given by

(10)

where the electron mass is introduced to make the matrix element dimensionless, and the sum runs over all states of the intermediate odd-odd nucleus. The energy denominators are given in terms of the energy of the initial, final and intermediate states by

(11)

Iii ET for Single- decay

In this section, we formulate an ET for the GT decays of parent odd-odd into daughter even-even nuclei. Our approach is valid for systems with low-energy spectra and electromagnetic transitions well reproduced by an ET written in terms of collective DOF, which at leading order (LO) represents a five-dimensional harmonic oscillator. The theoretical uncertainties due to omitted DOF can be propagated in the ET up to the nuclear matrix elements and decay half-lives, allowing for a comparison of the ET predictions with experimental data.

iii.1 ET for even-even and odd-odd nuclei

The ET developed in Refs. Coello Pérez and Papenbrock (2015b, 2016) describes the low-energy properties of spherical even-even and odd-mass nuclei in terms of collective excitations that can be coupled to an odd neutron, neutron-hole, proton, or proton-hole. The effective operators are written in terms of creation and annihilation operators, which are the DOF of the ET. These include:

  1. Collective phonon operators and , which create and annihilate quadrupole phonons associated with low-energy quadrupole excitations of the even-even core.

  2. Neutron operators and , which create and annihilate a neutron or neutron-hole in a single-particle orbital with total angular momentum and parity .

  3. Proton operators and , which create and annihilate a proton or proton-hole in a orbital.

Whether the fermion operators represent a particle or a hole depends on the odd-mass nucleus we want to describe. In Ref. Coello Pérez and Papenbrock (2016), silver isotopes with ground states were described both as an odd proton coupled to palladium cores and as an odd proton-hole coupled to cadmium cores. Both descriptions turned out to be consistent with each other. The above operators fulfill the following relations

(12)

While the creation operators are the components of spherical tensors, the annihilation operators are not. To facilitate the construction of spherical-tensor operators with definite ranks, we define annihilation spherical tensors with components , where and .

The Hamiltonian describing a particular even-even and adjacent odd-mass pair of nuclei at next-to-leading order (NLO) is given by Coello Pérez and Papenbrock (2016)

(13)

where , depending in which odd-mass nucleus we want to describe, and and are LECs that must be fitted to data. The LEC accompanying the LO term, , may be thought of as the energy of the collective mode. It scales as the excitation energy of the first excited state of the even-even system of interest. The energy scale , at which the ET breaks down, lies around the three-phonon level. Based on previous work Coello Pérez and Papenbrock (2015b), we will set in what follows, even though the breakdown scale might not be exactly the same for every pair of nuclei studied in this work. The effective operators of the theory are constructed order-by-order adding all relevant terms that correct the previous one by a positive power of .

The reference state of the ET represents the ground state of the even-even nucleus of interest. Multiphonon excitations of this state represent excited states in the even-even system. Of particular relevance for our work are one- and two-phonon excitations:

(14)

where in the notation , and are the total angular momentum of the state and its projection, and is the number of phonons. We define the coupling of two spherical tensors as in Ref Varshalovich et al. (1988), and refer to Ref. Rowe and Wood (2010) for a detailed description of the construction of multi-phonon excitations. We highlight that the ET introduced above reproduces the low-lying spectra and electromagnetic moments and transitions of vibrational medium-mass and heavy nuclei within the estimated theoretical uncertainties Coello Pérez and Papenbrock (2015b).

In a similar fashion, the ground states of adjacent odd-mass nuclei can be described as fermion excitations of the reference state . Even though several single-particle orbitals may be relevant, at LO in the ET it is assumed that only one orbital is required to describe the low-energy properties of the odd-mass system. The relevant single-particle orbital is inferred from the quantum numbers of the ground state of the odd-mass nucleus of interest. This assumption works well for odd-mass nuclei near shell closures with ground states Coello Pérez and Papenbrock (2016). In these systems, a reasonable agreement was found between the ET predictions and experimental data, regarding not only low-energy excitations but also electric and magnetic moments and transitions Coello Pérez and Papenbrock (2016).

In order to describe the allowed GT decays of parent odd-odd nuclei, we extend the collective ET of Refs. Coello Pérez and Papenbrock (2015b, 2016) and write the low-lying positive-parity states in the odd-odd nucleus as

(15)

where the fermion operators represent particles or holes depending on the odd-odd nucleus of interest. For example, can be described coupling a neutron and a proton-hole to a core, or coupling a neutron-hole and a proton to a core. The and labels in the odd-odd state indicate the coupling of the odd neutron and odd proton on top of the collective spherical ground state. The angular momentum and parity of the single-particle orbitals to be used are inferred from the quantum numbers of the low-lying states of the adjacent odd-mass nuclei. Therefore, the total angular momenta and parities of these orbitals must fulfill the relations , and for positive-parity states.

It is important to note that the LO ET calculation of single- decay of the ground-states of odd-odd nuclei does not require to explicitly construct their energy spectra, simplifying the calculations considerably. Contributions due to additional DOF relevant for excited states are taken into account in the uncertainty estimates associated with the LO results. We stress that these uncertainties must be tested whenever data is available, because they probe the validity of the power counting and the reliability of the LO calculations.

iii.2 Effective GT operator

Next we construct the operator corresponding to the GT operator in Eq. (4), in terms of the effective DOF. We write the most general positive-parity spherical-tensor operator of rank one capable of coupling the low-lying states of the parent odd-odd nucleus introduced in Eq. (15) to the ground, one- and two-phonon states of the daughter even-even nucleus represented in Eq. (14). At lowest order in the number of operators this operator is given by

(16)

where , and are LECs that must be fit to experimental data.

Let us discuss the effective operator of Eq. (III.2) in more detail. For the allowed GT decay of the odd-odd nucleus with neutrons and protons into the even-even nucleus with neutrons and protons, the odd-odd system is described within the ET as an even-even core coupled to a neutron and a proton-hole. For the decay to take place, the fermion annihilation operators must annihilate the odd neutron and proton-hole. This action represents the decay of a neutron in the odd-odd system into a proton, which then fills the proton-hole yielding the even-even system. An additional step in which the odd neutron fills a neutron-hole takes place if the annihilated neutron is part of the core; however, the ET cannot differentiate between the two processes. For the description of the or EC decay to the even-even nucleus with neutrons and protons, it is more convenient to describe the odd-odd nucleus in terms of the later even-even core coupled to a neutron-hole and a proton. In this case, the fermion annihilation operators annihilate the odd neutron-hole and proton, representing the conversion of a proton into a neutron that fills the neutron-hole. Again, the additional filling of a proton-hole by the odd-proton follows if the annihilated proton is in the core.

The reduced matrix elements of the effective GT operator in Eq. (III.2) between low-lying states of the parent odd-odd system in Eq. (15) and the daughter even-even ground, one- and two-phonon states in Eq. (14) are

(17)
(18)
(19)

where the subscripts and identify the ground and -phonon states of the daughter even-even nucleus, respectively. We also note that the LECs in Eqs. (17)–(19) implicitly take into account additional corrections to the GT operator in Eq. (4), such as a possible “quenching”.

iii.3 ET GT decay to ground and excited states

The first, second, and third terms of the effective GT operator in Eq. (III.2) couple states with phonon-number differences of zero, one, and two, respectively. Thus, they describe the decays from the ground state of the odd-odd nucleus to the ground, one- and two-phonon states in the even-even nucleus, respectively. Figure 1 schematically shows the case of the decays of into the , , and states of , identified as the ground, one- and two-phonon states.

The LECs , and encode the microscopic information of the nuclei involved in the decay. While the value of the LECs is not predicted by the ET, the power counting established in previous works Coello Pérez and Papenbrock (2015b, 2016) suggests scaling factors between them. This power counting is based on the assumption that at the energy scale where the ET breaks down, the matrix elements of every term of any effective operator scale similarly.

From this assumption and the effective Hamiltonian describing the even-even systems, it can be concluded that at the breakdown scale the matrix elements of an operator containing powers of operators scale as Coello Pérez and Papenbrock (2015b)

(20)

For more details, we refer the reader to Ref. Coello Pérez and Papenbrock (2015b).

Figure 1: Schematic representation of the relative size of the matrix elements for the decays of into the ground (), one- (), and two-phonon (, ) excited states of .

The power counting in Eq. (20) implies that at the breakdown scale the matrix elements of the different terms in the effective GT operator (III.2) scale as

(21)

and

(22)

The resulting relative sizes of the matrix elements for the decay are also shown schematically in Figure 1. The theoretical uncertainties for the above ratios have been estimated based on the expectation for the next-order LECs to be of natural size, so that a number of order one has a range from and an associated relative uncertainty. Finally, we emphasize that these theoretical uncertainty estimates must be tested by comparing the ET predictions to experimental data.

iii.4 ET GT decay to ground states and theoretical uncertainties

The matrix elements of single- decays to the ground state are set by the values of the LEC , see Eq. (III.2), which are to be fitted to experimental data. Within the ET the uncertainty of these matrix elements comes from two sources:

  1. Omitted terms in the effective GT operator that involve two operators and couple states of odd-odd and even-even nuclei with the same number of phonons. The matrix elements of these terms are expected to scale as

    (23)
  2. Next-to-leading order corrections to the ground state of odd-odd nuclei due to terms in the Hamiltonian that can mix states with phonon-number differences of one. These corrections are expected to scale as and are coupled to the even-even ground state by the second term of the effective GT operator, which contains an additional operator. Therefore, the corrections to the matrix elements also scale as

    (24)

As a result, the matrix element in Eq. (17) has an associated theoretical uncertainty estimated as

(25)

The uncertainty associated to the -value is estimated as the next-order contribution to the Taylor expansion of the logarithm in Eq. (7) with , that is,

(26)

where again we have assumed that .

Iv Results for single- decay

iv.1 GT decays to excited states

We begin our calculations by studying the decay and electron capture of spherical odd-odd parent nuclei with ground states into different ground and excited and states of the even-even daughter nuclei. This will show to which extent can the ET describe processes involving individual nucleons and whether the transitions scale as expected in the ET.

For each parent nucleus, the LEC can be fitted to the transition to the ground state. Then the GT decays into excited collective states are predicted by the ET according to the scaling factors in Eqs. (21) and (22):

(27)
(28)

We note that the ET naturally predicts the observed successive hindering reported in Ref. Sakai (1962) of the matrix elements for GT decays from , , and ground states of odd-odd nuclei into , , and states of the even-even daughter.

Figure 2 compares our ET predictions with experiment for GT and EC decay matrix elements for a broad range of medium-mass and heavy odd-odd nuclei with mass number from to . All parent nuclei have ground states and decay into excited , , and states of the corresponding even-even nuclei. Within the ET, the states are treated as one-phonon excitations of the even-even core, while the and states are considered to be two-phonon excitations. The ET results with uncertainties are calculated according to Eqs. (27) and (28) after adjusting the LEC to the matrix element of the decay to the ground state of the daughter nucleus. The same figure shows that most of the experimental data, including GT and EC decays, is consistent with the ET results. Inconsistencies are larger for the and two-phonon states where the ET matrix elements tend to be overestimated, especially for the decays into excited states around mass number . This is not unexpected because two-phonon states lie closer to the ET breakdown scale. In Table 1 we list the numerical values corresponding to all the theoretical and experimental matrix elements shown in Fig. 2.

Figure 2: Calculated ET matrix elements for GT (red bands) and EC (blue bands) decays from parent odd-odd nuclei with ground states into the (top pannel), (middle pannel), and (bottom pannel) excited states of the daughter even-even nuclei, compared to experiment (black circles) from Refs. Nichols et al. (2012); Singh (2007); Browne and Tuli (2010); McCutchan (2012); Gürdal and McCutchan (2016); Singh (2005); Farhan and Singh (2009); Tuli (2003); Singh and Hu (2003); Singh (2008); Frenne (2009); Blachot (2007); Frenne and Negret (2008); Blachot (2000); Gürdal and Kondev (2012); Lalkovski and Kondev (2015); Blachot (2012, 2010); Kitao (1995); Kitao et al. (2002); Tamura (2007); Katakura and Wu (2008); Elekes and Timar (2015); Singh (2001). For details see Table 1.
exp ET exp ET exp ET exp ET
Table 1: Experimental and calculated ET matrix elements for GT and EC decays from parent odd-odd nuclei with ground states (treated as a spherical collective core coupled to a neutron and a proton) into (one-phonon state), , and (two-phonon) excited states of the daughter even-even nuclei. The experimental values were calculated from the -values for the transitions to the , , , and states, taken from Refs. Nichols et al. (2012); Singh (2007); Browne and Tuli (2010); McCutchan (2012); Gürdal and McCutchan (2016); Singh (2005); Farhan and Singh (2009); Tuli (2003); Singh and Hu (2003); Singh (2008); Frenne (2009); Blachot (2007); Frenne and Negret (2008); Blachot (2000); Gürdal and Kondev (2012); Lalkovski and Kondev (2015); Blachot (2012, 2010); Kitao (1995); Kitao et al. (2002); Tamura (2007); Katakura and Wu (2008); Elekes and Timar (2015); Singh (2001). The theoretical uncertainties are estimated assuming all LECs to be of natural size.

iv.2 GT decays to ground states using GT transition strengths

In Sec. IV.1, we have used experimental data on single- decays to ground states to fit the value of and then predicted the matrix elements for transitions to excited states of the same nucleus. Next, we study whether it is possible to employ other data to fit the LECs and in turn predict the -decay matrix elements to ground states.

Besides weak processes, GT strengths studied in charge-exchange reactions (via the strong interaction) are also sensitive to the GT spin-isospin operator, because the zero-degree differential cross section of the reaction is proportional to the GT strength Watson et al. (1985)

(29)

Therefore, reactions such as or have the same form as decays, while reactions are related to -like transitions. The GT strengths and decays of isospin-mirror nuclei have been found to be consistent in medium-mass systems Molina et al. (2015).

In this spirit, we can use the GT transition strengths measured in charge-exchange reactions to fit the LECs of the effective GT operator in Eq. (III.2). In this way, we can predict the GT matrix elements for the transition to the ground states of spherical even-even nuclei. Figure 3 shows that the ET results for the GT matrix elements including the theoretical uncertainty, agree very well with experiment in three of the four cases where data is available. The experimental values are calculated from the -values of the corresponding EC decays from Refs. Singh (2007, 2008); Blachot (2010); Elekes and Timar (2015). The remaining results in Fig. 3 show ET predictions for single- decay of additional states, which are, however, excited states of the odd-odd system and decay via electromagnetic transitions.

Figure 3: Calculated GT matrix elements for the transition from the states of odd-odd nuclei to the ground states of even-even nuclei, using as ET input the GT transition strengths measured in charge-exchange reactions (blue bands), see also Table 2. The ET results are compared to experiment calculated from the -values of the corresponding EC decays (black circles) from Refs. Singh (2007, 2008); Blachot (2010); Elekes and Timar (2015).

The details of the ET predictions in Fig. 3 are given in Table 2, which lists the GT strengths measured in charge-exchange reactions with initial even-even and final odd-odd nuclei for –130 Popescu et al. (2009); Thies et al. (2012a); Frekers et al. (2016); Thies et al. (2012b); Akimune et al. (1997); Puppe et al. (2012). For each reaction, Table 2 gives the experimental partial GT strength to the lowest state of the odd-odd nucleus (the state expected to be well described by the ET). An exception is the case of the reaction, where all the GT strength below 500 keV was taken into account (as reported in Table IV of Ref. Thies et al. (2012a)), corresponding to three close-lying states. The resulting values for fit to the partial GT strength are given in Table 2 including the comparison of the ET results to the experimental -values.

exp ET
1