Weak Magnetism Correction to Allowed Beta-decay for Reactor Antineutrino Spectra

TL;DR

This paper evaluates the common approximation for weak magnetism correction in reactor antineutrino spectra, finding it valid for relevant beta-decays and that the main uncertainty stems from two-body effects.

Contribution

It clarifies the validity of the one-body weak magnetism approximation for fission fragment beta-decays and quantifies the associated uncertainty in reactor antineutrino spectra.

Findings

The approximation is valid for relevant beta-decays.

Uncertainty from one-body weak magnetism is less than 1%.

Main uncertainty arises from two-body meson-exchange currents.

Abstract

The weak magnetism correction and its uncertainty to nuclear beta-decay play a major role in determining the significance of the reactor neutrino anomaly. Here we examine the common approximation used for one-body weak magnetism in the calculation of fission antineutrino spectra, wherein matrix elements of the orbital angular momentum operator contribution to the magnetic dipole current are assumed to be proportional to those of the spin operator. Although we find this approximation invalid for a large set of nuclear structure situations, we conclude that it is valid for the relevant allowed beta-decays between fission fragments. In particular, the uncertainty in the fission antineutrino due to the uncertainty in the one-body weak magnetism correction is found to be less than 1%. Thus, the dominant uncertainty from weak magnetism for reactor neutrino fluxes lies in the uncertainty in the two-body meson-exchange magnetic dipole current.

Equations

$${\vec{\mu }}_V={\vec{\mu }}_s+{\vec{\mu }}_L+{\vec{\mu }}_{MEC}=\frac{{\mu }_v}{2M_N}\vec{\Sigma }+\frac{1}{2M_N}\vec{\Lambda }+{\mu }_{MEC}.$$

$$\frac{d\omega }{d{E}_e}$$

$$=$$

$${\delta }_{LS}^{j_f{j}_i}\equiv \frac{⟨J_f\mid \mid \vec{\Lambda }\mid \mid J_i⟩}{⟨J_f\mid \mid \vec{\Sigma }\mid \mid J_i⟩}\simeq -\frac{1}{2}$$

$${\delta }_{LS}^{j_f{j}_i}=$$

$${\delta }_{LS}^{--}=-(l+1),$$

$${\delta }_{LS}^{+-}=-1/2,$$

$$f(r)=V_0\left(1-0.5\frac{\rho }{{\rho }_0}\right),$$

$${\delta }_{LS}^{j_f{j}_i}=-0.42\pm 0.99,$$

Full text

I Introduction

There has been considerable interest recently in the expected magnitude and shape of the fission aggregate antineutrino spectra emitted from reactors. This interest has been driven by the fact that reevaluations mueller ; huber of the spectra for all four actinides (235U, 238U, 239Pu and 241Pu) contributing to reactor antineutrino fluxes led to a systematic increase in the expected number of antineutrinos above about 2 MeV, relative to earlier evaluations Schreckenbach ; vogel-1 , which, in turn, led to the so-called reactor neutrino anomaly anomaly . The changes in the evaluated spectra were in part due to changes in the treatment of sub-dominant corrections to nuclear beta decay, particularly the treatment of the finite-size and weak-magnetism corrections.

There are four sub-dominant corrections to beta decay that must be considered in calculating the antineutrino spectra. These are the recoil, radiative, finite-size, and weak-magnetism corrections. The recoil correction is quite small, and the radiative correction has been taken from the work of Sirlin sirlin ; sirlin-new , both before and since the occurrence of the anomaly. In the earlier work of Schreckenbach et al. Schreckenbach , the finite-size and weak-magnetism corrections were applied to the aggregate fission antineutrino spectrum as a single energy-dependent correction to the entire spectrum, whereas in the reevaluations mueller ; huber these corrections were applied for each end-point energy (or end-point energy range) contributing to the spectrum. In all cases, the finite-size and weak-magnetism corrections that were applied involve some level of approximation. For example, the finite-size correction is often taken to be the form appropriate for allowed beta decay transitions although it is well recognized Hayes14 ; hayes2 ; bnl that about 30% of the transitions making up the antineutrino spectra are forbidden. In the case of the 70% allowed transitions, the expression Hayes14 ; holstein for the finite-size correction has been found zemach16 to be reasonably accurate. For the weak magnetism correction, there is both a one-body and two-body contribution. As discussed in detail below, in case of the one-body weak-magnetism correction, both a spin $\vec{\Sigma}=\sum_{i}\tau_{i}^{\pm}\,\vec{\sigma}_{i}$ and orbital angular momentum $\vec{\Lambda}=\sum_{i}\tau_{i}^{\pm}\,l_{i}$ operator enter, where $\vec{\sigma}_{i}=2\vec{S}_{i}$ and $\vec{l}_{i}=\vec{x}_{i}\times\vec{p}_{i}$ for nucleon $i$. Thus, in general, the weak-magnetism correction is nuclear structure dependent. However, in most analyses of reactor antineutrino spectra Hayes14 ; vogel-1 , the nuclear structure dependence is simplified by assuming that matrix elements of the orbital angular momentum operator are proportional to matrix elements of the spin operator, i.e., it is assumed that $\langle J_{f}||\vec{\Lambda}||J_{i}\rangle=-\frac{1}{2}\langle J_{f}||\vec{\Sigma}||J_{i}\rangle$. The purpose of the present work is to check the validity of the latter nuclear structure assumptioni, particularly for allowed fission fragment beta-decays.

I.1 The Weak Magnetism correction

In earlier work zemach16 , we examined the usual approximations made for the finite-size correction to allowed beta decay by evaluating the Zemach moments using energy density functional theory, and found these approximations to be good  zemach16 . The approximations made for the weak-magnetism correction are of an entirely different origin, and need to be assessed separately. Weak magnetism is generally a small correction to beta-decay, that arises to first order from an interference term between the dominant Gamow-Teller contribution and the magnetic dipole contribution to the weak current. The magnetic dipole operator, $\vec{\mu}_{V}=\frac{1}{2}\int d^{3}x~{}\vec{x}\times\vec{J}_{V}(\vec{x})$, involves the sum of three terms in the vector current $\vec{J}_{V}(\vec{x})$, namely, the spin current, the orbital current, and meson-exchange currents.

[TABLE]

The label “MEC” indicates leading-order meson-exchange currents, which are dominated by pion-exchange, and whose form are not a subject of the current work. We note that, unless matrix elements of $\vec{\Sigma}$ are suppressed, the first term in eq.(1) tends to dominate because the vector magnetic moment is large compared to unity, i.e., $\mu_{v}=4.7$.

The first order contribution from the weak magnetism correction $\delta_{WM}$ to the beta-decay spectrum for allowed and and first forbidden decays is given in ref. Hayes14 , and for allowed GT transitions is,

[TABLE]

where $G_{F}(g_{A})$ is the Fermi (axial) coupling constant, $F(E,Z)$ is the relativistic point Fermi function, $E_{e}(m_{e})$ are the electron energy and mass, and $E_{0}$ is the end-point energy for the transition. For the sake of brevity we do not list other sub-dominant corrections such as recoil, radiative, and finite size corrections; they are discussed in the review hayes-vogel .

In several references vogel-1 ; mueller ; huber ; Hayes14 , a simple model was used in which the orbital angular momentum of the $ith$ nucleon $\vec{\ell}_{i}$ was replaced by $\vec{j}_{i}-\vec{\sigma}_{i}/2$, and it was argued that any change in the quantum numbers of the $\beta$-decaying nucleon would eliminate the $\vec{j}_{i}$ term, so that this term could be dropped. This simple model effectively assumes that

[TABLE]

in eq.(3). Examining the validity of this approximation, which replaces the fractional contribution from the orbital currents by “-1/2” in the expression for weak-magnetism correction, is the central focus of the present work. In addition, we examine whether matrix elements of the operator $\vec{\Sigma}$ truly dominate over those for the orbital angular momentum operator $\vec{\Lambda}$, for the set of transitions relevant to reactor antineutrino spectra.

II Nuclear Structure Studies of the Fission Fragment beta-decays

II.1 Single-particle matrix elements

We begin with a discussion of the simple case of pure single-particle matrix elements (SPME) for the spin and orbital currents involved in the weak-magnetism correction. In this case, the fractional orbital correction is,

[TABLE]

where “$nl_{i}j_{i}$” and “$nl_{f}j_{f}$” are the single particle orbits involved in the initial and final states, and $l_{i}=l_{f}=l$ for the allowed transitions. We introduce the following short-hand notation: with $j_{i}=l\mp 1/2$ and $j_{f}=l\pm 1/2$, we use the signs in these definitions to stand for $j_{i}$ and $j_{f}$ in $\delta_{LS}^{j_{f}j_{i}}$ (Eq. (4)), i.e., $\delta_{LS}^{+-}$ stands for $j_{i}=l-1/2$ and $j_{f}=l+1/2$, and etc.. From the analytical expressions for the $6-J$ symbols, listed in Ref. Edmonds60 , we find,

[TABLE]

Thus, if the overlap of the initial and final radial wave functions involved in the $\beta$-decay transition is close to unity, replacing the fractional orbital correction by “$-1/2$” can be a good approximation, when only one single-particle orbit from the initial and final state are involved in the transition and $j_{i}\neq j_{f}$ ($\delta_{LS}^{-+}$ and $\delta_{LS}^{+-}$). This is particularly true for spin-orbit partners.

II.2 Hartree-Fock-Bogoliubov calculations

Moving beyond the case of simple single-particle transitions requires a model to describe the structure of the fission fragments whose beta-decays make up the cumulative fission antineutrino spectra. In Fig. 1, we show the neutron and proton numbers for the dominant fission fragments contributing to the antineutrino spectra of 235U, 238U, 239Pu, and 241Pu. Displayed is the set of fission fragments listed in the ENDF/B-VII.1 nuclear database with cumulative fission yields $>\sim 0.01$, Q value $>\sim$ 3.0 MeV, half life less than 1 month, and those that are listed as decaying by allowed transitions. In all case, the large neutron excess means that the Fermi surfaces for neutrons and protons are quite different.

For all of these nuclei, any nuclear structure calculation has to involve a model-space truncation. To model these nuclei, we ran a series of Hartree-Fock-Bogoliubov (HFB) calculations. Compared to shell model calculations, HFB calculations can be realized in considerably larger model space. A main goal in the HBF study was to determine the effect of the pairing interaction on the nuclear wave functions, and the resulting effect on the ratio $\delta_{LS}^{j_{f}j_{i}}$. For nuclei close to the drip-line, the pairing interaction is found to scatter nucleon pairs from bound states to positive-energy orbitals, which in some cases leads to quenching of shell effects. For the HFB calculations, we used the Skyrme SLY4 parameterization Cha1998 for the particle-hole interaction. A standard zero-range density-dependent pairing force was used for the particle-particle channel,

[TABLE]

where $V_{0}$ is the interaction strength and $\rho_{0}$ was fixed at 0.16 fm*-3*. We used the Lipkin-Nogami (LN) method lipkin to restore particle number approximately. The numerical code HFODD hfodd was used for these calculations. We adopted the commonly used equivalent-spectrum cutoff of 60 MeV, applied in the quasiparticle configuration space, and the calculations were performed in a spherical basis of 14 major harmonic-oscillator shells. The value of $V_{0}$ was -272.76 MeV fm3, which gives an empirical neutron pairing gap for 120Sn of 1.245 MeV.

In Fig. 2, we show the “equivalent spectrum” of single-particle energies and the corresponding occupation coefficients, as defined in Ref. NPA1984 , from our HFB-LN calculations, for nuclei in the “$1g2d3s+1h11/2$” shell with $A<100$ Fig. 3 shows the same but for nuclei in the “$1h2f3p$” shells with $A>130$. Only even-even nuclei that are contained within the set of fission fragments in Fig. 1 are shown.

In this calculation we have assumed spherical shapes and studied the shell gaps in these nuclei. For the fission fragments of mass A $\sim 100$, there is a large shell gap between the $1g9/2$ and $1g7/2$ orbitals, shown in Fig. 2. When the nuclear interaction is turned on, $1g9/2$ valence protons remains in $1g9/2$ with large probability; even with the pairing interaction, the occupation probabilities in orbitals higher than $1g9/2$ is negligible. For neutrons, in Fig. 2 (c), the $1g9/2$ shell is nearly filled, and the valence $1g7/2$ and $2d5/2$ orbitals play a major role. Thus, the allowed beta-decay transitions for the lower mass fission fragments are dominated by neutron transitions from the $1g7/2$ orbital to the proton $1g9/2$ orbital. For typical fission fragments of mass $A>130$, Fig. 3, there is a large shell gap between $1h11/2$ and $1h9/2$ orbitals, and the beta decays are dominated by $1h9/2$ neutron transitions to the $1h11/2$ proton orbital.

II.3 Shell-model calculations

We also examined the allowed beta-decay between the fission fragments within the nuclear many-body shell model. We first note that many shell model truncation schemes that we examined, containing the most important contribution to the beta-decay transitions of interest and that are often adopted for the $A\sim 100$ and $A\sim 130-150$ regions of the nuclear mass table, automatically result in $\delta_{LS}^{j_{f}j_{i}}=-1/2$. This is because the only allowed $1^{+}$ transitions in such schemes involve transitions between spin-orbit partners, and, as shown in Sec. IIa, transitions between spin-orbit partners can only yield $\delta_{LS}^{j_{f}j_{i}}=-1/2$. For example, the jj45pna interaction mhj95 suitable to the $A\sim 100$ region, which is built on a 78Ni core, involves the $\pi 1f5/2$, $\pi 2p$ and $\pi 1g9/2$ proton orbitals, and $\nu 1g7/2$, $\nu 2d$, $\nu 3s$ and $\nu 1h11/2$ neutron orbitals. Similarly, for nuclei in the $A\sim 130-150$ mass region, the jj56 model space for use with the j56cdb interaction cdbonn-132sn , is built on 132Sn core, and involves the $\pi 1g7/2$, $\pi 2d$, $\pi 3s$ and $\pi 1h11/2$ proton orbitals, and the $\nu 1h9/2$ $\nu 2f$, $\nu 3p$ $\nu 3s$ and $\nu 1i13/2$ neutron orbitals.

Going beyond model spaces of this size for the heavier mass fragments ($A\sim 130$) was not possible in the present study. One of the issues is the lack of a reliable truncation schemes with corresponding effective two-body nucleon-nucleon interactions. Thus, by the choice of model space, none of our calculations predicted values for $\delta_{LS}^{j_{f}j_{i}}$ different from $-1/2$ for the $A\sim 130$ fragments.

For the $A\sim 100$ region, it was easier to handle larger model spaces, and we used glekpn interaction and model space, as described in Ref. glek . This model space includes five proton orbits ($\pi 1f_{7/2}$, $\pi 1f_{5/2}$, $\pi 2p_{3/2}$, $\pi 2p_{1/2}$ and $\pi 1g_{9/2}$) and five neutron orbits ($\nu 1g_{9/2}$, $\nu 1g_{7/2}$, $\nu 2d_{5/2}$, $\nu 2d_{3/2}$ and $\nu 3s_{1/2}$). This interaction is suitable for describing nuclei with $20<Z<50$ and $40<N<70$. In addition, it has been successfully applied glek to describe the first forbidden decay of a dominant contribution to fission antineutrino spectra, namely 96Y. To calculate the structure of the $A\sim 100$ fission fragments within this model space, we used the shell model code KSHELL kshell . This code corrects for spurious center-of-mass excitations by adding the center-of-mass Hamiltonian, multiplied by a large positive coefficient, to the nuclear Hamiltonian, thus pushing spurious states to very high excitation energies com . Studying the broad set of nuclei in the $A\sim 100$ regions shown in Fig. 1, including excited states, made it necessary for us to truncate the glekpn model space. This mainly involved freezing the proton configurations. For $Z<40$ nuclei, we took the $\pi 1f_{7/2}$ shell to be full, and only allowed two-particle excitations from $\pi 1f_{5/2}$, $\pi 2p_{3/2}$, and $\pi 2p_{1/2}$ shells into $\pi 1g_{9/2}$; For $Z>40$ nuclei, we froze $\pi 1f2p$ shell for positive parity states, and allowed one-particle excitations (from the $\pi 2p_{3/2}$ or $\pi 2p_{1/2}$ shells) for negative parity states. For the neutron configurations, we allowed one-particle excitations from $\nu 1g_{9/2}$ shell, and two-particles excitations from $\nu 2d_{5/2}$ shell, at most. The $\nu 1g_{9/2}$ and $\nu 2d_{5/2}$ orbitals have lower single-particle energies than other neutron orbitals in the glekpn interaction, which is consistent with the HFB-LN calculations shown in Fig. 2.

We calculated the beta decays for the allowed transitions to the lowest 20 positive parity states or lowest 20 negative parity states, depending on the parity of the parent nucleus. The energy distribution of the resulting value for the ratio $\delta_{LS}^{j_{f}j_{i}}$ for eight specific nuclei that contribute significantly to the fission antineutrino spectra is shown in Fig. 4. In this figure, 1947 predicted fission fragment transitions are shown, while there were an additional 43 calculated transitions found to lie outside the range of the figure, $-10.0$ to $10.0$. These outliers arose mostly because of the poor numerical precision possible for very weak transition matrix elements, $10^{-4}$ or smaller. Results are shown in the figure for positive-to-positive parity and negative-to-negative parity transitions separately. The model space for protons, which includes the $\pi 1g_{9/2}$ orbit, contains orbitals with different parities, while the neutron orbits are all positive parity. Thus, the different parity states are produced by configuration mixing between the protons. However, as can seen from the figure, the distribution of $\delta_{LS}^{j_{f}j_{i}}$ is very similar for the two parity cases.

We further collect the data for $\delta_{LS}^{j_{f}j_{i}}$ in Fig. 5, where we show the frequency of occurrence of a given value of $\delta_{LS}^{j_{f}j_{i}}$. Here there are 1897 data points for the range $-5.0\sim 5.0$, which covers $97.43\%$ of the transitions examined. If we average over all the data in Fig. 5, we find,

[TABLE]

where the error is estimated from the standard deviation.

We examined the transitions where $\delta_{LS}^{j_{f}j_{i}}$ deviated significantly from a value of -1/2 in detail, and found that this generally occurred only when matrix elements of $\vec{\Sigma}$ were weak. Since $\delta_{LS}^{j_{f}j_{i}}$ is the ratio of the matrix elements of orbital and spin currents, the absolute value of $\delta_{LS}^{j_{f}j_{i}}$ can become large and/or quite uncertain whenever the matrix elements of $\langle J_{f}||\vec{\Sigma}||J_{i}\rangle$ is less than $10^{-3}$. This issue is the dominant effect determining the standard deviation and quoted error for $\delta_{LS}^{j_{f}j_{i}}$.

II.4 Shell-model results for $\delta_{LS}^{j_{f}j_{i}}$ for a broader set of nuclei

To obtain a more physical understanding of the nuclear structure issues determining the magnitude of the ratio $\delta_{LS}^{j_{f}j_{i}}$, we studied a set of nuclei that do not contribute to the fission antineutrino spectra but that span a much broader range of nuclear masses. The results of this study, involving nuclei with mass ranging from 14 to 103, are summarized in Fig. 6. For this chosen set of nuclei, the neutron and proton number do not differ significantly, which allow many ways of forming allowed GT transitions. In particular, it allows a study of transitions between orbitals of the same $j$ quantum number. For all of these nuclei, except 14C, we restricted the shell model calculation to one major shell. The figure displays the value of $\delta_{LS}^{j_{f}j_{i}}$ for 3633 transitions, and the data are seen to vary in the range of $-50.0$ to $+50.0$. As can be seen from Fig. 7, the corresponding values for $\delta_{LS}^{j_{f}j_{i}}$ exhibit a very broad distribution, being significant for the entire range from $-10.0$ to $+10.0$. This situation is in sharp contrast to that for the fission fragments contributing to antineutrino spectra, where the distribution is much narrower, Fig. 5.

The broad and somewhat random pattern for $\delta_{LS}^{j_{f}j_{i}}$ seen for nuclei in which the neutrons and protons making up the beta-decay transitions are in the same shell is not completely surprising, given the analytical results of SPME for $\delta_{LS}^{j_{f}j_{i}}$ in Eq. (II.1) when $j_{i}=j_{f}$. For the general case involving configuration mixing and transitions between neutron and proton orbitals of the same $j$, there is no simple approximation for the quantity $\delta_{LS}^{j_{f}j_{i}}$, and a nuclear structure independent analytic expression for the orbital contribution to weak magnetism does not exist.

However, when large spin-orbit splitting are involved, and the neutron Fermi surface is very different from the proton one, the ratio $\delta_{LS}^{j_{f}j_{i}}$ tends to be dominated by values close to -1/2. The current work suggests that this situation describes well the strong GT transitions for both the A$\sim$100 and A$>$130 fission fragments, because those beta-decays mainly involve transitions between spin-orbit partners, $\pi g9/2$ to $\nu g7/2$ and $\pi h11/2$ to $\nu h9/2$, respectively. However, for weak transitions, particularly small branches to excited states of the daughter nucleus, matrix elements of $\vec{\Sigma}$ can be suppressed and the ratio of the matrix elements of the orbital to spin current becomes difficult to predict.

III The effect on the uncertainty in one-body WM correction on fission antineutrino spectra

Our nuclear structure calculations suggest that the value of $\delta_{LS}$ is close to $-1/2$, with a one standard deviation value of $\pm 1$. In this section, we examine the effect of this uncertainty on the one-body weak magnetism correction to allowed beta-decay. In Fig. 8, we show the ratio of the spectrum for a single beta-decay with different values of $\delta_{LS}^{j_{f}j_{i}}$ to that with $\delta_{LS}^{j_{f}j_{i}}=-1/2$. In this example the transition was assumed to be a pure GT one, with an end-point energy of $E_{0}=6.0+m_{e}c^{2}$ MeV. For all values of $\delta_{LS}^{j_{f}j_{i}}$ the spectra are normalized to unity. As expected, in all cases, the change in $\delta_{LS}^{j_{f}j_{i}}$ leads to a linear change in the shape of the spectrum, which crosses unity at $E_{0}/2$. This change is quite small, being $<2\%$ even when $\delta_{LS}^{j_{f}j_{i}}$ is taken to deviate from the mean more than three standard deviations.

In Fig. 9, we show the situation for the full aggregate antineutrino spectrum for 235U thermal fission, where the beta-decay end-point energies, branching ratios and the fission yields are taken from ENDF/B-VII.1. Again, even for very large ($>3$) standard deviations from the predicted mean value of $\delta_{LS}^{j_{f}j_{i}}$, the change in the shape of the spectrum is less than 2% at all energy of interest. If we restrict the uncertainty to two standard deviations, the uncertainty is less than 1%. The narrow distribution for $\delta_{LS}^{j_{f}j_{i}}$ in the case of fission fragments, Fig. 5, suggest that the uncertainty in the one-body weak magnetism contribution to fission antineutrino spectra is closer to $\sim 0.5\%$.

IV summary and conclusion

The weak magnetism correction to nuclear beta-decay involves three components, resulting from the spin, orbital and meson-exchange terms in the magnetic dipole operator. We have examined the often invoked approximation for the orbital contribution. This approximation assumes that the orbital contribution is proportional to the spin contrition, and that $\langle J_{f}||\vec{\Lambda}||J_{i}\rangle=-\frac{1}{2}\langle J_{f}||\vec{\Sigma}||J_{i}\rangle$. For a general beta-decay transition, this assumption is not found to be good and the one-body weak magnetism corrections require detailed nuclear structure calculations. However, in the case of the fission fragments that dominate fission antineutrino spectra the assumption is found to be a good approximation, and typically introduces less than a 1% uncertainty in the fission antineutrino spectra. This is because transitions between the fission fragments of interest are dominated by spin-orbit pairs, in which case $\delta_{LS}^{j_{f}j_{i}}=-1/2$. Contributions to weak magnetism from meson-exchange currents have not been examined and require additional study.

Acknowledgements.

We thank J.L. Friar for for very detailed and helpful discussions. X.B. Wang wishes to thank the National Natural Science Foundation of China under Grants No. 11505056 and No. 11605054 and China Scholarship Council (201508330016) for supporting his research. A. C. Hayes thanks the Los Alamos National Laboratory LDRD program. This work was partially supported under the U.S. Department of Energy FIRE Topical Collaboration in Nuclear Theory.