Effective field theory for vibrations in odd-mass nuclei

Heavy even-even nuclei exhibit low-energy collective excitations that are separated in scale from the microscopic (fermion) degrees of freedom. This separation of scale allows us to approach nuclear vibrations within an effective field theory (EFT). In odd-mass nuclei collective and single-particle properties compete at low energies, and this makes their description more challenging. In this article we describe spherical odd-mass nuclei with ground-state spin $I=\frac{1}{2}$ by means of an EFT that couples a fermion to the collective degrees of freedom of an even-even core. The EFT relates observables such as energy levels, electric quadrupole transition strengths, and magnetic dipole moments of the odd-mass nucleus to those of its even-even neighbor and allows us to quantify theoretical uncertainties. For isotopes of rhodium and silver the theoretical description is consistent with data within experimental and theoretical uncertainties. Several testable predictions are made.

Figure 1

NLO spectrum for the fermion in a $j=\frac{1}{2}$ orbital coupled to a quadrupole vibrator up to the two-phonon level in arbitrary units. The states labeled $I^{\pi }$ are displayed as long red and short blue lines for even-even and odd-mass nuclei, respectively. The centroids of the $I=J\pm j$ odd-mass states are shown as blue crosses.

Figure 2

Cumulative distributions of the $c_1$ (top) and $c_2$ (bottom) coefficients for the energies of states below breakdown in an ensemble containing the data of all studied Pd and Ag nuclei. These distributions, centered at ${\mu }_1$ and ${\mu }_2$, are approximated by Gaussian priors (shown as lines).

Figure 3

NNLO energy spectra of Ru/Rh systems. Rh is described as a proton in a $j^{\pi }={\frac{1}{2}}^{-}$ orbital coupled to a Ru core. Thick black lines denote states employed to fit the LECs while thin black lines denote states with a definitely known spin or a single tentative spin-parity assignment. Red crosses and shaded areas denote theoretical predictions and uncertainties, respectively.

Figure 4

NNLO energy spectra of Pd/Ag systems. Ag is described as a proton in a $j^{\pi }={\frac{1}{2}}^{-}$ orbital coupled to a Pd core. Thick black lines denote states employed to fit the LECs while thin black lines denote states with a definitely known spin or a single tentative spin-parity assignment. Red crosses and shaded areas denote theoretical predictions and uncertainties, respectively.

Figure 5

NNLO energy spectra of Cd/Ag systems. Ag is described as a proton hole in a $j^{\pi }={\frac{1}{2}}^{-}$ orbital coupled to a Cd core. Thick black lines denote states employed to fit the LECs while thin black lines denote states with a definitely known spin or a single tentative spin-parity assignment. Red crosses and shaded areas denote theoretical predictions and uncertainties, respectively.

Figure 6

LO (top), NLO (center), and NNLO (bottom) energy spectra of the ${}^{108}\mathrm{Pd}/{}^{109}\mathrm{Ag}$ system. The systematic improvement inherent to EFT approaches is evident.

Figure 7

Static $E2$ moments (top) and $E2$ transitions strengths (bottom) as functions of $Q\left(2_1^{+}\right)$. The uncertainty quantified from 68% DOB intervals is shown as error bands. Data for the ${}^{102}\mathrm{Ru}/{}^{103}\mathrm{Rh},{}^{106}\mathrm{Pd}/{}^{107}\mathrm{Ag}$, and ${}^{108}\mathrm{Pd}/{}^{109}\mathrm{Ag}$ systems are shown as diamonds, triangles, and circles, respectively.