Reference Quadrupole Moments of Transition Elements from Lamb Shifts in Muonic Atoms

Abstract

We present a novel method for accurately measuring the absolute electric quadrupole moments of light transition elements $(23 \leq Z \leq 30 )$. Our approach is based on performing precision muonic x-ray spectroscopy of the $2s-2p$ manifold, which is also referred to as the Lamb shift. These transitions are too weak to be detected with dispersive methods and too overlapping to be resolved by solid-state detectors. Here, we propose the use of cryogenic microcalorimeters, which possess high efficiency and excellent energy resolution in the relevant energy regime, coupled with state-of-the-art theoretical calculations. We demonstrate the feasibility of this approach by performing extensive calculations and realistic simulations. In this way, we establish that the uncertainty in the absolute moment, which is transferred to the quadrupole moments of all isotopes in the chain, could be reduced by up to an order of magnitude within a day of measurement. These precise reference quadrupole moments serve as valuable inputs for nuclear structure studies and for benchmarking state-of-the-art quantum chemistry calculations in open-shell elements.

Figures (5)

• Figure 1: Current status of the fractional accuracy in absolute reference quadrupole moments 2021-Stone and the potential improvement aimed based on the method presented here.
• Figure 2: Relative uncertainty in the measured $B$ for selected copper isotopes (square 2017-Cu, circle 2010-Cu and diamond 2011-Cu) and isomers (rightward triangle 2010-Cu, leftward triangle 2011-cu_isomer and downward triangle 2016-cu_68m) reported in the literature. The dashed line denotes the relative uncertainty of the $Q_{\text{ref}}$ for $^{63}\mathrm{Cu}$ from 2021-Stone.
• Figure 3: Estimation of the optimal muon implantation momentum for Cu for observing photons from the $2s-2p_{3/2}$ transition at $32\,$keV. The circles denote the transmission efficiency (TE, left-hand axis) from the implantation site. The triangles denote the available muon rate for each momenta 2023-pie1-gerchow and correspond to the right-hand axis. The squares denote the yield of photons that exit the target towards the detector. They are given in arbitrary units and are maximal for a momentum of $24$ MeV/c.
• Figure 4: Simulated spectrum of the area of interest in muonic $^{63}$Cu. The broken line shows the calculated $2s_{1/2~F'} \to 2p_{3/2~F}$ spectrum as individual Lorenzians with their natural linewidth. The blue solid line shows the total spectrum as the sum of Voigt profiles with the natural linewidths and a detector FWHM resolution of $10\,$eV. It includes the $12\to7$ ($31.1-32.3\,$keV) and $13\to7$ ($33.5-34.5\,$keV) manifolds as well. We also plot in green on top the total spectrum with a FWHM resolution of $500\,$eV befitting a low-energy HPGe detector.
• Figure 5: Results of the muon-induced background simulation. It is a stacked histogram of the energy deposition in all MMC pixels, decomposed into particle species. For photons, the spectrum is further split into those generated by the physics of a muonic atom and all other sources. The intensity is normalized to the number of detected muons in the tagging scintillator.