Nuclear Charge Radius of $^9$Be from Muonic Atom Spectroscopy Using a Microcalorimeter

Technical Summary: Nuclear Charge Radius of $^9$Be from Muonic Atom Spectroscopy Using a Microcalorimeter

Problem Statement
The precision of theoretical predictions for simple atomic systems is frequently limited by the uncertainty in the root-mean-square (RMS) nuclear charge radius, $r_c$. While muonic atom spectroscopy is a standard method for determining $r_c$ due to the muon's proximity to the nucleus, the technique has historically struggled with light nuclei ($Z=3–10$). For these elements, uncertainties in $r_c$ are up to two orders of magnitude larger than for lighter or heavier counterparts. This limitation stems from two factors: the experimental difficulty of performing muonic-atom laser spectroscopy above $Z=2$, and the insufficient energy resolution of conventional X-ray detectors, which cannot resolve the finite nuclear size contribution to atomic binding energies. Consequently, charge radii for light nuclei beyond helium are often derived from electron scattering, which relies on model-dependent analyses and suffers from incomplete knowledge of systematic effects. A specific example is the isotope $^9$Be, where the commonly accepted value of $r_c = 2.519(12)$ fm is based on such limited electron scattering data.

Methodology
This work introduces a high-precision approach utilizing muonic-atom X-ray spectroscopy with a detector array based on Metallic Magnetic Calorimeters (MMCs). The experiment was conducted at the $\pi$E1 beamline of the Paul Scherrer Institute (PSI), using a continuous beam of negative muons.

• Detector System: The team employed a maXs-30 (Micro-calorimeter Array for X-ray Spectroscopy) detector, developed at the Kirchhoff-Institut für Physik, Heidelberg University. This system features an $8 \times 8$ array of pixels, each equipped with a 20 $\mu$m-thick gold absorber, coupled to a two-stage SQUID readout. The detector operates at temperatures below 100 mK within a dilution refrigerator.
• Target and Beam: A high-purity ($>99\%$) beryllium disk (2.5 mm thick, 21.6 mm diameter), composed entirely of $^9$Be, was mounted at a 45-degree angle to minimize self-absorption. The beam momentum was set to 29.5 MeV/c, resulting in an estimated muon stopping rate of 5 kHz.
• Data Acquisition and Calibration: The experiment ran for 34 hours. To achieve sub-eV precision, the team implemented rigorous data reduction techniques:
  • Temperature Correction: Signal amplitude variations due to temperature drifts (a few mK) were corrected using dedicated temperature-sensitive channels and a robust linear model derived from 60 keV photons.
  • Coincidence Filtering: To distinguish muon-induced events from background, two spectra were constructed: one in coincidence with the muon trigger and one in anti-coincidence. This reduced the magnitude of calibration peaks in the muonic spectrum by a factor of 125.
  • Non-linearity Correction: Residual non-linearity in the detector response was quantified by fitting known calibration lines (from $^{241}$Am and X-ray fluorescence of La and Ba) and applying a quadratic correction to pulse amplitudes.
  • Spectral Analysis: The energy difference between the muonic $^9$Be $2p \to 1s$ transition and the Lanthanum $K\alpha_1$ calibration line was determined by simultaneously fitting the coincidence and anti-coincidence spectra using a Voigt function with exponential tails to model the instrumental lineshape.

Key Contributions and Results
The primary achievement of this work is the first determination of a nuclear charge radius using muonic X-ray spectroscopy with microcalorimeters.

• Transition Energy: The $2p \to 1s$ transition energy in muonic $^9$Be was measured with unprecedented precision:
  $$E_{2p \to 1s} = 33\,391.48(34) \text{ eV}$$
  This result is 30 times more precise than the previous best measurement.
• Nuclear Charge Radius: Using state-of-the-art calculations encompassing atomic and nuclear theory, the nuclear charge radius was extracted:
  $$r_c(^9\text{Be}) = 2.5506(51) \text{ fm}$$
  (Note: The abstract cites 51 fm uncertainty, while the text body cites 52 fm; the value is 2.4 times more precise than the commonly used electron scattering value).
• Comparison: The new value differs from the electron scattering value ($2.519(12)$ fm) by 2.3 times the combined uncertainties.
• Implications for Isotopic Chains: By combining this new reference value with differential radii from optical spectroscopy, the authors derived updated radii for the entire chain of Beryllium isotopes ($^7$Be, $^{10}$Be, $^{11}$Be, $^{12}$Be) and predicted radii for $^{10}$C and the excited state of $^{10}$B. These updated values show better alignment with recent ab initio nuclear lattice effective field theory (NLEFT) calculations, although a significant deviation remains for $^7$Be.

Significance
The paper claims that this measurement represents a breakthrough in resolving the "proton radius puzzle" context for light nuclei by providing a model-independent, high-precision benchmark. The use of MMCs overcomes the resolution limits of conventional detectors, allowing the extraction of nuclear size effects that were previously obscured. The authors note that the current accuracy is primarily limited by the precision of the reference calibration line (Lanthanum $K\alpha_1$) rather than the experimental setup itself. They suggest that with optimized detectors and improved reference line energies, sub-ppm accuracy is achievable. The results provide critical benchmarks for ab initio calculations in superallowed beta decay and help constrain the magnitude of finite-size contributions in nuclear theory.