1 Introduction

In a muonic atom, at least, one of the electrons is replaced by a muon. The energy levels of the system are strongly influenced by nuclear finite-size effects due to the strong overlap between the wave functions of the muon and the nucleus. Spectroscopy of muonic atom transitions can therefore provide information about the nuclear structure. The QUARTET (QUAntum inteRacTions with Exotic aToms) collaboration aims for high-resolution spectroscopy of light muonic atoms from lithium to neon. We intend to improve the precision of the absolute nuclear charge radii for these elements by factors of 3–20. For detailed information on the physics cases and theoretical considerations, we refer the reader to an accompanying publication [1].

At the Paul Scherrer Institute (PSI) in Switzerland, beam time has been granted for a proof-of-principle measurement in the fall of 2023, where as a first step, we will attempt to measure the transition energies in muonic lithium, beryllium, and boron. In addition, we want to observe the isotope shift in the muonic transitions between \(^6\)Li and \(^7\)Li as well as between \(^{10}\)B and \(^{11}\)B. A muon beam with a rate of 10 kHz and a momentum of 28 MeV \(c^{-1}\) directed at an exchangeable target inside a vacuum chamber is used to create muonic atoms. The muon is captured at a high principal quantum number n and cascades toward the ground state of the atom by emitting X-rays before decaying or being captured by the nucleus [2]. The most prominent muonic X-ray lines are in the energy range between 10 keV and 60 keV.

We plan to use a low-temperature X-ray detector based on metallic magnetic calorimeter (MMC) [3, 4] to measure the X-rays emitted by the muonic atoms. These highly linear detectors feature high resolving power over a wide energy range [5], providing a suitable technology for high-precision X-ray spectroscopy of light muonic atoms. For efficient placement of the relatively small detector close to the muon target, the detector will be integrated into a new refrigerator sidearm. In addition, existing muon, electron, and photon detectors from the muX experiment will be used for monitoring, validation and background suppression [6]. In the following, we will discuss the MMC-based detector, the new refrigerator sidearm, and the expected performance in more detail.

2 Detector Module for QUARTET

An MMC is a low-temperature micro-calorimeter based on a paramagnetic temperature sensor [5], typically operated in a dilution refrigerator at 20 mK and read-out using a two-stage SQUID (superconducting quantum interference device) circuit [7, 8]. MMCs are used in a wide variety of experiments [4, 9, 10]: They are characterized by high resolving power up to 6000 over a wide energy range [11, 12] while being highly linear with a well-understood energy calibration [3] and having a low energy threshold below 100 eV [4, 13]. They can be designed to have a high quantum efficiency in the energy range of interest up to nearly 100% [14] and can have a fast intrinsic response time up to 100 ns [15]. This makes MMCs attractive for high-resolution X-ray spectroscopy.

An MMC has a particle absorber designed to stop incoming particles of interest and a paramagnetic temperature sensor in thermal contact with the absorber. After a particle interacts with the absorber, the temperature of the detector increases proportionally to the absorbed energy. The sensor, located in a weak static magnetic field, has a temperature-dependent magnetization. The change of magnetization is measured as a change of magnetic flux, which is coupled to the input coil of a dc-SQUID with a superconducting coil. Thus, the screening current generated in the superconducting coil induces a flux change in the SQUID, resulting in a voltage change proportional to the energy of the incident particle.

The X-ray detector which we plan to use for the first QUARTET measurement is a maXs-30 (micro-calorimeter array for X-ray spectroscopy) originally designed for experiments at the heavy ion storage ring ESR at the GSI accelerator [3]. The 64-pixel two-dimensional detector array has a total active area of 4 mm \(\times\) 4 mm and is optimized for X-ray spectroscopy up to 30 keV with its 20 \(\upmu\)m thick gold absorbers. The detector is read out by 32 individual two-stage SQUID channels, where a single channel reads out two pixels of gradiometric design with signals of opposite polarity [14]. The simulated FWHM baseline energy resolution given the expected noise and signal height is 6 eV at a temperature of 20m K. The maXs-30 detector module to be used was developed and characterized as part of the detector development for IAXO [4]. The module, shown in Fig. 1, has 62 out of 64 operational pixels, shows a very linear response of 0.1% at 6 keV and achieves a mean energy resolution of 6eV FWHM at a detector temperature of 17 mK [4]. With the most prominent X-ray lines from the muonic atoms of interest in the energy range between 10 keV and 60 keV, the module is well suited for the proof-of-principle measurement.

Fig. 1
figure 1

The maXs-30 detector and eight first-stage SQUID chips of the detector module to be used for the proof-of-principle measurement at PSI [4]

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3 Dilution Refrigerator Sidearm

For efficient integration of the detector module to the muon target chamber, we have designed a new sidearm for a commercially available dilution refrigerator from Bluefors,Footnote 1 shown in Fig. 2. The sidearm consists of an aluminum vacuum shield with an attached Mu-metal shield, three aluminum thermal shields (connected to the 50 K, 4 K, and still temperature stages in the refrigerator), and an extension arm made out of a copper structure covered by a cylindrical niobium shield. The copper structure is attached to the bottom of the mixing chamber plate of the dilution refrigerator and can hold a detector module up to a cylindrical shaped volume of 8 cm in diameter and height. The mounting on the mixing chamber allows for a fine alignment in all three axes of at least 3 mm to align with respect to the outer shields. The different expansion coefficients of the materials have been taken into account so that the shields align when the detector is cold. The sidearm can be rotated to allow for different detector orientations. Two standard blind flanges on the vacuum shield can be replaced or removed for connection to other vacuum systems.

Fig. 2
figure 2

Design of the new sidearm for the dilution refrigerator. Multiple shields protect the detector from thermal radiation and magnetic field changes. The sidearm can be rotated and has two standard vacuum flanges. X-ray windows allow the measurement of external X-rays

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The vacuum shield and thermal shields are made of pure aluminum to minimize the amount of magnetic material near the detector. The Mu-metal, still, and niobium shields protect the detector from external magnetic fields and magnetic field changes. Once the side arm is installed and aligned, the entire inner copper frame with the prepared detector module can be removed as one piece and quickly reinstalled later. The thermal shields are fabricated to fit the refrigerator and require no further alignment. Quick assembly of the sidearm and its shields is critical when preparing for a measurement with limited beam time.

X-ray windows are used to allow external X-rays to enter and reach the detector. We plan to use an AP5 window from Moxtek with a diameter of 14 mm on the vacuum shield, while a total of three 6 \(\upmu\)m thick Mylar foils coated on both sides with 40 nm aluminum are planned for the three inner thermal shields. The detector mounted on the extension arm surpasses the mixing chamber plate by roughly 20 cm. The solid angle aperture of the detector is about \(6\times 10^{-2}\), limited by the outermost X-ray window on the vacuum shield, which is about 5 cm further away. The vacuum shield of the sidearm extends beyond the refrigerator frame, making it the outermost part and allowing it to be positioned close to the target chamber while leaving the necessary space to integrate germanium detectors, silicon detectors, and scintillators for monitoring, validation, and background suppression.

4 Expected Performance

The expected efficiency of the setup considering the aforementioned X-ray windows and the absorber thickness of the MMCs is shown in Fig. 3. The combined efficiency is reduced by the X-ray windows absorption below 10 keV and by the absorber thickness of the MMCs above 20 keV. The quantum efficiency of the absorber is above 50% up to 40 keV. The efficiency of the available X-ray window configuration is strongly dominated by the relatively thick Mylar foils. In the future, the thickness could be reduced to about 1 \(\upmu\)m, increasing the efficiency at low energies by more than an order of magnitude to 20% at 1 keV.

Fig. 3
figure 3

Expected efficiency considering the quantum efficiency of the detector (blue) and the transmission of the X-ray windows (green). The combined efficiency (red–brown, dashed) is above 20% between 2 and 50 keV. As an example, a calculated spectrum with the roughly estimated energies and intensities of the most prominent muonic \(^7\)Li lines is also shown, considering a finite detector response of 10 eV FWHM (gray). For the estimation, attenuation coefficients were taken from NIST [16], and the efficiency of the AP5 X-ray window was provided by Moxtek

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As an example, the expected muonic \(^7\)Li spectrum is shown in Fig. 3. The most prominent line from the 2p–1s transition is in the very high-efficiency region. At 19 keV, we anticipate an efficiency of 97%. We also estimate reasonable efficiencies for the 2p–1s line of a beryllium or boron target (59% at 33 keV for beryllium and 22% at % 53 keV for boron). The muon target will be about 15 cm away from the MMC detector array. Given the solid angle covered and the expected muon rate (10,000 s\(^{-1}\) triggered events), a rough estimate gives an expected rate on the order of 0.3 s\(^{-1}\) muon-induced X-ray events, with roughly 0.15 s\(^{-1}\) events at the MMC array from the 2p–1s transition, which is well below the maximum rate that the detector can reliably handle. Improving the radii by an order of magnitude requires a ppm precision on the line position. For the muonic 2p–1s transition of \(^7\)Li at 19 keV, this corresponds to a precision of 0.1 eV. For a detector with an FWHM energy resolution of 10 eV, a few thousand events are sufficient to achieve the required statistical precision. Given the expected rate of 0.15 s\(^{-1}\), this could be achieved with a few hours of measurement.

5 Conclusion

The QUARTET collaboration aims at performing high-precision X-ray spectroscopy of light muonic atoms from lithium to neon using a two-dimensional MMC array. For a proof-of-principle measurement with lithium, beryllium, and boron targets, we have prepared a new dedicated experimental setup with an already well characterized detector module. The main feature of this setup is a new refrigerator sidearm that allows for more efficient detector placement close to the muon target. We expect a suitable rate for de-excitation spectra for light muonic atoms. Combined with the excellent performance of the MMC array, we could exceed the current accuracy on the position of muonic X-ray lines already with the scheduled proof-of-principle measurement at PSI in October 2023. This will be the first use of MMCs to study exotic atoms.