Advanced muon-spin spectroscopy with high lateral resolution using Si-pixel detectors

Muon-spin spectroscopy at continuous sources has stagnated at a stopped muons rate of ~40 kHz for the last few decades. The major limiting factor is the requirement of a single muon in the sample during the typical 10 μs data gate window. To overcome this limit and to be able to perform muon-spin relaxation (μSR) measurements on millimeter-sized samples, one can use vertex reconstruction methods to construct μSR spectra. This is now possible thanks to the availability of very thin monolithic Si-pixel chips, which offer minimal particle scattering and high count rate. Here we present results from a Si-pixel based spectrometer that utilizes vertex reconstruction schemes for the incoming muons and emitted positrons. With this spectrometer we were able to obtain a first vertex reconstructed μSR (VR-μSR) spectrum. The unique capabilities and benefits of such a spectrometer are discussed.

💡 Research Summary

This paper addresses a long‑standing bottleneck in continuous‑beam muon‑spin spectroscopy (μSR), namely the limitation to a stopped‑muon rate of roughly 40 kHz imposed by the requirement that only a single muon be present in the sample during the typical 10 µs data‑gate window. The authors propose and demonstrate a novel spectrometer that replaces conventional scintillator‑based detectors with thin, high‑voltage monolithic active pixel sensors (HV‑MAPS), specifically the MuPix11 silicon‑pixel chips developed for the Mu3e experiment.

Each MuPix11 chip provides 23 µm spatial resolution and sub‑15 ns timing resolution while being only 50–100 µm thick, thereby minimizing multiple Coulomb scattering of both the incoming muon and the emitted positron. Four chips are assembled on a printed‑circuit “Quad” board; two Quads form an upstream detector set and two form a downstream set, giving four detection layers (L1–L4) positioned at ±10 mm and ±30 mm relative to the sample plane (z = 0). GEANT4‑based musrSim simulations show that the uncertainty in reconstructing the muon impact point (δµ) grows linearly with the spacing between the inner and outer layers at a rate of ~0.02 mm per mm, remaining below 0.5 mm for the chosen 20 mm spacing.

Data acquisition uses a custom front‑end board and DE5a‑NET FPGA to stream hit coordinates (x, y) and timestamps (t). The raw hit stream is filtered with the Corryvreckan framework to retain only valid coincidences in the upstream (L1+L2) or downstream (L3+L4) sets. A simple reconstruction algorithm identifies an incoming muon by a hit in L1 within a 2 mm radius of the beam collimator centre, extrapolates its trajectory to the sample plane, and then searches within a 13 µs time window for a positron track that extrapolates to the same sample location within a matching distance d_match (typically 1 mm). When such a pair is found, the time difference Δt is histogrammed to produce a μSR spectrum. Because the probability of two muons landing within 1 mm at a 40 kHz rate is only ~0.6 %, the algorithm effectively yields a one‑to‑one muon‑positron association with negligible accidental background.

The prototype was tested on a 6 mm diameter aluminum disc placed between two permanent magnets generating a transverse field of ~6.3 mT. The resulting μSR spectrum exhibits the characteristic muon precession signal with a frequency of 0.85 MHz and a damping rate of 0.32 µs⁻¹, matching measurements performed on the established GPS spectrometer at PSI. Notably, the Si‑pixel spectrometer achieved comparable statistical quality in only ~35 seconds of data taking, whereas the GPS required ~14 minutes. The background term B in the fit is essentially zero, confirming that vertex reconstruction eliminates uncorrelated events and permits extension of the data‑gate window without loss of time resolution.

To evaluate lateral resolution, a silver plate with cut‑out features ranging from 1.1 mm to 1.6 mm was measured. Vertices—defined as the intersection of a muon and its associated positron at the sample plane—were reconstructed with d_match values of 0.3 mm, 0.5 mm, and 1.0 mm. The vertex density maps clearly resolve features down to ~0.6 mm, and Gaussian fits to cross‑sections yield full‑width‑at‑half‑maximum (FWHM) values only ~0.4 mm larger than the true feature widths. The number of reconstructed vertices decreases with tighter d_match, but even at d_match = 1 mm the algorithm retains sufficient statistics at 40 kHz.

The authors conclude that the combination of thin HV‑MAPS Si‑pixel detectors and vertex reconstruction offers four major advances for μSR: (1) sub‑millimeter lateral resolution enabling studies of tiny or inhomogeneous samples, (2) the possibility of simultaneous measurements on multiple small specimens, (3) the ability to increase the usable muon rate by more than an order of magnitude while preserving time resolution, and (4) a dramatic reduction of accidental background, allowing longer data‑gate windows. Current limitations include operation in air (precluding cryogenic environments) and the lack of an integrated magnetic field. Future work will focus on vacuum operation with a cryostat, implementation of a three‑layer geometry for accurate vertexing in magnetic fields, and development of higher‑rate muon‑track separation algorithms. This proof‑of‑concept demonstrates that vertex‑reconstructed μSR can break the long‑standing rate ceiling and open new experimental possibilities for condensed‑matter and quantum‑material research.