Muons are particles with a spin of ½ that can be implanted into a wide range of condensed matter materials to act as a local probe of the surrounding atomic environment. Measurement of the muon’s precession and relaxation provides an insight into how it interacts with its local environment. From this, unique information is obtained about the static and dynamic properties of the material of interest. This has enabled muon spin spectroscopy, more commonly known as muon spin rotation/relaxation/resonance (μSR), to develop into a powerful tool to investigate material properties such as fundamental magnetism, superconductivity and functional materials. Alongside this, μSR may be used to study, for example, energy storage materials, ionic diffusion in potential batteries, the dynamics of soft matter, free radical chemistry, reaction kinetics, semiconductors, advanced manufacturing and cultural artefacts. This Primer is intended as an introductory article and introduces the μSR technique, the typical results obtained and some recent advances across various fields. Data reproducibility and limitations are also discussed, before highlighting promising future developments.
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Anderson, C. D. & Neddermeyer, S. H. Cloud chamber observations of cosmic rays at 4300 meters elevation and near sea-level. Phys. Rev. 50, 263–271 (1936).
Wulf, T. Über den Ursprung der in der Atmosphäre vorhandenen γ-Strahlung [German]. Phys. Z. 10, 811 (1909).
Hess, V. F. Über Beobachtungen der durchdringenden Strahlung bei sieben Freiballonfahrten [German]. Phys. Z. 13, 1084 (1912).
Garwin, R. L., Lederman, L. M. & Weinrich, M. Observations of the failure of conservation of parity and charge conjugation in meson decays — magnetic moment of the free muon. Phys. Rev. 105, 1415–1417 (1957).
Conversi, M., Pancini, E. & Piccioni, O. On the decay process of positive and negative mesons. Phys. Rev. 68, 232 (1945).
Blundell, S., De Renzi, R., Lancaster, T. & Pratt, F. Introduction to Muon Spectroscopy (Oxford Univ. Press, 2021).
Schenck, A. Muon Spin Rotation: Principles and Applications in Solid State Physics (Adam Hilger, 1985).
Yaouanc, A. & Dalmas de Réotier, P. Muon Spin Rotation, Relaxation, and Resonance (Oxford Univ.Press, 2010).
Nagamine, K. Introductory Muon Science (Cambridge Univ. Press, 2003).
Roduner, E. The Positive Muon as a Probe in Free Radical Chemistry (Springer, 1988).
Lee, S. L., Kilcoyne, S. H. & Cywinski, R. Muon Science: Muons in Physics, Chemistry and Materials: Proceedings of the Fifty First Scottish Universities Summer School in Physics, St Andrews, August 1998 (Scottish Univ. Summer School in Physics, 1999).
Walker, D. C. Muon and Muonium Chemistry (Cambridge Univ. Press, 1983).
Morenzoni, E. & Amato, A. in Lecture Notes in Physics (Springer, 2022).
Matsuzaki, T. et al. The RIKEN-RAL pulsed muon facility. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 465, 365–383 (2001).
Hillier, A. D., Lord, J. S., Ishida, K. & Rogers, C. Muons at ISIS. Philos. Trans. R. Soc. A Mathem. Phys. Eng. Sci. 377, 7 (2019).
Cook, S. et al. Delivering the world’s most intense muon beam. Phys. Rev. Accel. Beams 20, 030101 (2017).
Miyake, Y. et al. J-PARC muon facility, MUSE. Phys. Procedia 30, 46–49 (2012).
Hillier, A. D. et al. Developments at the ISIS muon source and the concomitant benefit to the user community. J. Phys. Conf. Ser. 551, 012067 (2014).
Giblin, S. R. et al. Optimising a muon spectrometer for measurements at the ISIS pulsed muon source. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 751, 70–78 (2014).
Abela, R. et al. Muons on request (MORE): combining advantages of continuous and pulsed muon beams. Hyperfine Interact. 120, 575–578 (1999).
Bakule, P. & Morenzoni, E. Generation and applications of slow polarized muons. Contemp. Phys. 45, 203–225 (2004). This work demonstrates low-energy muons.
Khasanov, R. et al. High pressure research using muons at the Paul Scherrer Institute. High. Press. Res. 36, 140–166 (2016).
Watanabe, I. et al. Development of a gas-pressurized high-pressure μSR setup at the RIKEN-RAL Muon Facility. Phys. B Condens. Matter 404, 993–995 (2009).
Hillier, A. D., Cottrell, S. P., King, P. J. C., Eaton, G. H. & Clarke-Gayther, M. A. High frequency measurements at a pulsed muon source: beating the pulse width! Phys. B-Condens. Matter 326, 275–278 (2003).
Cottrell, S. P., Johnson, C., Cox, S. F. J. & Scott, C. A. Radio-frequency techniques for muon charge state conversion measurements. Phys. B Condens. Matter 326, 248–251 (2003).
Fleming, D. G., Cottrell, S. P., McKenzie, I. & Ghandi, K. Rate constants for the slow Mu plus propane abstraction reaction at 300 K by diamagnetic RF resonance. Phys. Chem. Chem. Phys. 17, 19901–19910 (2015).
Kadono, R., Matsushita, A., Macrae, R. M., Nishiyama, K. & Nagamine, K. Muonium centers in crystalline Si and Ge under illumination. Phys. Rev. Lett. 73, 2724–2727 (1994).
Yokoyama, K. et al. The new high field photoexcitation muon spectrometer at the ISIS pulsed neutron and muon source. Rev. Sci. Instrum. 87, 12511 (2016).
Prokscha, T., Chow, K. H., Salman, Z., Stilp, E. & Suter, A. Direct observation of hole carrier-density profiles and their light-induced manipulation at the surface of Ge. Phys. Rev. Appl. 14, 014098 (2020).
Eshchenko, D. G., Storchak, V. G., Cottrell, S. P. & Morenzoni, E. Electric-field-enhanced neutralization of deep centers in GaAs. Phys. Rev. Lett. 103, 216601 (2009).
Hillier, A. D., Paul, D. M. & Ishida, K. Probing beneath the surface without a scratch bulk non-destructive elemental analysis using negative muons. Microchem. J. 125, 203–207 (2016).
Clemenza, M. et al. Muonic atom X-ray spectroscopy for non-destructive analysis of archeological samples. J. Radioanal. Nucl. Chem. 322, 1357–1363 (2019).
Hillier, A., Seller, K. I. P., Veale, M. C. & Wilson, M. D. Element specific imaging using muonic X-rays. JPS Conf. Proc. 21, 011042 (2018).
Yabu, G. et al. Imaging of muonic X-ray of light elements with a CdTe double-sided strip detector. JPS Conf. Ser. 21, 011044 (2018).
Hayano, R. S. et al. Zero-and low-field spin relaxation studied by positive muons. Phys. Rev. B 20, 850–859 (1979). This work presents a comprehensive but concise and still actual description of the relaxations in zero and transverse fields. The principle of the techniques and the standard relaxation function are introduced.
Rainford, B. D. & Daniell, G. J. μ-SR frequency-spectra using the maximum-entropy method. Hyperfine Interact. 87, 1129–1134 (1994).
McKenzie, I. The positive muon and μSR spectroscopy: powerful tools for investigating the structure and dynamics of free radicals and spin probes in complex systems. Annu. Rep. Sect. C 109, 65–112 (2013).
Heming, M. et al. Detection of muonated free-radicals through the effects of avoided level-crossing — theory and analysis of spectra. Chem. Phys. Lett. 128, 100–106 (1986).
Percival, P. W. et al. C-13 hyperfine coupling-constants in μC60. Chem. Phys. Lett. 245, 90–94 (1995).
Roduner, E., Reid, I. D., Ricco, M. & De Renzi, R. Anisotropy of 2-norbornyl radical reorientational dynamics in the plastic phase of norbornene as determined by ALC-μ-SR. Ber. Bunsen Ges. Phys. Chem. Chem. Phys. 93, 1194–1197 (1989).
McKenzie, I., Scheuermann, R. & Sedlak, K. How do strain and steric interactions affect the reactions of aromatic compounds with free radicals? Characterization of the radicals formed by muonium addition to p-xylene and 2.2 paracyclophane by DFT calculations and muon spin spectroscopy. J. Phys. Chem. A 116, 7765–7772 (2012).
Measday, D. F. The nuclear physics of muon capture. Phys. Reports Rev. Sect. Phys. Lett. 354, 243–409 (2001).
Ninomiya, K., Kubo, M., Strasser, P., Shinohara, A., Tampo, M., Kawamura, N. & Miyake, Y. Isotope identification of lead by muon induced X-ray and γ-ray measurements. JPS Conf. Proc. 21, 011043 (2018).
Suter, A. & Wojek, B. M. MuSRFit: a free platform-independent framework for mu SR data analysis. 12th Int. Conf. MuSpin Rotation Relax. Reson. 30, 69–73 (2012).
Pratt, F. L. WiMDA: a muon data analysis program for the Windows PC. Phys. B 289, 710–714 (2000).
Baker, P. J., Loe, T., Telling, M., Cottrell, S. P. & Hillier, A. D. in Proc. 14th Int. Conf. Muon Spin Rotation, Relaxation and Resonance (μSR2017) Vol. 21 JPS Conference Proceedings (Physical Society of Japan, 2018).
Luetkens, H. et al. The electronic phase diagram of the LaO1–xFxFeAs superconductor. Nat. Mater. 8, 305–309 (2009).
Drew, A. J. et al. Coexistence of static magnetism and superconductivity in SmFeAsO1–xFx as revealed by muon spin rotation. Nat. Mater. 8, 310–314 (2009).
Parker, D. R. et al. Control of the competition between a magnetic phase and a superconducting phase in cobalt-doped and nickel-doped NaFeAs using electron count. Phys. Rev. Lett. 104, 057007 (2010).
Hayward, M. A. et al. The hydride anion in an extended transition metal oxide array: LaSrCoO3H0.7. Science 295, 1882–1884 (2002).
Campbell, I. A. et al. Dynamics in canonical spin-glasses observed by muon spin depolarization. Phys. Rev. Lett. 72, 1291–1294 (1994).
Lovesey, S. W., Trohidou, K. N. & Karlsson, E. B. Muon spin relaxation in ferromagnets. 2. Critical and paramagnetic magnetization fluctuations. J. Phys. Condens. Matter 4, 2061–2071 (1992).
Uemura, Y. J. et al. Phase separation and suppression of critical dynamics at quantum phase transitions of MnSi and (Sr1–xCax)RuO3. Nat. Phys. 3, 29–35 (2007).
Uemura, Y. J., Harshman, D. R., Senba, M., Ansaldo, E. J. & Murani, A. P. Zero-field muon-spin relaxation in CuMn spin-glasses compared with neutron and susceptibility experiments. Phys. Rev. B 30, 1606–1608 (1984).
Yaouanc, A., Dereotier, P. D. & Frey, E. Zero-field muon-spin-relaxation depolarization rate of paramagnets near the Curie-temperature. Phys. Rev. B 47, 796–809 (1993).
Pratt, F. et al. Muon spin relaxation studies of critical fluctuations and diffusive spin dynamics in molecular magnets. Phys. B: Condens. Matter 404, 585–589 (2009).
Berlie, A., Terry, I. & Szablewski, M. A 3D antiferromagnetic ground state in a quasi-1D π-stacked charge-transfer system. J. Mater. Chem. C. 6, 12468–12472 (2018).
Adroja, D. T. et al. Muon spin rotation and neutron scattering investigations of the B-site ordered double perovskite Sr2DyRuO6. Phys. Rev. B 101, 13 (2020).
Yamauchi, H. et al. High-temperature short-range order in Mn3RhSi. Commun. Mater. 1, 43 (2020).
Dally, R. et al. Short-range correlations in the magnetic ground state of Na4Ir3O8. Phys. Rev. Lett. 113, 247601 (2014).
Rainford, B. D., Cywinski, R. & Dakin, S. J. Neutron and μ-SR studies of spin fluctuations in YMn2 and related alloys. J. Magn. Magn. Mater. 140, 805–806 (1995).
Le, L. P. et al. Searching for spontaneous magnetic order in an organic ferromagnet — μ-SR studies of β-phase p-NPNN. Chem. Phys. Lett. 206, 405–408 (1993).
Blundell, S. J. et al. μ+ SR of the organic ferromagnet p-NPNN — diamagnetic and paramagnetic states. Europhys. Lett. 31, 573–578 (1995).
Kojima, K. M. et al. Reduction of ordered moment and Neel temperature of quasi-one-dimensional antiferromagnets Sr2CuO3 and Ca2CuO3. Phys. Rev. Lett. 78, 1787–1790 (1997).
Sengupta, P., Sandvik, A. W. & Singh, R. R. P. Specific heat of quasi-two-dimensional antiferromagnetic Heisenberg models with varying interplanar couplings. Phys. Rev. B 68, 7 (2003).
Lancaster, T. et al. Magnetic order in the quasi-one-dimensional spin-1/2 molecular chain compound copper pyrazine dinitrate. Phys. Rev. B 73, 020410 (2006).
Pratt, F. L., Blundell, S. J., Lancaster, T., Baines, C. & Takagi, S. Low-temperature spin diffusion in a highly ideal S = 1/2 Heisenberg antiferromagnetic chain studied by muon spin relaxation. Phys. Rev. Lett. 96, 247203 (2006).
Manson, J. L. et al. Ag(nic)2 (nic=nicotinate): a spin-canted quasi-2D antiferromagnet composed of square-planar S = 1/2 Ag-II ions. Inorg. Chem. 51, 1989–1991 (2012).
Lancaster, T. et al. Quantum magnetism in molecular spin ladders probed with muonspin spectroscopy. New. J. Phys. 20, 103002 (2018).
Pratt, F. Superconductivity and magnetism in organic materials studied with μSR. J. Phys. Soc. Jpn. 85, 091008 (2016).
Amato, A. et al. Understanding the μSR spectra of MnSi without magnetic polarons. Phys. Rev. B 89, 184425 (2014).
Wang, R. Z. et al. Quantum griffiths phase inside the ferromagnetic phase of Ni1–xVx. Phys. Rev. Lett. 118, 267202 (2017).
Frandsen, B. A. et al. Volume-wise destruction of the antiferromagnetic Mott insulating state through quantum tuning. Nat. Commun. 7, 12519 (2016).
Kirschner, F. K. K. et al. Spin Jahn–Teller antiferromagnetism in CoTi2O5. Phys. Rev. B 99, 064403 (2019).
Pratt, F. L. et al. Magnetic and non-magnetic phases of a quantum spin liquid. Nature 471, 612–616 (2011).
Yaouanc, A. et al. Dynamical splayed ferromagnetic ground state in the quantum spin ice Yb2Sn2O7. Phys. Rev. Lett. 110, 127207 (2013).
Kirschner, F. K. K., Flicker, F., Yacoby, A., Yao, N. Y. & Blundell, S. J. Proposal for the detection of magnetic monopoles in spin ice via nanoscale magnetometry. Phys. Rev. B 97, 140402(R) (2018).
Brewer, J. H. et al. Observation of muon–fluorine hydrogen-bonding in ionic-crystals. Phys. Rev. B 33, 7813–7816 (1986).
Lancaster, T. et al. Muon–fluorine entangled states in molecular magnets. Phys. Rev. Lett. 99, 267601 (2007).
Wilkinson, J. M. & Blundell, S. J. Information and decoherence in a muon–fluorine coupled system. Phys. Rev. Lett. 125, 087201 (2020). This work demonstrates the use of muons to model fundamental decoherence effects in the solid state.
Aidoudi, F. H. et al. An ionothermally prepared S = 1/2 vanadium oxyfluoride kagome lattice. Nat. Chem. 3, 801–806 (2011).
Abdeldaim, A. H. et al. Realizing square and diamond lattice S = 1/2 Heisenberg antiferromagnet models in the α and β phases of the coordination framework, KTi(C2O4)2·xH2O. Phys. Rev. Mater. 4, 104414 (2020).
Mendels, P. et al. Quantum magnetism in the paratacamite family: towards an ideal kagome lattice. Phys. Rev. Lett. 98, 077204 (2007). This work shows that μSR has an unparalleled sensitivity to magnetic fields. This specificity is used in this article to discard even weak magnetic freezing in the first kagome-based quantum spin liquid candidate.
Majumder, M. et al. Breakdown of magnetic order in the pressurized kitaev iridate β-Li2IrO3. Phys. Rev. Lett. 120, 237202–237202 (2018).
Li, Y. et al. Muon spin relaxation evidence for the U1 quantum spin-liquid ground state in the triangular antiferromagnet YbMgGaO4. Phys. Rev. Lett. 117, 97201–97201 (2016).
Colman, R. H. et al. Spin dynamics in the S = 1/2 quantum kagome compound vesignieite, Cu3Ba(VO5H)2. Phys. Rev. B 83, 180416 (2011).
Pratt, F. L. Probing quantum critical spin liquids with μSR. JPS Conf. Proc. 21, 011002 (2018).
Gomilšek, M. et al. Kondo screening in a charge-insulating spinon metal. Nat. Phys. 15, 754–758 (2019).
Amato, A. Heavy-fermion systems studied by μSR technique. Rev. Mod. Phys. 69, 1119–1179 (1997). This work reviews the impact muons have had in the field of heavy fermion science.
Sonier, J. E., Brewer, J. H. & Kiefl, R. F.μSR studies of the vortex state in type-II superconductors. Rev. Mod. Phys. 72, 769–811 (2000). This work presents a superconductivity review on the impact of muons.
Tinkham, M. Introduction to Superconductivity 2nd edn (McGraw Hill, 1996).
Sonier, J. E. μSR studies of cuprate superconductors. J. Phys. Soc. Jpn. 85, 091005 (2016).
Khasanov, R. et al. Two-gap superconductivity in Ba1–xKxFe2As2: a complementary study of the magnetic penetration depth by muon-spin rotation and angle-resolved photoemission. Phys. Rev. Lett. 102, 187005 (2009).
Uemura, Y. J. Dynamic superconductivity responses in photoexcited optical conductivity and Nernst effect. Phys. Rev. Mater. 3, 104801 (2019).
Uemura, Y. J. et al. Magnetic-field penetration depth in Tl2Ba2CuO6+δ in the overdoped regime. Nature 364, 605–607 (1993).
Sigrist, M. & Ueda, K. Phenomenological theory of unconventional superconductivity. Rev. Mod. Phys. 63, 239–311 (1991).
Schenck, A. in Muon Science: Muons in Physics, Chemistry and Materials.Muon Science: Muons in Physics, Chemistry and Materials (eds Lee, S. L., Cywinski, R. & Kilcoyne, S. H.) 39–84 (Institute of Physics, 1999).
Luke, G. M. et al. Time-reversal symmetry breaking superconductivity in Sr2RuO4. Nature 394, 558–561 (1998).
Hillier, A. D., Quintanilla, J. & Cywinski, R. Evidence for time-reversal symmetry breaking in the noncentrosymmetric superconductor LaNiC2. Phys. Rev. Lett. 102, 117007 (2010).
Aoki, Y. et al. Time-reversal symmetry-breaking superconductivity in heavy-fermion PrOs4Sb12 detected by muon-spin relaxation. Phys. Rev. Lett. 91, 067003 (2003).
Maisuradze, A. et al. Evidence for time-reversal symmetry breaking in superconducting PrPt4Ge12. Phys. Rev. B 82, 024524 (2010).
Ghosh, S. K. et al. Recent progress on superconductors with time-reversal symmetry breaking. J. Phys. Condens. Matter 33, 28 (2021). This recent review on TRS breaking shows the impact of muon experiments in the discovery and understanding of unconventional superconductors.
Adachi, T. et al. Muon spin relaxation and magnetic susceptibility studies of the effects of nonmagnetic impurities on the Cu spin dynamics and superconductivity in La2–xSrxCu1–yZnyO4 around x = 0.115. Phys. Rev. B 69, 184507 (2004).
Zhang, J. et al. Discovery of slow magnetic fluctuations and critical slowing down in the pseudogap phase of YBa2Cu3Oy. Sci. Adv. 4, eaao5235 (2018).
Kaiser, C. T. et al. Li mobility in the battery cathode material Lix[Mn1.96LiO.O4]O4 studied by muon-spin relaxation. Phys. Rev. B 62, R9236–R9239 (2000).
Ariza, M. J., Jones, D. J., Roziere, J., Lord, J. S. & Ravot, D. Muon spin relaxation study of spinel lithium manganese oxides. J. Phys. Chem. B 107, 6003–6011 (2003).
Sugiyama, J. et al. Li diffusion in LixCoO2 probed by muon-spin spectroscopy. Phys. Rev. Lett. 103, 147601 (2009). This significant paper shows lithium diffusion observed by μSR and opens the field of the application of energy materials to μSR.
Medarde, M. et al. 1D to 2D Na+ ion diffusion inherently linked to structural transitions in Na0.7CoO2. Phys. Rev. Lett. 110, 266401 (2013).
Mansson, M. & Sugiyama, J. Muon-spin relaxation study on Li- and Na-diffusion in solids. Phys. Scr. 88, 068509 (2013).
Sugiyama, J. et al. Lithium diffusion in LiMnPO4 detected with μ+-SR. Phys. Rev. Res. 2, 033161 (2020).
Kadono, R. et al. Hydrogen bonding in sodium alanate: a muon spin rotation study. Phys. Rev. Lett. 100, 026401 (2008).
Sugiyama, J. et al. Desorption reaction in MgH2 studied with in situ μ+SR. Sustain. Energ. Fuels 3, 956–964 (2019).
Sugiyama, J. et al. Nuclear magnetic field in solids detected with negative-muon spin rotation and relaxation. Phys. Rev. Lett. 121, 087202 (2018).
Roduner, E. et al. Quantum phenomena and solvent effects on addition of hydrogen isotopes to benzene and to dimethylbutadiene. Ber. Bunsen Ges. Phys. Chem. Chem. Phys. 94, 1224–1230 (1990).
Nicovich, J. M. & Ravishankara, A. R. Reaction of hydrogen-atom with benzene — kinetics and mechanism. J. Phys. Chem. 88, 2534–2541 (1984).
Mezyk, S. P. & Bartels, D. M. Rate-constant and activation-energy measurement for the reaction of atomic-hydrogen with methanol, iodomethane, iodoethane, and 1-iodopropane in aqueous-solution. J. Phys. Chem. 98, 10578–10583 (1994).
Ghandi, K. et al. Muonium kinetics in sub- and supercritical water. Phys. B Condens. Matter 326, 55–60 (2003).
Percival, P. W., Brodovitch, J. C., Ghandi, K., McCollum, B. M. & McKenzie, I. H atom kinetics in superheated water studied by muon spin spectroscopy. Radiat. Phys. Chem. 76, 1231–1235 (2007).
Laidler, K. J. & Keith, J. Chemical Kinetics (Harper & Row, 1987).
Chandrasena, L. et al. Free radical reactivity of a phosphaalkene explored through studies of radical isotopologues. Angew. Chem. Int. Ed. 58, 297–301 (2019).
Veatch, S. L., Leung, S. S. W., Hancock, R. E. W. & Thewalt, J. L. Fluorescent probes alter miscibility phase boundaries in ternary vesicles. J. Phys. Chem. B 111, 502–504 (2007).
Hamley, I. W. Introduction to Soft Matter: Synthetic and Biological Self-Assembling Materials, Revised Edition (Wiley, 2007).
Safinya, C. R., Sirota, E. B., Roux, D. & Smith, G. S. Universality in interacting membranes — the effect of cosurfactants on the interfacial rigidity. Phys. Rev. Lett. 62, 1134–1137 (1989).
Roduner, E. Muons, soap, and drug delivery — an invitation to enter a new field of research. Phys. B Condens. Matter 326, 19–24 (2003). This work demonstrates the potential advances for ALC-μSR.
Scheuermann, R. et al. Partitioning of small amphiphiles at surfactant bilayer/water interfaces: an avoided level crossing muon spin resonance study. Langmuir 20, 2652–2659 (2004).
Kiefl, R. F. et al. 29Si hyperfine structure of anomalous muonium in silicon: proof of the bond-centered model. Phys. Rev. Lett. 60, 224–226 (1988).
Cox, S. F. J. et al. Experimental confirmation of the predicted shallow donor hydrogen state in zinc oxide. Phys. Rev. Lett. 86, 2601–2604 (2001). This work describes the pioneer contribution of μSR to the understanding of the role of hydrogen in semiconductors, namely when it incorporates the lattice structure either in a bond-centre configuration or as a shallow donor.
Hofmann, D. M. et al. Hydrogen: a relevant shallow donor in zinc oxide. Phys. Rev. Lett. 88, 045504 (2002).
Vilão, R. C. et al. Muonium donor in rutile TiO2 and comparison with hydrogen. Phys. Rev. B 92, 081202 (2015).
Brant, A. T., Yang, S., Giles, N. C. & Halliburton, L. E. Hydrogen donors and Ti3+ ions in reduced TiO2 crystals. J. Appl. Phys. 110, 053714 (2011).
Gorelkinskii, Y. V. in Hydrogen in Semiconductors II (Volume 61) Semiconductors and Semimetals Ch. 3 (ed. Nickel, N. H.) 25–77 (Academic Press, 1999).
Cox, S. F. J. Muonium as a model for interstitial hydrogen in the semiconducting and semimetallic elements. Rep. Prog. Phys. 72, 116501 (2009).
Gil, J. M. et al. Novel muonium state in CdS. Phys. Rev. Lett. 83, 5294–5297 (1999).
Yokoyama, K., Lord, J. S., Miao, J., Murahari, P. & Drew, A. J. Photoexcited muon spin spectroscopy: a new method for measuring excess carrier lifetime in bulk silicon. Phys. Rev. Lett. 119, 226601 (2017).
Alberto, H. V. et al. Slow-muon study of quaternary solar-cell materials: single layers and p–n junctions. Phys. Rev. Mater. 2, 025402 (2018).
Woerle, J., Prokscha, T., Hallén, A. & Grossner, U. Interaction of low-energy muons with defect profiles in proton-irradiated Si and 4H-SiC. Phys. Rev. B 100, 115202 (2019).
Jackson, T. J. et al. Depth-resolved profile of the magnetic field beneath the surface of a superconductor with a few nm resolution. Phys. Rev. Lett. 84, 4958–4961 (2000).
Suter, A. et al. Observation of nonexponential magnetic penetration profiles in the Meissner state: a manifestation of nonlocal effects in superconductors. Phys. Rev. B 72, 024506 (2005).
Kozhevnikov, V. et al. Nonlocal effect and dimensions of Cooper pairs measured by low-energy muons and polarized neutrons in type-I superconductors. Phys. Rev. B 87, 104508 (2013).
Stilp, E. et al. Controlling the near-surface superfluid density in underdoped YBa2Cu3O6+x by photo-illumination. Sci. Rep. 4, 6250 (2014).
Kozhevnikov, V., Suter, A., Prokscha, T. & Van Haesendonck, C. Experimental study of the magnetic field distribution and shape of domains near the surface of a type-I superconductor in the iIntermediate state. J. Supercond. Nov. Magn. 33, 3361–3376 (2020).
Morenzoni, E. et al. The Meissner effect in a strongly underdoped cuprate above its critical temperature. Nat. Commun. 2, 272 (2011).
Di Bernardo, A. et al. Intrinsic paramagnetic meissner effect due to s-wave odd-frequency superconductivity. Phys. Rev. X 5, 041021 (2015).
Flokstra, M. G. et al. Remotely induced magnetism in a normal metal using a superconducting spin-valve. Nat. Phys. 12, 57–61 (2016).
Flokstra, M. G. et al. Observation of anomalous meissner screening in Cu/Nb and Cu/Nb/Co thin films. Phys. Rev. Lett. 120, 247001 (2018).
Stewart, R. et al. Controlling the electromagnetic proximity effect by tuning the mixing between superconducting and ferromagnetic order. Phys. Rev. B 100, 020505(R) (2019).
Krieger, J. A. et al. Proximity-induced odd-frequency superconductivity in a topological insulator. Phys. Rev. Lett. 125, 026802 (2020).
Boris, A. V. et al. Dimensionality control of electronic phase transitions in nickel-oxide superlattices. Science 332, 937–940 (2011).
Suter, A. et al. Two-dimensional magnetic and superconducting phases in metal-insulator La2–xSrxCuO4 superlattices measured by muon-spin rotation. Phys. Rev. Lett. 106, 237003 (2011).
Suter, A. et al. Superconductivity drives magnetism in δ-doped La2CuO4. Phys. Rev. B 97, 134522 (2018).
Stilp, E. et al. Magnetic phase diagram of low-doped La2–xSrxCuO4 thin films studied by low-energy muon-spin rotation. Phys. Rev. B 88, 064419 (2013).
Maurel, L. et al. Nature of antiferromagnetic order in epitaxially strained multiferroic SrMnO3 thin films. Phys. Rev. B 92, 024419 (2015).
Langenberg, E. et al. Controlling the electrical and magnetoelectric properties of epitaxially strained Sr1–xBaxMnO3 thin films. Adv. Mater. Interfaces 4, 1601040 (2017).
Maurel, L. et al. Engineering the magnetic order in epitaxially strained Sr1–xBaxMnO3 perovskite thin films. APL. Mater. 7, 041117 (2019).
Dunsiger, S. R. et al. Spatially homogeneous ferromagnetism of (Ga, Mn)As. Nat. Mater. 9, 299–303 (2010).
Monteiro, P. M. S. et al. Spatially homogeneous ferromagnetism below the enhanced curie temperature in EuO1–x thin films. Phys. Rev. Lett. 110, 217208 (2013).
Saadaoui, H. et al. Intrinsic ferromagnetism in the diluted magnetic semiconductor Co:TiO2. Phys. Rev. Lett. 117, 227202 (2016).
Levchenko, K. et al. Evidence for the homogeneous ferromagnetic phase in (Ga, Mn) (Bi, As) epitaxial layers from muon spin relaxation spectroscopy. Sci. Rep. 9, 3394 (2019).
Drew, A. J. et al. Direct measurement of the electronic spin diffusion length in a fully functional organic spin valve by low-energy muon spin rotation. Nat. Mater. 8, 109–114 (2009).
Al Ma’Mari, F. et al. Beating the Stoner criterion using molecular interfaces. Nature 524, 69 (2015).
Moorsom, T. et al. Reversible spin storage in metal oxide–fullerene heterojunctions. Sci. Adv. 6, aax1085 (2020).
Hofmann, A. et al. Depth-dependent spin dynamics in thin films of TbPc2 nanomagnets explored by low-energy implanted muons. ACS Nano 6, 8390–8396 (2012).
Kiefl, E. et al. Robust magnetic properties of a sublimable single-molecule magnet. ACS Nano 10, 5663–5669 (2016).
Prokscha, T. et al. Depth dependence of the ionization energy of shallow hydrogen states in ZnO and CdS. Phys. Rev. B 90, 235303 (2014).
Hampshire, B. V. et al. Using negative muons as a probe for depth profiling silver roman coinage. Heritage 2, 400–407 (2019).
Ponting, K. B. M. The Metallurgy of Roman Silver Coinage: From the Reform of Nero to the Reform of Trajan (Cambridge Univ. Press, 2015).
Moreno-Suarez, A. I. et al. First attempt to obtain the bulk composition of ancient silver–copper coins by using XRF and GRT. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At. 358, 93–97 (2015).
Green, G. Understanding roman gold coinage inside out. J. Archaeol. Sci. 134, 105470 (2021).
Ninomiya, K. et al. Nondestructive elemental depth-profiling analysis by muonic X‑ray measurement. Anal. Chem. 87, 4597–4600 (2015).
Clemenza, M. et al. CHNET-TANDEM experiment: use of negative muons at RIKEN-RAL Port4 for elemental characterization of “Nuragic votive ship” samples. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom.Detect. Assoc. Equip. 936, 27–28 (2019).
Kohler, E., Bergmann, R., Daniel, H., Ehrhart, P. & Hartmann, F. J. Application of muonic X-ray techniques to the elemental analysis of archaeological objects. Nucl. Instrum. Methods Phys. Res. 187, 563–568 (1981).
Reidy, J. J., Hutson, R. L., Daniel, H. & Springer, K. Use of muonic X rays for nondestructive analysis of bulk samples of low Z constituents. Anal. Chem. 50, 40–44 (1978).
Reidy, J. J., Hutson, R. L. & Springer, K. Use of muonic X rays for tissue analysis. IEEE TAmuactionz NbuCteak Sci. NS-22, 1780–1783 (1975).
Hillier, A. D., Hampshire, B. V. & Ishida, K. in Handbook of Cultural Heritage Analysis (eds D’Amico S. & Venuti, V.) (Springer, 2020).
Ziegler, J. F. & Lanford, W. A. Effect of cosmic-rays on computer memories. Science 206, 776–788 (1979).
Sierawski, B. D. et al. Bias dependence of muon-induced single event upsets in 28 nm static random access memories. 2014 IEEE Int. Reliability Physics Symp. https://doi.org/10.1109/IRPS.2014.6860585 (2014).
Sierawski, B. D. et al. Muon-induced single event upsets in deep-submicron technology. IEEE Trans. Nucl. Sci. 57, 3273–3278 (2010).
Mahara, T. et al. Irradiation test of 65-nm bulk SRAMs with DC muon beam at RCNP-MuSIC Facility. IEEE Trans. Nucl. Sci. 67, 1555–1559 (2020).
Tang, J. Y. et al. EMuS muon facility and its application in the study of magnetism. Quant. Beam Sci. 2, 23 (2018).
Won, E. A proposed muon facility in RAON/Korea. JPS Conf. Proc. 2, 010110 (2014).
Williams, T. J. & MacDougall, P. G. J. Future muon source possibilities at the SNS (SNS, 2017).
Jing, H. T., Meng, C., Tang, J. Y., Ye, B. J. & Sun, J. L. Production target and muon collection studies for an experimental muon source at CSNS. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 684, 109–116 (2012).
Pan, Z. W. et al. Conceptual design and update of the 128-channel μSR prototype spectrometer based on musrSim. JJAP Conf. Proc. 7, 011303 (2019).
Kim, Y. J. Current status of experimental facilities at RAON. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. At. 463, 408–414 (2020).
Liu, Y., Rakhman, A., Long, C. D. & Williams, T. J. Laser-assisted high-energy proton pulse extraction for feasibility study of co-located muon source at the SNS. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 962, 163706 (2020).
Hillier, A. D. et al. in Proc. 14th Int. Conf. Muon Spin Rotation, Relaxation and Resonance (μSR2017) Vol. 21 JPS Conference Proceedings (The Physical Society of Japan, 2018).
Aiba, M. et al. Science case for the new high-intensity muon beams HIMB at PSI. Preprint at https://arxiv.org/abs/2111.05788 (2021).
Bonfà, P., Onuorah, I. J. & De Renzi, R. in Proc. 14th Int. Conf. Muon Spin Rotation, Relaxation and Resonance (μSR2017) Vol. 21 JPS Conference Proceedings (The Physical Society of Japan, 2018).
Herak, M. et al. Magnetic order and low-energy excitations in the quasi-one-dimensional antiferromagnet CuSe2O5 with staggered fields. Phys. Rev. B 87, 104413 (2013).
Moller, J. S., Ceresoli, D., Lancaster, T., Marzari, N. & Blundell, S. J. Quantum states of muons in fluorides. Phys. Rev. B 87, 121108 (2013).
Bernardini, F., Bonfa, P., Massidda, S. & De Renzi, R. Ab initio strategy for muon site assignment in wide band gap fluorides. Phys. Rev. B 87, 115148 (2013).
Moller, J. S. et al. Playing quantum hide-and-seek with the muon: localizing muon stopping sites. Phys. Scr. 88, 068510 (2013).
Bonfa, P. & De Renzi, R. Toward the computational prediction of muon sites and interaction parameters. J. Phys. Soc. Jpn. 85, 10 (2016).
Foronda, F. R. et al. Anisotropic local modification of crystal field levels in Pr-based pyrochlores: a muon-induced effect modeled using density functional theory. Phys. Rev. Lett. 114, 017602 (2015).
Bonfa, P., Sartori, F. & De Renzi, R. Efficient and reliable strategy for identifying muon sites based on the double adiabatic approximation. J. Phys. Chem. C 119, 4278–4285 (2015).
Blundell, S. J. et al. μSR study of magnetic order in the organic quasi-one-dimensional ferromagnet F4BImNN. Phys. Rev. B 88, 064423 (2013).
Huddart, B. M. et al. Magnetic order and enhanced exchange in the quasi-one-dimensional molecule-based antiferromagnet Cu(NO3)2(pyz)3. Phys. Chem. Chem. Phys. 21, 1014–1018 (2019).
Franke, K. J. A. et al. Magnetic phases of skyrmion-hosting GaV4S8–ySey (y = 0, 2, 4, 8) probed with muon spectroscopy. Phys. Rev. B 98, 054428 (2018).
Onuorah, I. J., Bonfa, P. & De Renzi, R. Muon contact hyperfine field in metals: a DFT calculation. Phys. Rev. B 97, 174414 (2018).
Liborio, L., Sturniolo, S. & Jochym, D. Computational prediction of muon stopping sites using ab initio random structure searching (AIRSS). J. Chem. Phys. 148, 9 (2018).
Sturniolo, S. & Liborio, L. Computational prediction of muon stopping sites: a novel take on the unperturbed electrostatic potential method. J. Chem. Phys. 153, 10 (2020).
Manson, J. L. et al. Cu(HF2)(pyz)2 BF4 (pyz = pyrazine): long-range magnetic ordering in a pseudo-cubic coordination polymer comprised of bridging HF2- and pyrazine ligands. Chem. Commun. 2006, 4894–4896 (2006).
Pant, A. D. et al. Characterization and optimization of ultra slow muon beam at J-PARC/MUSE: a simulation study. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 929, 129–133 (2019).
Nakamura, J. et al. Ultra slow muon microscope at MUSE/J-PARC. J. Phys. Conf. Ser. 502, 012042 (2014).
Miyake, Y. et al. Ultra slow muon project at J-PARC MUSE. J. Phys. Soc. Jpn. 2, 010101 (2014).
Prokscha, T. et al. The new μE4 beam at PSI: a hybrid-type large acceptance channel for the generation of a high intensity surface-muon beam. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 595, 317–331 (2008).
Morenzoni, E., Prokscha, T., Suter, A., Luetkens, H. & Khasanov, R. Nano-scale thin film investigations with slow polarized muons. J. Phys. Condens. Matter 16, S4583–S4601 (2004).
All authors acknowledge the support and access of the muon facilities around the world. In particular, the staff scientists and technical support teams over the years. F.B. acknowledges the support of the French Agence Nationale de la Recherche, under Grant No. ANR-18-CE30-0022. I.M. acknowledges the support of a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. A.D.H. acknowledges the support of an ERC grant (RACOM). H.A. acknowledges funds from FEDER (Programa Operacional Factores de Competitividade COMPETE) and from FCT- Fundação para a Ciência e Tecnologia, I. P. (Portugal) under projects UIDB/04564/2020, UIDP/04564/2020 and PTDC/FIS-MAC/29696/2017. L.S. acknowledges support from the National Natural Science Foundations of China, No. 12174065, and the Shanghai Municipal Science and Technology No. 20ZR1405300.
The authors declare no competing interests.
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FLAME instrument: https://www.psi.ch/en/smus/flame-project
ISIS muons: https://www.isis.stfc.ac.uk/Pages/muons.aspx
Low-Energy Muons Group: https://www.psi.ch/en/low-energy-muons
PSI muons: https://www.psi.ch/en/lmu
Super-MuSR instrument: https://www.isis.stfc.ac.uk/Pages/Super-MuSR.aspx
- Gyromagnetic ratio
The ratio of a particle’s magnetic moment to its angular momentum; for a muon, the gyromagnetic ratio is 2π × 135.5 MHz T–1.
The degree to which the spins are aligned along a particular direction.
Particles (either positively, neutral or negatively charged) that decay into a muon and muon neutrinos.
- Muon neutrinos
Neutrinos that are produced during the decay process of a muon.
The projection of the spin of a particle onto the direction of momentum. The helicity is right-handed if the direction of its spin is the same as the direction of its motion, and left-handed if it is opposite.
- Nuclear capture
A nucleus of an atom can capture a muon (µ–) after cascading down the muonic atom energy levels.
A particle accelerator that operates by accelerating charged particles with static magnetic fields and a high-frequency radio-frequency electric field, generally producing particles quasi-continuously, resulting in a continuous source of muons.
A particle accelerator that operates by accelerating charged in a closed loop where magnetic fields are used in a specific synchronized timing.
- Time resolution
The minimum time interval that can be measured with a specific technique.
An atom formed of a positive muon and an electron, analogous to hydrogen, with a very similar electronic structure but only one-ninth of its mass.
A process where a system reaches thermal equilibrium.
The response of a particle with spin in a magnetic field that is off-axis to the initial polarization. The relative orientation of the particles spin changes in response to the magnetic field, where the precession can be visualized by picturing a spinning gyroscope.
- Time-integral measurements
The integrated polarization of the muon is measured as a function of some external parameter, such as the applied magnetic field or the radio-frequency, which are scanned in a series of steps.
- Time-differential measurements
Measurements probing how the muon polarization evolves with time.
- Quadrupole magnets
Magnets with four pole pieces, generally used in doublets or triplets for focusing of a charged particle beam.
- Momentum slits
Slits that allow particles through with a particular momentum, often used in conjunction with a separator to ensure that only muons enter the instruments.
Two electrically charged plates and a dipole magnet with tuned electric and magnetic fields to act as a Wein filter. Often used in conjunction with momentum slits to ensure that only muons enter the instruments.
When a muon or pulse of muons is sent down different beamlines by using either a magnetic or electrostatic force kicking the particles by a determined angle.
- Lorentz force
A force on a point charge from a combination of both electric and magnetic fields.
- Larmor radius
The radius of the track of a charged particle in a magnetic field.
- Muonic atom
A negative muon captured and bound to an atom.
- Auger electrons
Auger electrons result from the response of an atomic shell with a core vacancy induced by an external (electron or photon) excitation. An Auger electron released from the outer shell of the excited atom while another outer shell electron relaxes to the core vacancy.
- Kα energies
The energy of an X-ray emitted when undergoing a transition between the 2p and 1s atomic energy levels
- Hyperfine structure
A small shift and/or splitting of the energy levels caused by interactions between the nucleus and the electron clouds, known as hyperfine interactions.
- Longitudinal relaxation
The depolarization of the muon spin along the longitudinal or z direction, parallel to the initial muon spin.
- Muoniated radicals
Radicals are molecules with an unpaired electron. Muoniated radicals contain a covalently bound Mu and are formed by Mu addition to an unsaturated bond.
- Muon hyperfine coupling constant
The strength of interaction between the magnetic moments of the muon and an unpaired electron.
- Nuclear hyperfine coupling constants
The strengths of interaction between the magnetic moments of a nucleus and an unpaired electron. They only occur for nuclei with non-zero spin.
- Cascade transition rates
The times taken for a negative muon to cascade between specific energy levels.
- Muon cascade probability
The probability of a particular transition between specific energy levels occurring.
- Cooper pairs
Pairs of electrons that are bound together by a weak force, and required for superconductivity.
- Abstraction reaction
This reaction involves the abstraction of an atom from a molecule. The abstracting species is usually a radical species itself. Mu may abstract a hydrogen atom from a C–H bond in saturated organic molecules, leading to the formation of a diamagnetic MuH molecule and a non-muoniated radical.
- Zero-point vibrational energy
The lowest energy of a quantum mechanical system, which is non-zero because of quantum mechanical fluctuations.
- High-dilution limit
The limit approached when the muon is in extreme dilution where there are no muon–muon interactions due to the scarcity of the muon.
- Soft error
An error in electronics where a bit has been changed by an external source, generally through some interaction with particles and/or radiation.
A large cell, generally associated with computation calculations, that is made up of many unit cells of the crystal structure.
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Hillier, A.D., Blundell, S.J., McKenzie, I. et al. Muon spin spectroscopy. Nat Rev Methods Primers 2, 4 (2022). https://doi.org/10.1038/s43586-021-00089-0
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