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Muon spin spectroscopy

Abstract

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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Fig. 1: Muon beamlines.
Fig. 2: Elemental emission of muonic X-rays.
Fig. 3: Muon spin spectroscopy (µSR) of CdS.
Fig. 4: Avoided level crossing muon resonance.
Fig. 5: Applications in copper-based systems.
Fig. 6: Li+ diffusion in battery materials.
Fig. 7: Muoniated radicals.
Fig. 8: Time windows and degrader curves.
Fig. 9: The calculated muon site in Pr2Sn2O7.

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Acknowledgements

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.

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Authors and Affiliations

Authors

Contributions

Introduction (A.D.H.); Experimentation (A.B., K.S., I.M., T.P., I.W. and A.D.H.); Results (K.S., I.M. and A.D.H.); Applications (S.J.B., F.B., L.S., I.U., J.A.W., I.M., H.A., T.P., I.W. and A.D.H.); Reproducibility and data deposition (A.D.H.); Limitations and optimizations (A.D.H. and A.B.); Outlook (A.D.H., S.J.B. and T.P.); Overview of the Primer (A.D.H.). All authors reviewed the content of the primer.

Corresponding author

Correspondence to Adrian D. Hillier.

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The authors declare no competing interests.

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Nature Reviews Methods Primers thanks Thomas Ashton, Pierre Dalmas de Reotier, Jingyu Tang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Related links

FLAME instrument: https://www.psi.ch/en/smus/flame-project

ISIS muons: https://www.isis.stfc.ac.uk/Pages/muons.aspx

J-PARC muons: https://j-parc.jp/researcher/MatLife/en/instrumentation/ms.html

Low-Energy Muons Group: https://www.psi.ch/en/low-energy-muons

Mantid: https://www.mantidproject.org/Main_Page

Muonsources: https://muonsources.org/

MuSIC: http://www.rcnp.osaka-u.ac.jp/RCNPhome/music/

NIST: http://physics.nist.gov/cuu/Constants/index.html

PSI: https://www.psi.ch/en/smus/instruments

PSI muons: https://www.psi.ch/en/lmu

Super-MuSR instrument: https://www.isis.stfc.ac.uk/Pages/Super-MuSR.aspx

TRIUMF: http://cmms.triumf.ca

Supplementary information

Glossary

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.

Spin-polarized

The degree to which the spins are aligned along a particular direction.

Pions

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.

Helicity

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.

Cyclotron

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.

Synchrotron

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.

Muonium

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.

Thermalize

A process where a system reaches thermal equilibrium.

Precess

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.

Separator

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.

Kicked

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.

Supercell

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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