Hyperfine Interact (2015) 230:35–55DOI 10.1007/s10751-014-1074-z

μ SR HOWTOMuon spin rotation/relaxation/resonance at TRIUMF

Jess H. Brewer

Published online: 30 October 2014© Springer International Publishing Switzerland 2014

Abstract I will first review the earlyhistory of muon spin rotation/relaxation/resonance(μSR), from its origins in elementary particle physicsthrough its applications in fundamen-tal physics, chemistry and materials science. Then I will review the more recentdevelopmentof μSR facilities, modern muon beams and advanced techniques that have made μSR anindispensibletool for molecular and materials science, focusing on (but not restricted to)those pioneered at TRIUMF.

Keywords Muon · Spin · Rotation · Relaxation · Resonance · TRIUMF

1 Early history of μSR

The history of μSR can be seen as a progression from fantasy (scenarios contradicting the“known laws of physics”) to science fiction [SF] (possible in principle, but impractical orheroic with existing technology) to routine science. In the days before particle accelerators,the muon (first misidentified as Yukawa’s meson [1]) was available only as a mysterioushigh energy cosmic ray that penetrates our protective atmospheric shield and most solidobjects in its path. Even then it provided an opportunity for Rasetti [2] to answer an out-standing question in physics (“Is the average effective magnetic field in iron H or B?”)by measuring the energy and deflection of cosmic ray muons traversing a magnetized ironplate.

Proceedings of the 5th Joint International Conference on Hyperfine Interactions and InternationalSymposium on Nuclear Quadrupole Interactions (HFI/NQI 2014) Canberra, Australia, 21–26September 2014

J. H. Brewer (�)Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canadae-mail: jess@phas.ubc.ca

36 J. H. Brewer

1.1 Fantasy

Through the first half of the 20th Century, one of the most sacred symmetry principlesin physics was parity (P): any reaction of elementary particles is exactly as likely as itsmirror image with r → −r. This would have rendered μSR (which relies completely onthe maximally P-violating decays of pions and muons), not even good SF. But as newaccelerators produced more exotic elementary particles, including strange ones like kaons,the impossible happened: the K+ meson was found to decay weakly into final states ofdifferent parity with different probabilities [3–5]. This “tau-theta puzzle” caused Lee andYang [6] to speculate in 1956 that perhaps P might not be a good symmetry in the contextof weak nuclear interactions.

This outrageous proposal was quickly tested by Wu et al. [7] in the beta decay of 60Coand by two groups [8, 9] in the π+ → μ+νμ and μ+ → e+νeνμ decays of pions and muons.All agreed: P symmetry was maximally violated in weak interactions — a property that canbe adequately understood by this glib description: (see Figs. 1 and 2) any ultrarelativisticDirac particle produced in a weak interaction will always have its momentum and angularmomentum antiparallel (negative or “left-handed” helicity); its antiparticle will always beright-handed, thus conserving the combined symmetry CP where C is charge conjugation.(Soon after, CP was also found to be violated in kaon decay; but that’s another story!)

In one of those three papers published in the same issue of THE PHYSICAL REVIEW,a method very similar to modern μSR was used; the authors of that paper [9] speculatedimmediately that, “It seems possible that polarized positive and negative muons will becomea powerful tool for exploring magnetic fields in nuclei. . . , atoms, and interatomic regions.”And so they have, but it took some time and effort.

1.2 Science fiction stories

Within a few years, the basic techniques of μSR — longitudinal field (LF) [10, 11] and weaktransverse field (wTF — see Fig. 3) [12, 13] — had been developed and applied to the studyof muonium [14] (the μ+e− or Mu atom, a light isotope of the H atom) in semiconductors[15] and (in Russia) Mu chemistry [16–19]; however, the low intensity of early muon beams(thousands of times less than today’s) made such measurements so tedious that their broadapplication was still the stuff of SF.

Fortunately, there was another fundamental physics application that spurred the develop-ment of μSR through the following decade and a half: the test of quantum electrodynamics(QED) provided by measurements of the muon’s magnetic moment [13, 21–28], its anoma-lous magnetic moment [29–32] and the Mu hyperfine splitting [24, 26, 33–36] to heroicaccuracy. These made for good SF, rewarding the community with some highly refined μSRtechniques. One iteration of the muon “g − 2” experiment at CERN also led to the stor-age ring technology that was later used to collect and cool antiprotons, a project that hasrecently led to the first observations of trapped atomic antihydrogen [37]. Improved accu-racy of muon “g − 2” measurements is still being sought today for tests of the “StandardModel” of elementary particles.

One of the problems with the QED tests was the depolarizing effect of certain media.Less muon polarization P0 means still longer measurements: the statistical precision is pro-portional to P0

√Nμ, where Nμ is the total number of muon decay events. In an effort to

understand and solve this problem [12], the theory of muon depolarization proposed byIvanter & Smilga [19, 38, 39] was extended [40–42] to include the rapid chemical reactionof paramagnetic Mu atoms to form diamagnetic compounds and/or radicals (paramagnetic

μ SR HOWTO 37

Fig. 1 The decay π+ → μ+ + νμ in the rest frame of the pion produces muons 100 % polarized opposite totheir momentum (“negative helicity” or “left-handed”) because conservation of linear and angular momentumwould require the other helicity to produce a “right handed” muon neutrino and the weak interaction nevermakes such things (The reaction in the mirror is forbidden)

molecules); this provided an opportunity to study Mu chemistry in liquids, still a thrivingfield today.

Another ambitious particle physics program was the search for spontaneous conversionof muonium into antimuonium [43] (allowed by some versions of lepton number conserva-tion). Since this process could only happen in vacuum, the problem became one of obtainingthermal Mu atoms in vacuum. The University of Arizona group under Bowen built the first“surface muon beam” at Berkeley (see Fig. 4) for this purpose [44]. As soon as it wassuccessfully commissioned, an opportunity was seen to use its low energy μ+ beam tostudy Mu chemistry in the gas phase (previously achieved only with backward muons athigh pressure, with great difficulty [45]) at near atmoshperic pressure; the first such exper-iments using surface muons [46] were carried out at Lawrence Berkeley Laboratory’s 184”Cyclotron.

During the SF era, many other heroic efforts at the old cyclotrons yielded μSR appli-cations in chemistry [40–42], semiconductor physics [47–50], quantum diffusion [51, 52],magnetic materials [53–57] and superconductors [58, 59]. The stage was set for modernμSR, but it was still impractical as a universal tool analogous to NMR, because typicalmeasurements took many hours for one spectrum and samples had to be thousands of timeslarger than those routinely measured today.

1.3 Routine science

The key to making μSR a practical tool was beam intensity, and this key was turned in themid-1970s when the first “meson factories” were commissioned at Los Alamos (LAMPF,now defunct) in 1972, Villigen (then SIN, now PSI) in 1974 and Vancouver (TRIUMF) in1974, soon followed by KEK/BOOM at Tsukuba in 1980 (moved since 2008 to J-PARCin Tokai) and RAL/ISIS near Oxford in 1987. Whereas μ+ stopping rates in large (up to1000 ml) samples were typically ∼ 103 μ+/s at the previous generation of accelerators, thedramatically higher primary beam intensity of meson factories like TRIUMF (≥ 100 μA at450 − 520 MeV) routinely produced surface muon beams [60] of ∼ 106 μ+/s that wouldstop in ≤ 1.4 mm of liquid or solid samples.

1.4 μ−SR

Because of its consistent simplicity, μ+SR (using positive muons) constitutes the vastmajority of μSR today; but it would be a mistake to ignore μ−SR (using negative muons), as

38 J. H. Brewer

Fig. 2 The decayμ+ → e+ + νe + νμ in the restframe of the muon tends to emitthe highest energy positrons (upto 52 MeV) preferentially alongthe muon spin direction becausethe spins of the neutrino-antineutrino pair emitted in theopposite direction cancel, leavingonly the positron to balance themuon’s angular momentum, andthe ultrarelativistic positron (likean antineutrino) can only beproduced with positive helicity.The “asymmetry” of the decaydistribution averages to 1/3 overall e+ energies

the latter gives access to completely different properties of matter. The primary differencesare:

1. Negative pions (π−) undergo strong nuclear capture before they can decay, so there areno “surface μ−” beams. All μ− beams are derived from π− decay in flight [see Fig. 4(LEFT)] and therefore have lower polarization (typically ∼ 80 %), lower luminosity(due to the extended “source”), higher momentum (typically ∼ 75 MeV/c) and greaterrange, requiring large, thick samples.

2. Negative muons (μ−) undergo weak nuclear capture after doing what any electron witha mass of ∼ 207me would be expected to do: cascading through many energy levels ofthe resultant muonic atom and finally taking up residence in a 1s state that is, in heavyelements, mostly inside the nucleus. This makes the muon lifetime a strong function ofZ, levelling off around 90 ns in the heaviest elements.

3. In the aforementioned atomic cascade, the μ− not only emits many muonic X-rays, italso can make spin-flip transitions that cause depolarization. In the best cases (such asmuonic carbon) only about 1/4 of the original μ− polarization remains; in some casesthere is no discernable polarization left.

4. If the nucleus has a spin of its own, the hyperfine coupling between the muon spin andthe nuclear spin is enormous — sometimes ∼ 1 MeV, and often enough to cause Augeremission of electrons from the host atom; once the ground state is reached, the muonspin becomes “locked” to that of the nucleus and the two precess together in either anF+ state or (for I > 1/2) an F− state, each of which has a characteristic frequencymuch smaller than that of the bare muon [61].

Thus μ−SR suffers many disadvantages and is far less popular than μ+SR; but the sameproperties give μ−SR some interesting capabilities of its own, mostly (but not exclusively)in particle and nuclear physics:

– Muonic X-rays reveal many subtle properties of nuclei (as well as a precision value forthe μ− mass) and can be used for nondestructive elemental analysis of materials [62].

– High energy muonic X-rays can be absorbed by the nucleus, leading to excited statesand fission with the muon still attached.

– Neutrons and fission fragments from nuclear muon capture provide information aboutthe nucleus itself as well as the weak interaction.

– Because the 1s state is so tiny, the muonic atom of atomic number Z acts chemicallylike a virtual transmutation of the host nuclei (Zeff = Z − 1) and so can be compared

μ SR HOWTO 39

Fig. 3 Simplest μ+SR configuration: wTF-μ+SR; the weak applied “transverse” field B is perpendicularto the beam direction and the initial muon polarization, so that the muon spins precess about B. A digital“clock” is started when the μ+ enters the sample and stopped by a trigger in the positron detector. Thetime interval is digitized and a count is added to the corresponding bin of the raw time histogram N(t) =N0 exp(−t/τμ)[1+A(t)] where τμ = 2.197 μs is the μ+ lifetime and A(t) is the asymmetry time spectrumanalogous to the free induction decay (FID) in NMR [20]

directly with (e.g.) the NMR of a Z − 1 impurity in a sample of Z. This has been usedto good effect in studies of paramagnetic impurity states by μ−SR in Si [63–65] andthe “hyperfine anomaly” [66] in Pd [67], Fe [68] and Ni [69].

– In helium it was possible to detect the muonium analogue α++μ−e− and study itschemical reactions [70]. By comparing its reaction rates with those of H and Mu it waspossible to study “isotopic effects” over an unprecedented factor of 36 in mass, leadingto important developments in reaction rate theory. An analogous paramagnetic species(Ne2+μ−e−) was detected in neon [71], but its chemistry has not yet been measured.

– Relativistic shifts of the bound μ− g-factor (predicted by the Breit equation for pointnuclei) have been measured in many elements [65, 72–74]. By the middle of the peri-odic table the shifts deviate from the simple theory due to the finite size of the nuclei:more and more of the muon wave function is inside the nucleus; by Pb and W, the rel-ativistic shift is essentially independent of Z. This may offer a new means of probingthe nuclear charge distribution [74].

– μ−SR is not restricted to muonic atoms with spinless nuclei; although the spectroscopyof F+ and/or F− states has not been widely utilized, many nuclei with spin yieldcharacteristic signals [74–79]. By utilizing the enormous hyperfine coupling in muonicbismuth [80], it was even possible to repolarize the muon through nuclear orientationof 209Bi.

Clearly the negative muon is a far more versatile probe than the more familiar μ+, usuallyinvolving emission of more than just decay electrons — e.g. X-rays, neutrons or fission

40 J. H. Brewer

Fig. 4 (TOP) The “Arizona beam” at LBL in 1973 [44] was the first intentional exploitation of what arenow known as “surface muons” — those which arise from positive pions decaying at rest in the “skin” ofthe production target. (RIGHT) Surface muons have the advantages of 100 % spin polarization, low range(∼ 140 mg/cm2) and good beam optics — the source is “imageable”, unlike for muons from pion decay inflight (LEFT)

fragments — but the same complexity makes μ−SR more difficult (and more difficult tointerpret) than μ+SR, which is why the generic term “μSR” is frequently used to refer onlyto the latter. I will say no more about μ−SR here, as this article is already abusing the “nopage limit” rule.

2 Muon beams

2.1 Stopping luminosity

The world’s inventory of stopped muons at any moment is at least 5 × 1012 muons persecond; unfortunately most are high energy cosmic rays that (a) are spread out over theEarth’s surface and (b) stop only reluctantly. This actually makes them quite useful for suchpurposes as “X-raying” pyramids [82] and volcanoes [83] or nondestructive testing of theiron in bridges; but to be useful for μSR, muon beams must be (1) high luminosity (flux perunit area) and (2) high stopping density (inverse range). The product of these two factorsis the stopping luminosity Lstop (μ+/g-s) which, along with the muon polarization and the

μ SR HOWTO 41

Fig. 5 The “stoppingluminosity” of muon beams hasincreased by a factor of ∼ 107

over the past half century and ispoised to rise even higher withthe advent of intenselaser-ionized muonium sources[81]

purity of the beam (ideally only muons), form the “figure of merit” for muon beams. Thefirst “backward muon” beams (from decay of pions in flight) required ∼ 100 g of sample tostop; today’s ultra-low-energy muons (LEM) from moderated muons are roughly the sameintensity but have a factor of ∼107 higher stopping luminosity (see Fig. 5).

2.2 To pulse or not to pulse?

2.2.1 CW facilities: Less of more

The high flux of the “meson factories” allowed much more rapid data acquisition, but atCW accelerators (SIN and TRIUMF) there is a limit on rates imposed by the fact that onlyone muon can be in the sample at a time — otherwise one cannot know which one decayed.Since the high time resolution of CW facilities depends upon starting a digital “clock”when a muon arrives and stopping it when that muon’s decay positron is detected, thislimit is unavoidable. For ordinary CW-μ+SR experiments these considerations limit usefulbeam rates to about 105 μ+/s or less, so that all but the “best” 10 % of the muon beam(those most highly focused on a bright central spot) must be discarded, usually using beamcollimators. As a result, experiments can now be done with surface muon beams on singlecrystal samples ∼ 1 mm across and ∼100 μm thick.

Later the 600 MeV ring cyclotron of PSI was upgraded to over 1 mA current, raising themaximum surface muon intensity to ∼ 1108 μ+/s. This allowed an even smaller fractionof the muons to be “used”, such as the one in ∼ 105 that trickles out of a solid rare gasmoderator to produce a source of almost thermal “low energy muons” (LEM) [84, 85]. Thismethod was first developed at TRIUMF [86, 87] but required the elevated intensity of PSIto be practical; now it is used there to investigate thin films and interfaces with μ+SR.

2.2.2 Pulsed facilities: More of less

Meanwhile the pulsed muon facilities at BOOM, ISIS and now J-PARC enjoyed a hugeadvantage in rate at the expense of highly segmented detectors and poor time resolution:When a pulse of high energy protons strikes a production target, many pions are produced

42 J. H. Brewer

Fig. 6 The “advantage factor” Apulsed of a pulsed muon facility over a CW facility like TRIUMF or PSIincreases with the number Nin of simultaneous “inputs” (muons and any forms of irradiation thereof) anddecreases with the number Nout of correlated “outputs” — positrons (or electrons in the case of μ−SR) andany other products of muon decay or capture. The improved time resolution of the CW facility is not includedin this reckoning

“at once” (over the duration of the pulse). These decay over the lifetime (26 ns) of the π+into muons that are collected into a beam line and delivered “all at once” to the sampleunder study. Thus there is in principle no rate limit, because there is no confusion aboutwhen the muons arrive. In practice, there is a little confusion: even if the primary beam werea perfect delta function, the muons would be spread out exponentially over 26 ns; and theprimary beam is not a delta function — typical pulse widths are 100-200 ns, so that the timeof arrival is uncertain by that amount, and the time resolution of time-differential (TD)-μSRsuffers accordingly. Moreover, with millions of muons decaying simultaneously, detectordead time would result in huge distortions of the time spectrum if the detectors were notbroken up into many segments, each of which intercepts only a small solid angle from thesample. Thus pulsed facilities are best for high statistics measurements at low transversefield or arbitrary longitudinal field. They also have two other advantages: first, because allthe muons arrive at once and then there is a long period of no beam at all, pulsed μSR hasvery low background at long times, allowing study of the muon’s polarization over times aslong as 10 muon lifetimes. Second, for the same reason, any irradiation of the sample withRF, microwave or lasers (for the purpose of true resonance studies) is vastly more efficientthan at CW facilities [88]. Not all samples will admit electromagnetic radiation, of course,but for a wide variety of materials one can overcome the time resolution limitation of pulsedfacilities by using a �/2 pulse of resonant RF to tip all the muon spins into the transverseplane coherently (see Fig. 17) and then following their free induction decay. I believe this isplanned at RAL/ISIS.

3 μSR methods

In the first decade of the Meson Factory Era, many new μ+SR techniques were developed(see Fig. 7) to augment the utility of muons in studies of molecular and materials science.Noteworthy examples were:

– Zero Field (ZF)-μSR — see Fig. 8. One of the first major accomplishments of μSRwas to realize experimentally [89] the theoretical predictions of Kubo and Toyabe

μ SR HOWTO 43

Fig. 7 Standard μ+SR acronyms: TF-μSR, LF-μSR and ZF-μSR are described in Fig. 8. FT-μSR (FourierTransform μSR) is used to determine the frequency spectrum from a time spectrum by a Fast Fourier Trans-form numerical algorithm. μALCR (Avoided Level Crossing Resonance), usually a time-integral technique,reveals longitudinal fields for which muon spin-flip transitions are resonant with some other (usually nuclearspin) transitions. RF-μSR is true magnetic resonance in which RF power drives transitions; the case shownis the Fourier transform of Mu atoms driven at the average of ν12 and ν23 frequencies, showing a sharptwo-photon peak. μSE (muon spin echo) is a hitherto underutilized method of using RF to explore the spec-troscopy of both muons and nearby particles with which they interact. In the latter diagram, ARRF refers tothe precession asymmetry in the Rotating Reference Frame (RRF), described in Fig. 11

[90] regarding spin relaxation in zero applied magnetic field. This has since becomea staple of μSR.

– Fast Fourier Transforms (FFT-μSR). This essential tool was of course developed assoon as μSR experiments revealed more than one simultaneous oscillation frequency.It has now been refined to include many ways of “massaging” the raw data to producemore visually appealing and informative figures, but because the FFT treats early times(with high statistics) the same as late times (with low statistics and thus more “noise”),Fourier transforms will always be a primarily qualitative tool. To extract all the infor-mation contained in the data, χ2-minimization fitting of time spectra is necessary, as inFig. 9(TOP).

– The “Spin Rotator” (SR) [60, 92]. An E × B velocity selector (see Fig. 10) not onlyremoves all positrons and other background from the μ+ beam but can also rotate themuon polarization into the plane perpendicular to the muon momentum; without thisfeature, it would be impossible to inject surface muons (whose maximum radius of

44 J. H. Brewer

Fig. 8 Standard μ+SR spectrometer configurations: TOP: conventional definitions of x, y and z directionsin terms of the muon beam and polarization directions. LEFT: LFz-μ+SR, with the applied “longitudinal”field along the beam direction (z); and wLFx -μ+SR, with the applied field along the (spin-rotated) initialmuon polarization Pμ ‖ z (the field must be weak or the muon momentum will be deflected). Both config-urations are also used for zero field (ZF)-μ+SR. RIGHT: xHT Fz-μ+SR, in which the applied field is stillalong the beam direction (so that the muons are not deflected) but the initial muon polarization has beenrotated 90◦ into the x direction by an E × B spin rotator (which also removes positrons from the beam); andzwT Fx -μ+SR, shown earlier in Fig. 3

curvature is 1 cm in a 10 T transverse field) into samples at high magnetic fields forTF-μ+SR experiments.

– High Transverse Field (HTF-μ+SR). The spin rotator allows TF experiments at fieldsup to 10 T; however, since one precession period of the free muon is only 1 ns at7.37796 T, at which field the radius of a positron orbit is ∼ 1 cm, such experimentsrequire extremely high time resolution as well as very small detectors in a miniaturizedspectrometer. They also necessitated the development of the Rotating Reference Frame(see Fig. 11) to reduce the large number of time bins and make visualization of multi-frequency signals feasible.

– Avoided Level-Crossing Resonance (ALCR) — in which the Zeeman transitions of theμ+ match energies with transitions of other spins coupled to the muon, either nuclear

μ SR HOWTO 45

Fig. 9 TOP: ZF-μ+SR time spectrum from a BaFe2As2 sample at 2 K, showing a time-domain fit to severalfrequencies. This is actually a rather simple case, but it serves to illustrate the idiosyncracies of the Fouriertransform. BOTTOM: plots of the Fourier Power P = R2 + I2, the Fourier Amplitude A = √

P , and theReal (R) & Imaginary (I) parts of the complex Fourier amplitude. Note the spurious broadening added toP and A by I, which includes the transform of a step function (the data starts abruptly at t = 0). For bestresolving power one must always plot R vs. frequency [91]

Fig. 10 Cartoon sketch of the surface muon Spin Rotator (SR) (also known as a DC Separator or a WienFilter) using crossed E and B fields perpendicular to the muon momentum to remove positrons from the beamand rotate the muon spin up to 90◦ relative to its original direction. Such SRs are essential for HTF-μ+SR

electric quadrupole transitions [93] (Q-ALCR) or nuclear hyperfine transitions [94](P-ALCR) coupled through the unpaired electron in radicals [95–98] (see K. Ghandi’sinvited talk at this conference) or muonium states in solids [94, 99]. See Figs. 12,13 and 14. The related technique of zero-crossing resonance (ZCR) can also be usedin magnetically ordered materials [100]. Because the ALCR technique can use time-integrated forward/backward decay rates, it works effectively at both pulsed and CWfacilities, regardless of macro time structure, and has no intrinsic rate limitations; thismade LAMPF the world’s most powerful facility for ALCR [101] for a brief time before

46 J. H. Brewer

Fig. 11 In high transverse magnetic field (HTF-μ+SR) the muon’s precession period can be shorter than1 ns, necessitating even smaller time bins (each accumulating only a few “counts”) in the histograms, whichtherefore have many bins and are expensive to fit; the resulting statistical precision in each bin is low, causinglarge “error bars” on each point. The Rotating Reference Frame (RRF) transformation (applied to the data“after the fact”) consists of simply multiplying the complex high frequency asymmetry time spectrum A(t) byexp(−i�RRFt), producing a spectrum with lower frequency oscillations, which allows coarser time binningand correspondingly smaller statistical error bars

Fig. 12 Quadrupolar μALCR (Q-ALCR) in Cu at 20 K [93], in which the Zeeman splitting of the μ+matches the electric quadrupolar splitting of the Cu nucleus, allowing “flip-flop” transitions, driven by thedipole-dipole interaction, that resonantly depolarize the muon

it was shut down. a fact that was demonstrated just before the μSR program (and even-tually the entire experimental program) was shut down in 1993, ending the domesticμSR effort in the USA, where it all began.

– RF Resonance μ+SR. In a strong longitudinal “holding field”, the μ+ retains its polar-ization despite the formation of muonium. However, RF power applied at the freemuon’s Larmor frequency only affects diamagnetic states. This allows study of veryslow formation or ionization of various states of Mu in different media, such as silicon[104], in which transitions between Mu0 ≡ Mu, Mu− and Mu+ ≡ μ+ take place at orbetween the tetrahedral site (where Mu is relatively vacuum-like) and the bond-centredsite (where its electron is shared covalently with a Si atom). The dynamics of thesetransitions are quite complex.

– Muonium RF Resonance. Developed first at microwave frequencies to measure the Muhyperfine interval (4.463 GHz) precisely [105], this technique was later used at RFfrequencies to excite transitions between triplet states in low magnetic field, includingmulti-photon resonances [106], but has since been largely unexploited.

μ SR HOWTO 47

Fig. 13 LEFT: Quadrupolar μALCR (Q-ALCR) in MnSi at various temperatures, showing the 5/2 − 3/2and 3/2 − 1/2 transitions. RIGHT: Temperature dependence of the resonance positions, showing criticaldivergence of the electric field gradient at Tc [102]

Fig. 14 LEFT: Paramagnetic μALCR (P-ALCR) occurs when “flipping” a third spin X that couples to themuon and/or electron causes the same change in energy as “flopping” the muon spin at a certain applied fieldBres. A resonant loss of muon polarization results at Bres. This method is widely used to identify the partnerX, its interaction strength J and, from that, the site and environment of the muon. RIGHT: P-ALCR in GaPand GaAs at 10 K [103]. Due to stochastic fluctuations in the muon beam, these early TRIUMF data weretaken in “differential” mode, where a small “toggle field” is added and subtracted from the applied field B0and the resulting difference in the positron asymmetry is plotted vs. B0

– Muon Spin Echo (μSE): When a TF-μ+SR signal “relaxes” by dephasing due to aspread of local magnetic fields, it can be “refocused” by a � pulse of resonant RFpower, just as in NMR. (See Fig. 17.) Moreover, when the dephasing is caused bydipole-dipole interactions with other particles (e.g. fluorine nuclei), the spins of thoseparticles can be refocused by a �F pulse at their resonant RF frequency. (See Fig. 18.)This technique was first demonstrated at TRIUMF [88] using asynchronously triggered

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Fig. 15 RF-resonance ofdiamagnetic states Mu− andMu+ ≡ μ+ in Si at hightemperature in LF [104],showing the various transitionsbetween sites and ionizationstates of muonium

Fig. 16 RF-resonance of muonium in Si at 10 K [106], in which the magnetic field is swept through theresonances where transition frequencies ν12 and ν23 match the RF frequency of 127 MHz. In between is thetwo-photon resonance where the frequency of the Sμ = 1 transition ν13 = ν12 +ν23 is 254 MHz. The latteris much narrower, due to cancelling of broadening effects on ν12 and ν23

RF pulses on individual muons as they entered the target. Obviously RF resonance isfar more efficient at pulsed facilities, where all the muons arrive at once and the pulsesare at times known in advance. It should also be possible to use resonant �/2 pulsesto “flip up” the spins of all the muons in a pulse simultaneously after they arrive withlongitudinal polarization (spread out over a pulse width of ∼ 100 ns) after which onecould observe high-frequency HTF-μ+SR precession.

– Muon Spin Imaging (μSI). The analogy with NMR naturally suggests the use of mag-netic field gradients and Fourier transforms to generate spatial maps of objects usingmuons, much as in MRI. This technique was proven in principle by an experiment atTRIUMF in 1994 [107] but μSI has several disadvantages relative to MRI: first, theshort muon lifetime dramatically reduces the frequency resolution of μSR, which trans-lates directly into reduced spatial resolution; second, the muon beam is not disperseduniformly over the sample, resulting in an “image” peaked in the centre like the beamprofile. However, an image of a “μ” cut out of aluminum on a substrate of depolarizingmedium was produced by this method (see Fig. 19).

– Electric Field (E-μSR). In liquids and “cryocrystals” (condensed gases at low tem-perature), as well as more conventional insulators and semiconductors, the μ+ createsmany free electrons as it slows down from several MeV to thermal energies; when thoseelectrons are sufficiently mobile (which turns out to be most of the time), they canbe attracted to the muon by its electric charge and form various μ+e− bound states,usually starting with weakly bound muonium (Muwb, in which the hyperfine couplingbetween muon and electron is very weak) and frequently ending with nearly vacuum-like Mu. This thermal process is referred to as “delayed muonium formation” (DMF)even though it often happens very fast (before the resultant Mu atoms can precess

μ SR HOWTO 49

Fig. 17 Direct Muon Spin Echoes (d-μSE) in LaF3, displayed in the RRF at 29.5 MHz (the muon Larmorfrequency in the applied field). LEFT: The muon spins are initially along the applied field; they are then“flipped up” by a (�/2)y RF pulse, at which point they begin to precess, relaxing by “dephasing” due tothe inhomogeneous field caused by 19F nuclei. After a time τ a longer RF �y pulse is applied; the muonssubsequently “refocus” and then relax again. RIGHT: This time the second �x pulse is applied 90◦ out ofphase with the first (�/2)y pulse, causing a positive echo in the x component. For a more detailed description,see Ref. [88]

appreciably in an applied transverse field) because there is usually also an epithermal“prompt” Mu fraction that thermalizes in the bound state.

Perhaps surprisingly, the μ+ always stops “downstream” (in the original direction ofthe muon’s momentum) of the electrons it liberated in its track. This is known becauseapplying an external electric field E in the direction of the muon beam drives the muonand electron apart, decreasing the resultant Mu fraction and correspondingly increasingthe diamagnetic fraction [108, 109].

Because the liberated electrons may collect at the anode, cancelling the applied fieldin the sample, it is necessary to periodically reverse E; the frequency of this reversalmust be on the order of kHz in materials such as GaAs, where a very large number ofparticularly mobile electrons is liberated by each incoming muon [111]. This methodhas been used to elucidate the process of Mu formation in solids [112] as well as toestimate the intrinsic mobility of electrons in materials where TOF measurements maybe dominated by trapping at dilute defects [113–117].

3.1 Methods developed at other labs

Other important μSR methods abound, many developed at other laboratories, such asthe stroboscopic technique (“Strobo-μSR”) developed at SIN by the ETH group [28] toprecisely measure the muon’s magnetic moment — and later utilized for numerous preciseKnight shift studies [118]. In Strobo-μSR one takes advantage of the RF time structure ofthe nominally CW isochronous cyclotron’s proton beam (50 MHz at SIN/PSI, 43 MHz at

50 J. H. Brewer

Fig. 18 Indirect Muon Spin Echo (i-μSE) in LaF3, displayed in the RRF at 29.5 MHz (the muon Larmorfrequency in the applied field). In this case the muons are introduced with their spins perpendicular to theapplied field, and immediately start precessing and relaxing. A �F pulse is then applied at the 19F Larmorfrequency, causing the fluorine spins to “refocus”, effectively bringing the muon spins along with them.Again, for a more detailed description, see Ref. [88].

Fig. 19 (a) A small aluminum “μ” rests on a sheet of depolarizing medium. (b) Reconstructed image of thetarget in (a), formed by rotating a strong magnetic field gradient in the plane of the target and taking Fouriertransforms of the resulting μ+SR signal at each orientation

TRIUMF), which results in the muons arriving in “bunches” with a very well-defined peri-odicity. If they stop in a magnetic field for which the muon’s Larmor frequency is an exactmultiple of the cyclotron frequency, each new bunch will arrive “in phase” with previous

μ SR HOWTO 51

−150 −100 −50 0 50 100 150

0.02

0.04

0.06

0.08

0.10

Dia

mag

netic

Asy

mm

etry

Electric Field (kV/cm)

Fig. 20 LEFT: Asymmetries (amplitudes) of muonium (circles) and diamagnetic (stars) signals as a func-tion of applied electric field for wTF-μ+SR in solid nitrogen, showing that muonium formation is suppressedby a positive electric field (driving the μ+ and e− apart) and actually enhanced by a negative one (pullingthem together) [108]. RIGHT: Diamagnetic asymmetry for wTF-μ+SR in GaP (squares) and GaAs (circles),showing that the initial weakly bound state of Muwb in GaP is more tightly bound than that in GaAs [110], asexpected from the effective masses of free electrons in each. Muwb formation is suppressed by both positiveand negative electric fields in this case, but there is still a slight anisotropy, indicating that most radiolysiselectrons are still behind the μ+

bunches and the asymmetry between simply scaled count rates in detectors on opposite sidesof the sample will show a resonance at the correct field. The advantage of this method is thatthere is no requirement to detect individual muons arriving (much as in a pulsed facility)and therefore no intrinsic limit on rate or statistics.

Because I have been asked to review μSR methods originally developed at TRIUMF, Iwill neglect the creations of our “competition” except for this example.

4 Accessibility

In the last decade or more, while new techniques are still emerging (LEM being an excel-lent example), the emphasis of the world’s μSR facilities has shifted to utilization efficiencyand making these tools more readily available to prospective users, many of whom are unfa-miliar with the demands of accelerators, beamlines, μSR spectrometers and such, but areattracted to the sensitivity and versatility of μSR. With these new standards of streamlining,automation and user support, μSR has entered a new era of Routine Science. In the future,this may be reckoned the true “beginning” of μSR’s contribution to science in general.

5 Further reading

This “HOWTO” review has concentrated on μSR methods developed at TRIUMF over thehistory of that laboratory, neglecting (a) most of the applications of those methods, particu-larly those covered in Prof. Ghandi’s contribution to this conference; and (b) μSR methodspioneered at other labs. For those wishing to acquire a more balanced and/or comprehen-sive knowledge of the history, methods, applications, successes and future potential of μSR,many excellent books [83, 119–125] and review articles [91, 112, 126–143] are available.

52 J. H. Brewer

Acknowledgments I am indebted to all the people who have worked to develop the capabilities of μSR,and particularly to the staff of the TRIUMF Centre for Molecular and Materials Science. I thank also allthe TRIUMF Directors who have contributed the lab’s resources to development and operation of its μSRfacilities, The National Research Council of Canada (which provides the funds for TRIUMF) and the NaturalSciences and Engineering Research Council of Canada, which funded the CMMS as well as all the CanadianμSR users over the years.

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