Muon Spin Rotation/Relaxation/Resonance

Published in Encyclopedia of Applied Physics, Vol. 11 (Mössbauer Effect to Nuclear Structure) pp. 23-53 (VCH Publishers, Inc., 1994).

Abstract

Quick Scroll - Table of Contents:

Introduction

1. Basic Muon Physics

   1. Muon Production

   2. Muon Decay

      1. Asymmetry Calibration.

   3. The ``Heavy Electron'' vs. the ``Light Proton''

      • Lifetime(s)
      • Muon Capture
      • Heterogeneous Signal
      • Lost Polarization
      • Hyperfine Effects

2. µSR Techniques

   1. Time-Differential µSR in Transverse Field

      1. The Basic Technique.

         • Frequency
         • Asymmetry
         • Initial Phase of precession
         • ``Relaxation''
         • Muonium Precession

      2. High Field.

         • Fourier Transforms
         • Rotating Reference Frames

      3. Paramagnetic States.

   2. Time-Differential µSR in Longitudinal and Zero Field

      1. Two-Counter Asymmetry.

         • The Experimental Asymmetry
         • The ``Corrected'' Asymmetry

      2. Zero Field.

      3. ``Relaxation.''

         • Nuclear Dipolar Relaxation
         • Motional Narrowing
         • Nuclear Dipolar Oscillations

      4. True Relaxation.

         • Longitudinal-Field Muonium Relaxation

   3. Time-Integral µSR

      1. Transverse Field I-µSRotation.

      2. Longitudinal Field I-µSRelaxation.

         • Experimental Asymmetry
         • Approximations

      3. Avoided Level Crossing Resonance.

         • Nuclear Quadrupolar µALCR
         • ``Muonated'' Radicals
         • Muonium in Semiconductors

   4. Muon Spin Resonance

      1. RF I-µSResonance.

      2. Muon Spin Echoes.

3. µSR Applications

   1. Muonium Chemistry

      1. Chemical Kinetics.

      2. Radicals.

      3. Molecular Structure.

   2. Condensed Matter Physics

      1. Hydrogen in Semiconductors.

      2. Quantum Diffusion.

         • Motion of the µ⁺ in Metals
         • Muonium Diffusion in Nonmetals

      3. Magnetism.

      4. Superconductivity.

         • Phase Diagrams
         • Magnetic Penetration Depth
         • Universal Correlations between T_c and n_s
         • Non-s-Wave Pairing?

4. µSR Facilities

   1. Logistics of Accelerator Experiments

   2. Standard µSR Hardware

      1. Detectors.

      2. Electronics.

      3. µSR Spectrometers.

      4. Target Vessels and Cryostats.

   3. µSR Software

5. Speculations

6. References

   1. Further Reading

Introduction

• µSR stands for Muon Spin Relaxation, Rotation, Resonance, Research or what have you. The intention of the mnemonic acronym is to draw attention to the analogy with NMR and ESR, the range of whose applications is well known. Any study of the interactions of the muon spin by virtue of the asymmetric decay is considered µSR, but this definition is not intended to exclude any peripherally related phenomena, especially if relevant to the use of the muon's magnetic moment as a delicate probe of matter.

Later sections will treat TF-µSR and other µSR techniques in some detail, but it is useful to begin with a qualitative phenomenological description just to establish some terminology. A crude apparatus for TF-µ⁺SR (µSR using positive muons) is pictured schematically in Fig. 1. The µ⁺ arising from pi+ decay at rest is perfectly spin-polarized as it enters the sample; later, it decays asymmetrically with the decay positron emitted preferentially along the muon spin direction.

• Figure 1. A simple transverse-field (TF)-µ⁺SR experiment: the µ⁺ beam enters from the left with its polarization antiparallel to its momentum. A magnetic field H is applied vertically, causing the µ⁺ spins to precess at the Larmor frequency \omega_µ = \gamma_µ H, where \gamma_µ / 2 \pi = 0.01355342 MHz/Oe. An incoming µ⁺ triggers the M counter, generating a start pulse for a fast time digitizer (``clock''), and an outgoing decay positron later stops the clock with a pulse from the E counter; for each such event the time interval is digitized and the corresponding bin in a discrete time spectrum is incremented.

N(t) = B + N₀ e^{-t/\tau_µ} [1 + A(t)] [Eq. 1]

where N₀ is an overall normalization, B is a time-independent background, \tau_µ = 2.197 microsec and A(t) is the corresponding asymmetry spectrum shown in Fig. 2 (bottom), which can be extracted numerically from N(t) as

A(t) = [N(t) - B \over N₀ ] e^{+t/\tau_µ} - 1 [Eq. 2]

Except for an empirical multiplicative constant, A(t) represents the time evolution of the muon polarization, much like a free induction decay (FID) signal in NMR.

• Footnote: In order to convert N(t) to A(t) one must know both B and N₀; fortunately, both constants can be extracted numerically from N(t) in cases where the period of µ⁺ Larmor precession is a negligible fraction of the muon lifetime. Treating N(t) as a continuous function (the actual discrete sums can easily be deduced from the integrals below) we may write

  N₀ = {S \tau_µ E_+ - R T \over \tau_µ² E_+ E_- - T²}

  B = {R \tau_µ E_- - S T \over \tau_µ² E_+ E_- - T²}

  where S = \int_{t_i}^{t_f} N(t) dt, R = \int_{t_i}^{t_f} N(t) \exp(t/\tau_µ) dt, E_± = ± [\exp(± t_f/\tau_µ) - \exp(± t_i/\tau_µ)] and T = t_f - t_i, the time interval over which the integrals are evaluated.

Figure 2.

• TOP: ``Raw'' time spectrum from a simple TF-µ⁺SR experiment: the overall exponential decay reflects the muon lifetime and the precession of the µ⁺ spins is manifest in the superimposed oscillations as the muon polarization sweeps past the positron detector.
• BOTTOM: TF-µ⁺SR asymmetry spectrum obtained from the raw time spectrum by subtracting any time-independent background and dividing out the exponential distribution of muon decay times.

With the advent of the Meson Factories in the 1970's came a hundred- to thousand-fold increase in the intensity of muon beams, dramatically accelerating the development of new experimental techniques and the discovery of new applications for µSR. When the potential of µSR became apparent, most Meson Factories invested in the upgrading of muon beamlines and µSR facilities that came on line in the early 1980's. Since then the techniques of µSR have been discovered by the chemistry and solid state physics communities and what was once an esoteric oddity has become one of the fastest growing areas of intermediate energy science.

Today's µSR is a standard magnetic resonance tool, almost exclusively devoted to disciplines rarely associated with subatomic physics -- primarily chemistry and condensed matter physics. A list of some of these areas of application will appear later in this article, but first it is necessary to describe the fundamental physics that makes µSR possible and to explain the basic techniques of its use.

BASIC MUON PHYSICS

Muon Production

\pi⁺ --> µ⁺ + \nu_µ ... or ... \pi¯ --> µ^- + \bar{\nu}_µ

from which the muon emerges (in the rest frame of the pion) with a momentum of 29.79 MeV/c and a kinetic energy of 4.119 MeV. The lifetime of a free charged pion is \tau_\pi = 26.? ns. Because the neutrino is only produced with negative helicity (spin antiparallel to momentum) and the antineutrino only with positive helicity -- a succinct description of P-nonconservation adequate for our purposes here -- the simultaneous conservation of linear and angular momentum forces the µ⁺ also to have negative helicity (and the µ^- positive helicity) in the rest frame of the pion. Thus muons emitted from pion decay at rest are also 100% spin polarized opposite to (for µ⁺) or along (for µ^-) the direction of their momenta. This is the greatest advantage of µSR as a magnetic resonance technique: whereas NMR and ESR rely upon a thermal equilibrium spin polarization, usually achieved at low temperatures in strong magnetic fields, µSR begins with a perfectly polarized probe, regardless of conditions in the medium to be studied. It also implies that muon spin degrees of freedom usually start their evolution as far from thermal equilibrium as conceivable.

Most µ⁺ beams today are literally emitted from pi+ decay at rest in the surface layer of the primary target where the pions themselves are produced by collisions of high energy protons with target nuclei -- hence the common mnemonic name, surface muons [ Bowen, 1985]. Unfortunately, this mode is not available for negative muons because a \pi¯ stopping in the production target almost always undergoes nuclear capture from low-lying orbitals of pionic atoms before it has a chance to decay. This problem was actually solved long before the surface muon beam was invented, when many fundamental physics experiments with muons were primarily concerned with the ``heavy electron'' behaviour of the µ<sup>-</sup>: in so-called ``conventional'' muon channels, pions are allowed to decay in flight down a relatively long straight section where the decay muons are collected by axial or alternating-gradient magnetic fields; the muons emitted ``backward'' in the pion rest frame have quite different momenta from those emitted forward (or from the pions themselves), and can thus be selectively extracted by a bending magnet. The disadvantages of such backward muon beams are their relatively higher momentum (usually ~ 50-100 MeV/c), their larger momentum spread (and therefore lower stopping density) and their much larger phase space (and therefore lower luminosity). Today they are rarely used for µ⁺SR, but for µ^-SR there is no alternative.

Muon Decay

The decay probability of the muon, illustrated in Fig. 3, depends upon the e^± energy x = \varepsilon_e/\varepsilon_{max (where \varepsilon_max = 52.83 MeV is the maximum possible total relativistic energy of the e^±) and the angle \theta between the muon spin direction and the direction of e^± emission as

dP(x,\theta) = E(x) [1 + a(x) \cos \theta] dx d(\cos \theta)

• Figure 3 . Angular distribution of the e⁺ from µ⁺ --> e⁺ + \nu_e + \bar{\nu}_µ decay: the asymmetry (anisotropy) of the distribution is 100% for the highest e^± energy \varepsilon_max = 52.83 MeV and zero (i.e., an isotropic distribution) for \varepsilon_e = \varepsilon_max/2; for \varepsilon_e --> 0 (not shown) the asymmetry is negative.

a(x) = ± {2x - 1 \over 3 - 2x }

and the normalized e^± energy spectrum has the form

E(x) = 2 x² ( 3 - 2x ).

Figure 4 shows the functions E(x)/2, a(x) and their product. Note that a changes sign at low energy; however, very few positrons are emitted with such low energies (see above) and those which are will usually not be detected (see below).

• Figure 4 . Solid lines: energy spectrum E(x)/2 (e^± energy \varepsilon_e = x \cdot 52.83 MeV) of e^± from normal µ± decay and \varepsilon_e-dependence of µ⁺ decay asymmetry a(x) [degree of correlation between e⁺ momentum and µ⁺ spin direction]. (For µ^- --> e^- \bar{\nu}_e \nu_µ, a has the opposite sign.) Dashed line: weighted µ⁺ --> e⁺ \nu_e \bar{\nu}_µ asymmetry spectrum, the product of E(x)/2 and a(x).

Asymmetry Calibration

< a > = \int_0^1 a(x) E(x) dx = ± {1 \over 3}

is never realized in practice (except by accident) for several reasons:

First, radiative corrections subtly distort the low-energy end of the spectrum [ Sachs et al., 1975].

Second, the low energy e^± are easily stopped in even thin layers of material such as the wrappings of scintillation counters; indeed, since this effect raises the average asymmetry, it is common practice to insert degraders between the muon stopping region and the e^± detectors, when feasible, up to an optimum range (e.g. ~3-4 cm of graphite) that maximizes the product A²N, where A is the empirical asymmetry (defined below) and N is the e^± rate reaching the detector. (One should generally use low-Z materials for degraders in order to mimimize bremsstrahlung and pair production, which can greatly confuse the issue, unless one wishes to use such effects to bypass strong magnetic fields -- see below.)

Third, a typical e^± detector intercepts a rather large solid angle and thus averages \cos \theta appreciably. The optimal detector geometry for general-purpose time-differential (TD)-µSR is a cube centered on the target where the muons stop, with each of the 6 faces a single detector. (Although in principle one could gain back some asymmetry by using position-sensitive detectors and weighting individual events according to the projections of the e^± track along different axes, the complications of counting statistics [not to mention the expense of processing such information at very high rates!] are not usually justified by the marginal improvement.)

Finally, because the decay e^± will follow a helical path in an applied magnetic field H (which is an essential part of many µSR experiments), the curling up of e^± orbits causes the efficiency of their detection to be a function of both their energy and the applied field strength; the apparent direction of emission may also be affected. At p_e=30 MeV/c (a typical e^± momentum) the radius of curvature \rho_e of an orbit perpendicular to an applied magnetic field H=1 T is exactly 10 cm, as given by the familiar formula

\rho [cm] = {p [MeV/c] \over 0.3 H [kOe] }.

This effect becomes problematic at high fields (H \gsim 5 T ==> \rho \lsim 2 cm) in which case µSR requires very small samples and detectors. Another possible approach to this problem is to intentionally generate bremsstrahlung photons from the decay e^± using a lead converter near the target and then reconvert the photons in front of the detector after they have traversed the high-field region.

Taken together, these systematic effects make the empirical asymmetry A in a given detector perplexingly dependent upon the thickness, geometry and material of target and detectors, not to mention magnetic field. Attempts to correct analytically for e^± energy etc. are usually only reliable to within a few percent.

Fortunately there are many materials (including most metals) in which the µ⁺ suffers negligible depolarization; thus the usual method for calibrating the asymmetry in µ⁺SR is to first measure a dummy sample of dimensions identical to those of the real sample, but made from aluminum or silver. If great care is taken to arrange the two samples in exactly the same position, this will determine the empirical maximum asymmetry A₀ to within a few percent of itself; typical values are 0.2 to 0.3. However, materials of the same nominal thickness (in g/cm²) will often exhibit different dE/dx and multiple scattering of the e^±, so that this calibration method cannot generally be trusted to better than ~1%.

In µ^-SR, as we shall see, such calibrations are much more difficult due to the loss of µ^- polarization in the initial cascade to the ground state of the muonic atom. To the author's knowledge there is no satisfactory solution for the A₀ calibration problem in µ^-SR. The usual method is to calibrate on graphite, which gives a residual µ^- polarization of about 17±2% of the initial muon beam polarization [ Kuno et al., 1984].

The ``Heavy Electron'' vs. the ``Light Proton''

In elementary particle physics, the muon is often described as a ``heavy electron'' in reference to its family resemblance to other leptons; it is the µ^- that is so designated. Indeed, the µ^- does just what one might expect of a heavy electron: it undergoes Coulomb capture into atomic orbitals analogous to those of electrons but 206.768 times closer to the nucleus and unrestricted by the Pauli exclusion principle; thus the muon quickly (usually in ~10^-14 s) cascades to the 1s ground state, emitting photons (known as muonic X-rays) and/or Auger electrons (ejected from low-lying orbitals in the same atom by muonic X-rays) on the way. Once there, the significant overlap between muon and nuclear wavefunctions [in heavy nuclei the muon spends most its time inside the mean nuclear radius] gives the weak interaction between the muon and the proton a chance to act, resulting in nuclear capture ([muon capture]) which appreciably reduces the µ^- lifetime (and the fraction of muons decaying to electrons for µ^-SR to detect).

By contrast, the positive muon avoids positively charged nuclei and will capture its own electron when it can, to form the hydrogenlike atom muonium (µ⁺e^-, usually abbreviated Mu). In fact, the qualitative behavior of the µ⁺ in matter resembles that of a light proton far more than that of its closer antilepton relative, the positron. The µ⁺ mass is 0.1126 times that of the proton but 206.768 times that of the positron; its magnetic moment is 3.1832 times that of the proton but only 0.48363% of the positron's. The Mu atom is almost identical (except in mass) to the hydrogen atom, whereas the positronium atom (e⁺e^- or Ps) has no nucleus and half the binding energy of H. Positronium also annihilates very quickly, whereas weak electron capture in muonium (µ⁺ + e^- --> \bar{\nu}_µ + \nu_e) only shortens the µ⁺ lifetime by ~1 part in 10¹⁰.

As a result, µ^-SR differs qualitatively from µ⁺SR:

Lifetime(s): The µ⁺ lifetime is independent of its environment while that of the µ^- depends strongly upon the Z of the nucleus it becomes attached to; a sample containing several elements will have as many µ^- lifetimes, the ratios of whose probabilities cannot be reliably estimated from the Fermi-Teller ``Z law'' [ Fermi and Teller, 1947] and therefore must be determined by fitting the time distribution of decay or capture products.

Muon Capture: A µ^- that undergoes nuclear capture ([muon capture]) does not produce a decay electron for detection in µ^-SR; the µ^-SR event rate per incident muon is thus proportional to the µ^- lifetime, which in high-Z elements can be as little as 4% of the free muon lifetime. This rate loss can often be offset by increasing the µ^- beam intensity, except when the sample contains both heavy and light nuclei.

Heterogeneous Signal: Whereas the µ⁺ produces only positrons and undetected neutrinos, a µ^- entering some high-Z samples like ²³⁸U is likely to cause emission of X-rays and Auger electrons during the atomic cascade, fission fragments and neutrons (on average several per muon) from either nuclear muon capture or nuclear internal conversion of muonic X-rays, as well as the occasional decay electron. The µ^-SR signal is therefore noisy and heterogeneous. This does make it rather interesting from the points of view of atomic and nuclear physics, but such complications are so far unwelcome in historically typical µ^-SR applications.

Lost Polarization: Spin-orbit coupling in the atomic cascade leaves the µ^- in its ground state 1s orbital with a dramatically reduced spin polarization, typically \lsim 20%. Because statistical uncertainties shrink as N^{-1/2 one must accumulate 25 times more events to achieve the same statistical accuracy in measuring a 5 times smaller signal. This seemingly irreducible disadvantage is the main limitation of µ^-SR applications and is the reason why µSR is often treated as synonymous with µ⁺SR. However, since µ^-SR offers complementary and unique information, it will always be an essential part of the µSR repertoire. There is also a chance that new techniques such as X-ray tagged µ^-SR may help overcome this disadvantage.

Hyperfine Effects: The hyperfine coupling between the muon and nuclear spins in nonzero-spin nuclei is enormous, often large enough to cause Auger emission of K and L electrons [ Winston, 1963]. The two hyperfine states F⁺ and F¯ of the 1s muonic atom (in which the muon spin is respectively parallel and antiparallel to the nuclear spin) therefore produce distinct µ^-SR signals, each with its own characteristic precession frequency, in such systems. This was long thought to limit useful µ^-SR to spinless elements, but quite a few nuclei with spin I > 1/2 have been found to exhibit potentially useful µ^-SR signals, albeit at reduced amplitudes [ Brewer, 1984a/b].

Taking into account the different lifetimes, cascade depolarizations and spin states of negative muons in different muonic atoms, we may generalize Eq. (?) to obtain the most general form for electron-triggered µ^-SR in a sample composed of different elements:

N(t) = B + \sum_i N°_i e^{-t/\tau_i} [1 + A_i(t) ]

in which N°_i = N₀ f_i (\tau_i/\tau_µ), where N₀ is an overall normalization, f_i is the fraction of muons captured on the ith type of nucleus and \tau_i is the µ^- lifetime in that species of muonic atom and hyperfine state; A_i(t) = A₀ P_i(t), where A₀ is a common maximal asymmetry factor and P_i(t) is the polarization of muons in the ith type of muonic atoms. For TF-µ^-SR, this polarization will have the form P_i(t) = P°_i G_i(t) \cos(\omega_i t + \phi_i), where P°_i is the polarization remaining in that atom's given hyperfine state after the cascade, G_i(t) is the corresponding relaxation function (see below), \omega_i is its characteristic precession frequency in the applied magnetic field and \phi_i is the initial phase.

In the interest of simplicity, further discussion of µ^-SR will be neglected in favour of µ⁺SR, which occupies most of the attention of the µSR community.

• Figure 5 . Coordinate system and labelling conventions for surface muon µSR experiments. Note that the superscript on the left indicates the direction of the incoming muon polarization while the subscript on the right indicates the direction of the applied field (if any); for longitudinal field (LF) and by continuation for zero field (ZF) both will always be the same, but for TF there are in principle two possible arrangements for each choice of super(sub)script. (For instance, ^zwTF_y vs. ^xwTF_y.) ``Weak'' (w) field means ``not strong enough to deflect the muon beam appreciably.''

µSR TECHNIQUES

Time-Differential µSR in Transverse Field

The Basic Technique

• Figure 6. Counter arrangements for ^xTF_z-µSR, in which the muon beam has been passed through a Wien filter to rotate the muon spins until they are perpendicular to their momenta. The momentum p_µ is undeflected in the magnetic field H = H z \parallel p_µ but the polarization P_µ precesses in the x-y plane.
• Figure 7. Counter arrangements for ^zwTF_x-µ⁺SR, in which the µ⁺ beam still has its polarization antiparallel to its momentum. The field must therefore be perpendicular to both p_µ and P_µ which restricts use with surface muon beams to weak fields H \lsim 100 Oe.

Frequency: The most obvious observable in a TF-µSR spectrum is the µon precession frequency \omega_µ = \gamma_µ B, which (thanks to our precise knowledge of \gamma_µ) is equivalent to the local magnetic field B at the muon. Since B may be affected by diamagnetism, paramagnetism and contact hyperfine interactions with polarized electrons (Knight shifts) in the medium, B is generally different from the applied field H₀; this difference is often the main focus of the TF-µSR experiment. An example is shown in Fig. 8.

• Figure 8. Positive muon Knight shift K_µ = (B_µ - H₀)/ H₀ (where H₀ is the applied magnetic field and B_µ is the field at the muon) as a function of temperature in a single crystal of antimony with the c axis parallel to the applied field (1.5 T). The precision (approximately ± 50 ppm) is typical of what can easily be achieved with routine methods in modest TF-µSR experiments; much higher precision is possible with refined techniques (see the later section on strobo -µSR).

Asymmetry: The second obvious parameter characterizing muon precession in a transverse field is the amplitude or asymmetry of the precession signal. As mentioned earlier, the absolute calibration of this amplitude is tricky but one can usually convert the initial amplitude into a residual polarization with an accuracy of a few percent. In metallic samples the initial polarization is generally consistent with 100\%, but in insulators or semiconductors, liquids or gases, some fraction of the muons (ranging from 0 to 100\%) either form muonium (Mu) or experience some other form of depolarization in the early (\lsim 1 ns) stages of thermalization in the sample.

The effect of Mu formation in wTF is to cause the muon polarization to precess in the opposite sense to that of muons in diamagnetic environments, roughly 103 times faster; this results in dramatic ``dephasing'' if the Mu atoms subsequently react at exponentially distributed times to enter diamagnetic states. If all this takes place within a few ns, all one can observe is the net effect on the subsequent diamagnetic µ⁺ precession signal, namely a reduction of asymmetry and a simultaneous shift of the apparent initial phase of precession. If the magnetic field is in the z direction and the initial µ⁺ polarization is in the x direction, we may define the complex muon polarization \tilde{P}(t) = P_x(t) + i P_y(t), in terms of which the overall polarization of an ensemble of muons starting as Mu atoms and reacting at a rate \Lambda to form some diamagnetic species is given in the low-field limit (averaging over high-frequency hyperfine oscillations) by

\tilde{P}(t) \approx ½ {\omega_Mu + Omega_µ \over \omega_Mu + Omega_µ - i\Lambda } e^{-(\Lambda + i\omega_Mu)t} \cr\noalign{\vskip5pt} + ½ [ { i\Lambda \over i\Lambda - (\omega_Mu + Omega_µ)} + { \Lambda² \over \Lambda² + \omega₀² } ] e^{iOmega_µ t}

where the diamagnetic precession frequency is \omega_µ and the muonium precession frequency is \omega_Mu \approx 103 Omega_µ in the opposite sense. This sort of simple residual polarization picture also describes many other delayed formation scenarios.

Initial Phase of precession: A more subtle aspect of the residual polarization picture is the shift of the apparent initial phase of the µ⁺ precession due to (e.g.) the formation of short-lived Mu atoms precessing in the opposite sense. Measurement of such phase shifts has often proved valuable in sorting out ``fast chemistry'' effects [ Brewer et al., 1974].

``Relaxation:'' Following thermalization of translational degrees of freedom, which may take as little as 100 ps in solids, the muon precession signal may still decrease in amplitude due to ``dephasing'' (T₂ effects) or true irreversible relaxation processes (T₁ effects). The latter are more often studied in longitudinal field (LF) where the distinction between T₁ and T₂ is not subject to so much semantic debate. (See later section on ``relaxation'' in LF-µSR.) Depolarization due to inhomogeneous magnetic fields often takes the form of a gaussian relaxation function \exp(-½ \sigma² t²) as shown in Fig. 9.

• Figure 9. Depolarization of muons in a sintered powder of superconducting La_1.85Sr_0.15CuO₄ at 10 K in an applied magnetic field of 150 Oe. The relaxation is due to local field inhomogeneities caused by flux exclusion and a vortex lattice in this type II superconductor; it is adequately described by a gaussian relaxation function in which \sigma \propto \lambda^-2 where \lambda is the London penetration depth. This has been much used for initial estimates of \lambda.

Muonium Precession: All the features just described for muon precession at the diamagnetic Larmor frequency \omega_µ = \gamma_µ H are also often seen for µonium precession in wTF-µSR experiments. The main differences are that \omega_Mu = \gamma_Mu H is roughly 103 times larger than \omega_µ in the same field H [due to the huge magnetic moment of the electron locked to the muon spin by the hyperfine interaction], that the sense of wTF Mu precession is opposite to that of the free µ⁺ [due to the opposite sign of the dominating e^- moment], that half the muon polarization appears lost in most experiments [due to the fast (4463 MHz for Mu in vacuum) hyperfine oscillations between the |\Uparrow\downarrow> and |\Downarrow\uparrow> states, where \Updownarrow refers to the electron spin and \updownarrow refers to that of the muon] and that \omega_Mu splits into two frequencies for H \gsim 20 Oe because the hyperfine interaction is finite. This phenomenon will be discussed in more detail below.

High Field

Fourier Transforms: Often, especially when high frequencies are involved, one would rather see a frequency spectrum than a time spectrum. The art of converting the latter into the former is subject to continuing evolution and perpetual debate, but virtually all µSR facilities include the fast fourier transform (FFT) in their standard data analysis tools. A complete account of the merits and hazards of such treatments is beyond the scope of this article, but it is useful to point out several ubiquitous features that are ignored at one's peril.

• Figure 10. Results of complex FFT of a quadrature µSR time spectrum with a slow relaxation rate: (a) through (d) show the power, the amplitude (square root of the power), the real and the imaginary parts of the frequency spectrum resulting from FFT of the ``raw'' complex time spectum over 10.24 microsec at a field of 1.489 T; (e) through (h) show the same quantities for the same data apodized by a gaussian envelope function with a 4 microsec time constant.

Second, because of the finite muon lifetime there are less statistics in the bins at late times and consequently the ``error bars'' in the asymmetry spectrum grow exponentially with time as \exp(+t/2\tau_µ). (See Fig. 2.) This introduces noise in an annoyingly non-uniform way, which may in principle be corrected for using not-so-fast fourier transform methods. These noise effects are always subject to suppression without limit (except for the finite patience of the experimenter) by simply taking more statistics; although the finite muon lifetime is responsible, it does not fix any absolute limits on precision -- only practical ones.

Third, because experimenters' patience is indeed finite, most µSR time spectra only extend to 10-20 microsec; thus the FFT is cut off abruptly at some time limit, introducing the ``ringing'' effect evident on the left side of Fig. 10. This can be suppressed by apodization -- multiplying the ``raw'' time spectrum by an envelope function that causes the product to go to zero gently before the abrupt end of the time range. As evident from Fig. 10, well-chosen apodization removes the ringing caused by FFT on a finite time interval but introduces a line-broadening effect that must be taken into account in the interpretation.

• Figure 11. Real part of frequency spectrum taken at a temperature of 6 K for positive muons in a single (3 mm × 3 mm × 0.2 mm) crystal of superconducting YBa₂Cu₃O_6.95 (a type II superconductor with T_c = 92 K) cooled in a field of 1.489 T applied along the crystalline c axis. The sharp peak at 202 MHz is due to muons missing the superconductor and stopping in a normal region where the field distribution is not broadened by the vortex lattice.

Rotating Reference Frames: For TF-µSR in high magnetic field (HTF-µSR), another essential tool is the rotating reference frame (RRF) transformation, an example of which is shown in Fig. 12. The following description will omit the details of transforming discrete spectra and give only a simplified treatment in terms of an idealized continuous time spectrum. In fact, the RRF transformation works best on orthogonal pairs of spectra -- as for instance when the magnetic field acts in the z direction and detectors are situated in the ± x and ± y directions defining the plane of precession of the muon polarization -- where a complex asymmetry spectrum \tilde{A}(t) can be defined as (e.g.) \tilde{A}(t) = A_x(t) + i A_y(t). In this case the RRF transformation is simply

\tilde{A}_RRF(t) = \tilde{A}(t) e^{-i \Omega t}

where \Omega is the RRF frequency -- chosen arbitrarily to produce a transformed \tilde{A}_RRF(t) with the desired characteristics. For instance, if the muons precess at a high frequency \omega_µ, one may select \Omega slightly smaller than \omega_µ to produce a \tilde{A}_RRF(t) which varies relatively slowly with time; the resulting complex spectrum can be packed (n_p original time bins \to one coarser time bin) in which case the statistical uncertainties of individual bins are reduced by a factor roughly equal to \sqrt{n_p}.

• Figure 12. The first 4.5 microsec of a µSR time spectrum taken at roughly 0.2 T in the lab frame where the data are recorded and in the rotating reference frame (RRF) to which it has been transformed numerically for display and fitting. The RRF frequency of 21.2 MHz is chosen to produce a slow precession signal for visual clarity. Because the original spectrum is real (no orthogonal detector arrays) the maximum amplitude in each of the complex RRF spectra (real part: circles; imaginary part: triangles) is actually a factor of two smaller than in the lab frame. The solid line in the RRF spectra is a fit to three frequencies; a single frequency (or even two) would result in a much poorer fit to the data.

In the event that a single spectrum contains several signals at drastically different frequencies \omega_i, separate RRF transformations at frequencies \Omega_i \approx \omega_i can be used to isolate each signal for easy fitting.

It is also possible to perform a RRF transformation on a pure real spectrum (i.e., one without orthogonal detector axes) if care is taken to select \Omega and n_p so as to ``bin out'' the spurious signal at the RRF frequency [ Riseman and Brewer, 1990].

Paramagnetic States

• Figure 13. Breit-Rabi diagram showing the energy levels of a system of two spin-1/2 particles of opposite sign and different magnetic moments -- e.g., muonium -- as functions of the reduced field x = H/H₀ where H₀ (1585 Oe for Mu in vacuum) is a characteristic hyperfine field. For the purpose of illustration, unphysical values of moments and coupling constants have been used. The hyperfine frequency \nu₀ = \omega₀/2\pi has the value 4463 MHz for muonium in vacuum. In zero field the three triplet (J=1) eigenstates |1>, |2> and |3> are degenerate and the singlet (J=0) ground state |4> is \hbar\omega₀ lower in energy. At high reduced field (x \to \infty) the eigenstates are |1> \to |\Uparrow\uparrow>, |2> \to |\Uparrow\downarrow>, |3> \to |\Downarrow\downarrow> and |4> \to |\Downarrow\uparrow>. (\Updownarrow refers to the electron spin and \updownarrow refers to that of the muon.) Note that state |1> is lower energy than state |2> above H_c = \omega₀(\gamma_e - \gamma_µ) / (2 \gamma_e \gamma_µ) [16.386 T for Mu in vacuum].

• Figure 14. Frequency power spectrum of positive muons in a pure silicon crystal at 10 K and 75 Oe transverse field showing the diamagnetic (D) signal, the rapidly-diffusing muonium (Mu) signals and the signals from so-called ``anomalous'' muonium (Mu*) which has been shown to be a muonium atom localized on a Si-Si bond center. Note vertical scale change at 60 MHz. After [ Brewer et al., 1973].
• Figure 15. Frequency power spectrum for positive muons in powdered buckminsterfullerite (crystalline C₆₀) at room temperature and 108 Oe [ Kiefl et al., 1992b, showing simultaneously the signals from muons in the C₆₀Mu^. radical (R₁₂ and R₃₄) and the endohedral muonium (Mu@C₆₀) atom (Mu₁₂ and Mu₂₃). In paramagnetic molecules with nuclear moments, such radical signals cannot be seen in low field due to nuclear hyperfine broadening.

Time-Differential µSR in Longitudinal and Zero Field

• Figure 16. Longitudinal-field µSR counter arrangement for LF_z-µ⁺SR, in which the µ⁺ beam is polarized antiparallel to its momentum. As H = H z \parallel p_µ \parallel P_µ any magnetic field strength may be utilized.
• Figure 17. Longitudinal-field µSR counter arrangement for LF_x-µSR, in which the muon beam arrives spin-rotated by a Wien filter. Since the field is now perpendicular to P_µ, use with surface muon beams is restricted to weak fields H \lsim 100 Oe.

Two-Counter Asymmetry

\epsilon_{U,D} = efficiency of U or D e detector

B_{U,D} = backgound in said e detector

(measured using ``t<0 bins'')

A_{U,D} = intrinsic asymmetry of e detector

[Count rate ~(1 ± A_{U,D}) for muons fully polarized along (opposite) detector symmetry axis.] P_x(t) = muon polarization along axis.

Thus N_{U,D}(t) = B_{U,D} + N₀ \epsilon_{U,D} [1 ± A_{U,D} P_x(t)] where N₀ is a common normalization.

The Experimental Asymmetry a(t) is then obtained from either

a(t) = { [N_U(t) - B_U] - [N_D(t) - B_D] \over [N_U(t) - B_U] + [N_D(t) - B_D] } \qquad or

a(t) = { (1-\alpha) + (1+\alphaß) A_U P_x(t) \over (1+\alpha) + (1-\alphaß) A_U P_x(t) }

where \alpha = { \epsilon_D \over \epsilon_U } \quad and \quad ß = { A_D \over A_U }.

Thus (1-\alpha) / (1+\alpha) is the ``baseline'' asymmetry for totally unpolarized muons.

The ``Corrected'' Asymmetry: In addition to the obvious ``baseline shift'' there is also a more subtle distortion in a(t) for \alphaß \ne 1: a plot of a(t) looks as if it has a nonlinear scale for the abcissa. To completely remove these distortions one must somehow independently determine \alpha and ß (usually by fitting data taken in wTF with the same geometry) and then apply the correction

A_U P_x(t) = { (\alpha-1) + (\alpha+1) a(t) \over (\alphaß+1) + (\alphaß-1) a(t) } .

These corrections apply equally to TF-µSR asymmetry spectra formed from opposing pairs of detectors; in fact these are usually used to fit for \alpha. However, ß must be determined from simultaneous fits to the opposing ``raw'' spectra N_{U,D}(t) in TF. It is not unusual to assume ß = 1, although in principle one should always determine this empirical parameter as accurately as possible.

Zero Field

``Relaxation''

As noted by [ Kubo and Toyabe, 1966], these distinctions are blurred and the terminology becomes less useful as the applied field becomes comparable to the local fields and eventually goes to zero. Basically, if the components of local fields transverse to the applied field are non-negligible in the vector sum forming the total field at the probe, then even in this classical picture the relaxation phenomena become quite complicated and are still being worked out today [ Dalmas-de-Reotier et al., 1992]. A broad review of this subject is impossible here, but a few examples can help to illustrate the potential of ZF- and LF-µSR.

Nuclear Dipolar Relaxation: In the presence of strong electric field gradients, nuclei with electric quadrupole moments (such as Cu) exert an effective classical dipolar field on the muon:

B_dip = \hbar \gamma_n J_q [3(r\cdotq)r - q]/r³

where \gamma_n is the nuclear gyromagnetic ratio, J_q is the component of nuclear spin along the electric field gradient direction q and r = r/r where r is the vector of length r from the muon to the nucleus. The muon usually has several near neighbor nuclei generating a broad and nearly isotropic distribution of nuclear dipolar fields. Assuming a gaussian distribution of internal fields with random orientation leads to a muon polarization function [ Hayano et al., 1979]

g_zz^KT(t) = {1\over3} + {2\over3} ( 1 - \Delta² t² ) e^{- ½ \Delta² t²} }

which at early times (t « \Delta^-1) approaches a simple gaussian form G_zz(t) ~\exp[-\Delta² t²] . The interpretation of \Delta is defined by

\Delta²/\gamma_µ² = ½ ( < B_x²> + < B_y²> ) ;

thus \Delta is \gamma_µ times half the mean squared internal field in the plane perpendicular to the initial muon polarization (taken in this notation to be along the z direction).

• Figure 18. Zero-field (ZF) and weak (7 and 18 Oe) longitudinal field (wLF)-µ⁺SR in a single crystal of pure copper at 45 K with the muon polarization initially along the < 111 > axis of the crystal. At this temperature the muons are almost static in the Cu lattice (hop rate \lsim 0.1 inv microsec). The solid lines show fits to the exact spin hamiltonian between the muon in an octahedral interstitial site and its six nearest neighbor Cu spins; the dashed line shows a fit to a simple gaussian Kubo-Toyabe function (?).

Motional Narrowing: Figure 18 shows a famous example of nuclear dipolar relaxation of muons in copper metal for ZF and wLF. At the temperature of 45 K the muons are almost perfectly static in the Cu lattice; at higher temperatures they diffuse by thermally activated ``hopping'' between adjacent octahedral interstitial sites, causing a reduction of the relaxation rate and a change of its shape toward a slow exponential decay (often referred to as ``motional narrowing'' in reference to the analogous phenomena in NMR where one observes a resonance lineshape whose width is proportional to the relaxation rate). Considerable literature is devoted to the mathematics of ``dynamicizing'' static relaxation functions [ Kehr, 1978; Hayano et al., 1979; Celio, 1987]. At lower temperatures the muons again begin to hop as quantum mechanical tunneling becomes important; this phenomenon (and others like it) is an important illustration of quantum diffusion in the presence of dissipation [ Luke et al., 1991], a major area of application of µSR, which provides a light interstitial probe ideally suited to testing theories of quantum dissipation [ Kagan and Klinger, 1974; Kondo, 1986; Yamada, 1986; Kagan and Prokof'ev, 1991; Kagan and Prokof'ev, 1992].

Nuclear Dipolar Oscillations: In Fig. 18 we also see a slight difference between the simple Kubo-Toyabe function (?) and the exact quantum mechanical solution of the coupled equations of motion of the muon and its six nearest neighbor Cu spins [ Celio, 1986]. This reflects the fact that the nuclear moments are not simply static dipoles producing a field at the muon site, but also precess in the dipolar field due to the muon. Such a classical picture quickly becomes useless in picturing the actual evolution of such spin systems, particularly when the nuclei are few in number and have no electric quadrupole interactions, as in the case of the FµF¯ ion formed when positive muons are implanted into any ionic fluoride compound such as LiF (shown in Fig. 19).

• Figure 19. Zero-field µ⁺SR asymmetry spectrum in LiF at 89 K, showing the oscillatory behavior of the FµF¯ ion as a nearly-isolated system of 3 spin-1/2 moments coupled by dipole-dipole interactions. Inset:the rock salt crystal structure showing the µ⁺ site (black dot) between two F¯ ions (large spheres), which are actually pulled in slightly toward the µ⁺ [ Brewer et al., 1986].

True Relaxation

T₁^-1 = { 2 \delta² \tau_c \over 1 + Omega_µ² \tau_c² }

which leads to a ``T₁ minimum'' T_1(min) = Omega_µ/\delta² at \omega_µ \tau_c = 1.

• Figure 20. Longitudinal relaxation of muonium atoms in solid nitrogen at 10 K (diamonds), 9 K (triangles), 8 K (circles) and 6 K (squares) for an applied LF of 8 Oe.

Longitudinal-Field Muonium Relaxation: In very low field, the triplet component of the muonium atom spin system can be treated as a single spin-1 particle with approximately the magnetic moment of an electron. In this case Eq. (?) applies as well for Mu as for the diamagnetic µ⁺ (with \omega_Mu \approx 103 Omega_µ substituted for \omega_µ) and we may see the behavior shown in Fig. 20 as the Mu hop rate \tau_c^-1 changes with temperature.

In higher fields the Mu spin system has 4 eigenstates and transitions among levels become more complicated. Nevertheless an analogous system of rate equations [ Yen, 1988] can be used to extract the Mu hop rate as a function of temperature, as shown in Fig. 21. This has allowed precise measurements of the quantum diffusion of muonium in insulators [ Kadono, 1990], an important test ground for theories of quantum dissipation (see above).

Figure 21.

• (a) Longitudinal relaxation rate T₁^-1 of muonium atoms in a pure GaAs crystal as a function of field and temperature.
• (b) The deduced Mu hop rate \tau_c^-1 as a function of temperature. [ Schneider et al., 1992]

Time-Integral µSR

Transverse Field I-µSRotation

• Figure 22. Resonance signals in a strobo -µSR experiment [ Camani et al., 1978].

This stroboscopic µSR technique was developed to make a precise measurement of the muon's magnetic moment [ Camani et al., 1978] and has since been adapted to muon Knight shift measurements [ Gygax et al., 1984], where its unlimited rate gives it an advantage over TD-µSR methods. ``Strobo-µSR,'' as it is called, is best suited to situations where the muon precesses at a single well-defined frequency and where any relaxation is either uninteresting or fast enough (\gsim 0.454 µs<sup>-1</sup>) to be easily unfolded from the ``natural linewidth.'' Splittings of less than this linewidth cannot be unambiguously resolved by strobo-µSR. However, single frequencies (and thus, e.g., muon Knight shifts) can be routinely measured with a precision of \lsim 5 ppm by this method.

Longitudinal Field I-µSRelaxation

R = rate of arrival of muons

\epsilon_± = efficiency of e detector along(+) or opposite (-) µ polarization

B_± = backgound rate in said e detector

A_± = intrinsic asymmetry of e detector [Count rate ~(1 ± A_±) for muons fully polarized along (opposite) detector axis of symmetry.]

G_zz(t) = longitudinal relaxation function

The number of counts in time T » \tau_µ is

N_± = B_± T + RT\epsilon_± ± RT\epsilon_± A_± {\cal L}(G_zz),

where {\cal L}(G_zz) = \int_0^\infty e^{(-t/\tau_µ)} G_zz(t) { dt \over \tau_µ }

is the Laplace Transform of G_zz(t).

Experimental Asymmetry:

{\cal A} = { N_+ - N_- \over N_+ + N_- } \qquad or

{\cal A} = { b_- + (1-\alpha) + (1+\alphaß) A_+ {\cal L}(G_zz) \over b_+ + (1+\alpha) + (1-\alphaß) A_+ {\cal L}(G_zz) }

where b_± = { B_+ ± B_- \over R\epsilon_+ }, \alpha = { \epsilon_- \over \epsilon_+ }, ß = { A_- \over A_+ }.

Approximations: B_± \approx 0, A_± \approx A and \epsilon_+ \approx \epsilon_-, giving \alpha \approx ß \approx 1, \qquad b_± \approx 0 \qquad and

{\cal A} \approx { (1-\alpha) \over (1+\alpha) } + { 2 \over (1+\alpha) } A_+ {\cal L}(G_zz)

where (1-\alpha) / (1+\alpha) is the ``baseline'' asymmetry for totally unpolarized muons.

Avoided Level Crossing Resonance

This is particularly useful when one is looking for conditions of enhanced relaxation, the most common being those values of the applied magnetic field for which the muon Zeeman splitting matches a transition between energy levels of the nuclear spins, leading to the possibility of a ``flip-flop'' of spins in which the muon and the nucleus simultaneously change levels with no change in total energy.

Such conditions are referred to as either ``level crossing resonance'' (LCR) or ``avoided level crossing'' (ALC) depending mainly upon one's laboratory affiliation; the two acronyms refer to the same phenomena but one emphasizes the fact that levels never actually cross in quantum mechanics.

Nuclear Quadrupolar µALCR: The first such resonance studied in µSR was actually observed with TD-µSR methods following a suggestion by A. Abragam [ Abragam, 1984]. In that experiment [ Kreitzman et al., 1986], the µ⁺ Zeeman splitting at 81 Oe matches the (mostly) electric quadrupolar splitting between m_I=± 3/2 and m_I=± 1/2 levels of the spin-3/2 Cu nuclei, causing the resonant relaxation shown in Fig. 23.

• Figure 23. Longitudinal relaxation rate of muons in a single crystal of copper with the applied field (and the muon polarization) along the < 111 > crystalline axis [ Kreitzman et al., 1986].

``Muonated'' Radicals: Shortly after the discovery of nuclear quadrupolar µALCR it was realized that a much stronger resonance could occur through the hyperfine interactions of unpaired electrons with both the µ⁺ and the nuclear spins. The quantum mechanics of this system is considerably more complicated, but the vague notion of level-matching is still applicable: when the energy difference between two states in which both the muon and the nuclear spins have flipped approaches zero, resonant muon depolarization can occur.

One of the first applications of this principle to paramagnetic systems was in the case of the C₆F₆Mu^. radical, a paramagnetic molecule formed by addition of a Mu atom to a double bond in the hexafluorobenzene ring [ Kiefl et al., 1986].

• Figure 24. Field-differential µALCR in the C₆F₆Mu^. radical [ Kiefl et al., 1986].

Many such radicals have now been studied using this method, resulting in much detailed information about both the structure and the dynamics of these molecules, whose difference from the analogous radicals formed by H atom addition is often insignificant. Several new species never observed in any other way have been discovered using µALCR spectroscopy.

Muonium in Semiconductors: Another paramagnetic system amenable to study by these methods is the muonium-like center in semiconductors, where the unpaired electron may have hyperfine interactions not only with the µ⁺ but also with neighboring nuclei of the lattice. The resulting spectroscopy can be very rich, as illustrated in Fig. 25, and reveals the muonium site as well as a great deal of information about its electronic wavefuction. This has been used to obtain most of what is known about isolated atomic hydrogen in Si, GaAs, GaP and CuCl [ Kiefl et al., 1992a.

• Figure 25. Muonium-nuclear ALCR spectra in a single crystal of CuCl [ Schneider et al., 1990]. The Mu^I (low temperature) and Mu^II (high temperature) muonium states have the same hyperfine interactions with neighbouring nuclei but slightly different \omega₀ values.

Muon Spin Resonance

RF I-µSResonance

For the higher frequencies the same apparatus may be used at lower fields to study paramagnetic species such as muonated radicals. An interesting variation of this is seen in Fig. 26.

• Figure 26. Muonium resonance in mildly p-type Si at 10 K in a 127 MHz RF field, showing power-broadened resonances at the two muonium transition frequencies \nu₁₂ and \nu₂₃ (see Fig. 13) as well as a sharp two-photon resonance at the \nu₁₃ frequency.

One advantage of RF-µSR is that it can identify final states of muons in situations where stochastic processes render impossible the usual means of identification of these states by their distinctive time spectra. A typical example is the delayed formation of diamagnetic molecules following initial formation of Mu atoms: in TF-µSR the early Mu precession quickly dephases the muon polarization so that no signal can be seen from the diamagnetic final state unless the reaction times are short compared to the Mu frequency; in strong LF, however, not only is the Mu polarization ``held'' by the applied field but the population of the final state as a function of time can be determined by delaying the RF pulse (at the diamagnetic resonance frequency) relative to the time of arrival of the muons.

Perhaps the most exciting feature of RF-µSR is its ability to reveal the time evolution of the muon polarization during the RF irradiation, a capability denied to conventional magnetic resonance techniques which detect the same sort of electromagnetic signal as they use to drive the probe spins. This is a consequence of the fact that µ⁺SR detects high energy positrons whose registration by the counters is unaffected by the RF fields.

Muon Spin Echoes

As mentioned earlier, µSR detection techniques allow observation of the time evolution of the muon polarization during the RF irradiation. This can be especially informative in the case of microsecE pulse sequences. An example is shown in Fig. 27.

• Figure 27. Muon spin echo in the rotating reference frame during a (\Pi/2)_y-\tau-\Pi_y pulse sequence (shaded regions) where the muons arrive initially polarized along H₀ = H₀ z and the detectors are in the x and y directions. The small signal observed along the y axis indicates that the RF was slightly off resonance. The inset shows the geometry of the positron detectors and the polarization during the \Pi/2 pulse. [ Kreitzman et al., 1988].

µSR APPLICATIONS

Muonium Chemistry

Chemical Kinetics

Once the differences between Mu and H chemistry are well understood (as is now the case for many types of reactions), measurements of Mu reactivity (which are often easy) can be used to reliably predict the reactivity of H under circumstances where H atoms cannot even be detected. This is now entirely feasible but has not yet attracted much attention from the ``practical'' side of the chemistry community.

Radicals

Molecular Structure

Condensed Matter Physics

Hydrogen in Semiconductors

• Figure 28. The RF-µ⁺SR diamagnetic resonance amplitude in pure silicon, showing evidence for thermally activated transitions between various charge states and lattice sites of the µ⁺: Mu_BC = ``anomalous'' muonium (also called Mu*) localized at a Si-Si bond-centred site; Mu_T = quasi-free Mu atom diffusing rapidly between tetrahedral sites; µ⁺_BC and µ⁺_{T indicate freshly-ionized muons at the corresponding sites [ Hitti et al., 1994].

Quantum Diffusion

Motion of the µ⁺ in Metals: In impure or imperfect metallic crystals, the µ⁺ tends to trap at defects or at least be localized in their vicinity; the resultant temperature dependence of motional narrowing effects, used to measure µ⁺ mobility, can be quite complicated [ Moslang et al., 1983; Petzinger, 1982]. In pure, well-annealed crystals, however, one often observes the intrinsic interactions between the µ⁺ and the lattice. The hopping of positive muons in various pure metals has been studied extensively by means of ZF- and wLF-µ⁺SR measurements such as those shown in Fig. 18 [ Luke et al., 1991] as well as by TF-µ⁺SR motional narrowing experiments. These studies have revealed a qualitatively consistent temperature dependence typified by the classic example shown in Fig. 29: at high temperatures (above about 80 K in the case of Cu) the muon exhibits semiclassical thermally activated hopping over the energy barriers between adjacent interstitial sites; at lower temperatures (below about 20 K for Cu) the µ⁺ actually hops faster as T decreases, due to the increasing probability of quantum tunneling between sites, as the disorder due to lattice vibrations is reduced. The power law \tau_c^-1 \propto T^{-\alpha is predicted by theory [ Kondo, 1992] to have a much smaller exponent \alpha for screened positive muons in metals than in insulators, due to the additional dissipation in electronic degrees of freedom (often referred to as ``electron drag'' effects). At still lower T the intrinsic disorder of even pure high quality crystals begins to inhibit tunneling once again. Similar results have been observed in aluminum using indirect methods involving trapping of the µ⁺ at impurities.

• Figure 29. Temperature dependence of the µ⁺ hop rate \tau_c^-1 in pure copper crystals [ Luke et al., 1991].

Muonium Diffusion in Nonmetals: In ionic crystals, the µ⁺ tends to form a hydrogen bond with the most negative species in the lattice, such as F¯ or O^-- ions. Muon diffusion is thus suppressed until quite high temperatures, typically -sim 200 K [ Kiefl et al., 1990]. If the muon forms muonium, however, the Mu atom is more or less decoupled from local electric fields and usually diffuses freely through the lattice; interestingly, Mu diffusion is quite similar in both covalent and ionic nonmetals. An example of Mu diffusion in the semiconductor GaAs was shown earlier in Fig. 21 [ Schneider et al., 1992]; a startlingly similar result was obtained [ Kiefl et al., 1989] in an ionic insulator (KCl), as shown in Fig. 30.

• Figure 30. Temperature dependence of Mu hop rate in KCl [ Kiefl et al., 1989].

• Figure 31. Temperature dependence of the muonium hopping rate in solid nitrogen cryocrystals [ Storchak et al., 1994].

Magnetism

In magnetically ordered materials the muon (µ⁺ or µ^-) generally samples the local magnetic field at some preferred site or sites and reveals that field through its characteristic Larmor precession frequency [ Seeger and Schimmele, 1992]. The local field B_µ is composed of several contributions:

B_µ = H₀ + B_L + B_dip + B_HF

where H₀ is the externally applied field (if any), B_L is the Lorentz field produced by magnetic ``charges'' induced on the interior of a hypothetical spherical cavity around the muon site due to the average bulk magnetization of the medium, B_dip is the dipolar field due to all the microscopic moments within the Lorentz cavity and B_HF is a hyperfine field transmitted to the muon by its Fermi contact interactions with conduction electrons. The last two components are usually the focus of experiments on magnetically ordered systems, where B_µ is usually measured with H₀ = 0, as well as measurements of the muon Knight shift K_µ = (B_µ - H₀)/H₀ in paramagnets.

Interpretation of these fields requires a knowledge of the muon site, including zero-point motion and any tunneling between nearby sites, and can be quite difficult; nevertheless, in many cases it is easier than trying to sort out the ``core polarization'' effects of various interacting atomic electrons that transmit the hyperfine field to the nucleus in (e.g.) NMR.

One attractive feature of µSR in magnetism studies is that measurements of internal magnetic fields can be performed on unaligned polycrystalline or powdered samples almost as easily as on perfect single crystals. In zero applied field, if all the local fields have the same magnitude, the only effect of random directions is to reduce the muon precession amplitude to 2/3 of the value that would be observed if the fields were all perpendicular to the muon polarization. Thus ZF-µSR, which requires a minimum of apparatus, is often adequate for very detailed investigations of the temperature dependence of magnetic order and dynamics. Moreover, using ZF-µSR one can measure B_µ in crystals exhibiting antiferromagnetism or even more exotic forms of magnetic order as easily as in simple ferromagnets. A classic example is shown in Fig. 32 for the intermetallic compound MnSi, where the magnetic field due to itinerant electron moments forms a long-wavelength spiral with a small component along the axis of the spiral -- so-called helimagnetic order.

• Figure 32. Temperature dependence of the magnitudes of the internal fields at two different µ⁺ sites in a single crystal of the itinerant helimagnet MnSi.

• Figure 33. Comparision of neutron spin echo (NSE), ZF-µSR and ac-susceptibility measurements on the spin glass CuMn(5 at.\%), showing the time dependence of the static order parameter \xi (related to the fraction of Mn moments that have not yet moved appreciably) at several temperatures below the nominal glass transition at T_g = 27.4 K [ Uemura et al., 1984].
• Figure 34. Muon spin relaxation rate T₁^-1 due to fluctuating local fields in the antiferromagnetically frustrated two-dinensional Kagome lattice (shown inset) of SrCr₈Ga₄O₁₉ [ Uemura et al., 1994]. The increase of T₁^-1 with decreasing T is a manifestation of the gradual slowing down of Cr spin fluctuations; the saturation of this effect below T_g shows that the Cr moments never freeze even as T \to 0. This is in marked contrast with the situation in dilute spin-glasses like CuMn, where all fluctuations cease as T \to 0.

Superconductivity

Phase Diagrams: All of the copper oxide superconductors show either antiferromagnetic (AFM) or spin glass-like (SGL) order very close to (and sometimes overlapping) the superconducting (SC) phase, as first demonstrated for YBa₂Cu₃O_x by µSR experiments in 1988. Since then the temperature vs. doping phase diagrams of most HTSC materials have been mapped out using µSR.

• Figure 35. First TF-µSR measurement of the temperature dependence of the µ⁺ dephasing rate \sigma (proportional to \lambda^-2, the inverse square of the magnetic penetration depth, which is in turn proportional to the superconducting carrier density n_s) in a high-T_c cuprate superconductor -- in this case an unaligned sintered powder sample of La_1.85Sr_0.15CuO₄ in an applied field of 0.4 T [ Aeppli et al., 1987].

Magnetic Penetration Depth: The first application of µSR to HTSC was in measuring the magnetic penetration depth \lambda by fitting the high transverse field (HTF)-µSR time spectrum to a simple gaussian relaxation function and thus estimating the second moment of the frequency distribution due to the vortex lattice of magnetic flux penetrating a type II superconductor in the mixed state. Figure ? shows the earliest results, obtained in January 1987 [ Aeppli et al., 1987]. Since then an enormous body of data has been accumulated by this method, which is still the most reliable way to estimating \lambda in sintered, random powders. However, as evident in Fig. 11, for well-characterized single crystals one finds a lineshape much less symmetric than a gaussian, as predicted by London and later Gorkov models of the vortex lattice. The ability to make such measurements and interpret them correctly in terms of \lambda and \xi is a recent achievement of µSR, which is now poised to exploit these capabilities to legitimize (or, in some cases, debunk) the many published µSR results worldwide interpreting the gaussian-fitted linewidths (for sintered powder or aligned HTSC samples) in terms of \lambda.

Universal Correlations between T_c and n_s: A collection of such results has been assembled by Y.J. Uemura to produce the plot shown in Fig. 36, commonly known as an ``Uemura plot.'' [ Uemura et al., 1989] The ``universal'' linear dependence of T_c on n_s in the underdoped (less than optimal n_s) regime gives way to a ``turnover'' and eventual linear decrease of T_c with increasing n_s in the overdoped regime [ Niedermayer et al., 1993].

• Figure 36. Correlations between the superconducting carrier density n_s (as measured by the µ⁺ dephasing rate \sigma) and the superconducting transition temperature T_c for a wide variety of ``exotic'' superconductors including many HTSC; conventional BCS superconductors have higher n_s with lower T_c.

Non-s-Wave Pairing? Whether it be crudely characterized in terms of a gaussian relaxation rate \sigma or more accurately represented by the second moment of a lineshape, the muon dephasing rate due to the vortex lattice in a type II superconductor is proportional to the inverse square of the London penetration depth, \lambda^-2, which in turn is proportional to the superconducting carrier density n_s. Thus a measurement of the temperature dependence of \sigma is (apart from a normalization) also a measurement of the excitation of normal carriers across the superconducting energy gap which characterizes the pairing mechanism. A conventional BCS ``s-wave'' superconductor has a prescribed T-dependence of n_s, variations from which are taken as evidence for unconventional pairing mechanisms. While all this should, in the author's opinion, be taken with a grain of salt, still µSR has been an important contributor to the debate over the still-mysterious mechanism for HTSC. While the µSR data for the 18 K superconductor K₃C₆₀ are consistent with s-wave BCS superconductivity, the µSR results on certain organic and heavy-fermion superconductors show a T-dependence strongly suggestive of nodes in the gap [momentum-space directions in which the superconducting gap vanishes] [ Uemura et al., 1992]. A very similar T-dependence has been observed for n_s in various samples of YBa₂Cu₃O_6.95 -- most notably in high-quality single crystals -- as shown in Fig. 37.

• Figure 37. Temperature dependence of the µ⁺ dephasing rate \sigma (proportional to the superconducting carrier density n_s) in single crystals of YBa₂Cu₃O_6.95 in a TF-µSR experiment at 0.5 T (triangles) and 1.5 T (squares). An ``s-wave'' BCS superconductor is expected to show very weak T-dependence near T=0; in contrast, these data indicate an almost linear decrease from T=0.

µSR FACILITIES

Logistics of Accelerator Experiments

Furthermore, because a typical µSR experiment involves so many diverse technologies (from magnetic beam optics to charged particle detectors to magnets and cryostats to fast electronics and extensive on-line computer software) it makes an excellent training ground for students, who, once having mastered the techniques of µSR, can apply them almost unaltered to a wide variety of scientific topics.

Standard µSR Hardware

Detectors

While scintillation counters are simple, reliable, versatile and have excellent performance in the time domain, they have a number of frustrating features: First, even if the scintillator itself is small the light guides are bulky and inflexible (flexible fibre optics can be used but so far only at the expense of timing degradation and/or loss of light collection efficiency); this becomes increasingly inconvenient as increasing stopping luminosity reduces the scale of µSR samples and experiments. Second, most photomultiplier tubes are very sensitive to magnetic fields and must be magnetically shielded and/or removed to field-free regions by long light guides which again reduce time resolution. Third, scintillation counters have only crude position sensitivity, which is generally of secondary importance to µSR experiments, but could be useful. For example, by distinguishing one incoming muon from another and matching the outgoing positrons to the muons from which they arise, one may overcome the pile-up rate limitation normally imposed by allowing only one muon in the sample at a time; this technique has been demonstrated but never widely applied.

Potentially promising alternatives include modern proportional counters and solid-state barrier detectors, but neither of these have the required time resolution in their standard configurations; more development work is needed.

Electronics

• Figure 38. Schematic electronics logic diagram for two ``channels'' of a typical time-differential (TD)-µSR experiment (see Fig. 1).

Similar care must be taken to reject ``second E'' events in which a (possibly accidental) count in an E counter renders the identity of the stop pulse ambiguous. This function is handled internally in the LRS 4204 TDC, which is now the standard µSR clock primarily because it provides this function along with excellent time resolution and the ability to directly increment time bins in a CAMAC Histogramming Memory, thus relieving the data acquisition computer of all event-processing duties except periodic histogram and scaler readout.

There are numerous improvements and adaptations of this basic TD-µSR arrangement, of course, as well as several entirely different electronics setups for time-integral (I)-µSR techniques.

µSR Spectrometers

Target Vessels and Cryostats

µSR Software

Speculations

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Further Reading

The Proceedings of all six International Conferences on µSR to date have been published in Hyperfine Interactions as special Volumes 6 (1979), 8 (1981), 17-19 (1984), 31 (1986), 63-65 (1990) and 85-87 (1994). These (and any subsequent Proceedings) are the ideal sources for state-of-the-art developments in µSR.