In this section we provide an overview of the technical aspects of gSR spectroscopy without going into details. The aim is to bring the reader to some understanding of what is involved in a ~SR experiment and what is meant by the termini technici which he will encounter when studying original ~tSR literature.
2.1. General aspects
Basically, ~SR is the measurement of the temporal development of the spatial orientation of the spins of muons which have been implanted in the material o f interest with all spins initially fixed in one direction (complete muon spin polarization). The three names covered by the acronym ~tSR, namely muon spin rotation, relaxation or resonance, refer loosely to different means of observation.
When speaking of Muon Spin Rotation one emphasizes the measurement of coherent Larmor precession of the ensemble of muon spins in the magnetic field present at the site of the muons embedded in the sample. The spin rotation frequency is a direct measure of the magnitude of this field. To produce a precessional motion the field must have a component perpendicular to direction of the muon spin. Usually, this can be achieved by applying an external magnetic field in this direction, and Muon Spin Rotation is often synonymous with transverse field ~tSR.
Muon Spin Relaxation refers to the observation of incoherent motions of the muon spins which result in a loss of polarization with time. This will occur if the magnetic field sensed by the ensemble of implanted muons is broadly distributed. I f the local field each muon sees in addition fuctuates randomly during a muon's life we observe what is called "dynamic depolarization", but also a stationary distributed field causes depolarization by phase incoherence ("static depolarization"). These two cases must be clearly distinguished. The situation corresponds to the two relaxation times T1 (spin- lattice) and T2 (spin-spin) in NMR. Muon Spin Relaxation measurements can be carried out without observing spin rotation and thus are possible in zero applied field or with a longitudinally applied field (i.e., a field applied parallel to the muon spin direction at the moment of implantation). Longitudinal field measurements are the most appropriate way to obtain a clear distinction between static and dynamic muon spin depolarization. Muon Spin Relaxation hence mostly refers to zero or longitudinal field ~tSR.
The term Muon Spin Resonance defines a NMR-type technique. In the presence of a static external field one induces muon spin flips by the application of resonant radio frequency (Kitaoka et al. 1982, Kreitzman 1990, Hampele et al. 1990, Nishiyama 1992, Scheuermann et al. 1997, Cottrell et al. 1997) or microwave field (Kreitzman et al.
1994). The resonance condition is detected via a loss of muon polarization. As in NMR, frequency shifts and linewidth are the sensitive parameters.
There are still other means to carry out ~tSR experiments. Most notably among them is the so-called (Avoided) Level Crossing Resonance (LCR or ALC, Kreitzman 1986, Kiefl 1986, Heming et al. 1986, Kiefl and Kreitzman 1992, Leon 1994). In this review we will
~tSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 63
discuss only Muon Spin Rotation/Relaxation measurements since the other techniques have not yet been applied to the materials under consideration. LCR has its main impact on muon radical chemistry. Muon Spin Resonance experiments are most important for the study of semiconductors.
In principle, ~SR can be performed either with positive or negative muons. The latter behave like heavy electrons and will be captured into Bohr orbits by the atoms in the sample material. They will quickly reach the l s ground state, whose orbital radius, however, is comparable to nuclear radii. Negative muons thus sense predominantly the effects of the protonic charge distribution and the result is primarily of interest to nuclear physics. In addition, the W will be captured quickly by the nucleus, meaning that its life time is considerably shortened compared to the free muon lifetime (-2.2 ~ts), especially for atoms with large Z. Some aspects of WSR on oxygen are of interest to the study of high-T~ superconductors (Nishida 1992) but in general, condensed matter physics or chemistry studies are carried out exclusively with positive muons. They act chemically like ions of a light isotope of hydrogen. As will be discussed in some detail further below, the 9+ comes to rest at a site between the atoms (interstitial site in crystalline materials), since it is repelled by their nuclear charge. In consequence, it senses the magnetic field present near, but outside, the atoms. In the following, when speaking of ~SR without further specification we always refer to positive muons.
The magnetic field at the site of the muon in a magnetic compound is, at least in part, created by the dipole moments on neighboring paramagnetic atoms or ions. As stated, we can gain information on the magnitude of the field from the muon spin rotation frequency and also (to a limited degree) on the spatial arrangement of the moments. From muon spin relaxation times we gain knowledge on the dynamics and the couplings of these atomic moments. Consequently, one of the important features of ~tSR is its capability to serve as a microscopic probe of magnetism.
Competing microscopic methods in magnetism are the hyperfine methods such as M6ssbauer spectroscopy, Perturbed Angular Distribution, NMR or EPR. All these techniques measure the magnetic field at the nucleus of a probe atom, this being quite often the magnetic atom in a compound. EPR, like optical spectroscopy, actually measures the disturbance of electronic states by the electron-nuclear hyperfine coupling, and the hyperfine coupling constant is completely dominated by the magnetic field at the nucleus. ~tSR senses the interstitial field, a quite different quantity. Furthermore, the muon normally possesses no electron shell of its own in conductors and also in most magnetic non-metals. This is another basic difference from the hyperfine methods, where the field at the nucleus is either entirely created, or at least strongly influenced by, the electronic shell of the probe atom. In certain materials, notably in semiconductors, the muon may capture an electron forming muonium, which, however, is too sensitive to magnetic fields to be useful in the study of magnetic materials (see also sect. 2.4).
Relative to M6ssbauer spectroscopy, which is probably the most important hyperfme technique for the study of magnetic materials, one has no limitation in temperature in
~SR (no Lamb-M6ssbauer factor). Also, ~tSR can be applied to liquids and gases, but
64 G.M. KALVIUS et al.
this is of course of little consequence in magnetism. A comparison between Mrssbauer spectroscopy and ~tSR can be found in Asch et al. (1988a).
One of the advantages of gSR compared to NMR is the possibility of measuring relaxation rates of paramagnets in true zero field. The extremely high resolving power of microwave resonance techniques often causes the signal to be lost in disordered magnetic systems (strong inhomogeneous line broadening), while 9SR, since it is mainly sensitive to interstitial dipolar fields rather than nuclear hyperfme (contact) fields, is still capable of detecting a spectrum. Finally, it should be pointed out, that for ~SR no limitation exists as to the type of atoms contained in the sample material.
Clearly, the most important method for gaining information on intrinsic properties of magnets is neutron scattering, which is primarily a probe of long-range correlations. ~SR, a local probe, cannot directly obtain a spin structure the way neutron diffraction can, but the dipolar contribution to the field at the muon site (see sect. 3.1) is strongly dependent on the spatial arrangement of the surrounding moments. Hence the gSR spectrum provides a consistency test on any spin structure derived by neutrons. Differences may well exist, however, because gSR is a much more local probe, which needs no large coherence length. Consequently, gSR is particularly sensitive to short-range order and other forms of disordered magnetism.
Magnetic scattering of photons can in principle provide similar information on magnetic structures as neutron diffraction. With neutrons, the cross sections for nuclear (lattice structure) and magnetic (spin structure) scattering are roughly of equal magnitude, but the magnetic photon scattering cross section is many orders of magnitude smaller than that for charge scattering. Only the development of powerful synchrotron X-ray sources has allowed this obstacle to be overcome and studies of magnetism via X-ray scattering have emerged in recent years (see, for example, Isaacs et al. 1989 and Hannon et al. 1989).
A special case is X-ray resonant magnetic scattering (XRMS) where the photon energy is set exactly on an absorption edge energy. This enhances the magnetic scattering cross section by six orders of magnitude or more. The Miv and Mv absorption edges of the actinides are particularly favorable, so this technique has been used largely for the study of actinide compounds to date. XRMS has the additional advantage that it is both element and site (through the slight variation of edge energy) sensitive. This method can be a competitor for ~tSR because it also allows the detection of weak magnetism (i.e., moments of the order 10-2~B or less) as found, for example, in heavy-fermion compounds (Isaacs et al. 1990, 1995). In contrast to gSR, it is at present impossible to deduce the size (even the order of magnitude) of the moment from XRMS data. Nonetheless, the spin structure involving such small moments can be found. It also allows (especially in conjunction with a related method, X-ray magnetic dichroism), at least in principle, separation of orbital and spin contributions to the moment, which is a rather fundamental question of actinide magnetism, gSR can give no information in this respect.
There is also a considerable difference in the time window for the study of spin- dynamical processes when comparing ~tSR to other methods (see fig. 1). A brief discussion of the different time windows in the various nuclear techniques has been given by Dattagupta (1989). It is of special importance that ~tSR, as shown in fig. 1, bridges the
~tSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS Remanence
= •
ac S u s c e p t i b i l i t y
65
p, S R
P A C
M6ssbauer
Neutrons
I ' I ' I ' I ' I I ' I ' I ' I ' I ' I
10 .4 10 -2 100 102 104 106 10 s 10 l° 1012 10 T M 10 T M
Fluctuation Rate 1/'% (s -1)
Fig. 1. Time scales o f various tech- niques to study magnetism. 1/r is the fluctuation rate of the field sensed by the nuclear probes or the rate of spin fluctuations.
gap between neutron scattering on the one side, and bulk magnetic measurements (i.e., ac susceptibility) on the other (Uemura 1989, Karlsson 1995). It is possible to relate the muon spin relaxation rate to the general scattering function S(q, co) (Lovesey et al. 1990).
~SR as a largely local method integrates over q space. This is of course a serious loss of information. On the other hand, gSR measures at co ~ 0, a region not accessible to neutron scattering.
Studies in magnetism are but one application of ~SR. Other fields where ~tSR has given important information is the diffusion of light interstitials, especially with regard to quantum diffusion in metals, semiconductors and insulators (Kehr et al. 1982, Kondo 1986, Kadono 1990, Prokof'ev 1994, Storchak et al. 1996, Karlsson 1996). Other very active fields are applications to chemistry with emphasis on chemical kinetics especially in connection with radical formation (Brewer et al. 1975, Walker 1983, Fischer 1984, S.EJ. Cox and Symons 1986, Roduner 1990, 1999, Fleming and Senba 1992) and also the study of hydrogen states in semiconductors (Chow et al. 1995). ~tSR in life sciences is discussed by Nagamine (1999) in an article on Exotic applications ofmuons.
From the point of view of the physics involved, ~tSR is closely related to NMR (Slichter 1978). De Renzi (1999) offers in a recent review a detailed comparison between NMR and ~SR using well selected examples. From the point view of apparatus and detection schemes, gSR is more akin to nuclear or particle physics experiments. Reviews covering the gSR technique are numerous (e.g.: Brewer et al. 1975, Karlsson 1982, 1995, Chappert 1984, Schenck 1985, S.EJ. Cox 1987, Schatz and Weidinger 1992, Davis and Cox 1996, S.L. Lee et al. 1999), but since the technique has advanced, the older ones have lost some of their relevance. Tri-annually, a special international conference on ~tSR is held. Its proceedings were published in the journal Hyperfine Interactions up to the 1996 conference in Nikko (Japan). Proceedings of the 1999 conference (Les Diablerets, Switzerland) have appeared in Physica B (Roduner et al. 2000). Although ~SR emerged in the early 1970s, its application to magnetism became more widespread in the 1980s,
66 G.M. KALV1US et al.
Table 1
Some properties of the muon
Property Symbol Value Units
Rest mass m~ 206.77 m e (electron masses)
0.1126 mp (proton masses)
105.66 MeV/c 2
Mean life r~ 2.197 ~ts
Charge Q~ ±e (elementary charge)
Spin S~ h/2
g-factor g~ 2.00233
Magnetic moment #~ 4.84 x 10 3 /gB (Bohr magnetons)
8.88 #N (nuclear magnetons)
3.18 #p (proton moment)
Gyromagnetic ratio yv/2z 135.54 MHz/T
particularly with respect to spin-dynamical studies. Disordered and frustrated magnets, spin glasses, critical phenomena, Kondo-lattice and heavy-fermion systems, the interplay o f magnetism and superconductivity, especially in high-Tc superconductors and related materials, are some o f the subjects being actively studied with ~tSR.
2.2. Properties o f the muon
The muon, when discovered in cosmic ray studies by Neddermeyer and Anderson (1938), was thought to be the Yukawa meson which transmits the strong forces between nucleons, but about 10 years later it became apparent that it is not the particle postulated by Yukawa (which is in fact the pion or z[ meson). Within the standard model, the muon is a fundamental particle. As such it is a point-like object without internal structure. With its neutrino it forms the second generation o f the family o f leptons, the first family being the electron and the third the tauon, always together with their neutrinos. Fundamental particles come in particle-antiparticle pairs. By convention, the ~+ is the antiparticle and strictly speaking, we are concerned with "antimuon spin rotation/relaxation". A short review o f the history o f the muon has recently been given by T.D. Lee (1994). Due to its past, the muon is still on occasion referred to as the ~t meson. This is o f course wrong since mesons are composite particles containing a quark and an antiquark. The properties o f the muon o f interest here are summarized in table 1.
One finds the muon to be ~200 times heavier than the electron and ~10 times lighter than the proton. The finite mean life o f the muon is long from the point o f view o f timing resolution o f modern electronic circuits and poses no detection problem. The muon carries electric charge and spin. Consequently it has a magnetic dipole moment which is small compared to the moment o f the electron but larger than the protonic moment. The size o f its magnetic moment makes the muon an ideal probe for the relatively weak interstitial
gSR STUDIES OF RARE-EARTH AND ACTINIDE MAGNETIC MATERIALS 67 fields in materials. The Larmor precession frequency of S~ in the field Bg at the muon site is given by:
fg = ( y J 2 g ) . Bg. (1)
The ~tSR signal can typically be followed over 5 r~ (i.e., ~11 ~ts). Assuming that S~
must make at least one half turn in this period, one obtains a minimum observable field value on the order of a mT. Good ~tSR spectrometers can resolve a precession period of 2 - 5 ns -1, putting the maximum observable field in the range of 3 - 5 T. Similar arguments put the limits for muon spin relaxation rates between 0.001 and 100 gs -1 under favorable conditions. It must be emphasized that the latter numbers refer to the decay rate of the muon spin polarization. In magnetism one is interested in the fluctuations of the magnetic moments in the sample which drive, via magnetic coupling, the muon spin depolarization.
The average fluctuation period of the moments provide the correlation time for the local field at the muon, which causes muon spin relaxation. Observed relaxation times and the underlying correlation times need to be connected by an appropriate model. We shall discuss these aspects whenever the need arises.
2.3. Muon generation and decay
Muons take part in numerous elementary particle reactions. For gSR applications, the decay of pions is the source used. That means for positive muons
zg + --* ~t + + v~ with r~ = 26 ns. (2)
Pions in turn are produced in medium energy nucleon-nucleon collisions. For example:
p + p --+ ~+ + p + n . (3)
Typically, one directs a beam of protons with a kinetic energy in the range 0.5-1 GeV onto a target of light nuclei such as Be or C.
Pion decay is mediated by the weak interaction and full parity violation comes into play.
In particular, the neutrino (here v~) always has left-handed chiral symmetry, meaning that its spin Sv (being h/2 like that of the muon) is oriented in the opposite direction to its linear momentum (Pv). The pion has spin zero. Conservation of momentum in its decay (eq. 2) requires that the neutrino (%) and the muon (~t +) are ejected 180 ° apart in the rest frame of the pion. Since the orientation of Sv is fixed to (Sv TJ, Pv), the same must hold for the muon (S~ Tl p~). We thus get perfectly spin polarized muons with their spin directed opposite to their line o f flight. The situation is illustrated in fig. 2. Under certain circumstances (to be discussed in the next section) the pion rest frame is identical with the laboratory frame ("surface muon" beam). The linear momentum given to the muon from pion decay at rest is 29.8 MeV/c which corresponds to a kinetic energy of 4.1 MeV
68 G.M. KALVIUS et al.
~t + /~+ V ~t
$ = 0
m
Fig. 2. Spin and momentum o f decay products o f the pion seen in the pion rest frame.
The decay of muons once more involves weak interactions and full parity violation.
For the positive muon one has
g+ --+ e + + ve + 9~. (4)
The situation here is more complex than in pion decay since there is now a three particle final state. The easily detectable signature of muon decay is the emitted positron (e+).
As in nuclear B-decay (also governed by the weak interaction), the kinetic energy of the positron (electron) is distributed continuously between E = 0 and E = Emax, where Emax is the decay energy arising from the mass difference of initial and final particles. In the case of muons, Emax = 52.3 MeV. The prominent effect of parity violation in muon decay is the anisotropic spatial distribution of the emitted positrons (see Commins 1973). The probability dP(6), E, t) of finding a positron with energy between E and E + dE at a time between t and t + dt in solid angle dO, located at the angle 6) with respect to the muon spin at the moment of decay is given by
dP - C(E)e -t/T~ W(E, 6)). (5)
dO dE dt
C(E) is the energy distribution function and W(E, 6)) is the angular distribution function of the decay positrons. The latter can be expressed as
W(E, 6)) = 1 + ao(E) cos 6), (6)
and is the quantity essential for ~SR. The factor a0 is called the initial asymmetry and is strongly dependent on positron energy. The positron detectors used in ~SR usually are not sensitive to the energy of the particle detected and thus one integrates from energy zero to Emax with C(E) as the weight function: this gives a0 = ½. In practice, the initial asymmetry is often somewhat smaller (typically a0 ~ 0.2) due to a number of experimental conditions. The resulting angular distribution is shown as a polar diagram in fig. 3.
Even for a0 -- 0.2 one has a substantial spatial asymmetry. It means that a detector counts 40% more positrons when the muon spin is oriented in its direction compared