Inhomogeneous quantum diffusion of muons in solids

This article deals with the problems raised when a muon(muonium) quantum diffusion in a crystal is highly inhomogeneous. It is shown how static disorder arising from the crystal doping influence the diffusion process and drastically changes both the time decay of the polarization function and the temperature dependence of the depolarization rate. The spin depolarization of muons moving in a spatially inhomogeneous defect potential and trapping of particles by the long-ranged traps is studied in detail. Most attention is given to the particle localization and delocalization phenomena resulting in the two-component behavior of muon polarization at low temperature. Finally, the experimental data on muon depolarization in insulators KCl, GaAs, N₂ and superconducting metals Al, V are analyzed.     

The positive muon and the positron as probes of defects

Conclusions A brief but hopefully general comparison has been made between muons and positrons as probes for the study of defects in metals. Since muon experiments are not only more demanding in manpower, cost and availability than positron experiments, they should be carefully designed in light of the knowledge that the muon is extremely sensitive to both intrinsic and extrinsic defects. Initial experiments should provide estimates of the muon diffusion coefficients as a function of sample temperature. High temperature hydrogen diffusion measurements do provide quidelines, although so far most of the observations of muon trapping have been made at low sample temperatures where hydrogen diffusion data do not exist. Given that the diffusion constant is known as a function of temperature, high-purity Fe²⁶ after low temperature electron irradiation is therefore a good candidate to study with muons. Since the defect type and concentration can be controlled in electron irradiated samples, such investigations could confirm the stated values of the diffusion constants in Fe thereby providing a new method for evaluating diffusion coefficients below Stage III in different metals.All the samples should be initially characterized by other less costly techniques to obtain, where possible, the concentration and type of extrinsic and intrinsic defects. Both transverse and longitudinal measurements should be made to unravel the question of different diffusion mechanisms versus defect-, impurity- or self-trapping.Provided that the full capabilities of the muon are utilized and coupled with complementary techniques, i.e., positron annihilation, the muon will constitute a useful new probe in deepening our understanding or defects in metals.Work supported by the US Dept. of Energy under contract DE-AC02-76CH00016.     

New results on diffusion in fcc metals

The Kondo model for the diffusion of light particles in metals has provided a satisfactory explanation for the low temperature diffusion rates for muons in the fcc metals Cu and Al. Explicit experiments which show the strong dependence of muon behaviour on the presence of conduction electrons have now been performed belowT=1 K in Al. Challenging new diffusion studies on fcc metals such as Pt are also presented.     

Muon spectroscopy for studying magnetism and protons and lithium dynamics in spinel manganese oxides

Polarised positive muons can be implanted into any type of material and rapidly thermalize, then the local magnetic environment dictates the evolution of muon spin vectors and provokes the muon depolarisation. The muon spin relaxation (μSR) technique provides interesting information on magnetism and spin dynamics in spinel lithium manganates insertion compounds. In this work, we compare the behaviour of muons into a lithium-rich spinel manganese oxide and its lithium extracted product. The chemical extraction of lithium from Li_1.33Mn_1.67O₄, where all the manganese is Mn^IV, is essentially a lithium by proton ion exchange process to give a protonated manganese oxide with spinel structure, H⁺–MnO₂. Muons clearly have showed the presence of protons in H⁺–MnO₂, and the movement of lithium ions or protons at increasing temperatures in both samples. Muons are quasi-static in these compounds, and they are located both in ‘regular’ lithium and proton sites and also in interstitial sites of the spinel structure, these latter being used during diffusion of lithium ions. Below 50 K, static muons behave as in a paramagnet, where Mn magnetic spins are slowing down and ordering near 6 and 14 K in the protonated and lithiated spinels, respectively.     

Positive muon as a probe of magnetization density in ferromagnetic metal vacancies

The hyperfine field at a positive muon site trapped in a ferromagnetic metal vacancy is calculated and found to be more positive than that in the interstitial position. It is suggested that the muons can be used to enrich our knowledge of the electron spin distributions in metal vacancies that so far has not been possible.     

Influence of lattice relaxation and zero-point motion on the hyperfine field at muons in ferromagnetic metals

Results of model calculations are given which show the large influences of the zero-point motion of the muon and of the lattice relaxation on the hyperfine fields at muons in ferromagnetic metals. The change in the spin density induced by substituting a magnetic host ion in the vicinity of the muon by a nonmagnetic impurity is estimated.     

Positive muon behavior in KCl with and without F centers

Positive muon behavior in KCl containing F centers has been studied. The muon spin depolarization rate showed a maximum near 120 K, which was not found in pure KCl. This is probably due to the fact that free positive muons are trapped by F centers in KCl. However, the binding energy between a positive muon and an F center is not large, so that muons detrap again above 150 K.     

Radio-frequency muon spin resonance (RFμSR) experiments on condensed matter

In this paper we present an overview of the radio-frequency muon spin resonance (RFμSR) technique, an analogue to continuous-wave NMR, and an introduction to time-integral (TI) and time-differential (TD) RFμSR on muons in diamagnetic or in paramagnetic environments. The general form of the resonance line for TI-RFμSR as well as the expression for the time-dependence of the longitudinal muon spin polarization at resonance are given. Since RFμSR does not require phase coherence of the muon spin ensemble, this technique allows us to investigate muon species that are generated by transitions from, or in the course of reactions of, a precursor muon species even if in transverse-field (TF) μSR measurements the signal is lost due to dephasing. This ability of RFμSR is clearly demonstrated by measurements on doped Si. In this example, at low temperatures, a very pronounced signal from a muon species in diamagnetic environment has been found in RFμSR measurements, whereas in TFμSR experiments only a very small signal from muons in diamagnetic environment could be detected and a large fraction of the implanted muons escaped detection. These findings could be interpreted in terms of the delayed formation of a diamagnetic muonium-dopant complex, and, due to the large diamagnetic RFμSR signal, the RFμSR technique is a unique tool to study how the variation of parameters and experimental conditions such as illumination affects formation and behavior of these complexes. First results obtained on illuminated boron doped Si are reported. However, as illustrated by the example of experiments on the muonated radical in solid C₆₀, results from conventional TI-RFμSR cannot always be interpreted unambiguously since different parameters, namely the fraction of muons forming the investigated muon species, the longitudinal and the transverse relaxation rates, have similar effects on height and shape of the RFμSR resonance line. These ambiguities, however, may be resolved by collecting time-differential data. With this extension RFμSR becomes a very powerful complementary method to TFμSR in the studies of dynamic effects.     

Electronic structure, dynamics, and metastability of muonium in semiconductors

Muonium centers are light hydrogen-like centers formed when positive muons are stopped in crystalline semiconductors. Detailed information on the hyperfine structure, dynamics and metastability of muonium are obtained using a combination of muon spin rotation or relaxation, muon level-crossing resonance and related methods. The expected close similarity to hydrogen, especially with regard to electronic structure, is important since the equivalent information on isolated hydrogen is either less detailed or completely absent. There are also interesting differences between muonium and hydrogen. In particular muonium dynamics are expected to exhibit enhanced quantum mechanical effects since the muon has only 1/9th the proton mass. In this paper we review the current status of experiments.     

Observation of “decoupled” diamagnetic muon states in Alkali halides by muon spin resonance

The states of positive muons in KCl, NaCl and KI were studied with the muon spin resonance method under a 3 kG decoupling longitudinal field, revealing a considerably larger fraction of diamagnetic muon state than observed by the conventional spin rotation method. The origin of this fraction, which increases with temperature, is attributed to a muonium to muon transition in solids.     

Missing fractions in heavy gases: Xe and CCl4?

The spin polarization of positive muons thermalized in Xe has been measured as a function of pressure up to 4660 Torr (6.1 atm) by the muon spin rotation (MSR) technique. At 4660 Torr, triplet muonium (F=1, M=1) accounts for about 40% of the initial muon polarization and no significant signal from diamagnetic muons has been observed. The unexpectedly slow recovery of the polarization in Xe at high pressures is discussed in conjunction with similar results seen in CCl₄ and CHCl₃ vapors.     

Effects of electron correlations,dynamical screening and plasmon excitations on muon diffusion in metals

An investigation is made of the effects of electron-electron interaction on muon diffusion in metals. It is shown that electron-electron correlation plays an important role in the motion of muons. The equation-of-motion method is used to calculate the correlation function. It is shown that electron correlations effectively reduce the muon hopping rate at low temperatures. It is also shown that the effect of dynamic screening increases the hopping rate. We found that due to plasmon excitation, the hopping rate is reduced by a factor which can be as larger as one to two orders of magnitude.     

Measurement of the depolarization rate of positive muons in copper and aluminum

Positive muon spin rotation experiments for polycrystalline Cu and Al from 19K to temperatures near the melting points are reported. At low temperatures, the depolarization associated with localization of the muons at octahedral interstitial sites is seen in Cu, while in A? only slight depolarization is observed below 250K. At high temperatures, no evidence for trapping of positive muons at vacancies in thermal equilibrium is found for either metal. It is concluded that the muons either diffuse too slowly to find vacancies or, if they do find vacancies, are bound too weakly to remain trapped.     

On zero-field anisotropic static Kubo-Toyabe relaxation functions

For muons which have thermalized in the allowed stopping sites of a given crystal, zero-field static Kubo-Toyabe spin dynamics is expressed in terms of the crystal frame spherical harmonic coefficients of the local classical random magnetic field distributions associated with each stopping site. The resulting muon spin polarization involves nine observable relaxation functions which are the counter frame spherical tensor expansion coefficients of the second-rank dynamic motion tensor. These relaxation functions can be measured simultaneously using the skewed field technique for the same experimental conditions, that is, with a single apparatus and with a single crystal-counter frame orientation. The local field distributions of, in general, arbitrary symmetry are classical approximations to the magnetic field interactions between the spin of the muon and the spins of the nuclei associated with each site at which the muon has stopped. They are characterized by equating their moments with the quantal moments generated by the quantal magnetic fields. The observable consequences of the anisotropic second-order moments associated with the planar sites of hexagonal crystals are used as an illustration.     

Muon level crossing resonance in niobium

A useful guide to detecting quadrupole level crossing resonance (QLCR) spectra is to search in the vicinity of B_E\cdot(\gamma_n/\gamma_μ), where B_E is the onset magnetic field for the decoupling of the quadrupole interaction, as seen in muon spin rotation linewidths. More detailed predictions of the positions and intensities of the resonances require numerical simulations taking account of the local geometry of the muon site. We present such simulations for muons adjacent to nuclei of spin 9/2 and demonstrate a pronounced dependence on the anisotropy of the quadrupole coupling tensor. Simulations for the specific cases of muons located at the octahedral and tetrahedral interstices in niobium metal are compared with the experimentally detected spectrum in a polycrystalline sample. This revised version was published online in August 2006 with corrections to the Cover Date.     

Muon diffusion and level crossing in copper

We have studied the quantum diffusion of positive muons in pure copper over the temperature range 12 mK≤T≤150 K using weak longitudinal field μSR. Below 150 K, this technique has proved to be the most sensitive to the muon hop rate. Our final results for the behaviour of the muon hop rate are well explained within the framework of theories for the quantum diffusion of light interstitials in metals of Kondo, Yamada and others. In addition, the use of level-crossing resonance has allowed us to measure the electric quadrupole interaction strength (and sign) of the copper nuclei, ω_Q= −3.314(7) μS⁻¹. These results have enabled us to show that the muon occupies the same octahedral site at all the temperatures studied, ruling out the possibility of metastable muon sites contributing to any significant portion of the muon polarization.     

Zero field isotropic Muonium Kubo — Toyabe relaxation functions

Static zero field Gaussian Kubo — Toyabe relaxation functions for muons in isotropic muonium atoms are presented. That is, as with diamagnetic muons, an average of the spin dynamics of a muon in an isolated isotropic ground state muonium atom is taken over an isotropic Gaussian continuous classical local random magnetic field distribution. This motion approximates the exact quantal spin dynamics generated by the dipole-dipole interactions between the muonium atom and the surrounding nuclear spins associated with the site at which the muonium atom has stopped. Expressions are derived for triplet muonium only since, in general, singlet muonium is not observed. For normal nuclear spins and ground state muonium, the resulting relaxation functions are identical to the standard diamagnetic function (except for a shift in the time scale).     

Magnetic dynamics of dilute iron nano-clusters in silver films from Mössbauer spectroscopy and muon spin rotation

A silver film containing nanometer size clusters of iron (nominal conc. 1 at%) has been studied by Mössbauer spectroscopy and Low-Energy Muon Spin Rotation. Below about 20 K spin glass freezing due to interparticle interactions is found from both methods. Whereas Mössbauer spectra are insensitive to the fast fluctuations of cluster moments above spin glass freezing temperature, muon spin rotation in magnetic fields applied perpendicular to the polarized muon spins allows tracing the fluctuations of superparamagnetic moments. The temperature dependence of the damping of the muon spin rotation signal shows Arrhenius behavior between 10 to 100 K. Depending on the assumed shape of damping the activation energy of superparamagnetic fluctuations of cluster moments ranges between about 20 K ·k _B and 40 K ·k _B . Above about 120 K muon spin depolarization indicates diffusion and trapping of muons.     

Production of pulsed ultraslow muons and first μSR experiments on thin metallic and magnetic films

Standard muon spin rotation (μSR) spectroscopy implants 4 MeV spin-polarized positive muons to investigate the bulk properties of matter. Success in producing epithermal muons opens interesting possibilities for studying ultrathin films, interfaces, and even surfaces. At the ISIS Facility, Rutherford Appleton Laboratory (Chilton, UK), we have produced a pulsed ultraslow muon beam (E< 20 eV) and have performed the first μSR experiments. Due to the pulsed feature, the implantation time is automatically determined and, by adjusting the final muon energy between about 8 and 20 eV, depth slicing experiments are possible down to monolayers distances. We describe slicing experiments across a 20 nm copper film on quartz substrate with evidence for a 2 nm copper oxide surface layer. A preliminary experiment on a hexagonal cobalt film suggests the existence of muon precession in the local magnetic field. The results are discussed in relation to the morphological features of the film.     

Muon spin resonance — A variant of magnetic resonance

Spin polarized positive muons injected in matter serve as magnetic probes for the investigation of various properties. The evolution of muon spin polarization rests on the same basis as in conventional magnetic resonance techniques. The background of the technique, different variants of the experimental set-up, and potential and limitations of the muon as a probe are described.