The use of positive muons in metal physics

The present status of work in metal physics by the new method of “muon spin rotation” is reviewed. This spectroscopy is based on the spin interactions of positive or negative muons and resembles NMR as far as the interpretation of interactions in metals is concerned. The positive muon behaves in several respects as a light isotope of hydrogen in metals. Local properties like site symmetry, local magnetic field, dynamic effects from surrounding spins as well as effects from the diffusion of the particle itself can be measured with high sensitivity.A brief review of the technical aspects is given. The problems of diffusion of light positive particles in metals are discussed, with regard to specific mechanisms at low temperatures, trapping of muons by impurities, etc. The local electronic structure around this kind of impurity in normal metals as well as ferromagnets has been subject to a large nnumber of studies. Other applications include the interaction of muons with other kinds of defects, the study of metal hydrides and measurements on the dynamics of spin glasses.     

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.     

Diffusion and localization of muons at very low temperatures

Conclusion The measurements of muon diffusion below 2 K have provided many new data and ideas in the metals Cu and Al, although the understanding is far from complete. Especially intriguing are theAlMn data with their very clear temperature and concentration dependence over a wide temperature range. My feeling is that we now approach well-defined situations where the basic properties of muon diffusion and localization can be seen. Still I would hesitate to draw any definite conclusions regarding the existence of e. g. coherent motion, and I strongly hope that the existing theories of low-temperature diffusion can be further developed, hopefully yielding less extreme results for muons in the very low temperature range.     

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.     

Negative muon capture in very light atoms

The transition rates for unbound muons to be captured into atomic bound states are calculated as functions of (1) incident muon center-of-mass energy, (2) muon principal quantum number n, and (3) muon (final) angular momentum l, for the hydrogen, helium, and lithium atoms. These rates reflect differences in electron binding energies. At muon energies of several hundred electron volts, lithium K-shell electrons are more likely to be ejected than the L-shell electron, while this behavior is reversed for energies ? 10 eV. However, in each case when the capture rate is folded with a muon stopping power function, the result is that more than half of the unbound muons are absorbed above 75 eV. Implications for experiments which look at muon transfer processes are noted.     

Self-trapping of positively charged particles in metals

It is shown that the existence of a metastable state in which positrons in metals are “self-trapped” by strong interaction with the lattice gives rise to an anomalous temperature dependence in positron annihilation properties. The “intermediate” temperature variation of the shape of the annihilation photon line discovered by MacKenzieet al. is well accounted for by this mechanism; alternative interpretations in terms of thermal expansion effects may be refuted. This result calls for considerable revision of some of the published monovacancy formation energies obtained from positron annihilation measurements. Approximate criteria for the existence and the metastability of a selftrapped state of positively charged particles in metals are given. It is found that metastable self-trapping may occur for positrons; hydrogen isotopes and positive muons should be self-trapped in configurations that are always stable relative to the Bloch-wave states of these particles.     

Empirical information on quantum diffusion

The applicability of the quantum theory of diffusion to experiments on muon motion in selected metals is examined. Thereby the conventional picture of immediate self-trapping of the muon is employed. Small-polaron hopping of muons at intermediate temperatures seems to be established. There are indications for coherent diffusion in several metals at low temperatures. The quantitative behavior of the diffusion coefficient or transfer rate at low temperatures found in Al and Cu is in disgreement with the theoretical predictions.Many discussions with O. Hartmann, E. Karlsson, L.O. Norlin, T.O. Niinikoski, D. Richter, J.M. Welter, and A. Yaouanc are grateful acknowledged.     

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.     

On knock-on electrons of cosmic-ray muons in condensed matter (at 12° N)

The dependence of knock-on electron probability on the momenta of muons producing the knock-on electrons in condensed matter has been investigated by using a counter controlled cloud chamber. A search has also been made on the nature of the energy spectrum of the knock-on electrons from muons in the momentum region (0.80±0.05) to (1.70±0.05) Bev/c. It appears from the results of the measurements that there is a very weak dependence of knock-on electron probability on the primary muon momentum in the investigated muon spectral region (i.e. (0.80±0.05) to (1.70±0.05) Bev/c). The angular distributions and hence the energy spectra of the emergent electrons are found to be independent of muon momentum.     

Diffusion of hydrogen in metals

The present status of our understanding of the diffusion of hydrogen in metals, both experimental and theoretical, is reviewed. Discussions are focused on the mechanism of diffusion of hydrogen isotopes H, D and T in f.c.c. and b.c.c. metals; the positive muon (μ ⁺) is referred to where appropriate. An up-to-date compilation of diffusion data as a function of temperature and isotope mass has been made, and a clear distinction in general diffusion behaviour in f.c.c. and b.c.c. metals is noted. Subsequently, the results obtained from the Gorsky effect, nuclear magnetic resonance and quasi-elastic neutron scattering that provide information on elementary jump processes are discussed.

A conceptual framework of the quantum diffusion of light interstitials in metals is given, including the recent Kondo theory that emphasizes the crucial importance of particle-conduction electron interactions in the diffusion process, especially at low temperatures. It is shown with the help of recent estimates of the tunneling matrix element that the overall feature of diffusion of hydrogen isotopes in b.c.c. metals as well as μ ⁺ in f.c.c. metals can be explained consistently within the frame presented here.

Finally, recent advances in the diffusion studies on hydrogen in b.c.c. metals are described. They include a re-analysis of quench-recovery experiments that manifested nearly athermal diffusion of H, D and T in Ta at low temperatures, and an enormous enhancement of the diffusivity under stress (superdiffusion) observed for H and D in V.     

Sites and diffusion for muons and hydrogen in titanium hydrides

Low temperature sites for muons implanted in TiHx have been found to be a mixture of interstitial and substitutional sites, with substitutional occupancy determined by the probability that a muon in an interstitial site will have a vacant nearest neighbor substitutional site. As with ZrHx, activation from the interstitial site is observed below 300 K. From the depolarization rate in the substitutional site, the muon likely displaces the neighboring H atoms by about 0.1 A. Diffusion for the substitutional muons occurs above room temperature with an activation of about 0.38 eV, which is less than the 0.505 eV for hydrogen vacancy motion observed by NMR. To explain this the muon transition rate to a vacancy must be less than that of hydrogen.     

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.     

μ+SR in imperfect solids

The use of ⁺SR as a probe of point defects in metals is discussed. The importance of separating the effects due to the presence of the muon from those intrinsic to the defect-host system is stressed. It is suggested that studies of magnetic impurities in non-magnetic hosts may yield data which are relatively probe-independent. A theoretical analysis of the depolarization of muons coupled to magnetic impurities is outlined. It is pointed out that measurements of the field dependence of the depolarization rate can give information about impurity spin correlation times.Work performed under the auspices of the U. S. Department of Energy.     

Energy spectrum of cosmic ray muons above 10 TeV according to BUST data

The energy spectrum of cosmic ray muons in the range of several TeV to PeV obtained through the analysis of multiple interactions of muons (the pair meter technique) in the Baksan Underground Scintillation Telescope (BUST) is presented. Results are compared with prior BUST data on the muon energy spectrum measurements and data of other experiments, along with calculations for different muon spectrum models.     

Comparison between positive muon research and positron annihilation in the study of crystalline defects

Conclusions Positive muon experiments are quite useful to study open space crystalline defects measured at lower temperatures. They are powerful to study radiation damage or quenched-in defects or plastic deformation. By controlling the energy of muons, they can be used for the detection of defects at different depths by nondestructive testing.     

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.     

Application of the Nuclear Reaction Analysis Online Technique to Study the Diffusion of Deuterium in Metals

Nuclear reaction analysis online technique has been applied to study the diffusion of deuterium in metals. Investigations ensuring the application of the new method have been performed. These investigations include the development of a device for diffusion annealing of samples in the chamber of an accelerator and an algorithm for taking into account the effect of radiation defects on the diffusion coefficients. Test measurements of the diffusion coefficients of deuterium in nickel in the temperature range from 130 to–60°С have been performed. For negative temperatures, experimental data on the diffusion of a hydrogen isotope in a metal have been obtained for the first time by a direct method and it has been shown that the online nuclear reaction analysis provides reliable data.     

Longitudinal muon spin relaxation in the organic metal polypyrrole

Longitudinal and zero field measurements of the muon spin relaxation function are reported for muons implanted in the conducting polymer polypyrrole. The observed muon relaxation can be accounted for by interaction with polaron excitations in the conducting polymer. Evidence is found for polaron localisation at temperatures below ∼ 50 K. The temperature dependence of the polaron mobility derived from the muon data is consistent with direct electrical conductivity measurements.     

Atmospheric effects in the intensity of cosmic ray muon bundles

We analyze experimental data on cosmic ray muon bundles collected with a DECOR coordinate detector. Substantial variations in the intensity of the events are observed during the experiment. These variations are found to be caused by changes in atmospheric conditions. This study is the first to obtain experimental estimates of the temperature and barometric coefficients for muon bundles. It is shown that the observed effect can be explained by changes in the side distribution function of EAS muons.     

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.