Measuring macroscopic diffusion of positive muons in metals

Conclusions We have discussed some schemes for measuring the diffusion coefficient for positive muons in metals, using diffusion over macroscopic distances. The method offers some promise of yielding diffusion information over wider temperature ranges and a greater variety of metals than the presently used technique. The major difficulty in applying the laminate scheme will be fabricating suitable, well-characterized targets that are thick enough to stop a significant fraction of the ⁺ beam. The use of a surface (i.e., 4 MeV) ⁺ beam appears most promising, but this brings with it formidable technical problems in heating and cooling the targets. We may hope that these problems will be attacked vigorously and effectively in the near future.Research performed under the auspices of the U.S. Department of Energy.     

Diffusion of positive muons and other light particles in metals

The paper gives a survey of the principal techniques available for the experimental determination of the diffusivityD ⁺(T) of positive muons in crystals (Gurevich technique, trapping, longitudinal muon-spin relaxation, transverse muon-spin relaxation in superconductors) and discusses their strengths and weaknesses. The main theoretical ideas of the quantum theory of diffusion are outlined and the distinction between different mechanisms is emphasized. It is argued that at high temperaturesT the so-called adiabatic regime with a preexponential factor of the diffusivity of the order of magnitude _D d ² ( _D = Debye frequency of the host crystal,d=jump distance of the muons) always exists. In the fcc metals and in the case of¹H in Nb it is followed by a so-called Flynn-Stoneham regime at intermediate temperatures, whereas for ⁺ in Nb and-Fe such a regime is not observed. Instead, in these cases the adiabatic regime appears to go over directly to the few-phonon regime of incoherent tunnelling between adjacent ground states, leading to the one-phononD ⁺ ~T law at low temperatures.The metal best-studied with regard to muon diffusion,-Fe, is used to illustrate the theoretical analysis of experimental results in some detail. In an Appendix the theoretical expressions required for the quantitative determination ofD ⁺ by the Gurevich technique are collected.     

On the hyperfine field values measured at positive muons in metals

The enhancement of conduction electron density at the position of a positive muon implanted into a metal is calculated in a simple model. Some problems in the interpretation of muon spin frequencies in ferromagnets are pointed out.     

Study of the trapping of positive muons by extended defects in aluminum

PulsedSR spectra were measured with surface ⁺ in neutron irradiated aluminum as a function of specimen temperature above room temperature. It was found that the depolarization rate of ⁺ increases and the precession frequency decreases as the specimen temperature is raised. This fact was attributed to the trapping of ⁺ by radiation induced voids and suggests the importance of ⁺-surface interactions in the case of voids.     

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.     

Quantum diffusion of muons in metals

We consider the influence of electrons on the quantum diffusion of muons in metals. The temperature dependence of the muon-diffusion coefficient is found.     

Quantum diffusion of positive muons

Recent studies of the diffusion of positive muons in metals, particularly aluminium, are reviewed. At low temperatures, quantum tunnelling is an important process for the mobility of the muons and experiments aimed at the study of tunnelling in the presence of phonons and conduction electrons are discussed. The concept of quantum diffusion is introduced and the conditions for quantum diffusion and quantum propagation (i.e. band-like motion) in normal and superconducting metals are compared.     

The localization and thermal diffusion of positive muons in niobium

The depolarization rate for spin polarized ⁺ particles implanted into a high purity niobium crystal was studied as a function of temperature. The results were analyzed in terms of nuclear dipolar field inhomogeneity due to the host⁹³Nb, and the local field averaging effect of the muon's motion. An analysis is presented in terms of the structure of the muon wavefunction and parameters characterizing the muon diffusion.Supported in part by the U. S. National Science FoundationSupported by the U. S. Energy Research and Development AdministrationSupported by the U. S. National Aeronautics and Space Administration     

The anomalous magnetic moment of positive and negative muons

The anomalous g-factor a ≡ (g?2)/2 has been measured for muons of both charges in the Muon Storage Ring at CERN. The two results, a_μ⁺ = 1165910(12) × 10^?9 and a_μ^? = 1165936(12) × 10^?9, are in good agreement with each other, and combine to give a mean a_μ = 1165922(9) × 10^?9, which is very close to the most recent theoretical prediction 1165921(10) × 10^?9. For the experimental results, the total statistical and systematic error is given. The measurements thus confirm the remarkable QED calculation plus hadronic contribution, and serve as a precise verification of the CPT theorem for muons.     

The spin turning in ferromagnetic Gd studied by positive muons

We have studied the muon precession frequency in a ferromagnetic single crystal of Gd metal. The overall features of our findings are compatible with earlier results on polycrystalline material. In the temperature region between 245 and 220 K where the Gd magnetization starts to turn away from the c-axis, we observe an increase in the muon depolarization rate, and a complex precession signal which can be separated into two frequency components meaning that spin turning does not occur simultaneously in different parts of the sample (domains). From these more detailed data follows that previously obtained values forB _fc andB _dip can not both be correct. Two explanations for our new result are possible: EitherB _fc undergoes a change around 230 K which is directly coupled to the spin turning angle, or the value of the dipolar field contribution used in the earlier evaluation is too low. This imposes some uncertainty as to the value of the angle at the onset of spin turning derived fromSR frequencies.     

Muon method for structural defects investigation in ferromagnetic metals

Equations describing depolarization of positive muons implanted in a pure crystalline ferromagnetic metal with structural defects are derived. In particular cases, solutions of these equations are obtained. In terms of the proposed model some experimental consequences are discussed.     

Electric field gradient for interstitial positive muons in fcc metals: An AB Initio calculation of size effect

The electric field gradients (EFGs) resulting from interstitial point defects in fcc metals have been investigated. The defect induced charge density, used to evaluate the valence effect EFG, is calculated self-consistently in the density functional formalism. An ab initio calculation of the size effect EFG is carried out for a positive muon at an octahedral site in the fcc lattice in the elastic continuum model. The components of the strain field tensor are evaluated assuming the lattice of dressed point ions interacting through the screened Coulomb potential. No adjustable parameter has been used. The theoretical results are in good agreement with the experimental values within experimental uncertainties. It is emphasized that both the strain and conduction electron contributions are equally important in the estimation of the electric field gradient.     

Local fields at positive muons in magnetic substances

The various contributions to the local field at a positive mouon in ferro- and antiferromagnetic substances are discussed. Formulas for the calculation of the dipolar field tensors are derived. The influence of the oscillation of the muon around its equilibrium site is analyzed.Supported in part by the Swiss National Science Foundation.