Ralph Eric Turner

The Analysis of Muonium Hyperfine Interaction Measurements of Thermal Rate Constants for Addition Reactions

A new model is developed for the analysis of muon spin relaxation (μSR) measurements of muonium (Mu) reaction rates with other chemical species. The kinetics component of the model is specifically developed for addition reactions but can be extended to direct reactions. The polarization component of the model assumes that the adduct formed possesses a single, effective spin precession frequency. The complete model is solved to obtain an analytic expression for the μSR signal as a function of time. Both a time‐ordered sequence method and a Boltzmann spin equation method give equivalent solutions. These solutions are analyzed under various kinetic conditions. It is concluded that the coefficient controlling the decay of the μSR signal is closely related to the adduct formation rate constant, i.e., the high pressure limit of the apparent bimolecular addition rate constant. In the most favorable case, the decay constant gives the adduct formation rate constant directly at all pressures of buffer gas.

Determination of very slow μ+ hop rates in Cu by LLF-μSRhop rates in Cu by LLF-μSR

Muon spin relaxation in low (weak) longitudinal magnetic field (LLF-μSR) provides a means of independently determining the static dipolar width Δ characterizing the μ+ lattice site and the correlation time τc for μ+ hopping, in a manner that is nearly model-independent for τc and especially accurate in the near-static limit (τc>τμ). The advantages of this method are illustrated by its application to muon hopping in Cu near the τc maximum around 50 K.

Thermal hopping ofμ+ between “FμF” centres in NaF

Strongly hydrogen-bonded diamagnetic “FμF” centres are formed by theμ+ in a wide variety of fluoride crystals. Hydrogen atoms are expected to form similar “FHF” complexes. Through the “motional narrowing” of the zero-field muon relaxation function in NaF, we have observed an Arrhenius temperature dependence of the dissociation rate of theFμF complex, yielding a binding energy of 1700 (200) K for theμ+ in theFμF centre.

On the pressure dependence of the polarization of diamagnetic muons and muonium in pure noble gases: Is there a missing fraction?

The amplitudes of the signals in μSR exhibit pressure dependencies which are associated with the stopping dynamics of muonium atoms and diamagnetic muon species observed when muons are thermalized in pure noble gases. To explain this effect, a set of coupled rate equations, with time dependent rates and based upon quantal Boltzmann equations, have been developed to describe the spin dynamics for the thermalization of the two species. These, by definition positive, rates depend upon time through the translational single particle density operators associated with each species. Thus, to exactly solve the spin dynamics, the coupled kinetic Boltzmann equations for the stopping process must also be solved. Furthermore, the rate equations also contain spin dynamics generated by the muonium hyperfine interaction. It is the presence of this hyperfine interaction which leads to the loss of polarization for low pressure gases. The coupled quantal rate equations have been solved for a model of the stopping dynamics i...

A comparison of local and global single Gaussian approximations to time dynamics: One‐dimensional systems

The detailed calculation of the dynamics of a chemical system is usually not considered due to the size and cost of the computation. It is thus useful to examine various approximation methods. Such methods first need to be tried out on simple systems, like one‐dimensional motion. Here two approaches to approximating the solutions of the Schrodinger and von Neumann equations by single time‐dependent Gaussians are explored and contrasted, explicitly for one‐dimensional barrier penetration. The first approach, in which no tunneling occurs, is local in nature and characterized by an expansion of the equations of motion to second order about the average position of the Gaussian wave packet or about the average position and momentum of the Gaussian Wigner function. This approach was first introduced by Heller [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. Here both Heller’s approach and a more general truncation method are considered. Indeed tunneling can be incorporated if second‐order terms in the quantal vo...

Muonium in bulk fused quartz: Test case for RAHD relaxation theory

Muonium (μ+e−) in bulk fused quartz is a unique system in that theμ+ spin polarization (in the muonium state) relaxes almost entirely via random anisotropic hyperfine distortions (RAHD). As such, this system provides an excellent test case for a new RAHD spin relaxation theory. This theory is quantitatively compared to static zero field data and the functional characteristics in both the high field and dynamic limits are considered as well.

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.

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.

On the observables associated with zero and intermediate applied external field muon spin rotation

For zero and intermediate applied external fields, the dynamics of the static spin polarization associated with muons which have thermalized in the allowed stopping sites of a given solid is expressed in terms of counter and crystal frame spherical harmonic tensor expansions of the second-rank dynamical motion tensor. Coefficients of these expansions, usually termed relaxation functions, are related to observable longitudinal, co-planar transverse and perpendicular transverse relaxation functions which can be measured simultaneously using the skewed field technique. Since the motion tensor is of second rank then, in general, nine observable relaxation functions exist. This stands in contrast to the traditional high-field longitudinal and transverse techniques employed in muon spin rotation (ΜSR) experiments wherein only three of the nine observable relaxation functions are non-zero. These traditional experiments, usually employing different apparatus, are carried out separately. Even when the other six relaxation functions are identically zero in zero and intermediate applied field experiments, they still contain information about the stopping probabilities or the dynamical symmetries associated with the muon sites, that is, that such symmetries are not present or that the sites with these symmetries are orientated in such a way that the statistical sums of their relaxation functions are zero. Thus these relaxation functions can then be used asexperimental confirmation of expected stopping probabilities associated with one type of site for which there are different orientations or for aligned sites with appropriate dynamical symmetries.

Charge exchange of muons in gases: Experimental implications from rate theory

The effects of the charge exchange process on muon spin dynamics have been investigated using a density operator formalism with special interest placed upon the diamagnetic muon and paramagnetic muonium signals observed after thermalization. In the charge exchange region the dynamics of the spin density operator is assumed to be determined by the muonium hyperfine interaction and by electron capture and loss processes for muons. Analytical expressions are obtained for the amplitudes and phases of the diamagnetic muon and paramagnetic muonium signals as a function of the duration of the charge exchange region,tc, which is inversely proportional to the number density of the moderating gas. The theoretical signals exhibit three features which have, as yet, to be experimentally observed, namely: (i) that the amplitudes associated with the muonium Larmor frequency and with the hyperfine frequency are not, in general, equal, (ii) that all the amplitudes are, in general, damped oscillatory functions oftc (temperature/pressure) and (iii) that phase jumps occur when an amplitude decreases to zero and then increases with falling pressure. Fits to the experimental argon data are discussed in light of the above points.