Muon Species Research Articles

Muons in sulphur

Longitudinal-field repolarisation measurements of positive muons in sulphur suggest the existence of two paramagnetic muon species, consistent with a molecular radical and interstitial muonium. Avoided level-crossing measurements reveal a very weak signal at low temperatures (<100 K) , ascribed to the Δ M=1 resonance of a radical with a hyperfine constant of 233±5 MHz . We report further μSR experiments in sulphur, which attempt to shed light on the dynamics of muon charge states in this material. Above 200 K, transverse-field (TF) measurements show two diamagnetic signals, one of small amplitude and low relaxation λ, the other with high λ and a T-dependent amplitude. From 200 to 300 K the amplitude of the “fast” component increases rapidly, suggesting that the species it represents is formed in a thermally activated process from a paramagnetic precursor. Its relaxation follows a similar trend, but further investigation shows that λ for this component falls above 300 K while the amplitude remains essentially constant up to the β-phase transition. The amplitude drops in the β-phase and the fast relaxation disappears in the melt, where most of the polarisation is found in a slowly relaxing signal. While no TF radical signals have been found at any temperature, very recent low-field measurements at low temperatures show features consistent with an extremely quickly relaxing prompt muonium fraction, a possible precursor for the “fast” diamagnetic component.

A vacancy-related muon species in crystalline silicon

A new muonium centre, termed Mu V, with weak hyperfine interaction has been discovered recently in crystalline silicon in longitudinal field quenching μSR (LFQ) experiments (Schefzik et al., Solid State Commun. 107 (1998) 395). The signatures of this species were found in intrinsic, but not in doped samples (dopant concentration larger than 10 16 cm −3 ). The centre is not formed promptly but results from a reaction in which normal muonium transforms into the novel species. Since from LFQ experiments one can obtain only a rough estimate of the hyperfine tensor of Mu V it is now determined from zero-field (ZF) μSR experiments. The hyperfine interaction is found to be axially symmetric around the 〈1 1 0〉 crystallographic axis with small hyperfine parameters. Properly rescaled it essentially agrees with the hyperfine tensor of the hydrogen centre VH, which has been discovered recently by Bech Nielsen et al. (Phys. Rev. Lett. 79 (1997) 1507) and which has been attributed to hydrogen trapped in vacancies. Accordingly Mu V is interpreted as muonium trapped in a vacancy, in agreement with the interpretation given on the basis of the earlier LFQ data.

Oxygen-related anisotropic muonium centres in crystalline silicon

Abstract A new muonium centre, termed Mux, with anisotropic hyperfine interaction has been discovered in crystalline silicon in transverse-field muon spin rotation experiments. Its hyperfine tensor is very close to that of bond centred muonium (MuBC). The new muonium centre has been observed only in some of the many samples investigated. A correlation between dopant and oxygen concentration in the investigated samples and the observation of Mux has been found, which indicates that Mux might be related to a oxygen-muoniuni centre. According to the hyperfine tensor of Mux the new muonium species is attributed to an oxygen-MuBC complex. The Mux signal is affected by applied electrostatic fields in contrast with the signal of all other known muon species.

Evidence for a novel muon species in crystalline silicon

Investigations of positive muons ( μ +) in crystalline silicon employing the longitudinal field-quenching (LFQ) technique give strong evidence for the existence of a novel paramagnetic muon species with a small anisotropic hyperfine interaction. It adds to the list of known muon species, i.e., normal and anomalous muonium and the diamagnetic muon species. The signatures of the novel species are found in intrinsic but not in doped samples (dopant concentration about 10 16 cm −3). The novel species is not formed promptly but results from a reaction in which normal muonium transforms into the novel species. The reaction rate constant at 10 K was found to be about 10 6 s −1. The hyperfine coupling of the novel paramagnetic muon species corresponds, after rescaling, to that of a hydrogen center termed VH, which has been discovered recently by Bech Nielsen et al. ( Phys. Rev. Lett., 79, 1997, 1507) and which has been attributed to hydrogen trapped in vacancies. The LFQ data are tentatively interpreted in terms of trapping of normal muonium in vacancies that are created during the slowing-down of implanted muons close to the end of their stopping tracks.

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.

Time‐differential radio‐frequency muon spin resonance (TD‐RFμSR) technique at a pulsed muon beam

Longitudinal‐field μSR methods, e.g., radio‐frequency μ+ spin resonance (RFμSR), are well suited to investigate dynamic processes that destroy the phase coherence of the muon spin ensemble. Additional information on relaxation processes of the muon species under investigation is obtained from time‐differential (TD) data acquisition.

Muon studies in organic conductors

This paper contains a brief review of muon studies of organic conductors with emphasis on polymers. The species created by implanted muons in both semiconducting polymers and polymers doped to metallic values of conductivity are described. Both diamagnetic centres and paramagnetic radicals are formed in the semiconducting materials. The mechanisms of interaction of the muon with the polymers and the longitudinal relaxation of the muon species are discussed. In highly conducting polymers the muons are primarily present in diamagnetic states. Implanted muons have also been used to study both organic chargetransfer salts and Rb3C60in the superconducting state.

Positive muons in spin-crossover complex [Fe(NCS)2(phen)2

Diamagnetic muon species in a spin-crossover complex diisothiocyanatobisphenanthrolineiron(II) were investigated: two species with different spin relaxation functions and rates were observed over the range from 21 to 293 K. The sum of their yields changed at about the spin transition temperature of the complex, where the spin state of the complex changes from paramagnetic to diamagnetic when it is cooled down across the temperature. The results suggest that paramagnetic muon species could exhibit diamagnetic muon-like μSR signals in paramagnetic environment.

Applications of μSR to the study of fullerenes

A review is given of the current status of μSR results on muonium states in the pure fullerenes C60 and C70 as well as on the alkali doped series of C60. In particular the results of studies of the hyperfine interaction of the muonated radicals are discussed in order to determine their electronic distribution and the rotational ordering and dynamics of the fullerenes. Unique information on the alkali doped C60 species is obtained from studies of the endohedral muonium species.

Electric field dependence of muonium atom formation in solid nitrogen

Muon spin precession signals arising from both muonium (μ +e - or Mu) and diamagnetic muon species have been studied in condensed molecular nitrogen in the temperature range 10–78 K. Muonium is formed both in “prompt” epithermal processes and in “delayed” convergence of the thermalized μ + with an electron from the muon ionization track. The latter process is strongly correlated with changes in the electron mobility in solid N 2; it is inhibited by an electric field E ∼ 3.5 kV/cm applied in the direction of the uonn motion and enchanced by a comparable electric field in the opposite direction, indicating that the μ + stops on average about 60 nm “downstream” of the e - liberated in its last ionization of the medium.

Field dependence of the longitudinal muon polarization for anomalous muonium.

An unusual longitudinal-magnetic-field dependence of the muon polarization for anomalous muonium (${\mathrm{Mu}}^{\mathrm{*}}$) in polycrystalline semiconductor targets is predicted, and has been observed for Si. As a function of field, the experimentally measured muon polarization at 53 K exhibits a cusp, i.e., a discontinuous jump in the slope from negative to positive, at 340(5) mT, in good agreement with theory. This occurs because the effective field on the ${\mathrm{\ensuremath{\mu}}}^{+}$ vanishes at this external field as the angle between it and the ${\mathrm{Mu}}^{\mathrm{*}}$ symmetry axis approaches 90\ifmmode^\circ\else\textdegree\fi{}. At 200 K no cusp is seen because ${\mathrm{Mu}}^{\mathrm{*}}$ has been thermally annihilated. Fractional populations of the different muonium species are derived from the 53-K data. All ${\mathrm{Mu}}^{\mathrm{*}}$ and other anisotropic centers, including molecular radicals, should exhibit such a cusp, allowing the possibility of observing such species even in disordered or amorphous solids where spectroscopic detection might be impossible. Integral counting (and therefore very high data rates) can be used for these observations.

A μSR study on chemical behaviors of positive muons in crystalline tris/acetylacetonato/metal/III/ complexes

Chemical behaviours of positive muons implanted in crystalline tris/acetylacetonato/chromium/III/, iron/III/, and cobalt/III/ were investigated by means of the muon spin relaxation technique. The muon spin relaxation function was exponential in tris/acetylacetonato/iron/III/ and Gaussian in tris/acetylacetonato/cobalt/III/, suggesting that the muon spin relaxation may be ascribed to the interaction between the muon spin and a fluctuating electronic spin in the former and between the muon and a static nuclear spin in the latter. The yield of diamagnetic muon species was found to be nearly unity in these three complexes. Based on the analysis of the relaxation function, it is likely that the positive muon implanted in the cobalt/III/ complex will reside at a distance of about 1.7 A from the cobalt atom.

Muon chemistry and µ.S.R.

An understanding of the experimental muon spin rotation technique is provided. The temperature dependence of the p-hyperfine coupling constants for muonic radicals formed in [2H1]ethene, [2H3]ethene and [2H4]ethene is analysed and compared to the ones for the corresponding protonic species. The barrier height restricting internal rotation is also studied for the same range of radicals. Finally, the family of ions and radicals of the type RXY (R = H,Mu; X,Y = C,O,N) is discussed with regard to their vibrational frequencies which provide a measure of the extent of the large amplitude vibrations for the muonic species.

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 in which the rates, taken as square box functions of time, describe the charge exchange regime wherein muonium is both formed and ionized by subsequent collisions. Two post charge exchange extensions of this model are now considered. Following the charge exchange region, in the loss model, it is assumed that the rate of muonium formation is zero while the ionization rate is not. On the other hand, the capture model assumes that the ionization rate is zero while the muonium formation rate is not. Fits are presented to the available data for argon, krypton, xenon and neon. Since argon has both diamagnetic and paramagnetic signals then independent fits to each may be compared. A single set of fitting parameters has been found which describes both signals. This single fit requires a further extension of the models. In particular, a missing fraction must be assumed! The missing fraction model is also required when fits are made to the krypton and xenon data. On the other hand, the fit to the neon data is inconclusive. This is the first suggestion that a missing fraction may exist in the pure noble gases. Such missing fractions have been well established in condensed phases.

Muonic species

The large amplitude vibrations arising in muonic species are examined through a non-adiabatic study of the muonium substituted hydrogen molecular ion Mu + 2. Steric effects in organic muonic radicals are described and the muon spin rotation (μSR) spectra for radicals formed in butene-2-4-ol illustrated in comparison with deuterated ethenes. The proposal for a contribution of electronic origin to the isotope effect is questioned.

Partial spin depolarization of muonium in ice

Muon spin precession signals arising from both muonium and a diamagnetic muon species have been studied in single crystal of ice over the temperature range 90–263 K. Conversion of initial signal amplitudes to fractions of muon polarization reveals that for temperatures above 200 K part of the original muon polarization is unaccounted for. Such a missing fraction is well known for liquid water. but was not found in earlier work on polycrystalline ice. Muonium signals in Polycrystalline ice were reanalyzed using a non-exponential decay function appropriate to the powder pattern spectrum. Smaller muonium fractions were found. consistent with those determined for single crystall, thus confirming the existence of the missing fraction. The origin of the missing fraction is discussed, and it is proposed that non-reactive spin exchange encounters between muonium and hydrogen atoms may be the cause.

The inclusion of d-type Gaussian functions in the analytic method for the calculation of electrostatic molecular potentials. Interaction of a proton or a positive muon with carbon monoxide

The analytic method for the calculation of electrostatic molecular potentials is extended to allow for the inclusion of d-type Gaussian functions in the molecular basis set. The method is illustrated with the isopotential maps for the species CO, HCO+ and HOC+ and the sensitivity of such maps to the choice of basis set is highlighted.The family of ions and radicals of the type HXY (X, Y = C, O, N) is discussed with particular regard to the effect of isotopic substitution of protium by muonium. Vibrational frequencies are calculated to provide a measure of the extent of the large-amplitude vibrations anticipated in the muonic species.

Muonic isotope effects and non-adiabatic natural orbitals for the isotopically substituted hydrogen molecular ion

The meaning of the Born–Oppenheimer approximation with respect to muonic species is discussed. The value for D0, the dissociation energy from the lowest rovibrational level, is calculated to be 255.758 kJ mol–1 for H+2, in comparison with an observed value for D0 of ⩽ 255.765 kJ mol–1; for Mu+2 the value for D0 is calculated to be 229.82 kJ mol–1. Expectation values of 〈R〉 are given for non-adiabatic wavefunctions.Nuclear and electronic natural orbitals are illustrated for the molecules T+2, H+2, Mu+2, MuT+ and MuH+.

Observation of two distinct diamagnetic muon signals in the liquid phase using selective paramagnetic relaxation

Using selective paramagnetic relaxation, two distinct diamagnetic muon signals have been detected in solutions of manganous nitrate hexahydrate in acetone and in methanol. This is the first time that separate signals have been observed for chemically diamagnetic muon species in the liquid phase.

Charge exchange of muons in gases. II. Application of kinetic equations to the pressure dependence of muon-spin-resonance signals

Kinetic theory has recently been used to describe the spin dynamics associated with the stopping process encountered by highly energetic muons as they thermalize in a single-component gas. The result is a pair of rate equations that describe the dynamics encountered by the diamagnetic and paramagnetic muon species. This dynamical system has motion generated by the hyperfine interaction of the paramagnetic muon species (muonium) and by rates which are products of time-dependent rate constants for the electron-capture and -loss processes with the number density of the moderating gas. These rates are positive by definition. Using a finite-width-step-function approximation to the time dependence of the rates, analytic solutions of these equations are obtained and related to the muon spin polarization of the diamagnetic and paramagnetic species. Line shapes are obtained for the amplitudes of these polarizations as a function of the length of the charge exchange regime ${t}_{c}$. This time duration is inversely proportional to the number density of the moderating gas. The line shapes have two general features, namely, (i) they are constants when the duration of the charge exchange region is short (high-number densities), and (ii) they have resonances for long durations (low-number densities). Also, in general, the amplitudes of the "singlet" (muonium hyperfine frequency) term and the "triplet" (muonium Larmor frequency) term are not equal. Fits to the available experimental noble-gas data are presented using the rate constants and the time duration of the charge exchange region as parameters. The theoretical predictions suggest further experimental studies be made, in particular, to see whether the resonances are experimentally resolvable or not.