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The quasi-mobility function of charge carriers with a specified energy for describing their dynamics using the kinetic equation is studied in the important case of two-term isotropic approximation. In the stationary case, the quasi-mobility function is independent of the source function of charge carriers and makes it possible to calculate the integral mobility. The correlation between the quasi-mobility and parameters of the system is analyzed. It is proved that this characteristic does not generally describe the contribution of charge carriers with a specified energy to the integral mobility. In the case of almost elastic scattering, the quasi-mobility, as is known, can have a clear physical meaning; however, in the case of the scattering of charge carriers at acoustic phonons in a solid, this quasi-mobility interpretation is found to be incorrect due to the specific features of the collision integral and the form of the quasi-mobility function.

The discrepancies between the quasi-elastic and inelastic approaches to the calculation of the electron and hole mobilities in diamond at low temperatures when the carrier scattering from acoustic phonons becomes significantly inelastic have been numerically estimated. The calculations showed that the mobility described by a close-to-equilibrium distribution function differs several times from that obtained within the quasi-elastic approach even at 20 K. The results obtained are important for interpreting the low-temperature electrical experiments on high-purity diamond single crystals.

The results of μ SR study of the behavior of shallow acceptor centers (AC) in diamond, silicon, and germanium are presented. It was found that the muonic atom, which imitates acceptor in semiconductors, is formed in the paramagnetic state in silicon and germanium at low temperatures whereas in diamond it is formed in the diamagnetic state. The hyperfine interaction constant for the aluminium AC in silicon was estimated for the first time. It was shown that the main contribution to the relaxation of the magnetic moment of the AC is due to interaction of the AC with the phonon via the Roman process in non-degenerated silicon at temperatures ⩽ 50 K . In the case of germanium, there is an experimental evidence for a change in the contribution of different phonon processes to the relaxation of the AC at temperature ∼ 10 K . The hole capture rate by the ionized boron acceptor in diamond was determined.

Polarized negative muons were used to study the dynamics of non-equilibrium charge carriers in diamond. From the behavior of the muon polarization it is concluded that the acceptor center μ B in diamond is formed in the diamagnetic state (with the probability close to unity) and then, by capturing a hole, turns to the paramagnetic state. The hole capture rate by an ionized acceptor center μ B − in synthetic diamond was found to depend on temperature as 1/ T α , where α = (1.5–2.5) at T > 80 K.

The mobilities of charge carriers in boron-doped diamond at low temperatures are estimated by numerically solving the Boltzmann equation for a steady-state nonequilibrium velocity distribution function of the carriers with allowance for their scattering by phonons. Estimates show that, at temperatures of up to 100 K, charge carriers with a negative mobility exist over a fairly broad (on the order of 100 K) range of their energies. In the steady-state case, the absolute mobility integrated over the distribution function turns out to be positive, but there are grounds to suppose that, in the case of an unsteady pulsed source of charge carriers, it may become negative.

We consider the negative conductivity of electrons in semiconductors excited by a picosecond laser pulse at low temperatures, due to the inelastic electron-phonon collisions. For the first time, the dependence of the deformation potential on the phonon wave number is taken into account. This dependence significantly changes the region of negative electron conductivity as a function of the phonon temperature.

We consider the negative mobility of electrons in diamond excited by a picosecond laser pulse at low temperatures, due to the inelastic electron-phonon collisions. For the first time the dependence of the deformation potential on the phonon wave number has been taken into account. This dependence changes significantly the region of the negative electron mobility as a function of the phonon temperature.

Kinetic equation has been solved analytically for electron density evolution in the positive muon track formed in solid argon. Muon polarization dependence on time due to muonium formation was estimated numerically. The hot (non-thermalized) electrons were shown to contribute to muonium fast formation. Additionally, it was shown that in the case of applied external electric field, the electron mobility estimations on the basis of μSR experiments could contradict to well-known data for thermal mobility. This is due to both electron heating by external electric field and muonium formation in the non-thermalized electron capture by muon.

A theoretical analysis is given to describe the evolution of the distribution function of non-equilibrium charge carriers, which were formed due to the capture of negative muon by a carbon atom. The electrons and holes thermalization by interaction with acoustic phonons is considered. Diffusion and mobility coefficients were calculated for non-equilibrium charge carriers in diamond at low medium temperatures (below 20 K) and the appearance of a negative mobility is predicted for some energies and temperatures. Results of numerical calculations of charged cloud evolution in diamond are presented. It is shown that at low temperatures the acceptor center μB is rapidly formed in a paramagnetic state due to collecting of holes at the charged center. The results explain the experimental negative muon polarization dependence versus temperature.

The results of investigating the behavior of the boron impurity in synthetic diamond and the effect of uniaxial static pressure on the aluminum acceptor in silicon are presented. The data on the hyperfine interaction and on the relaxation of the magnetic moment of Al acceptor in silicon were obtained. It was found that the uniaxial stress changed both the absolute value and the temperature dependence of the relaxation rate of the acceptor. Unlike the case in silicon, in CVD diamond the paramagnetic shift in the muon spin precession frequency was not observed within the accuracy of the measurements, and a missing fraction of muon polarization was found at T

The effect of uniaxial compression on the behavior of shallow aluminum acceptor centers in silicon has been studied. The μAl impurity atoms were created by implanting negative muons into silicon single crystals doped with phosphorus to 1.6×1013 cm−3 (sample 1) and 1.9×1013 cm−3 (sample 2). The muon polarization was studied in the temperature range 10–300 K. Measurements were performed in a magnetic field of 2.5 kG oriented perpendicularly to the muon spin. The samples were oriented so that the selected crystal axis ([111] and [100] in samples 1 and 2, respectively), the magnetic field, and the initial muon-spin polarization were mutually perpendicular. External pressure applied to the sample along the indicated crystal axis changed both the absolute value of the acceptor magnetic-moment relaxation rate and the character of its temperature dependence.

This work is aimed at creating a theoretical basis for analyzing and interpreting the results of μSR experiments in doped nondegenerate semiconductors at temperatures below 50 K, when the influence of kinetic processes at the endpoint of a muon stopping track on the behavior of its polarization is substantial. The effects related to the formation of free electrons and holes in a solid-state plasma at a muon track endpoint are shown to be responsible for the mere possibility of observing negative muon spin precession at the muon frequency in doped nondegenerate semiconductors at low temperatures. The Vangsness-Bloch equations are generalized to the case of parameters varying with time. A theory based on these generalized equations allowed us to interpret more correctly the available experimental results of mSR studies of semiconductors with the use of negative muons. We showed that the μSR method could be used to obtain information about the cross sections of exchange scattering of electrons and holes by impurity centers in the region of energies inaccessible to the other measurement techniques and to estimate the cross section of capture by a solitary charged Coulomb center at virtually all charge carrier concentrations and temperatures. Under the conditions when the Debye radius is larger than the mean distance between charged particles but smaller than the Thomson radius, the capture (recombination) cross section is described by a temperature dependence qualitatively different from that predicted by current theory.

Processes of ionization of shallow acceptor centers (ACs) in silicon are studied. In crystalline silicon samples with phosphorus (1.6×1013, 2.7×1013, and 2.3×1015cm−3) and boron (1.3×1015cm−3) impurities, μAl impurity atoms were produced by implantation of negative muons. It is found that thermal ionization is the main mechanism for ionizing the Al acceptor impurity in both p-type and n-type silicon with an impurity concentration of ≲1015cm−3 at T>45 K. The thermal ionization rate of Al ACs in Si varies from ∼105 to ∼106s−1 in the temperature range 45–55 K.

A system consisting of two interacting particles, one of which has the angular momentum 1/2 and the other of which has an arbitrary angular momentum (a muonium-like system) is considered. The possibility of reconstruction of all the elements of the spin density matrix with the help of the μSR method is shown. For entangled states (Einstein-Podolsky-Rosen and Schrodinger cat states), the time dependence of components of the polarization vector (μSR photographs) is demonstrated. It is shown that, to identify these states, two μSR photographs suffice.

The theory was proposed to explain the results of different experiments on investigation of acceptor center in silicon. It was shown that previous disagreement between the results of different experiments was connected with incorrect interpretation of experimental results. All experiments could be explained in assumption that the most reliable estimation of hyperfine constant for μ Al acceptor center in silicon is ∼25 MHz. Proposals for further experimental investigations of μ Al acceptor center in longitudinal magnetic field are given.

The behavior (in the temperature range of 4– 300 K ) of the polarization of the negative muon spin for more than 20 n- and p-type crystalline silicon samples with impurity concentrations from ∼10 12 to ∼10 20 cm −3 made it possible (a) to infer the mechanisms for the relaxation of the magnetic moment of the Al acceptor in Si at different impurity concentrations, including the concentration region above the critical concentration corresponding to the Mott (insulator-to-metal) transition; (b) to find the value of the hyperfine interaction constant for the acceptor and to estimate the density of the hole wave function on the nucleus of the impurity atom.

The interactions of the aluminum acceptor impurity in silicon are investigated using polarized negative muons. The polarization of negative muons is studied as a function of temperature on crystalline silicon samples with phosphorus (1.6×1013 cm−3) and boron (4.1×1018 cm−3) impurities. The measurements are performed in a magnetic field of 4.1 kG perpendicular to the muon spin, in the temperature range from 4 to 300 K. The experimental results show that, in phosphorus-doped n-type silicon, an μAl acceptor center is ionized in the temperature range T>50 K. For boron-doped silicon, the temperature dependence of the shift of the muon spin precession frequency is found to deviate from the 1/T Curie law in the temperature range T ≲ 50 K. The interactions of a μAl acceptor that may be responsible for the effects observed in the experiment are analyzed.

A study is made into the temperature dependence of residual polarization of negative muons in crystalline silicon with the concentration of impurity of the n-and p-types ranging from 8.7×1013 to 4.1× 1018 cm−3. The measurements are performed in a magnetic field of 1000 G transverse to the muon spin, in the temperature range from 4.2 to 300 K. The form of the temperature dependence of the relaxation rate v of the magnetic moment of the μAl0 acceptor in silicon is determined. For a nondegenerate semiconductor, the relaxation rate depends on temperature as v ∝ Tq (q ≈ 3). A variation in the behavior of the temperature dependence and a multiple increase in the relaxation rate are observed in the range of impurity concentration in excess of 1018 cm−3. The importance of phonon scattering and spin-exchange scattering of free charge carriers by an acceptor from the standpoint of relaxation of the acceptor magnetic moment is discussed. The constant of hyperfine interaction in an acceptor center formed by an atom of aluminum in silicon is estimated for the first time: |Ahf (Al)/2π| ∼ 2.5×106s−1.

Track electron kinetics is shown to be important for the correct understanding of μSR experiments. The quasi-neutrality restoring time for the region of muon or meso-ion location is investigated theoretically and numerically in different models relating to positive and negative muons. Simple approximate formulae for numerical results are presented. This time is found to be much less than the muon's lifetime for light gases and Ne at a liquid phase pressure. The foregoing makes it possible that meso-ion and electron interaction leads to the formation of Mu or muonium-like systems.

μSR histograms obtained from different regions of a target have to be slightly different since magnetic fields are inhomogeneous. Calculation of a correlation function of these histograms allows to estimate the correlation radius of inhomogeneities. This method was named μSR 2 . Application of the μSR 2 method for estimating the sizes of magnetic domains is analyzed. Results of the computer simulation of a μSR 2 experiment to measure domain sizes are given. It is shown that domain sizes can be estimated within the accelerator operating time allocated for a conventional μSR experiment.

A phenomenological derivation of the Wangsness–Bloch–Redfield equation for the density matrix of muonium-like systems is given in this paper. In the case of the isotropic relaxation, it is shown that the corresponding relaxation term is described by only two independent constants. A typical form of the density matrix of the isotropic hyperfine interaction is given, and it is shown that the density matrix is decomposed into diagonal blocks. Relaxation rates corresponding to the different blocks can differ by orders of magnitude.

Results on the temperature dependence of the residual polarization of negative muons in silicon with phosphorus ( 4.5×10 18 , 2.3×10 15 , and 3.2×10 12 cm −3 ) and aluminium (2.4×1018 and 2×10 14 cm −3 ) impurities are presented. The muon spin rotation (μSR) experiments were carried out in a magnetic field of 0.2 T and in the temperature range 4.2–300 K. In all investigated samples a relaxation of the muon spin and a shift of the spin-precession frequency were observed. The frequency shift (relative to the room-temperature value) amounts to 7×10−3 at 15 K. In the sample with a high concentration of phosphorus impurity ( 4.5×10 18 cm −3 ) damped and undamped components of the muon spin polarization were observed at T K . Hyperfine interaction between the magnetic moments of the muon and that of the electron shell of the muonic atom (acceptor centre – μAl) is estimated on the basis of the muon spin precession frequency shift data. The temperature dependence of the spin-lattice relaxation rate of the magnetic moment of the shallow acceptor centre in silicon in the absence of external stress is determined for the first time. It is found that the relaxation rate is well approximated by the power function ν(T)=CTq, where the parameter q lies between 2 and 3.

In the temperature range below 30 K a negative muon in Si forms an acceptor centre. The temperature dependencies of the transverse field relaxation rate and the paramagnetic shift of the muon spin rotation frequency are analysed. It is shown that results of experiments with negative muons in Si may be explained by phonons interacting with the electron moment. The hyperfine constant of the acceptor centre is estimated.

Temperature-dependent remanent polarization of negative muons in a silicon crystal doped with phosphorus (3.2 × 1012, 2.3 × 1015, and 4.5 × 1018 cm−3) and aluminum (2 × 1014 and 2.4 × 1018 cm−3) was examined. Measurements were made over the temperature range 4–300 K in a magnetic field of 2000 G perpendicular to the muon spin. Temperature dependence of the relaxation rate was determined for the magnetic moment of a shallow Al acceptor center in a nondeformed silicon sample, and the hyperfine interaction constant was estimated for the interaction between the magnetic moments of muon and electron shell of the muonic mAl atom in silicon.

The residual polarization of negative muons has been studied for phosphorus-doped and antimony-doped silicon crystals. The measurements were carried out in a transverse magnetic field of 0.1 T over the temperature region 4 K-300 K. The ionized and neutral states of the pseudo-acceptor were observed in antimony-doped silicon for the first time. The rate of transition from the neutral to the ionized state of the acceptor was found to be equal to over the temperature range 4 K-12 K. The estimated rates of relaxation of the magnetic moment of the acceptor-centre electron shell are and in phosphorus-doped silicon and and in antimony-doped silicon at 4 K and 15 K respectively. The experimental results obtained are interpreted in terms of spin-lattice relaxation of the acceptor magnetic moment and of the acceptor-donor pair formation.

The residual polarization of negative muons in crystal silicon samples with phosphorus (P: 1.6×1013 cm−3) and antimony (Sb: 2×1018 cm−3) impurities is investigated. The measurements are made in a 1000 G magnetic field oriented in a direction transverse to the muon spin in the temperature range 4–300 K. The relaxation rate and shift of the precession frequency in the silicon sample with the phosphorus impurity are measured more accurately than previously. It is found that in antimony-doped silicon the acceptor center µ A1 at temperatures below 30 K can be in both ionized and neutral states. The experimental data are interpreted on the basis of spin-lattice relaxation of the magnetic moment of an acceptor center, formation of acceptor-donor pairs, and recombination of charge carriers at the acceptor. Preliminary measurements showed a nonzero residual polarization of negative muons in germanium.

The dependence of the residual polarization of negative muons in n‐type Si with impurity concentration (1.6\pm 0.2)\times 1013\ cm-3 on temperature in the 10–300 K range has been investigated. Measurements were carried out in external magnetic field of 0.08 T transverse to the muon spin. Muon spin relaxation and frequency shift were observed at temperatures below 30 K. The relaxation rate at 30 K is equal to 0.25\pm 0.08\,μ s-1. The frequency shift at 20 K is equal to 7\times 10-3. Both the relaxation rate and the frequency shift grow with decrease of temperature. Below 30 K the relaxation rate is well described by the dependence \varLambda=bT-q, where q=2.8.

It is theoretically shown that in rare gases at P < /100 atm the external electric fields of the order of 100 V/cm could significantly change the value of electron density in the region of μ- or μ+ stopping point and affect the muonic atoms formation in plasma‐chemical reactions. The kinetic model of processes which might explain the results of μSR‐experiments for μ- in Ne and Ar both with and without external electric field is proposed.

The existence of an opportunity for research on the spatial‐time correlation of magnetic fields is demonstrated. A summation method for the time correlation function of μSR signals containing information about fluctuations of the muon polarization is proposed for use with an experimental installation in which the muon decay point is recorded. A description of the S.PNIP μSR2 installation, formerly used in experiments related to the search for rare muon decays, is also given.

When a negatively charged muon is captured by a silicon atom, the atom is transformed into a solitary acceptor center similar to an Al atom. An external electric field influences the formation process of the neutral acceptor center (A.C.). It is shown in this article that the behaviour of the muon polarization vector changes appreciably in electric fields with intensities E\,\gtrsim\,10 V/cm. We estimate the muon spin relaxation rate \varLambdaμ in the so‐called “dirty limit”; for example, interaction between an internal electric field from charged impurities and a nonzero dipole moment of the A.C. is taken into account. A phonon mechanism is proposed to explain the temperature dependence of \varLambdaμ. We also estimate the value of the paramagnetic shift of the muon spin precession frequency \delta\omega/\omega0 which is also temperature dependent.

We investigate the model describing a muon tunneling between two equivalent positions in (magnetic) insulators. Crystals are assumed to have concentrations such that the muon tunneling states in a double well potential can be considered as real during the muon lifetime \tauμ. In the simplest model there are only two such states: the ground state and the first excited state. Our aim is to calculate lifetimes \tau1(T) and \tau0(T) of these muon tunneling states resulting from interaction between the muon and phonons of the sample. In the case of a distorted double well potential, all matrix elements of \hatĤ(ph)int are finite. This leads to rescaling of the first and second orders of the perturbation theory compared with the results for the muon in “pure” samples. We explain the vanishing of some part of the oscillation frequencies of the muon polarization vector in the framework of a muon‐phonon interaction model: for example, it is necessary to take into account that \tau1(T) decreases more rapidly than \tau0(T) when the temperature increases. If the muon tunnels through a set of equivalent interstitial positions in a unit cell, the experimental picture of the polarization vector behaviour can change drastically under the action of various strain fields.

The possibility ofµF− formation due to electron attachment to a XeµF molecule inµSR-experiments with a gaseous Ne/Xe mixture is considered. This model explains the results of experiments in Ar and Ne with and without an external electric field.

The residual polarization of negative muons in n-type silicon with impurity density (1.6±0.2) · 1013 cm−3 is investigated as a function of temperature in the range 10–300 K. The measurements are performed in an external magnetic field of 0.08 T oriented transversely to the spin of the muons. Relaxation of the muon spin and a shift of the precession frequency are observed at temperatures below 30 K. The relaxation rate at 30 K equals 0.25±0.08 μs−1. The shift of the precession frequency at 20 K equals 7 · 10−3. Both the relaxation rate and the shift of the precession frequency increase as the temperature decreases. At temperatures below 30 K the relaxation rate is described well by the relation Λ=bT−q, where q=2.8±0.2.

An approximate analytical solution is obtained for the kinetic equation for electrons in the field of a Coulomb centre formed by a muon in the end of its track. It is shown that thermalization times and could differ by several orders of magnitude, depending on the behaviour of transport cross sections. The fraction of non-thermalized electrons returning to the muon and forming muonium (P Mu ) depends on pressure and temperature. Therefore,μSR experiments give us an opportunity to obtain information on the distribution function of track electrons.

The dependence of the residual polarization of negative muons in p-type Si on temperature in the 4.2–270 K range has been investigated. Measurements were carried out in external magnetic field of 0.08 T transverse to the muon spin. The impurity concentration in the sample was 2 · 1013 cm−3. Muon spin relaxation was observed at temperatures below 30 K. The relaxation rate atT=30 K is equal to 0.18±0.08μs−1. The relaxation rate grows with the decrease of temperature and at 4.2 K exceeds 30μs−1. The value of the residual polarization at zero timeP(t=0) is constant within the investigated temperature range.

Muon spin rotation/relaxation/resonance (MuSR) technique for studying matter structures is considered by means of a recently introduced probability representation of quantum spin states. A relation between experimental MuSR histograms and muon spin tomograms is established. Time evolution of muonium, anomalous muonium, and a muonium-like system is studied in the tomographic representation. Entanglement phenomenon of a bipartite muon-electron system is investigated via tomographic analogues of Bell number and positive partial transpose (PPT) criterion. Reconstruction of the muon-electron spin state as well as the total spin tomography of composed system is discussed., 20 pages, 4 figures, LaTeX, submitted to Journal of Russian Laser Research

The Mu formation mechanism due to Coulomb interaction of a muon with track electrons is considered. It is shown that taking into account the external electric field must essentially change the spin polarization behavior in liquid helium. The theory developed is consistent with experiments.

It is shown that using position-sensitive detectors in μSR experiments to determine the muon stopping site in a target permits one to study correlation effects in μSR time histograms produced by the decay of muons stopping in the same domain, i.e. to obtain time correlators of μSR histograms of decays from a small region. These correlators contain information on the spatial correlation of magnetic fields in the sample under study. The proposed method (μSR2-technique) allows measuring correlation radii (r c ) down to 10−5 cm in a bulk sample. Among interesting physical phenomena occuring overr c≥3×10−6 cm are, for instance, long wavelength fluctuations of the order parameter near the phase transition point in ferromagnets and antiferromagnets and magnetic field correlations in magnet domains and spin glasses. One may use this method also on heavy-current accelerators producing pulsed muon beams to investigate the variation in time of spatial correlations in magnets, spin glasses and superconductors.

Possibilities of using negative muons are discussed: (i) for understanding the behaviour of wave functions of outer electron shells in the vicinity of a nucleus and (ii) for investigation of the exchange effects in solid state physics. It is noticed that the effective life time of μ bound in light atoms (nuclear spin i≠O) depends on the state of the outer electron shell. This fact can be used for measurement ofK-capture rates from different components of hyperfine structure of a mesic atom and for estimation of the weak interaction constant.

The measurement of the muon polarization and the muon spin relaxation rate in a ferromagnet in zero external magnetic field allows the determination of such parameters as the initial magnetic texture and the magnetic inhomogeneity of the target material. The power of the μSR-method for a number of targets made of iron alloys is illustrated.

This review discusses a number of questions related to the use of the muon method for the study of metals. The possibilities of the method for the study of normal metals, superconductors, and ferromagnetic materials are analyzed. A systematic comparison of theory and experiment is made. It is shown that the method permits determination of the local magnetic fields at crystallographic sites, the type of magnetic structure in complicated ferromagnetic crystals (rare earth metals), analysis of the magnetic texture (grain orientation) of electrical steels, diagnosis of the interstitial sites occupied by a proton or muon, determination of the diffusion rate and the degree of distortion of the crystal lattice, and study of phase transitions in rare earth metals and spin glasses. In normal metals the muon method permits determination both of the muon diffusion rate and of the nature of diffusion (classical or quantum), the Knight shift in the muon, the charge state of a proton or muon in the metal, and the distortion of the crystal lattice. For superconductors the method permits information to be obtained which is inaccessible by other means, particularly determination of the ratio of the normal and superconducting phases and study of the dynamics of destruction and appearance of the superconducting state. The prospects for development of the muon method for study of the properties of metals are demonstrated.