Muon states in metals: Recent progress

We report on our results in two interesting questions related to muon spin rotation studies in condensed matter: (i) energetics of muons in metals, including lattice relaxation and zero point motion in self-trapping phenomena, and (ii) systematics of Knight shifts and hyperfine fields.In the former topic, a comprehensive theory is developed which entails the construction of the muon potential energy field in terms of the effective-medium or quasi-atom theory first introduced by Zaremba, Stott, NØrskov and Lang. The muon wave function is then solved by numerical (3-D) relaxation techniques. From this the forces exerted by the muon on the neighbouring lattice atoms are calculated, and the ensuing relaxations are evaluated by lattice Green's function techniques. These in turn modify the potential energy field, and the calculation is iterated to self-consistency. We search for the stable trapping sites in bcc and fcc metals, calculate self-trapping energies, diffusion barriers and excitation energies. Other hydrogenic imputies are also considered, and isotopic effects in e.g. heats of solution are investigated.In the latter topic, the spin-density functional theory is applied, including in the Knight shift calculation both the contact spin density and the diamagnetic shielding. The lattice potential is described in terms of the spherical solid model. A systematic behaviour as a function of the electron density and the host valency is found in good agreement with the experimental results.     

On the spectrum of the F-centres in the neighbourhood of charged dislocations

Upon deformation of alkali halide crystals an electrical voltage can be observed between suitably placed electrodes due to the movement of charged dislocations. When the specimen is coloured, light induces electrical relaxation, the dislocation photoconduction. We analysed the spectrum of dislocation photoconduction as a spectrum of F-centres situated in the neigh-bourhood of charged dislocations. It is difficult to segregate the two competing effects on the F-spectrum, namely the electric and the strain field of the dislocation, so when we discussed our experimental data we had to assume that the change was caused by either the strain or the electric field. In this paper we discuss this problem theoretically by means of the method of moments. Comparing the expressions of the changes of the first and second moments for the strain field with the analogous ones for the Coulomb field of a charged dislocation jog we see that for every reasonable distance between the dislocation and the F-centre the effect of the strain field is the dominant one. The change of moments can be fitted to experimental data.The authors express their gratitude to I.Tarján and R.Voszka for their interest in this work and I.Gaál for his constructive criticism, he also directed our attention to the dislocation quadrupole field as to a possible third effect.     

What is the effective stress energy of particles created from the vacuum?

When spontaneous particle creation occurs in a strong gravitational field, it seems clear on physical grounds that the particle creation must back-react on the gravitation field. It is generally believed that in the semiclassical approximation this effect can be described by assigning an effective stress energy to the created particles, which acts as a source of the gravitational field via Einstein's equation. In this essay, I discuss an axiomatic approach for determining the renormalized value of this effective stress energy.This essay was awarded the third prize for 1977 by the Gravity Research Foundation.     

Numerical study on an isolated vortex of the type-II superconductor in the presence of a twin plane

Based on the Ginzburg-Landau theory and using the numerical relaxation approach, we have carried out a numerical analysis of an isolated vortex structure for a type-II superconductor in the presence of a twin plane (being a normal region) situated parallel to the external magnetic field. Meanwhile, the lower critical field is also calculated and is found to decrease as compared to the twin-free case, which is consistent with the intuitive theoretical understanding and the fact that twins are easy channels for the flux penetration.     

Reaction,trapping, and multifractality in one-dimensional systems

In the first part of this paper, we present two variants of the A+AA and A+AP reaction in one dimension that can be investigated analytically. In the first model, pairs of neighboring particles disappear reactively at a rate which is independent of their relative distance. It is shown that the probability density (x) for a nearest neighbor distance equal tox approaches the scaling form(x) c exp(–cx/2)/(cx)^1/2 in the long-time limit, withc being the concentration of particles. The second model is a ballistic analogue of the coagulation reaction A+A A. The model is solved by reducing it to a first-passagetime problem. The anomalous relaxation dynamics can be linked in a direct way to the fractal time properties of random walks. In the second part of this paper, we discuss the complications that arise in systems with disorder. We present a new approach that relates first-passage-time characteristics in a one-dimensional random walk to properties of random maps. In particular, we show that Sinai disorder is a borderline case for the appearance of multifractal properties. Finally, we apply a previously introduced renormalization technique to calculate the survival probability of particles moving on the line in the presence of a background of imperfect traps.     

Mössbauer spectra of the Γ8 quadruplet in the presence of spin-lattice relaxation

The influence of crystal field fluctuations on Mössbauer hyperfine spectra is analyzed. These fluctuations lead to violation of cubic symmetry of the environment, to a time-dependent splitting of a quadruplet and, as a consequence, to a temporal modulation of the hyperfine interaction. It is found that fluctuations of E-type do not lead to a complete averaging of the hyperfine interaction in the fast relaxation limit, unlike the case of e.g. magnetic relaxation and any other type of crystal field fluctuations. Moreover, for ions with J=7/2 in the fast relaxation limit the relaxation spectra of ₈ are identical to the spectra of the ₈ doublet.     

Energy criterion in the interpretation of dielectric relaxation

The universal fractional-power-law frequency dependence of relaxation seen in most solid dielectrics has the unique property that the ratio of macroscopic energy lost per radian to energy stored is independent of frequency. In a recent publication we have shown that this energy criterion is satisfied if the energy loss in every microscopic dipolar reversal is independent of the rate of reversals. The present paper derives for the first time a quantitative relationship between the macroscopic energy loss per radian and the microscopic loss , thus providing a justification of the energy criterion approach. The relationship between this new frequency-domain interpretation of relaxation processes and the currently accepted theories of universal behaviour is discussed.     

Anisotropic Lattice Gases

We have studied lattice gases with a particle-conserving dynamic rule that involves two principal parameters. One of them has two limiting values that correspond, respectively, to a large, saturating constant field, which induces a positive particle current, and to a random field (zero net current). Varying the other parameter, either particle attractions or repulsions perpendicular to the field are simulated. The nature of ordering is shown to be independent of the value for the field parameter. In particular, the two indicated limiting cases of the latter lead to the same order-parameter critical behavior, consistent with 1/3, in the presence of a linear interface for attractions in two dimensions. Some qualitative features of the time relaxation are briefly described.     

Random walks in a random field of decaying traps

We study random walks on ^d (d 1) containing traps subject to decay. The initial trap distribution is random. In the course of time, traps decay independently according to a given lifetime distribution. We derive a necessary and sufficient condition under which the walk eventually gets trapped with probability 1. We prove bounds and asymptotic estimates for the survival probability as a function of time and for the average trapping time. These are compared with some well-known results for nondecaying traps.     

The dynamics of water in nanoporous silica studied by dielectric spectroscopy

Broad-band dielectric spectroscopy is used to investigate the dynamics of hydration water on the surface of the cylindrical pores of a nanostructured silica material (MCM-41, with pore diameter of 3.2 nm) at various hydrations, in the temperature range 250-150 K. We focus our attention on orientational relaxations that shift from 0.5 MHz at 250 K to less than 1 Hz at 150 K. The measurements distinguish the relaxation of the hydroxyl groups at the surface of silica from the orientational dynamics of hydration water which strongly depends on the degree of hydration. Although it is significantly faster than the dynamics of water in ice, the orientational relaxation of non-freezing water has an activation energy comparable to that in ice when the hydration layer is complete and approximately two-molecule thick.     

On the time evolution of transmittivity in magnetic fluids

When a magnetic fluid is subjected to a magnetic field, a part of the magnetic particles in the fluid agglomerates to form chains. Thus, the ferrofluid becomes optically anisotropic. In this work we describe optically observed patterns in some magnetic fluid films in applied parallel magnetic fields and optical effects of these, especially the optical transmittance. The most interesting experimental observation is that concerning the time dependence of relative transmittivity . For kerosene base ferrofluids relax rapidly at coupling and decoupling magnetic field, but for a transformer-oil magnetic fluid the relaxation times can attain (5–10) minutes, depending on the intensity of applied magnetic field.     

A generalized Kubo-Toyabe formula for muon spin relaxation in crystals with uniaxial symmetry

The Kubo-Toyabe semiclassical formula, describing the time development of the polarization of a particle in zero external field at a lattice site with cubic local environment, is generalized for uniaxial site symmetry. The relaxation function and, in particular, its first moments and long time asymptotics obtained in a closed form depend on the angle between polarization and the crystalc-axis and are shown to vary sensitively with the asymmetry of the field distribution at the particular muon site. Besides the exact uniaxial variant of the Kubo-Toyabe relaxation function, an approximate simple interpolation formula is also derived, which is correct for both short times and in its long time asymptotics. The two parameters (, ₁) in the uniaxial formulae can be determined by using the observed values of the second momentM ₂ for two different crystal orientations.     

Positive muons in bismuth: determination of their sites,detection of a large local lattice contraction and giant associated electric field gradients,evidence for short and long range diffusion

Using a high purity Bi single crystal the temperature and orientation dependence of the zero and transverse field muon spin relaxation rate has been studied in detail. The results imply that the ⁺ occupies one of the two possible interstitial sites in the distorted rhombohedral crystal structure of Bi below 10 K and the other site above 80 K. At both sites the nearest neighbor Bi atoms are found to be shifted towards the ⁺ by 10% of their nominal distance, implying a large local lattice contraction. In concomitance extremely strong electric field gradients are manifest at the nn Bi nuclei. An almost temperature independent reduced relaxation rate in the temperature range from 20 K to 60 K is interpreted in terms of short range diffusion along a limited chain of alternating types of sites. Above 100 K both long range and short range diffusion are indicated.     

Photovoltage and photo-induced charge trapping in porous silicon

Spectral and kinetic dependencies of photovoltaic effects in porous silicon-crystal substrate structures have been studied. Both as-prepared and aged in air samples were used. It is shown for the first time that besides the photovoltage component connected with a depleted region in the silicon substrate at the interface with the porous layer there is a photovoltage due to porous silicon itself. It is established that the electron states with the relaxation time about several minutes are located on the pores surface. The properties of these states changed with the thermal annealing of structures. The superslow hole traps with a relaxation time of about several hours were registered in the aged samples. These traps are located in the oxide on the silicon skeleton surface of the porous silicon. An energy diagram based on the data is suggested that explains the appearance of photovoltaic effects in the investigated structures.     

Dynamics of a domain wall coupled to thermally agitated spins: A model of a rare earth ferrite-garnet

The nonlinear mobility of a domain wall in an idealized model of a RE ferrite-garnet is considered, the thermally agitated RE sublattice being described by the Landau-Lifshitz-Bloch equation [4] with longitudinal relaxation terms. It is shown that in some field intervals the DW velocity is small and governed by the longitudinal relaxation of the RE ions. This effect may be observed in sufficiently low magnetic fields near the compensation point of ferrites. If the dynamics of the RE spins is dissipationless, there is a non-analytic contribution to the DW damping due to the wake in the RE subsystem.     

A study of solutions to the aesthetic field equations

We have found hundreds of solutions to the integrability equations in aesthetic field theory. The behavior of the solutions to the aesthetic field equations depends on which solution to the integrability equations we take. From computer runs down a coordinate axis we have found a type of solution where we have a maximum and a minimum, as well as the field going to zero at large distances along both directions. This kind of solution is quite prevalent. We call this type of solution a pulse solution. We have found the pulse solution in two and three dimensions as well as four dimensions. It appears regardless of whether certain symmetries are present or absent. We have taken a two- or three-dimensional and made a four-dimensional theory from it with the use of a four-dimensionale i. This process we call imbedding. We have found imbedding has not affected the overall characteristics of the solution in the cases we considered. We were able to change the character of the solutions to some degree by altering the magnitude of some of the gammas—but this did not lead to solutions with significantly more wiggles. We also found an example of an oscillatory solution. The oscillations occurred in too regular a pattern to give a realistic model for basic behavior. However, this solution indicates that aesthetic field theory has more structure then we have ever seen before. We also obtained a solution in which errors took over so fast that the computer was literally helpless in telling us what is going on. In other solutions the field appears to increase without bounds. Whether this is due to singularities or to the presence of large numbers is not clear.     

Hadron fields from anisotropie space and their interaction

Hadron fields are constructed from constituent fields in the anisotropic microdomain, regarded as Finslerian, which were discussed in an earlier paper. The general one-particle hadron states are formulated. The many-particle hadron states are also formed as direct products of one-particle states. Then the field theory of hadrons in the macrodomain is discussed and formal calculations are made for the reaction amplitude of the meson-baryon interaction and compared with that of our previous model. It is found that the amplitude is dependent on that of the interaction together with a factor arising from the rearrangement of the constituents. This factor provides an extra momentum dependence that leads to the energy-dependent coupling which makes it possible to apply perturbation technique in strong interactions, as discussed in our earlier papers.     

Hamilton and the Law of Varying Action Revisited

According to history texts, philosophers searched for a unifying natural law whereby natural phenomena and numbers are related. More than 2300 years ago, Aristotle postulated that nature requires minimum energy. More than 220 years ago, Euler applied the minimum energy postulate. More than 200 years ago, Lagrange provided a mathematical proof of the postulate for conservative systems. The resulting Principle of Least Action served only to derive the differential equations of motion of a conservative system. Then, 170 years ago, Hamilton presented what he claimed to be a general method in dynamics. Hamilton's resulting Law of Varying Action was supposed to apply to both conservative and non-conservative systems and was supposed to yield either the differential equations of motion or the integrals of those differential equations. However, no direct evaluation of the integrals of motion ever resulted from Hamilton's law of varying action. In 1975, a scant 29 years ago, following five years of controversy with engineer mechanicians, Dr. Wolfgang Yourgrau, Editor, Foundations of Physics, published my first paper based on Aristotle's postulate, without mathematical proof. That and subsequent papers present, through applications, a true general method in dynamics. In this essay, I present the mathematical proof that is missing from my 1975 and subsequent papers. Six fundamental integrals of analytical mechanics are derived from Aristotle's postulate. First, however, Hamilton must be revisited to show why his H function and his force function prevents the law of varying action from being the general method in dynamics that he claimed it to be. I have found that Hamiltons Law of Varying Action (HLVA), as Hamilton presented it, cannot be applied to systems for which the force function is non-integrable. In 1972, Dr. B.E. Gatewood and Dr. D.P. Beres (then a graduate student) discovered that the end-point term associated with the principle of least action does not vanish. I named the new equation, the general energy equation. In 1973, because I was doing with it what Hamilton claimed could be done with HLVA, I simply assumed that this new equation was HLVA. I gave the new equation the misnomer HLVA. In 2001, I learned that I had made a grave mistake. I found that HLVA is at most a special case of the general energy equation. My interpretation of Aristotle's postulate permits one to by-pass the differential equations of motion completely for both conservative and non-conservative systems (no calculus of variations).     

Isothermal relaxation of non equilibrium charge in a dielectric (the case of weak charge-carrier trapping)

The features of the isothermal discharge of a dielectric containing single-energy traps after a voltage pulse V₀(t_u) is applied to the specimen, during which charge carriers (electrons) are injected into the volume of the dielectric (monopolar injection), are considered. The initial stage of the discharge is analyzed at instants comparable with the time of flight of free carriers, and also the discharge under conditions of self-consistent variation of the internal field and the volume charge in the dielectric when the external field is switched off and only the pulling field e₁ s e₀ remains. Analytical expressions are obtained which describe the change in the total current, the displacement current, and the conductivity, and also the variation of the total charge with time, by means of which one can estimate experimentally the main parameters of the material: E_t, v,, n_t.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 5, pp. 27–32, May, 1984.