Probabilistic bootstrap percolation

In bootstrap percolation, sites are occupied with probabilityp, but those with less thanm occupied first neighbors are removed. This culling process is repeated until a stable configuration (all occupied sites have at leastm occupied first neighbors or the whole lattice is empty) is achieved. Formm ₁ the transition is first order, while form<m ₁ it is second order, withm-dependent exponents. In probabilistic bootstrap percolation, sites have probabilityr or (1–r) of beingm- orm-sites, respectively (m-sites are those which need at leastm occupied first neighbors to remain occupied). We have studied the model on Bethe lattices, where an exact solution is available. Form=2 andm=3, the transition changes from second to first order atr ₁=1/2, and the exponent is different forr<1/2,r=1/2, andr>1/2. The same qualitative behavior is found form=1 andm=3. On the other hand, form=1 andm=2 the transition is always second order, with the same exponents ofm=1, for any value ofr>0. We found, form=z–1 andm=z, wherez is the coordination number of the lattice, thatp _c=1 for a value ofr which depends onz, but is always above zero. Finally, we argue that, for bootstrap percolation on real lattices, the exponents and form=2 andm=1 are equal, for dimensions below 6.On leave from Universidade Federal de Santa Catarina, Depto. de Fisica, 88049, Florianópolis, SC, Brazil     

Phase transitions in a ferromagnetic semiconductor. III

We investigate the ferromagnetic phase transition in a heavily doped semiconductor or semimetal containing magnetic ions. The coupling between conduction electrons and magnetic ions is treated in the molecular field approximation. We show that the system undergoes a transition to an ordered state for arbitrarily small concentrations of the magnetic ions and the conduction electrons. In the limitJ/? _F?1 we obtain the Ruderman-Kittel result for the Curie temperature, and forJ/? _F?1 a generalization of Thompson’s strong coupling result. (J is the exchange coupling constant between conduction electrons and magnetic ions and? _F is the kinetic Fermi energy at absolute zero.) The intermediate range is evaluated numerically.     

Die Winkelverteilung der Protonen der Reaktion B11 (d,p 0,1 B12 im Energiebereich von 0,7 ... 1,7 MeV

Angular distributions of the protons in this low-energy region have been measured by only few groups for obvious reasons: the energy of the emitted protons is very low. It will be shown that by using non-supported targets of B¹¹ with very little oxygen contamination the angular distributions can be obtained between approximately 5 and 165 degrees (cm.,p ₀-group) and between 20 and 165 degrees (cm.,p ₁) with high accuracy. The Butler-fits givel=1,r _B =5.2 fm forp ₀ andl=1,r _B =4.7 fm forp ₁. All angular distributions seem to exhibit strong participation of non-stripping mechanism.     

The structure of single vortices in type II superconductors

This paper reports the first results of a program for the investigation of the detailed electronic and magnetic structure of type II superconductors. The structure of a single flux bundle or vortex in an infinite superconducting specimen is investigated, assuming weak coupling and a spherical Fermi surface of the material. The equations governing the gap parameterΔ(r) and the vector potentialA(r) are suitably scaled and simplified using the symmetry of the system. These equations can be solved numerically only; for their solution the asymptotics ofΔ(r) andA(r) far from the center of the vortex are required. These asymptotics are derived here by properly linearising the equations; they are found to be of the typer ^?1/2 exp (?r/rin0) wherer ₀ is determined by temperature and material parameters. Implicit equations forr ₀ are derived and discussed. We find that for materials withκ-values less than 1.7 (e.g. pure niobium)r ₀ becomes complex at low temperatures; i.e. the deviation ofΔ(r) andA(r) from their values at infinity is a damped oscillation as function ofr.     

Thermal conductivity of large-grain niobium and its effect on trapped vortices in the temperature range 1.8–5 K

Experimental investigation of the thermal conductivity of large grain and its dependence on the trapped vortices in parallel magnetic field with respect to the temperature gradient \(\nabla T\) was carried out on four large-grain niobium samples from four different ingots. The zero-field thermal conductivity measurements are in good agreement with the measurements based on the theory of Bardeen–Rickayzen–Tewordt (BRT). The change in thermal conductivity with trapped vortices is analysed with the field dependence of the conductivity results of Vinen et al for low inductions and low-temperature situation. Finally, the dependence of thermal conductivity on the applied magnetic field in the vicinity of the upper critical field H _c2 is fitted with the theory of pure type-II superconductor of Houghton and Maki. Initial remnant magnetization in the sample shows a departure from the Houghton–Maki curve whereas the sample with zero trapped flux qualitatively agrees with the theory. A qualitative discussion is presented explaining the reason for such deviation from the theory. It has also been observed that if the sample with the trapped vortices is cycled through T _c, the subsequent measurement of the thermal conductivity coincides with the zero trapped flux results.     

The 1/r expansion for the critical multiple well problem

We consider the critical multiple well problem $$H = - \Delta + \sum\limits_{i = 1}^n {V(x - rx_i )} ,$$ where ?Δ+V(x) has a zero energy resonance. We prove that all eigenvalues and resonances ofH tending to zero as 1/r ² are analytic in 1/r. We give an explicit equation for the lowest nonvanishing coefficient in the 1/r expansion for any of these eigenvalues or resonances and observe thatH has infinitely many resonances tending to zero. Forn=2 andn=3, we compute the coefficients explicitly and forn=2, we also give the next coefficient in the 1/r expansion.     

Veneziano model for nucleon nucleon and nucleon antinucleon scattering. I

The thermal conductivity has been measured on two niobium samples of different purity (ratio of mean free path to coherence length 220 and 10, respectively), in the mixed state, with heat current parallel (k _∥) and normal (k _⊥) to the magnetic field. The experimental results are compared with theories; the range of validity is discussed. Maki's relation (k-k _n) ～ (H _c2 -H)^1/2 could be verified only for the purer sample in the field range (H _c2 -H)?140 Oe. The relationsk _∥/k ₁>1 forT?T _c andk _∥/k ₁<1 forT?T _c were found to be valid for the whole range of fields in the mixed state.     

Influence of the kinematical constraints on the transverse momentum correlation features

The influence of the kinematical constraints on the transverse momentum correlation features in multiparticle processes is studied by means of thedN/(d r ₁ d r ₂) distribution,r ₁ andr ₂ being the transverse momenta of the particles entering in each pair. Using a limited transverse momentum phase space we obtain for high energy reactions a simple analytical expression fordN/(d r ₁ d r ₂). This expression, similar to a phenomenological parametrization proposed for analyzing¯pp annihilation data, served to define a correlation parameter, the non vanishing value of which is due to the energy-momentum conservation constraints.     

Stability of a rotating pinch

The problem of pinch stabilisation is of interest in connection with plasma confinement and has been the subject matter of various investigations in recent years, (Kruskal andSchwarzschild, Taylor, Shafranov). The theoretical investigations in the past have been restricted to static equilibrium configurations. It is of some interest to explore whether an initial motion of plasma tends to make the pinch more, or less stable than when the fluid velocity is zero. Keeping this in mind let us investigate theaxisymmetric stability of an infinitely long cylindrical plasma column of radiusR rotating uniformly with angular velocityΩ about theZ-axis, the fluid being assumed to the incompressible, inviscid, and of infinite electrical conductivity. The plasma carries a uniform axial current of densityj ₀ and a uniform axial magnetic fieldH ₁. The field outside is taken to beH ₂ inZ-direction and a toroidal component which is continuous across the boundary.     

Fe-induced magnetic transition in YBa2(Cu1−x 57Fe x )3O6 by NQR and Mössbauer spectroscopy

Nuclear quadrupole resonance spectroscopy (around 30 MHz) on the chain site Cu(1) nuclei in oxygen deficient YBa₂(Cu_1?x Fe _x )₃O₆ doped with different amounts of⁵⁷Fe (x≤0.01) reveal an onset of magnetic order at low temperatures represented by a symmetrical doublet contribution to the nuclear quadrupole resonance (NQR) spectrum. The onset temperatureT _N2 depends on the concentration of Fe reaching 130 K forx=0.01. The splitting forx=0.01 at 100 K corresponds to a net internal field of 0.09 T with a distribution of ≈0.08 T representing about 70 percent of the Cu(1) nuclei.⁵⁷Fe and⁵⁷Co Mössbauer spectroscopy at 4.2 K with and without an external magnetic field of 5 T revealed that belowx=0.00015 Fe spins are decoupled from the Cu(2) moments in the antiferromagnetic state. Results are interpreted in terms of the magnetic model structure suggested by Kadowaki et al. [1].     

Magnetic field dependence of the thermal conductivity in EuxSr1-xS

The thermal conductivity κ of two Eu_xSr_1-x single crystals (x = 0.25 and 0.54) was measured between 1.5 and 25 K. In magnetic fields of ≈ 7 T κ is enhanced for temperatures up to 20 K with respect to κ(B = 0). At 1.5 K, where the relative increase κ(B)/κ(0) is largest, this ratio is 1.5 for x = 0.54 and ≈ 4 for x = 0.25. Two possible mechanisms for this effect, i.e. freezing of phonon scattering by magnetic excitations in a magnetic field, and heat transport by field-induced magnons, are discussed.     

Coulomb energies and nuclear radii in the energy density formalism

The relation between Coulomb displacement energies,ΔE _c , andΔr=r _n -r _p , the difference between the rms radii of neutrons and protons in nuclei, is investigated within the energy density formalism (EDF). The variational equation, obtained by minimizing the Coulomb plus symmetry energies, is solved assuming the symmetry interaction is a simple functional of the local nuclear matter density. Varying parameters of the model, rather unique relation betweenΔE _c andΔr is obtained (within ±50 keV).ΔE _c isindependent ofr _ex, the rms radius of the excess neutrons distribution. Using the experimental values ofr _p and adjusting the model to reproduce the recent data onΔr (Δr∽~0.05 fm for⁴⁸Ca and²⁰⁸Pb), which are significantly smaller than those obtained from current Hartree-Fock calculations, the calculatedΔE _c agree with the experimental results. Using the value ofΔr～0.05 fm and the experimental values ofr _ex, a small compression (<0.02 fm) of the proton core in the analogue state relative to its parent state emerges, thus contributing an additional electrostatic term to the Coulomb displacement energy. The size of this relative core-compression effect depends on the values assumed forΔr andr _ex, it increases with the decreasing ofΔr and the increasing ofr _ex. IfΔr～0.05 fm the effect is large enough to remove the long standing Coulomb energy anomaly. The main result of the present work is the correlation betweenΔE _c andΔr, suggesting that the difficulties of current Hartree-Fock calculations in reproducing isotope shifts ofr _p , the small value ofr _n ?r _p and the values ofΔE _c may all be different manifestations of some missing residualp n effective interaction.     

Magnetic ordering in Mg−Zn ferrites

The magnetic ordering of a series of magnesium-zinc ferrite, Zn_0.3Mg _x Fe_2.7?x O_4±δ (0.5≤x≤1.1; 0≤δ≤0.2) has been investigated using Mössbauer measurements in the temperature range 295–620 K. The samples were found to be magnetic at room temperature with a hyperfine field at each site which increases with iron content. The Curie temperature was also observed to increase in a similar manner. The slope of this increase forB _hf andT _c is steeper forx≤0.6 thanx≥0.7. It has also been observed that Mg²⁺ substitution by Zn²⁺ in MgFe₂O₄ affects the magnetic ordering and the internal hyperfine field. The Curie temperature decreases by ～200 K andB _hf by ～20%.     

Photoionization of the Xe atom and Xe@C<Subscript>60</Subscript> molecule

Photoionization of the Xe atom and Xe@C₆₀ molecule have been studied usingthe random phase approximation with exchange (RPAE) method. The Xe atom was described byrelaxed orbitals including overlap integrals. The C₆₀ fullerene has beenrepresented by an attractive short range spherical well with potentialV(r), given byV(r) =  ?V ₀ forr _i  < r < r _o ,otherwise V(r) = 0 wherer _i andr _o are respectively, the inner and outerradii of the spherical shell. The time independent Schrödinger equation was solved usingboth regular and irregular solutions and the continuous boundary conditions atr _i andr _o . The results demonstrate improvementto previous calculations for both the Xe atom and Xe@C₆₀ molecule and comparevery well with the recent experimental data.     

Transient magnetic fieldg-factor measurements of 2 1 + in64, 66, 68, 70Zn

The implantation perturbed angular correlation technique (IMPAC) has been used to determine the magnetic dipole moments of the first excited 2⁺ states in the stable even-even Zn-isotopes. Transient magnetic field measurements for⁵⁶Fe in ferromagnetic Gd at 77 K have been performed for recoil velocities up tov/v ₀～8. The result confirms the earlier reported linear velocity dependence. Theg-factors are deduced to be 0.46±0.10, 0.47±0.11, 0.46±0.14 and 0.30±0.07 for^{64, 66, 68}Zn and⁷⁰Zn respectively. The results are compared with theoretical predictions.     

Mössbauer spectroscopic study of FeAl1−x Co x

We report the results of a Mössbauer study of the alloy sytem FeAl_1?x Co _x forx ≥ 0.3 at temperatures down to 83 K. Magnetic splitting is observed forx ≥ 0.35 at all temperatures. However, forx=0.3, no splitting is observed at room temperature, and superparamagnetic behavior occurs at LN₂ temperature. The magnetically split spectra are fitted each with a distribution of hyperfine fields and the average hyperfine field \(\bar B_{hf} \) as a function of temperature is obtained. The variation of \(\bar B_{hf} \) withT is explained using the model of magnetic clusters with collective magnetic excitations from which the saturation hyperfine field and the magnetic anisotropy energy for these clusters are obtained. Also, the results are discussed using the model of random atomic distributions, and the agreement between the calculated and the experimentally obtained distributions of hyperfine fields is found improve asx increases.     

A dispersion analysis of pion photoproduction from nucleons. II

The paper gives an account of numerical results forl=0 andl=1 multipole amplitudes of pion photoproduction. They have been obtained by solving integral equations which originate from fixed angle dispersion relations that have been deduced in an earlier paper. In this Part II presented are solutions of truncated integral equations for photoproduction amplitudes, which take into account coupling with the most important transitionsM ₁₊ ⁽³⁾ (in the isovector case) andE ₀₊ ⁽⁰⁾ (in the isoscalar case). Discussed are implications of such an approach and a comparison of results obtained by a different method is being made.     

Experimentally favoured potential for mesons

We show first that a natural potential to be used in a semirelativistic wave equation for theS=1,J=J _max light and heavy mesons is, forr?1 Fermi,V(r)=ar+b/r withb??π/2, forr?1 Fermi,V(r)=cln(r/r ₀). Next we show that this particularb value favours in fact a QCD-like potential forr?1 Fermi so that a logarithmic potential is only an effective one in a smallr region.     

A tilted rotatory frame method for the measurement of nuclear spin diffusion coefficients in solids doped with paramagnetic centers: Mn-Doped CaF<Subscript>2</Subscript>

Nuclear magnetic resonance (NMR) experiments recording the recovery of the magnetization of the nuclei in one phase, following the excitation of the nuclei in the other phase, is a classical way of studying blends inhomogeneous at the nanometer scale. Interpretation of the time recovery in terms of the spatial dimension requires knowledge of the two-phase spin diffusion coefficientsD ₀. A new method of measurement of_{D 0} is proposed on the basis of variable angle-tilted rotatory frame relaxation in homogeneous samples doped with paramagnetic centers. The choice of the tilt angle allows one to finely balance the direct relaxation by the paramagnetic center and the spin diffusion. The shape of the relaxation is analyzed with the solution for the diffusion-limited regimeM(t)/M(0)=exp[?(r ₂ t)^1/2?r ₁ t] andD ₀ then calculated fromr ₁,r ₂ and the concentration of paramagnetic centers. Conditions where reliable results can be obtained both theoretically and numerically are explored. The method has been implemented and applied to polycrystalline Mn-doped CaF₂ leading toD ₀=540±60 nm²/s, in agreement with existing values on this model compound.