Structures of distorted phases and critical and noncritical atomic displacements of elpasolite Rb<Subscript>2</Subscript>KInF<Subscript>6</Subscript> during phase transitions

The structures of all three phases of the Rb₂KInF₆ crystal have been determined from the experimental X-ray diffraction data for the powder sample. The refinement of the profile and structural parameters has been carried out by the technique implemented in the DDM program, which minimizes the differences between the derivatives of the calculated and measured X-ray intensities over the entire profile of the X-ray diffraction pattern. The results obtained have been discussed using the group-theoretical analysis of the complete order-parameter condensate, which takes into account the critical and noncritical atomic displacements and permits the interpretation of the experimental data obtained previously. It has been reliably established that the sequence of changes in the symmetry during phase transitions in Rb₂KInF₆ can be represented as $ Fm\bar 3m\xrightarrow[{0,0,\phi }]{{11 - 9\left( {\Gamma _4^ + } \right)}}{{I114} \mathord{\left/ {\vphantom {{I114} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}}} \right. \kern-\nulldelimiterspace} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}} $ Fm\bar 3m\xrightarrow[{0,0,\phi }]{{11 - 9\left( {\Gamma _4^ + } \right)}}{{I114} \mathord{\left/ {\vphantom {{I114} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}}} \right. \kern-\nulldelimiterspace} {m\xrightarrow[{\left( {\psi ,\phi ,\phi } \right)}]{{11 - 9\left( {\Gamma _4^ + } \right) \oplus 10 - 3\left( {X_3^ + } \right)}}{{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-\nulldelimiterspace} {n1}}}} .     

Coherent pion production in neutrino and antineutrino interactions on nuclei of heavy freon molecules

Studying the coherent diffractive production of pions in neutrino and antineutrino scattering off the nuclei of freon molecules we have observed for the first time in one experiment all three states of the isospin triplet of the axial part of the weak charged and neutral currents. For the corresponding cross sections we derive $$\begin{array}{*{20}c} {\sigma _{coh}^v (\pi ^ + ) = (106 \pm 16) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ {\sigma _{coh}^{\bar v} (\pi ^ - ) = (113 \pm 35) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}and} \\ {\sigma _{coh}^v (\pi ^0 ) = (52 \pm 19) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ \end{array} $$ . Comparing our data with theoretical predictions based on the standard model of weak interactions we find reasonable agreement. Independently from any model of coherent pion production we determine the isovector axial vector coupling constant to be |β|=0.99±0.20.     

Asymptotic behavior of Mayer cluster sums for the one-dimensional Ising model

The properties of the high-field polynomialsL _n (u) for the one-dimensional spin 1/2 Ising model are investigated. [The polynomialsL _n (u) are essentially lattice gas analogues of the Mayer cluster integralsb _n (T) for a continuum gas.] It is shown thatu ^?1 L _n (u) can be expressed in terms of a shifted Jacobi polynomial of degreen?1. From this result it follows thatu ^?1 L _n (u); n=1, 2,... is a set of orthogonal polynomials in the interval (0, 1) with a weight functionω(u)=u, andu ^?1 L _n (u) hasn?1 simple zerosu _n (v); v=1, 2,...,n?1 which all lie in the interval 0<u<1. Next the detailed behavior ofL _n (u) asn→∞ is studied. In particular, various asymptotic expansions forL _n (u) are derived which areuniformly valid in the intervalsu<0, 0<u<1, andu>1. These expansions are then used to analyze the asymptotic properties of the zeros {u _n (v); v=1, 2,...,n?1}. It is found that $$\begin{array}{*{20}c} {u_n (v) \sim \tfrac{1}{4}({{j_{1,v} } \mathord{\left/ {\vphantom {{j_{1,v} } n}} \right. \kern-\nulldelimiterspace} n})^2 [1 - ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {12}}} \right. \kern-\nulldelimiterspace} {12}})n^{ - 1} + ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {700)( - 3 + 2j_{1,v}^2 )n^{ - 4} }}} \right. \kern-\nulldelimiterspace} {700)( - 3 + 2j_{1,v}^2 )n^{ - 4} }}} \\ { + ({{j_{1,v}^2 } \mathord{\left/ {\vphantom {{j_{1,v}^2 } {20160)(40 + 4j_{1,v}^2 - j_{1,v}^4 }}} \right. \kern-\nulldelimiterspace} {20160)(40 + 4j_{1,v}^2 - j_{1,v}^4 }})n^{ - 6} + \cdot \cdot \cdot ]} \\ {u_n (n - v) \sim 1 - ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } 4}} \right. \kern-\nulldelimiterspace} 4})n^{ - 2} + ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } {48)( - 2 + j_{0,v}^2 )n^{ - 4} }}} \right. \kern-\nulldelimiterspace} {48)( - 2 + j_{0,v}^2 )n^{ - 4} }}} \\ { + ({{j_{0,v}^2 } \mathord{\left/ {\vphantom {{j_{0,v}^2 } {2880)(2 + 9j_{0,v}^2 - 2j_{0,v}^4 )n^{ - 6} + \cdot \cdot \cdot }}} \right. \kern-\nulldelimiterspace} {2880)(2 + 9j_{0,v}^2 - 2j_{0,v}^4 )n^{ - 6} + \cdot \cdot \cdot }}} \\ \end{array} $$ asn→∞v fixed, wherej _k,v denotes thevth zero of the Bessel functionJ _k(^z)     

On (n,p) cross sections for 14 MeV neutrons

Using older compilations and recent data the (n, p) cross sections for neutron energies between 14 and 15 MeV have been collected and revised critically. The experimental data can be represented phenomenologically by the formula $$\log _{10} ({{\sigma _{np} } \mathord{\left/ {\vphantom {{\sigma _{np} } {mb}}} \right. \kern-\nulldelimiterspace} {mb}}) = 0.2 + 0.4A^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - 4.6{{(N - Z)} \mathord{\left/ {\vphantom {{(N - Z)} {A^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} }}} \right. \kern-\nulldelimiterspace} {A^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} }}$$ . The compound part of the (n, p) reactions is described by a statistical model; the direct reactions are taken into account semiempirically.     

Superconductive superheating field for finiteκ

We have calculated analytically the superheating fieldH _sh for bulk superconductors, correct to second order in. We find , which agrees well with numerical computations for<0.5. The surface order parameter is , and the penetration depth is .     

Laser-Rf double-resonance studies of the hyperfine structure of metastable atomic states of55Mn

The hyperfine structure (hfs) of the metastable atomic states 3d⁶4s⁶ D _{1/2, 3/2, 5/2, 7/2, 9/2} of⁵⁵Mn was measured using theABMR- LIRF method (atomicbeammagneticresonance, detected bylaserinducedresonancefluorescence). The hfs constantsA andB, corrected for second order hfs perturbations, could be derived from these measurements. The theoretical interpretation of these correctedA- andB-factors was performed in the intermediate coupling scheme taking into account the configurations 3d ⁵4s ², 3d ⁶4s and 3d ⁷. Examining the influence of the composition of the eigenvectors on the hfs parameters \(\left\langle {r^{ - 3} } \right\rangle ^{k_s k_l } \) it was found, that for the configuration 3d ⁶4s the two-body magnetic interaction should be considered in the calculation of the eigenvectors. Investigating second order electrostatic configuration interactions and relativistic effects and using calculated relativistic correction factors we obtained for the nuclear quadrupole moment of the nucleus⁵⁵Mn a value ofQ=0.33(1) barn, which is not perturbed by a shielding or antishielding Sternheimer factor. The following hfs constants have been obtained: $$\begin{gathered} A\left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 882.056\left( {12} \right)MHz \hfill \\ A\left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 469.391\left( 7 \right)MHzB\left( {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = - 65.091\left( {50} \right)MHz \hfill \\ A\left( {{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 436.715\left( 3 \right)MHzB\left( {{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = - 46.769\left( {30} \right)MHz \hfill \\ A\left( {{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 458.930\left( 3 \right)MHzB\left( {{7 \mathord{\left/ {\vphantom {7 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 21.701\left( {40} \right)MHz \hfill \\ A\left( {{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 510.308\left( 8 \right)MHzB\left( {{9 \mathord{\left/ {\vphantom {9 2}} \right. \kern-\nulldelimiterspace} 2}} \right) = 132.200\left( {120} \right)MHz \hfill \\ \end{gathered} $$      

Self-diffusion in α-Fe2O3 natural single crystals

Measurements of¹⁸O self-diffusion in hematite (Fe₂O₃) natural single crystals have been carried out as a function of temperature at constant partial pressure a_{O 2}=6.5·10^?2 in the temperature range 890 to 1227 °C. The a_{O 2} dependence of the oxygen self-diffusion coefficient at fixed temperature T=1150 °C has also been deduced in the a_{O 2} range 4.5·10^?4 - 6.5·10^?1. The concentration profiles were established by secondary-ion mass spectrometry; several profiles exhibit curvatures or long tails; volume diffusion coefficients were computed from the first part of the profiles using a solution taking into account the evaporation and the exchange at the surface. The results are well described by $$D_O \left( {{{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } s}} \right. \kern-\nulldelimiterspace} s}} \right) = 2.7 \cdot 10^8 a_{O_2 }^{ - 0.26} \exp \left( { - \frac{{542\left( {{{kJ} \mathord{\left/ {\vphantom {{kJ} {mol}}} \right. \kern-\nulldelimiterspace} {mol}}} \right)}}{{RT}}} \right)$$ From fitting a grain boundary diffusion solution to the profile tails, the oxygen self-diffusion coefficient in sub-boundaries has been deduced. They are well described by $$D''_O \left( {{{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } s}} \right. \kern-\nulldelimiterspace} s}} \right) = 3.2 \cdot 10^{25} a_{O_2 }^{ - 0.4} \exp \left( { - \frac{{911\left( {{{kJ} \mathord{\left/ {\vphantom {{kJ} {mol}}} \right. \kern-\nulldelimiterspace} {mol}}} \right)}}{{RT}}} \right)$$ Experiments performed introducing simultaneously¹⁸O and⁵⁷Fe provided comparative values of the self-diffusion coefficients in volume: iron is slower than oxygen in this system showing that the concentrations of atomic point defects in the iron sublattice are lower than the concentrations of atomic point defects in the oxygen sublattice. The iron self-diffusion values obtained at T>940 °C can be described by $$D_{Fe} \left( {{{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } s}} \right. \kern-\nulldelimiterspace} s}} \right) = 9.2 \cdot 10^{10} a_{O_2 }^{ - 0.56} \exp \left( { - \frac{{578\left( {{{kJ} \mathord{\left/ {\vphantom {{kJ} {mol}}} \right. \kern-\nulldelimiterspace} {mol}}} \right)}}{{RT}}} \right)$$ The exponent - 1/4 observed for the oxygen activity dependence of the oxygen self-diffusion in the bulk has been interpreted considering that singly charged oxygen vacancies V _O ^? are involved in the oxygen diffusion mechanism. Oxygen activity dependence of iron self-diffusion is not known accurately but the best agreement with the point defect population model is obtained considering that iron self-diffusion occurs both via neutral interstitals Fe _x ⁱ and charged ones.     

Gleichzeitige Messung von Hyperfeinstruktur,Stark-Effekt und Zeeman-Effekt des39K19F mit einer Molekülstrahlresonanzapparatur

An electric Molecular-Beam-Resonance-Spectrometer has been used to measure simultanously the Zeeman- and Stark-effect splitting of the hyperfine structure of³⁹K¹⁹ F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. The observed (Δm _J =±1)-transitions were induced electrically. Completely resolved spectra of KF in theJ=1 rotational state have been measured. The obtained quantities are: The electric dipolmomentμ _e l of the molecul forv=0,1 and 2; the rotational magnetic dipolmomentμ _J forv=0,1; the difference of the magnetic shielding (σ_⊥ ? σ_∥) by the electrons of both nuclei as well as the difference of the molecular susceptibility (ξ_⊥ ? ξ_∥). The numerical values are

$$\begin{array}{*{20}c} {\mu _{e1} = 8,585(4)deb,} \\ {\frac{{(\mu _{e1} )_{\upsilon = 1} }}{{(\mu _{e1} )_{\upsilon = 0} }} = 1,0080,} \\ {{{\mu _J } \mathord{\left/ {\vphantom {{\mu _J } J}} \right. \kern-\nulldelimiterspace} J} = ( - )2352(10) \cdot 10^{ - 6} \mu _B ,} \\ {(\sigma _ \bot - \sigma _\parallel )F = ( - )2,19(9) \cdot 10^{ - 4} ,} \\ {(\sigma _ \bot - \sigma _\parallel )K = ( - )12(9) \cdot 10^{ - 4} ,} \\ {(\xi _ \bot - \xi _\parallel ) = 3 (1) \cdot 10^{ - 30} {{erg} \mathord{\left/ {\vphantom {{erg} {Gau\beta ^2 }}} \right. \kern-\nulldelimiterspace} {Gau\beta ^2 }}} \\ \end{array} $$     

Der Einfluß der Wärmebewegung auf die Breite einer Mössbauerlinie über den Dopplereffekt zweiter Ordnung

AtT=0 a perfect Mössbauer line has natural line widthΓ=?/τ _n . However, with rising temperature the width increases. The reason of the line broadening is the second order Doppler effect which causes a stochastic frequency modulation of theγ-radiation, reflecting the thermal motion of the Mössbauer atom. Following Josephson in treating the second order Doppler shift as a mass changeΔM=E _n/c² of theγ-emitting atom caused by the loss of nuclear excitation energy E _n , and using the well known relaxation formalism for calculating theγ-frequency spectrum, the line broadeningΔ Γ is evaluated within the framework of harmonic lattice theory. For a parabolic lattice frequency spectrum with Debye-temperature Θ one obtains $$\Delta {\Gamma \mathord{\left/ {\vphantom {\Gamma \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma } = \left( {{{\tau _n } \mathord{\left/ {\vphantom {{\tau _n } {\tau _c }}} \right. \kern-\nulldelimiterspace} {\tau _c }}} \right) \cdot \left( {{{E_n } \mathord{\left/ {\vphantom {{E_n } {Mc^2 }}} \right. \kern-\nulldelimiterspace} {Mc^2 }}} \right) \cdot F\left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right),where\tau _c = {{\rlap{--} h} \mathord{\left/ {\vphantom {{\rlap{--} h} k}} \right. \kern-\nulldelimiterspace} k}\Theta $$ is the correlation time of the lattice vibrations. The functionF(T/Θ) may be expanded in powers ofT/Θ, yielding $$F\left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right) = 9720\pi \left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right)^7 forT<< \Theta $$ and $$F\left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right) = 2.7\pi \left( {{T \mathord{\left/ {\vphantom {T \Theta }} \right. \kern-\nulldelimiterspace} \Theta }} \right)^2 forT > > \Theta $$ , respectively. Although unavoidable, the line broadening is obviously too small to be observable by means of the present experimental technique.     

Search for double electron capture of 106Cd

A search for double electron capture of ¹⁰⁶Cd was performed at the Modane Underground Laboratory (4800 m w.e.) using a low-background and high-sensitivity multidetector spectrometer TGV-2 (Telescope Germanium Vertical). New limits on β ⁺/EC, EC/EC decays of ¹⁰⁶Cd were obtained from preliminary calculations of experimental data accumulated for 4800 h of measurement of 10 g of ¹⁰⁶Cd with enrichment of 75%. They are > 9.1 × 10¹⁸ yr, > 1.9 × 10¹⁹ yr for transitions to the first 2⁺, 511.9 keV excited state of ¹⁰⁶Pd, and > 1.3 × 10¹⁹ yr, > 6.2 × 10¹⁹ yr for transitions to the ground 0⁺ state of ¹⁰⁶Pd. All limits are given at 90% C.L. The text was submitted by the authors in English.     

Study of the mechanisms of pre-equilibrium nuclear reactions in the microscopic approach

The mechanisms of pre-equilibrium nuclear reactions are investigated within the Statistical Multistep Direct Process (SMDP) + Statistical Multistep Compound Process (SMCP) formalism. It has been shown that from an analysis of linear part in such dependences as $$\ln \left[ {{{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} \mathord{\left/ {\vphantom {{\frac{{d^2 \sigma }}{{d\varepsilon _b d\Omega _b }}} {\varepsilon _b^{1/2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon _b^{1/2} }}} \right]upon\varepsilon _b $$ and $$\ln \left[ {{{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} \mathord{\left/ {\vphantom {{\frac{{d\sigma ^{SMDP \to SMCP} }}{{d\varepsilon _b }}} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right. \kern-\nulldelimiterspace} {\frac{{d\sigma ^{SMDP} }}{{d\varepsilon _b }}}}} \right]upon{{U_B } \mathord{\left/ {\vphantom {{U_B } {\left( {E_a - B_b } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {E_a - B_b } \right)}}$$ one can extract information about the type of mechanism (SMDP, SMCP, SMDP→SMCP) and the number of stages of the multistep emission of secondary particles. In the above approach, we have discussed the experimental data for a broad class of reactions in various entrance and exit channels.     

Weber potential from finite velocity of action?

The Weber potential energy U for charges q and q' separated by the distance R is U = (qq'/R)[1 – (dR/dt)²/2c²]. If this potential arises from a finite velocity c of energy transfer Q', where the retarded rate of transfer from q' to q is dQ(t-R/c)/dt = Q'[1 – (dR/dt)/c] and where the advanced rate from q to q' is dQ(t+R/c)/dt = Q'[1 + (dR/dt)/c], then the resultant time-average root-mean-square action is given by . Identifying Q' with the Coulomb potential energy qq'/R, the Weber potential is obtained. Using the same argument, Newtonian gravitation yields a corresponding Weber potential energy, qq'/R being replaced by ( - Gmm'/R).     

Neutrino production of dimuons

Neutrino interactions with two muons in the final state have been studied using the Fermilab narrow band beam. A sample of 18v _μ like sign dimuon events withP _μ>9 GeV/c yields 6.6±4.8 events after backgroud subtraction and a prompt rate of (1.0±0.7)×10^?4 per single muon event. The kinematics of these events are compared with those of the non-prompt sources. A total of 437v _μ and 31 \(\bar v_\mu \) opposite sign dimuon events withP _μ>4.3 GeV/c are used to measure the strange quark content of the nucleon: \(\kappa = {{2s} \mathord{\left/ {\vphantom {{2s} {\left( {\bar u + \bar d} \right) = 0.52_{ - 0.15}^{ + 0.17} \left( {or\eta _s \frac{{2s}}{{u + d}} = 0.075 \pm 0.019} \right) for 100< E_v< 230 GeV\left( {\left\langle {Q^2 } \right\rangle = {{23 GeV^2 } \mathord{\left/ {\vphantom {{23 GeV^2 } {c^2 }}} \right. \kern-0em} {c^2 }}} \right)}}} \right. \kern-0em} {\left( {\bar u + \bar d} \right) = 0.52_{ - 0.15}^{ + 0.17} \left( {or\eta _s \frac{{2s}}{{u + d}} = 0.075 \pm 0.019} \right) for 100< E_v< 230 GeV\left( {\left\langle {Q^2 } \right\rangle = {{23 GeV^2 } \mathord{\left/ {\vphantom {{23 GeV^2 } {c^2 }}} \right. \kern-0em} {c^2 }}} \right)}}\) using a charm semileptonic branching ratio of (10.9±1.4)% extracted from measurements ine ⁺ e ^? collisions and neutrino emulsion data.     

A rigorous upper bound in electrostatics on a random lattice ensemble

We consider a Kirchhoff network on a random two-dimensional lattice with links and weights as previously specified, and a circular boundary of radiusR. We show rigorously that the resistance between the central point and the boundary, averaged over all placements of the remaining sites with site density ?, is bounded above by $$\begin{array}{*{20}c} {(4\pi )^{ - 1} [\ln (4\pi \rho R^2 ) + 1] + 16[\tan ^{ - 1} 5^{ - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} + 5^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} /(\sqrt 5 + 1)^2 ]} \\ { \simeq (4\pi )^{ - 1} \ln (4\pi \rho R^2 ) + 12.0.} \\ \end{array} $$      

Universal estimate of the gap for the Kac operator in the convex case

The aim of this paper is to prove that ifV is a strictly convex potential with quadratic behavior at ∞, then the quotient μ₂/μ₁ between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L²(? ^m ) by exp ?V(x)/2 · exp Δ_x · exp ?V(x)/2 where Δ_x is the Laplacian on ? ^m satisfies the condition: $${{\mu _2 } \mathord{\left/ {\vphantom {{\mu _2 } {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}} \right. \kern-\nulldelimiterspace} {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}$$ where σ is such that HessV(x)≥σ>0.     

Influence of a strong longitudinal static electric field on free carrier faraday effect in an n-type InSb at room temperature at submillimeter wave frequencies

The expression for free carrier Faraday rotation and for ellipticity , as the function of the applied parallel static electric field and static magnetic field for a given value of wave angular frequency and electron concentration N₀, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as , where , , and . Here m^* and e denote the effective mass and charge of electron, respectively. _g is the forbidden bandgap of semiconductor. v₀ is the carrier drift velocity, which is a non-linear function of E₀ in high field condition. A possibility of a simple way of determining the non-linear v₀ vs E₀ characteristics of semiconductors by the measurement of Faraday rotation is also discussed.     

Light quark masses in quantum chromodynamics and chiral symmetry breaking

We derive model independent lower bounds for the sums of effective quark masses \(\bar m_u + \bar m_d \) and \(\bar m_u + \bar m_s \) . The bounds follow from the combination of the spectral representation properties of the hadronic axial currents two-point functions and their behavior in the deep euclidean region (known from a perturbative QCD calculation to two loops and the leading non-perturbative contribution). The bounds incorporate PCAC in the Nambu-Goldstone version. If we define the invariant masses \(\hat m\) by $$\bar m_i = \hat m_i \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^{{{\gamma _1 } \mathord{\left/ {\vphantom {{\gamma _1 } {\beta _1 }}} \right. \kern-\nulldelimiterspace} {\beta _1 }}} $$ and <F ²> is the vacuum expectation value of $$F^2 = \Sigma _a F_{(a)}^{\mu v} F_{\mu v(a)} $$ , we find, e.g., $$\hat m_u + \hat m_d \geqq \sqrt {\frac{{2\pi }}{3} \cdot \frac{{8f_\pi m_\pi ^2 }}{{3\left\langle {\alpha _s F^2 } \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} $$ ; with the value <α _u F ²?0.04GeV⁴, recently suggested by various analysis, this gives $$\hat m_u + \hat m_d \geqq 35MeV$$ . The corresponding bounds on \(\bar m_u + \bar m_s \) are obtained replacingm _π ² f _π bym _K ² f _K . The PCAC relation can be inverted, and we get upper bounds on the spontaneous masses, \(\hat \mu \) : $$\hat \mu \leqq 170MeV$$ where \(\hat \mu \) is defined by $$\left\langle {\bar \psi \psi } \right\rangle \left( {Q^2 } \right) = \left( {{{\frac{1}{2}\log Q^2 } \mathord{\left/ {\vphantom {{\frac{1}{2}\log Q^2 } {\Lambda ^2 }}} \right. \kern-\nulldelimiterspace} {\Lambda ^2 }}} \right)^d \hat \mu ^3 ,d = {{12} \mathord{\left/ {\vphantom {{12} {\left( {33 - 2n_f } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {33 - 2n_f } \right)}}$$ .     

Formation of a narrow baryon resonance with positive strangeness in K + collisions with Xe nuclei

The data on the charge-exchange reaction K ⁺Xe → K ⁰ pXe′, obtained with the bubble chamber DIANA, are reanalyzed using increased statistics and updated selections. Our previous evidence for formation of a narrow pK ⁰ resonance with mass near 1538 MeV is confirmed. The statistical significance of the signal reaches some 8 (6) standard deviations when estimated as $ {S \mathord{\left/ {\vphantom {S {\sqrt B \left( {{S \mathord{\left/ {\vphantom {S {\sqrt {B + S} }}} \right. \kern-0em} {\sqrt {B + S} }}} \right)}}} \right. \kern-0em} {\sqrt B \left( {{S \mathord{\left/ {\vphantom {S {\sqrt {B + S} }}} \right. \kern-0em} {\sqrt {B + S} }}} \right)}} $ . The mass and intrinsic width of the Θ⁺ baryon are measured as m = 1538 ± 2 MeV and Γ = 0.39 ± 0.10 MeV.