830 nm emission in sodium vapor

We demonstrate that the intriguing 830 nm coherent emission, which is observed when sodium vapor is pumped with a high-power pulsed laser tuned near the 3S4D two-photon transition, is due to an axially phase-matched six-wave mixing process. This conclusion is based upon the observation of emission near 584 nm, which is coupled to the 830 nm emission in the six-wave mixing process: ₁+₂=2_L–_4D4P –_4P3D . In addition, we have observed coherent emission near 1.16 m, which is due to an analogous process involving cascade through the 4S (as opposed to the 3D) state. We calculate the wavelengths of all photons involved in these processes using the standard formulas of parametric wave-mixing theory, and show that they can be predicted to within experimental uncertainties. Finally we report observations of significant blue shifts of the 830 nm and 1.16 m emissions in a mixed sodium-potassium vapor. These shifts can be readily understood by considering the effect of the potassium on the frequency-dependent refractive index of the vapor. Due to these results, other recent interpretations of the 830 nm emission as stimulated excimer emission on the Na₂ 1³ _g ⁺ 1³ _u ⁺ band must now be rejected.     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.     

Asymptotics of decay of correlations for lattice spin fields at high temperatures. I. The Ising model

We find the asymptotic decrease of correlations < _{A +y} , _B >,yZ ^v +1, |y|, in the Ising model at high temperatures. For the case when monomials _A and _B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension . For even monomials _A , _B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) =1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary . Here we use besides the saddle-point method, some arguments from scattering theory.     

Non-commutative spheres

LetA be the irrational rotation algebra, i.e. theC ^*-algebra generated by two unitariesU, V satisfyingVU=e ²ⁱ UV, with irrational, and consider the fixed point subalgebraB under the flip automorphismUU ^–1,VV ^–1. We prove thatB is an AF-algebra.Dedicated to Professor Huzihiro Araki on the occasion of his 60'th birthday     

Spectral duality for planar billiards

For a bounded open domain with connected complement inR ² and piecewise smooth boundary, we consider the Dirichlet Laplacian- on and the S-matrix on the complement ^c . We show that the on-shell S-matricesS _k have eigenvalues converging to 1 askk ₀ exactly when-- has an eigenvalue at energyk ₀ ² . This includes multiplicities, and proves a weak form of transparency atk=k ₀. We also show that stronger forms of transparency, such asS _{k 0} having an eigenvalue 1 are not expected to hold in general.     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Nonstationary Markov chains and convergence of the annealing algorithm

We study the asymptotic behavior as timet + of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt + , a nonstationary Markov chain may exhibit phase transitions. Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC/log(t/t ₀) ast+, with a computable constantC andt ₀ the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t^–1+) for some 0<1, with =0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal annealing schedule. This optimal choice reflects the competition between two physical effects: (a) The adiabatic effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the super-cooling effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly.     

Symmetries of physical theories

A physical theory is, by definition, a complete orthomodular atomic lattice having the covering property. GivenL a quantum logic andS _L the set of all its states, a theorem is proved which asserts that, if certain reasonable assumptions concerningS _L are satisfied, then for any bijective convex mappingU: S _LS_L, satisfying also certain physically meaningful conditions, there exists a unique automorphismV: L L such thatU(p)=p oV ^–1 for allp S _L.     

Finite-size effects in a field-theoretic model with long-range exchange interaction

We present a systematic approach to the calculation of finite-size (FS) effects for anO(n) field-theoretic model with both short-range (SR) and long-range (LR) exchange interactions. The LR exchange interaction decays at large distances as 1/r ^d+2–2,0⁺,0⁺. Renormalization group calculations ind=d _u – are performed for a system with a fully finite (block) geometry under periodic boundary conditions. We calculate the FS shift of the critical temperature and the FS renormalized coupling constant of the model to one-loop order. The universal scaling variable is obtained and the FS scaling hypothesis is verified.     

Relativistic Hypernuclei: Old Problems and New Prospects

The first experiments with relativistic hypernuclei (Bowen; Khorozov and Lukstins) were done many years ago. They demonstrated not only that such extremely difficult experiments are manageable but also their great advantage, a possibility of observing and studying independently the production and weak decay of hypernuclei: the points of the production and decay of relativistic hypernuclei are separated by tens of centimeters instead of some microns in classical experiments. At the same time these first experiments revealed a huge problem with selection of the proper trigger.Recently we proposed to explore a unique feature of the ⁹Be nucleus: after removing a neutron from its ground state several groups of alpha-particles appear from different excited states of a residual nucleus ⁸Be. Detection of the correlated pair of -particles produced in a vacuum volume at a distance of some 40 cm from target is an unambiguous signal of nonmesonic decay of hypernucleus ¹⁰ Be(++n+n) or ¹⁰ Be(++n+p). In this particular case it is possible to take exclusive decay rates (on different excited states of ⁸Be^*). This open a way for a phenomenological analysis of matrix elements of the four-baryon weak interaction.The experiment is approved for new accelerator Nuclotron at JINR, Dubna.     

Analyticity properties of eigenfunctions and scattering matrix

For potentialsV=V(x)=O(|x|^–2–) for |x|,x³ we prove that if theS-matrix of (–, –+V) has an analytic extension to a regionO in the lower half-plane, then the family of generalized eigenfunctions of –+V has an analytic extension toO such that for |Imk|<b. Consequently, the resolvent (–+V–z ²)^–1 has an analytic continuation from ⁺ to {kOImk|<b} as an operator from _b ={f=e ^–b|x| g|gL ₂(³)} to _–b . Based on this, we define for potentialsW=o(e ^–2b|x|) resonances of (–+V, –+V+W) as poles of and identify these resonances with poles of the analytically continuedS-matrix of (–+V, –+V+W).The author would like to thank the Institute for Advanced Study for its hospitality and the National Science Foundation for financial support under Grant No. DMS-8610730(1)     

Energy-Momentum Properties and the Spin of Short-Lived Particles

The space–time translation property of a stable particle is characterized by a time-like Lorentz vector (E, _k ). We show in this contribution that unstable particles are, in addition, characterized by a space-like Lorentz 4-vector of uncertainties, or spreads, ( E, k). This is true for unstable states created in formation-, in production-, and in decay-experiments. The space-like nature of the spread vector causes a nonzero momentum spread to be present in all Lorentz frames so that there is no Lorentz frame in which the unstable particle is entirely at rest. With the space-like spreadvector ( E, k) in addition to the time-like (E, _k ), also the rotation property of an unstable particle is affected, and unstable states have an uncertainty in their spin. This means neighboring spin states are occupied in addition to the original spin. Experiments are discussed that show a principal limitation of the accuracy of spin measurement from finite lifetimes. Wave functions for unstable particles are discussed, and we show in the example of a short-lived spin-0 state that the appearance of a spin neighbor in the amplitude is proportional to the inverse lifetime.     

π-Meson absorption by three-particle nuclei

The reactions ⁺ +³He 3p and ^– +³H 3n are considered with both two- and three-nucleon pion-absorption taken into account. The pion-absorption with all nucleons of the nucleus involved and the interference of the two absorption mechanisms are shown to be of the same importance as a pure two-particle absorption in nuclei. The latter cannot be considered dominating as it used to be.     

Pion absorption and proton k.o. reaction in4He

The experimental results obtained by the Torino—Dubna—Frascati collaboration in the study of the ⁺ absorption in⁴He and of the⁴He ³H knock-out reactions induced by ⁺ and ^– mesons are presented. The analysis of the data shows that the absorption by a quasideuteron is the principal mechanism as in the case of the ⁺ absorption in light nuclei, that the isobaric resonance excitation is dominant in the proton k.o. reactions, that the pion interaction on quasi-free³H is not negligible. The quasi-free processes appear to play a major role in the nonelastic pion interaction on nuclei, but the clustering effects are also important.Presented at the symposium Mesons and Light Nuclei, Liblice, Czechoslovakia, June 1981.It seems to be possible to conclude that in the future important nuclear information will be extracted in the accurate study of the different channels of pion interactions with nuclei.     

Two-neutrino positron double-beta decay of <Superscript>106</Superscript>Cd for the 0<Superscript>+</Superscript> → 0<Superscript>+</Superscript> transition

The two-neutrino positron double-beta decay of ¹⁰⁶Cd for the 0 ⁺ 0⁺ transition has been studied in the Hartree-Fock-Bogoliubov model in conjunction with the summation method. In the first step, the reliability of the intrinsic wave functions of ¹⁰⁶Cd and ¹⁰⁶Pd nuclei has been tested by comparing the theoretically calculated results for yrast spectra, reduced B(E2: 0⁺ 2⁺) transition probabilities, quadrupole moments Q(2⁺) and gyromagnetic factors g(2⁺) with the available experimental data. In the second step, the nuclear transition matrix element M₂ and the half-life T_1/2² for the 0 ⁺ 0⁺ transition have been calculated with these wave functions. Moreover, we have studied the effect of deformation on the nuclear transition matrix element M₂.     

μSR studies in the heavy fermion systems UNi2Al3, UPd2Al3 and U2PtC2

We have performed Muon Spin Relaxation (SR) measurements of several heavy fermion systems: UNi₂Al₃ (single crystal), UPd₂Al₃ (single crystal) and U₂PtC₂ (polycrystal). ZF -SR measurements of UNi₂Al₃ show magnetic order below the Néel temperatureT _N=4.6K, with the muon precession frequency(T 0)=2.5MHz at low temperatures. The local field distribution obtained at low temperatures in UNi₂Al₃ resembles that of the incommensurate spin density wave (SDW) system (TMTSF)₂PF₆, suggesting an incommensurate spin arrangement in UNi₂Al₃. TF -SR measurements give the penetration depth(T 0)=6100Å for UPd₂Al₃ and(T 0)=7000Å for U₂PtC₂ respectively. Since UPd₂Al₃ and U₂PtC₂ are clean superconductors (/l-0), we can combine n _s/m^* with the Sommerfeld constant n _s ^1/3 m^* to calculate the Fermi temperatureT _F ^3/4–1/4, which represents the characteristic energy scale of the superconducting carriers. We obtainT _C/T_F 0.01 for both UPd₂Al₃ and U₂PtC₂, comparable to other exotic superconductors, but quite different from simple metal BCS superconductors, whereT _C/T_F 0.01.     

Finite size effects in critical dynamics and the renormalization group

Systems representable as a time-dependent Ginzburg-Landau model with nonconserved order parameter are considered in a block (V=L ^d) geometry with periodic boundary conditions, both for space dimensionalitiesd4 andd=4–. A systematic approach for studying finite size effects on dynamic critical behavior is developed. The method consists in constructing an effective reduced dynamics for the lowest-energy (q=0) mode by integrating out the remaining degrees of freedom, and generalizes recent analytic approaches for studying static finite size effects to dynamics. Above four dimensions, the coupling to the other (q0) modes is irrelevant and the probability densityP(,t) for the normalized order parameter=d^d x(x,t)/V satisfies a Fokker-Planck equation. The dynamics is equivalently described by the Langevin equation for a particle moving in a ||⁴ potential or by a supersymmetric quantum mechanical Hamiltonian. Dynamic finite size scaling is found to be broken, e.g. the order parameter relaxation rate varies at the bulk critical temperatureT _c, as (T _c, L)L ^–d/2 asL. By contrast, ford<4, the coupling to the other (q0) modes cannot be ignored and dynamic finite size scaling is valid. The asymptotic behavior of correlation and response functions can be studied within the framework of an expansion in powers of ^1/2. The scaling function associated with is computed to one-loop order. Finally, the many component (n) limit is briefly considered.     

A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases

Using the properties of the Jordan curve, the following theorem on the heteroclinic tangency in orientation-preserving two-dimensional maps is proved: LetT :R ² R ² be a one-parameter family ofC ¹ diffeomorphisms andJ=DetDT be such that 0<J1 or 1J<. LetW _u ⁿ be the unstable manifold of a hyperbolicn-cycle andW _s ^m the stable manifold of a hyperbolicm-cycle. Suppose that for< _c ,W _u ⁿ andW _s ^m have no common points, and that for> _c ,W _u ⁿ andW s/m have a transversal heteroclinic point. Then at= _c ,W _u ⁿ andW _s ^m are in the first asymptotic heteroclinic tangency except for the following three cases: (1)n=m; both cycles are without reflection. (2)m=2n; then- andm-cycles are with and without reflection, respectively; (3)n=2m; then- andm-cycles are without and with reflection, respectively.     

Particle polarizations in decays of5HeΛ and4HΛhypernuclei

Nucleon polarizations are examined in the hypernucleus decays⁵He, ⁴He + p + _–,⁴H, ³H + p + _–, and⁴H, ³He + n + _–. The effect of the strong interaction in the final state is taken into account. A significant role is predicted for the resonance interaction in the p-⁴He system in the formation of the proton polarization in the⁵He decay. It is also shown that in the⁴H, ³H + p + _– decay the proton-polarization components should attain significant magnitudes, while in the charge-exchange decay⁴H, ³He + n + _–, the neutron polarization is expected to be not too large.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 100–103, October, 1988.     

On the indistinguishability of classical particles

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2)³ are reduced to points in the phase space, the partition functionTr{exp(–) for N identical bosons (fermions) approaches (2)^–3N(N!) ... d³r₁ d³p₁ ... d³r_N d³p_N exp(–H). The two factors, (2)^–3N and (N!)^–1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles.