Screening of charge in a semiinfinite electron gas

Within linear response and the self-consistent field approximation an equation for the screening of a chargee ^iwt δ(r?r ₀),r ₀=(0, 0,z ₀) by an electrongas confined to the half-spacez>0 is derived. From this 3 cases are discussed: 1. Application to a homogeneous electron gas bounded by an infinite potential. 2. Thomas-Fermi approximation. 3. Image potential approximation.     

The oscillator representation and the stability of three-body Coulomb systems

Within nonrelativistic quantum mechanics the Wick-ordering method, called the oscillator representation, is suggested for calculating the energy spectrum for a wide class of potentials allowing the existence of a bound state. As test cases, anharmonic (V(r)=r ²) and screened Coulomb potentials are considered. In particular, the method is applied to three-body Coulomb systems to obtain the dependence of the bound-state energy on the masses and charges of the particles. The calculations of the bound-state energies for the moleculesH ^–=(pee),H ₂ ⁺ =(ppe), (e ^–e^–e⁺) and (pp), (dd), (dt) prove the accuracy of the zeroth approximation to be better than one per cent. For the three-body Coulomb system with charges +, –, – and arbitrary masses the region of stability is determined. For the systems (pe ^–C⁺), (A ⁺e^–e⁺), and (pB ^–e^–) the critical masses are calculated to beM _c=1.945m_e,M _A=4.350m_e andM _B=1.575m_e. It turns out that the system (pe ^–e⁺) is unstable.     

Analyticity properties and a convergent expansion for the inverse correlation length of the low-temperatured-dimensional Ising model

We show that the inverse correlation lengthm(z) of the truncated spin-spin correlation function of theZ ^d Ising model with + or — boundary conditions admits the representationm(z) = –(4d–4)ln z(1–_d1) + r(z) for smallz=e ^–, i.e., large inverse temperatures is ad-dependent analytic function atz = 0, already known in closed form ford = 1 and 2; ford = 3 b_n can be computed explicitly from a finite number of the Z^d limits of z = 0 Taylor series coefficients of the finite lattice correlation function at a finite number of points ofZ ^d.     

Potential surfaces for time-like geodesics in the Curzon metric

In this paper we deduce the general pattern of the potential surfaces for time-like geodesics in the Curzon metric. We find that for fairly small energies and orbital angular momenta, the time-like geodesics group into two sets; the geodesics of one set tend to thez-axis asR=(r²+z²)^1/2 0,R=0 being a directional singularity, the others tend to ther-axis. At low energies these two sets are detached but they merge together as the energy increases. Stable circular motion is confined to thez = 0-plane and an energy threshold for stationary motion exists and is equal (per unit of rest-mass energy) to 0.945, a value almost indistinguishable from that in the Schwarzschild space-time.     

Hyperfine structure investigations of some odd levels of Co I by level crossing and optical methods

Using the level crossing technique the ratios and absolute values of the hyperfine structure (hfs) constants of the levelsz ⁴F_9/2 andz ⁴F_7/2 of the configuration 3d ⁷4s4p of Co I were measured:z ⁴ F _9/2: ¦A¦=(811±12)MHz; ¦B¦=(48±93) MHz;B/A=?0.06±0.11 A>0; B<0z ⁴ F _7/2: ¦A¦ = (659 ±11)MHz; ¦B¦=(33±84)MHz;B/A=?0.05±0.13 A>0; B<0. In addition the hfs constants of three other excited levels of Co I could be determined by optical methods:z ⁴ F _9/2:A=525±26 MHz;B=200 MHzy ⁴ F _9/2:A=300±30 MHz;B=?500 MHzy ⁴ G _11/2:A=315±20 MHz;B=400 MHz. The experimental results are compared with known experimental and also with theoretical values which where calculated using the parametric potential method.     

On the Entropy of Schwarzschild Space-Time

In a previous paper by Pollock and Singh, it was proven that the total entropy of de Sitter space-time is equal to zero in the spatially flat case K=0. This result derives from the fundamental property of classical thermodynamics that temperature and volume are not necessarily independent variables in curved space-time, and can be shown to hold for all three spatial curvatures K=0,±1. Here, we extend this approach to Schwarzschild space-time, by constructing a non-vacuum interior space with line element ds ²=e^2λ(r) dt ²?e^?2λ(r) dr ²?r ²(dθ ²+sin² θd? ²), where $\mathrm{e}^{2{\lambda }(r)}=-\frac{1}{2}(1-\frac{r^{2}}{R_{0}^{2}})$ , which matches onto the vacuum exterior Schwarzschild metric in such a way that e^2λ and d(e^2λ )/dr are both continuous at the Schwarzschild radius R ₀=2M. Then we show that the volume entropy is equal to A/4, where $A\equiv 4\pi R_{0}^{2}$ is the area of the apparent horizon, as found by Hawking.     

Observation of instabilities in a Paul trap with higher-order anharmonicities

Systematic measurements of the relative ion number stored in a Paul trap within the stability diagram given by the solution of the equation of motion reveal many lines, where only few or no ions can be confined. The observations can be explained by the presence of perturbations from higher-order components in the trapping potential, which is a quadrupole potential in the ideal case. The resonances follow the equation (n _r /2) _r + (n _r /2) _z = 1,n _r +n _z =N, where 2N is the order of the perturbation,n _r ,n _z are integer and _r , _z are stability parameters of the trap. The experiments were performed on H⁺ and H ₂ ⁺ ions, which are detected after a storage time of 0.3 s by ejection from the trap.     

Variation of Dipole Moment Function of O-H Bond of t-Butanol with the Nature of Solvent and Its Temperature

The intermolecular influence on dipole moment function is evaluated for O-H bond of t-butanol in different nonpolar solvents at temperatures ranging from 10° to 60°C employing IR band intensities of fundamental and first overtone bands. Two sets of dipole moment derivatives have been calculated corresponding to – and +- combinations of the transition moment matrix elements R₁₀ and R₂₀, the values for ++ and -+ combinations are equal in magnitude to those for - and +- combinations, respectively and opposite in signs. In general the dipole moment derivatives increase on lowering the temperature as well as with increasing molecular interactions with the solvent molecules. Dipole moment plots with dimensionless coordinate ξ [=(r-r_e)/r_e, where r and r_e are internuclear distances during vibration and at equilibrium, respectively] are reported for various systems considered. It is found that for +- combination the dipole moment maximum shifts to higher internuclear distances when polarisation of the solute molecules is increased by lowering of temperature or increase in molecular interactions between solute and solvent molecules. A reverse trend is observed for – combination.- The OH band of t-butanol vapor has been measured.     

Proton shielding function for hydrogen chloride

For a diatomic molecule the nuclear shielding constant σ(ξ) of either nucleus can be expanded as a power series in the relative displacement from equilibrium ξ. Thus the nuclear shielding function is

where ξ=(r-r _e)/r _e with r the actual bond length and r _e the equilibrium bond length. The σ_e ⁽ⁱ⁾ are molecular parameters. By experimental observation of the temperature dependence of the proton magnetic shielding of hydrogen chloride gas it is possible, after allowing for intermolecular effects, to obtain the values σ_e ⁽⁰⁾=32·48 (±0·33) p.p.m. and σ_e ⁽¹⁾=-100 (±24) p.p.m. for the coefficients of the proton shielding function. Using this data it is possible to show that the isotope shift between H³⁷Cl and H³⁵Cl is about 0·001 p.p.m. By a comparison with earlier results for molecular hydrogen it would appear that in some instances differences in vibrational and rotational averaging may alter chemical shifts between different compounds by amounts considerably larger than the experimental error in chemical shift measurement.     

On rational solutions of Yang-Baxter equations. Maximal orders in loop algebra

In 1982 Belavin and Drinfeld listed all elliptic and trigonometric solutionsX(u, v) of the classical Yang-Baxter equation (CYBE), whereX takes values in a simple complex Lie algebrag, and left the classification problem of the rational one open. In 1984 Drinfeld conjectured that if a rational solution is equivalent to a solution of the formX(u,v)=C ₂/(u–v)+r(u,v), whereC ₂ is the quadratic Casimir element andr is a polynomial inu,v, then deg _u r=deg _v r1. In another paper I proved this conjecture forg=sl (n) and reduced the problem of listing nontrivial (i.e. nonequivalent toC ₂/(u–v)) solutions of CYBE to classification of quasi-Frobenius subalgebras of g. They, in turn, are related with the so-called maximal orders in the loop algebra of g corresponding to the vertices of the extended Dynkin diagramD ^e (g). In this paper I give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofD ^e (g) with coefficient 2 or 3. In particular I will find all solutions forg=o=(5) and some solutions forg=o(7),o(10),o(14) andg ₂.     

Zur Chronologie des alten Ägypten

The velocityv of the propagation of discharge along the anode of a self-quenchingG—M-counter is a function of total pressureP, pressure of the quenching gasP _D, radius of the cathoder _a and of the anoder _i andV _ü the difference between working- and starting-potential. For the mixtures argon-methylal, argon-alcohol and helium-alcohol isv=v ₀·exp[k·(V _ü/V _e)^1/2] withv ₀ the velocity at the starting potentialV _e v ₀=(a+b·P _D/P)·V _n ^1/2 ·exp [(c?d·P_D/P·V _n ^?1/2 ] andV _n=V _e·(lnr _a/r _i)^?1.k, a, b, c andd are characteristical constants of the filling gas.     

The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity

We demonstrate that the six vertex model (XXZ spin chain) with =(q+q^-1)/2 and q^2N=1 has an invariance under the loop algebra of sl₂ which produces a special set of degenerate eigenvalues. For =0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.     

A theoretical study of BrCN in the Π electronic ground state: Large amplitude bending motion

We have calculated the three-dimensional potential energy surfaces for the 1 ²A′ and 1 ²A″ states of BrCN⁺ at the MR-SDCI_DK+Q/[QZP-ANO-RCC (Br, C, N)] level of theory, where MR-SDCI_DK means ‘multi-reference single and double excitation configuration interaction calculation with Douglas-Kroll Hamiltonian.’ These ab initio potential energy surfaces have a common minimum (corresponding to the state) at a linear equilibrium structure with r_e(Br-C) = 1.735 Å and r_e(C-N) = 1.199 Å. Variational RENNER calculations yield a zero-point averaged structure (with the structural parameters calculated as expectation values over rovibrational wavefunctions) with 〈r(Br-C)〉₀ = 1.739 Å, 〈r(C-N)〉₀ = 1.204 Å, and 〈∠(Br-C-N)〉₀ = 172(4)°. A severe Fermi resonance between 2ν₂ and ν₃ has been found theoretically for the ²A″ potential energy surface. Comparing the ab initio zero-point averaged structure with a recent, experimentally derived r₀ structure, it is concluded that the effects of large-amplitude bending motion should be taken into account explicitly in the process of deriving the r₀ structure from the experimental values of the rotational constants. The electronic structure of BrCN⁺ has also been discussed.     

Regularisierende Deltafunktionen

In the Laurent Schwartz theory of distributions the integral∫ δ(r)f(r)dτ _r is only defined for a certain class of test functions. Unfortunately, in physics, we obtain test functionsf(r) likee ^ikr /r, e ^?kr /r, ?(e ^ikr /r) and so on, which are not belonging to the Laurent Schwartz class (§1). Here, we want to extend the class of test functions so that it includes physically meaningful functions with poles of finte order inr=0. For this purpose we replaceLaurent Schwartz's axiomatic method defining theδ-distribution by a constructive one consideringδ(r) as a given set of sequences of functions (§2). First we prove that the redefinedδ-function satisfy the equations, axiomatically assumed byLaurent Schwartz (§3). Then we obtain well defined and finite results even in the case of test functions with poles atr=0 (§4). The Fourier components of the newδ-function are given (§5). Finally we show why theδ-function is Lorentz invariant (§6).     

Investigation of the dependence of the e<Subscript>0</Subscript>-electron yield on the radioactive decay location in <Superscript>64</Superscript>Cu sources

The yields of near-zero-energy e₀ electrons from sources of different thickness in the ⁶⁴Cu decay have been measured. At small thicknesses of the ⁶⁴Cu source, the e₀-electron yield Y _e sharply increases and is qualitatively described by the dependence Y _e ～ r ^?2, where r is the distance from the point of charge formation to the source surface.     

Power series expansions of quantum-mechanical potential surfaces

Power series expansions of quantum-mechanical potential surfaces for H₂O and LiH₂ are generated using conventional valence displacement coordinates and alternative distance [(r-r _e)/r] and angle [(sin (θ/2) — sin (θ_e/2))/sin (θ_e/2)] variables. Power series in the new variables are demonstrated to be superior in terms of the quality of the fit to the energygeometry data, predicted equilibrium geometry and energy, and stability of expansion coefficients.     

On the Determination of the Equilibrium Structure of the PH3 Molecule

Values of the r_e and α_e equilibrium structural parameters of the PH₃ molecule have been determined on the basis of experimental data on the fundamental bands of the PH₂D and PHD₂ species only, without any knowledge of the force-constant-type parameters of the PH₃ molecule. The functions of the equilibrium rotational parameters A^e, B^e, and C^e have been determined. These functions are totally independent of the values of any resonance interaction parameters. Then these functions have been used to determine r_e and α_e constants. The obtained values r_e=(1.416 776±0.000 164) Å and α_e=(93.56±0.29)° differ a little from those found in the earlier literature.     

Nonlinear Optical Characteristics of BSO and BGO Photorefractive Crystals in Visible and Infrared Ranges

Nonlinear refraction, nonlinear absorption and optical limiting in photorefractive crystals Bi₁₂SiO₂₀(BSO) and Bi₁₂GeO₂₀(BGO) at the wavelengths of 1064 and 532 nm were investigated. It was shown that both BSO and BGO crystals possess by positive nonlinear refraction in two investigated spectral ranges (n ₂ ^BSO=(2.5 ± 0.5)× 10^–12 esu, n ₂ ^BGO=(6.3 ± 1.3)× 10^–12 esu at equals 1064 nm; n ₂ ^BSO=(4.4 plusmn; 0.9)× 10^–12 esu, n ₂ ^BGO=(7.4 ± 1.5)× 10^–12 esu at = 532 nm). The nonlinear absorption was due to three-photon absorption at the wavelength of 1064 nm ( ⁽³⁾ _BSO=(2.5 ± 0.8)× 10^–20cm³W^–2, ⁽³⁾ _BSO=(4.4 ± 1.3)× 10^–20cm³W^–2) and two-photon absorption at the wavelength of 532 nm ( ⁽²⁾ _BSO=(2 ± 0.4)× 10^–9cm W^–1, ⁽²⁾ _BGO=(3.7 ± 0.7)× 10^–9cm W^–1).     

Behavior at small distances and low temperatures of the ion-ion distribution function of the two-dimensional Coulomb gas

We compute upper and lower bounds for the canonical ion-ion distribution functiong ₁₁ ^(N) (r) of the two-dimensional Coulomb gas for smallr and 1<<2, where is the plasma parameter. Both bounds are proportional tor ^2-/(-1), which proves thatg ₁₁ ^(N) (r) behaves asr ^2-, as conjectured by Hansen and Viot. We conjecture that, in the thermodynamic limit,g ₁₁(r) ~ 2(-1)^-1(r/a)^2-, wherea=(n)^–1/2 is the mean interionic distance. We also compute the canonical one-body distribution function with one pair (+,–) in a disk, for anyr and any.