A comment on Hönl-London factors

The literature on the Hönl-London factors of diatomic molecules, the factors in intensity formulas dependent on the rotational quantum numbers, is somewhat confused for the case of perpendicular transitions. We derive a formula, Eq. (23), for allowed singlet-singlet transitions between levels of definite parity, and show that it is consistent with standard sum rules. As an illustration, the formula is applied to the calculation of lifetimes in the B¹Π_u state of Na₂, showing that the corrected formula gives results in good agreement with experimental measurements.     

Hot electron transverse conductivity in quantizing magnetic fields

In this work the general expression of the electron transverse conductivity tensor of an electron-phonon system being in crossed strong electric and quantizing magnetic fields is considered starting from the Kubo-Kalashnikov formula. An explicit formula for the hot electron transverse conductivity σ _xx is obtained and it is compared to a Titeica-type formula with the temperature of electrons replaced by an effective electron temperature depending on the electric field.     

Model independent fitting to theZ line shape

An improved model independent formula to fit theZ line shape parameters is presented. The accuracy of the QED radiative corrections used in the formula is estimated, as is the absolute accuracy of the formula itself. The precision of the formula is compared with existing fitting formulae, in terms of cross sections produced and in terms of theZ parameters returned from a fit. The errors which the LEP-SLC experiments can expect to achieve on theZ parameters are examined as a function of total integrated luminosity. The formula presented here can fit theZ parameters with high accuracy: below 1 MeV for theZ mass, the order of 1 MeV for total width, and below 0.04% for the hadronic and leptonic peak cross sections. Such biases will remain much below the expected experimental uncertainties from LEP and SLC.     

A contribution to the temperature dependence of the tunnel current of metal-dielectric-metal structures

Assuming the tunnel transfer of electrons, a general formula is derived for the current flowing between the metal electrodes of a metal-dielectric-metal structure. The formula is numerically solved for a rectangular barrier without image forces at absolute zero and several higher values of temperature. Though the temperature dependence of the tunnel current is very weak, it is possible to use it for determining the dielectric parameters. The metal-dielectric work function can be found from the maximum of the curvej _T /j ₀ vs.V, the thickness of the dielectric layer from the value ofj _T /j ₀ at zero voltage.     

Macroscopic approach to the Casimir friction force

The general formula is derived for the vacuum friction force between two parallel perfectly flat planes bounding two material media separated by a vacuum gap and moving relative to each other with a constant velocity v. The material media are described in the framework of macroscopic electrodynamics whereas the nonzero temperature and dissipation are taken into account by making use of the Kubo formulas from non-equilibrium statistical thermodynamics. The formula obtained provides a rigorous basis for calculation of the vacuum friction force within the quantum field theory methods in the condensed matter physics. The revealed v dependence of the vacuum friction force proves to be the following: for zero temperature (T = 0) it is proportional to (v/c)³ and for T > 0 this force is linear in v/c.     

A Variant of the Mukai Pairing via Deformation Quantization

Let X be a smooth projective complex variety. The Hochschild homology HH_?(X) of X is an important invariant of X, which is isomorphic to the Hodge cohomology of X via the Hochschild?CKostant?CRosenberg isomorphism. On HH_?(X), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH_?(X). The latter pairing on HH_?(X) is intimately linked to the study of Fourier?CMukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu??s Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quantization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the ??Noncommutative Riemann?CRoch?? implies the classical Riemann?CRoch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties.     

Exact analytical formula for Voigt spectral line profile

An analytical formula for the Voigt spectral line profile has been derived. The obtained formula is the solution of the second-order differential equation of the Voigt profile. The obtained results using the new analytical formula are in perfect agreement with the published results obtained by numerical calculation methods for a wide range of the line damping parameter (a) . The accuracy of the obtained formula is found to be 100% for all ranges of the line damping parameter (a).     

Skyrme force and the mass formula

Coefficients of the mass formula related to the property of the nuclear surface are calculated from the Skyrme forces SII, SIII and SV defined by the Orsay group. Comparing with Myers' mass formula parameters, it is found that the effective stiffness Q for separation of the neutron and proton surfaces is too large by a factor of about two for SIII. The discrepancy can also be stated by saying that the surface symmetry energy calculated from SIII is too small, and related to this is a too thin neutron skin for N > Z nuclei. The SV force gives a much too small value for Q. The SII force, with a repulsive term somewhere intermediate between SIII and SV gives a better value for Q. The strong density dependence of SIII is needed, however, to give good single-particle spectra (or the effective mass large enough). The SIII force has a density dependence between unlike nucleons only (between neutrons and protons). In this paper we modify the Skyrme force so as to allow for a density dependence also between like nucleons. A many-body theoretical justification for this modification is given in the paper. The mass formula parameter Q is reproduced with our new force maintaining a sufficiently strong density dependence. The neutron skin thickness is increased and the ⁴⁰Ca-⁴⁸Ca isotope shift of the charge radius is also well reproduced with this new force. A change of the density dependence from a linear to a one-third power dependence improves the agreement with the compressibility, and also affects the neutron-proton distributions in the surface.     

Final-state summation in photoelectron diffraction

A formula was derived to perform the summation over the l_c±1 final states of the dipole transition matrix element and the scattering path operator. This formula shows that often it seems more convenient to speak about l_c+1 and l_c±1 channels instead of the l_c+1 and the l_c−1 channel. Using this formula and restricting the considerations on the first order scattering and the plane wave approximation, explicit formulas were given for the photoelectron intensity, the circular dichroism and the spin polarization in terms of products of the vectors involved in the problem: photon wavevector, photon polarization, magnetization, electron wave vector, and orientation of the spin detector. Additionally the scattering part contains the position vectors of the scatterers with respect to the emitter. These formulas are discussed as well for magnetic as for nonmagnetic samples and for s, p, d and f core states. General expressions are given for the peak rotation. Also it is possible to write down explicitly the dependence on absolute value and the phases of the complex radial matrix elements.     

On the quantum density of states and partitioning an integer

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one-dimensional harmonic oscillator potential, it is well known that the asymptotic (N→∞) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p^s(n), the latter being the number of partitions of n into a sum of sth powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d^s(n) for distinct partitions. We find that the distinct square partitions d²(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos-Lehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.     

On the fermion mass-spectrum of Kaluza-Klein supergravity

A general setup for the harmonic expansions of D = 11 supergravity about classical compactifying solutions is given. Mass-matrices for the fermion mass-spectrum are derived both in the case of Freund-Rubin backgrounds and of Englert type solutions. The mass-matrix is explicitly given in terms of the eigenvalues of invariant operators on the 7-manifold. In the Freund-Rubin case the mass formula is very easy, while in the Englert case it is quite complicated and requires special techniques. In the second part of the paper applications to special manifolds for the Freund-Rubin background are considered, postponing the application of the general Englert formula to future publications. In particular, the round 7-sphere with the complete fermion spectrum and the SU(3) × SU(2) × U(1) isometric M^pqr-spaces are considered. For this latter the condition for the existence of a spinor structure (p = odd number) is established, deriving the explicit spectrum and the mass-eigenvalues for one of the two sectors of which it is composed. This relies on a general distinction between longitudinal and transverse fermion particle-modes which is emphasized in the first part of the paper. In this paper the longitudinal spectrum of M^pqr cosets is derived, postponing the transverse one to future work.     

Theory of higher-order stark effect of symmetric top molecules

The Stark effect formula for symmetric top molecules, which is correct to the fifth order of an external electric field, is given. The formula is applicable to linear molecules without or with electronic angular momenta coupled in Hund;s case a. Numerical tables are given for the special case of linear molecules without any electronic angular momentum. The effect of centrifugal stretching is considered to the first order. It is shown that our perturbation formula, which is correct to the sixth order of an external electric field, is good even when μE/B is about 1.     

The microwave spectrum of trifluoroacetonitrile in excited vibrational states

The microwave transitions J → J + 1 have been observed in the symmetric top molecule trifluoroacetonitrile, CF₃CN, for molecules excited into several of the low-lying vibrational states. Measurements made in the degenerate states v₇ = 1 v₈ = 1 have been fitted to the formula derived by Grenier-Besson and Amat (1) for transitions J → J + 1 in degenerate vibrational states of molecules with C_3v symmetry. The measurements for the state v₈ = 1 have been extended to 150 GHz where it is shown that the above formula becomes inadequate to describe accurately the observed spectrum.     

The quantum theory of nucleon correlation in finite nuclei. II. Higher order correlation effects and additive pair approximation

In the first part of this article it was shown that the variational solution of the Schroedinger equation of a finite Fermion system can be written as a finite sum of A terms (for A particles) the first of which is the Hartree-Fock energy, while the rest represent the correlation effects. In the first part explicit formulas for the 2-particle correlation energy were given. In this paper explicit formulas are given for the higher order correlation energies. It is shown that two different models can be developed depending on the orthogonality condition used. Beginning with the 4th order effects the “linked” and “unlinked” correlation terms are separated. An exact formula is given for the case in which only the 2-particle effects, linked and unlinked are taken into account. The “additive pair approximation” in which the correlation energy is given as the sum of 2-particle energies is investigated and it is shown to be related to the exact formula by a clearly defined set of approximations. Various possible applications of the model are discussed.     

Nonlinear superposition formula for SUSY SG/MKdV equations revisited

The derivation of nonlinear superposition formula is reexamined for the N=1 supersymmetric sinh-Gordon equation. It is shown that this formula results from the spatial part of Bäcklund transformation only and therefore can be associated with the supersymmetric MKdV equation.     

Refinement of a formula for decay after weak coherent excitation of a sphere

A new formula has been given recently by A.A. Svidzinsky and M.O. Scully to describe the temporal evolution of the excitation function in a large sphere satisfying the Markov condition after excitation by a single photon. This formula is based on a physically reasonable Ansatz from which differential equations are inferred for the undetermined radial functions in the Ansatz. The solution to these differential equations leads to the formula for β. Numerical calculations from this formula yield a value ∼10% for the maximum probability of occupancy of secondary excited states.In this paper, we refine the formula of Svidzinsky and Scully by allowing the radial functions in the Ansatz to depend on the angular index l of the spherical Bessel functions. By using the Debye formula for the asymptotic behavior of j_l(u) for large l as well as u, we obtain differential equations in each angular sector, similar to theirs but with a dependence on l. The solution to these equations yields our improved formula, from which we calculate 17.1% for the maximum probability of secondary excited states.     

The application of fractional derivatives in stochastic models driven by fractional Brownian motion

In this paper, in order to establish connection between fractional derivative and fractional Brownian motion (FBM), we first prove the validity of the fractional Taylor formula proposed by Guy Jumarie. Then, by using the properties of this Taylor formula, we derive a fractional Itô formula for H∈[1/2,1), which coincides in form with the one proposed by Duncan for some special cases, whose formula is based on the Wick Product. Lastly, we apply this fractional Itô formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional Brownian motion. The case that the drift, volatility and risk-free interest rate are all dependent on t is also discussed.