Two-parameter Formula Research Articles

Two-parameter formula for ground-band energy spectra of A = 120-200 mass region nuclei

This paper presents a two-parameter formula which works best for an Anharmonic Vibrator (AHV), i.e., soft and rigid rotor nuclei in the A = 120-200 mass region. A comparison with the other two-parameter formulae, i.e., ab, Brentano Soft Rotor Formula (SRF), Ejiri and Variable Moment of Inertia (VMI) model, shows that this formula gives a better fit with the experimental data.

Validity of the single-term formula for ground band energies in light Xe–Gd nuclei

A two-parameter single-term energy formula EJ=aJb is used to study the energy–spin relationship in the ground bands of even–even light Xe–Gd (68⩽N⩽78) nuclei. This formula works better for soft nuclei as well as for deformed nuclei. We also compared it with other two-parameter formulae, i.e. Ejiri, ab, pq and Brentano et al soft rotor (SRF) formulae. The power index b and the coefficient a are fairly constant, independent of the number of levels for spin J⩽12+. The variation of average and with Z, N and with the valence nucleon pair product NpNn is illustrated. Extension of the formula to EJ=aJ(b+cJ) is also tested.

Recursive Formulas Related To The Summation Of The Mobius Function

Let M(x)=∑⌊x⌋k=1μ(k) be the Mertens function, where μ is the Mobius function. The Mertens conjecture, that |M(x)|<x√, was disproved indirectly by A. M. Odlyzko and H. J. J. te Riele [J. Reine Angew. Math. 357 (1985), 138–160; MR0783538 (86m:11070)]. The authors present several formulas and identities for M(x) with the aim of aiding the search for an explicit counterexample to the Mertens conjecture. The paper's main goal is to reduce the number of summands in the calculation of the Mertens function. To this end, the authors use previously known identities for M(x) and combinatorial arguments to give a two-parameter formula (Theorem 10) for calculating M(x) involving previously known values and an additional term depending mainly on residues. No explicit calculations are made, but several specialized versions of this formula are derived; the formulas involved require no more than x/3 summands, depending on the number of previously calculated values.

From correlation-consistent to polarization-consistent basis sets estimation of NMR spin–spin coupling constant in the B3LYP Kohn–Sham basis set limit

Abstract Based on B3LYP spin–spin coupling constants (SSCC) of several molecules calculated with cc-pV x Z, cc-pCV x Z, cc-pCV x Z-sd and cc-pCV x Z-sd+ t basis sets, a reasonably fit, using the two-parameter formula, to the Kohn–Sham complete basis set limit (CBS) is shown. Improvement in the CBS values going from cc-pV x Z to the most elaborated cc-pCV x Z-sd+ t basis set family is observed: standard deviation for all data drops from 33.7 to 23.1, and from 6.0 to 4.8 Hz after excluding problematic 1 J (F,H) and 1 J (F,C). Calculation of water’s 1 J (OH) using B3LYP/cc-pCV x Z and B3LYP/pcJ- n significantly improved the FC term convergence.

Computational forces in the Linpack benchmark

Dimensional analysis reduces a complicated ten-parameter formula for the execution time of the Linpack benchmark to a simpler two-parameter formula. These two parameters are ratios of software forces and hardware forces that determine a self-similarity surface. Machines move along paths on this surface as the problem size and the number of processors change. Two machines scale the same way, they move along the same path, if they have the same hardware forces. To design efficient algorithms, the programmer must produce software forces large enough to overcome the hardware forces. Modern machines have larger hardware forces than older machines and are harder to program.

Measuring nanopore size from the spin–lattice relaxation of CF 4 gas

The NMR 19F spin–lattice relaxation time constant T 1 for CF 4 gas is dominated by spin–rotation interaction, which is mediated by the molecular collision frequency. When confined to pores of approximately the same size or smaller than the bulk gas mean free path, additional collisions of molecules with the pore walls should substantially change T 1. To develop a method for measuring the surface/volume ratio S/ V by measuring how T 1 changes with confinement, we prepared samples of known S/ V from fumed silica of known mass-specific surface area and compressed to varying degrees into cylinders of known volume. We then measured T 1 for CF 4 in these samples at varying pressures, and developed mathematical models for the change in T 1 to fit the data. Even though CF 4 has a critical temperature below room temperature, we found that its density in pores was greater than that of the bulk gas and that it was necessary to take this absorption into account. We modeled adsorption in two ways, by assuming that the gas condenses on the pore walls, and by assuming that gas in a region near the wall is denser than the bulk gas because of a simplified attractive potential. Both models suggested the same two-parameter formula, to which we added a third parameter to successfully fit the data and thus achieved a rapid, precise way to measure S/ V from the increase in T 1 due to confinement in pores.

PROOF OF THE VARIATIONAL PRINCIPLE FOR A PAIR HAMILTONIAN BOSON MODEL

We give a two-parameter variational formula for the grand-canonical pressure of the Pair Boson Hamiltonian model. By using the Approximating Hamiltonian Method we provide a rigorous proof of this variational principle.

Threshold and saturation properties of the field-induced twist cell with two parameters weak anchoring boundaries

On the basis of two-parameter formula of weak surface coupling anchoring energy of nematic liquid crystals proposed by Zhao et al recently, the general torque equilibrium equation and boundary conditions of the director have been obtained and the threshold field as well as the saturation field of the field-induced twist cell have been analysed for three kinds of configurations, i.e. homogeneous, splay and Pi cells formed by different rubbing conditions and pretilt angles. The results indicated that the polar anchoring has no effect on the threshold field. It is determined only by the twist anchoring and pretilt angle. But, the polar anchoring and twist anchoring are coupling with each other and have a great influence on the saturation field. The formulae for calculating the threshold field and saturation field are given. These results will be very useful in understanding surface physics and the design of liquid crystal cells.

Importance of Configuration Interaction for Accurate Atomic Data: Fluorescence Yields of K-Shell Vacancy Lithium-Like Ions

We demonstrate that the inclusion of configuration interaction (CI) results in significant values for the K-shell fluorescence yields of Li-like ions, which are zero in a single-configuration approach. Modeling codes for simulating supernova remnants under nonequilibrium ionization conditions or photoionized plasmas such as active galactic nuclei or X-ray binaries need to be updated accordingly. A two-parameter fitting formula for the fluorescence yields has been developed. The generality of important CI effects on atomic calculations is pointed out. Subject headings: atomic data — atomic processes — line: formation — X-rays: general

Importance of Configuration Interaction for Accurate Atomic Data: Fluorescence Yields of K-Shell Vacancy, Lithium-like Ions

We demonstrate that the inclusion of configuration interaction (CI) results in significant values for the K-shell fluorescence yields of Li-like ions, which are zero in a single-configuration approach. Modeling codes for simulating supernova remnants under nonequilibrium ionization conditions or photoionized plasmas such as active galactic nuclei or X-ray binaries need to be updated accordingly. A two-parameter fitting formula for the fluorescence yields has been developed. The generality of important CI effects on atomic calculations is pointed out.

New nomogram for foetal weight estimation based on Hadlock's two-parameter formula.

To develop and validate a graphical method for ultrasonic foetal weight estimation with predictive qualities comparable to state-of-the-art computational methods. Study data were obtained from 3839 consecutive singleton pregnancies with ultrasound examination within 7 days of delivery. We translated the well established Hadlock 2-parameter formula based on abdomen circumference and femur length into a nomogram [corrected]. We compared the measured foetal weight with estimations obtained from this new nomogram and three other methods. Measured by the foetal weight percentage error the new nomogram underestimates the foetal weight on average by 2.5 % (9.86) (mean [SD]) with uniform results over the complete birth weight spectrum. By the same measurement the Hadlock 4-parameter formula underestimates the foetal weight by 1.2 % (9.3) whereas the Hansmann nomogram overestimates the foetal weight by 14.5 % (16.1) The difference between the estimated foetal weight percentage error obtained from the new nomogram and from the Hadlock 2-parameter formula is 0.06 % (0.27) and hence clinically irrelevant. The new nomogram has predictive qualities comparable to state-of-the-art computational methods and is thus not only as reliable but also easy to use in situations when computers are not available. It can be recommended for foetal weight estimation over the whole spectrum of birth weight.

Analytical Representation of the Curves of Liquid–Vapor Phase Equilibrium for Saturated Hydrocarbons

A unified two-parameter approximation formula is used to derive analytical dependences for P(T) curves of liquid–vapor phase equilibrium of saturated hydrocarbons CnH2n + 2 (alkanes) with numbers 1 ≤ n ≤ 20 from methane (n = 1) to eicosane (n = 20). Two approximation options are given, namely, that involving the use of individual values of parameters for each alkane, and that in the form of a unified formula in which the only parameter characterizing an individual component is the number n.

Energy dependence of the total cross section for electron scattering by chloromethanes in the energy range 0.5–10 keV

The energy dependence of the total cross section for electron scattering by methane and its chloride compounds has been studied by considering the latest experimental results and those predicted by the Born-Bethe theory. A simple analytical expression depending on the number of electrons and the polarizability of the target is proposed for these molecules. The energy dependence given by this formula shows an asymptotic behavior for increasing energy in better agreement with the Born-Bethe theory. The inconsistency with the Born-Bethe theory of the two-parameter formula given recently by Karwasz et al. has also been shown.

Neutron fibres: a three-dimensional analysis of bendinglosses

The confined propagation of slow neutrons along waveguides of asmall cross section (fibres), discussed theoretically some time ago anddemonstrated experimentally by other authors later, is analysed further.Motivated by those experiments, a three-dimensional quantum-mechanicaltreatment of the associated bending losses is presented. An approximate(two-parameter) formula is derived for the transmission coefficientΥ of the curved fibre, which displays its explicit dependence onthe curvature radius Rcu, for 0<Rcu< + ∞. Byadjusting one of the parameters (b), consistently with the theoreticalanalysis, and by fitting the other, the approximate formula describes allmeasured data for Υ. The parameter b (which accounts for theexperimental data for Υ at large Rcu) allows theestimation of the number of effectively excited modes in the fibre.

Total cross sections for electron scattering on chloromethanes: Formulation of the additivity rule

Total cross sections for electron scattering on CH 3Cl, CH2Cl2 , and CHCl3 have been measured by an absolute method in the ~75‐4000!-eV energy range. The overall experimental error is below 5%. A formulation of the additivity rule is proposed. The molecular cross sections are approximated by a Born-like twoparameter formula. We show that the low-energy parameter is correlated to the molecular polarizability. We show also that the high-energy parameter for a given molecule can be expressed as the sum of the high-energy parameters of the constituent atoms. The model has been successfully verified for two groups of halomethanes: the CH4 ,C H 3Cl, CH2Cl2 , CHCl3 , and CCl4 series and the CF4 ,C F 3Cl, CF2Cl2 , CFCl3 , and CCl4 series. The model has been successfully extended to molecules containing Si and S atoms, such as H2S, SF6 , SiH4 , and SiF4 . @S1050-2947~99!10102-1# PACS number~s!: 34.80.Bm, 34.80.Gs Chlorine substituted methanes play an important role in atmospheric chemistry @1#; their time in the Earth’s troposphere is several tens of years. Free Cl radicals formed by ultraviolet-induced dissociation act as catalysts in the chain of reactions leading to the destruction of the stratospheric ozone @2#. In particular, chloromethane (CH3Cl), used in refrigerators and organic chemistry, is the most abundant halocarbon present in the atmosphere. CH3Cl is also the most frequently studied halocarbon in electron-scattering experiments.

Iterative relationship in two-parameter formulas for rotational spectra and a universal equation

An iterative relationship connecting a series of two-parameter formulas for rotational spectra is revealed. It is also pointed out that various two-parameter formulas are merely solutions of a universal equation with different powers.

Channel Dependence of Dielectronic Recombination Rate Coefficients of Fe25+ Ion

In the frame of quantum defect theory, the state-to-state dielectronic recombination rate coefficients of Fe25+ ion are calculated by a simple relativistic configuration interaction method. The calculated results are further expressed with a two-parameter fitting formula, and then the dependence of fitting parameters on radiative decay final channel is studied.

Comparison of the Harris andabexpressions for the description of nuclear superdeformed rotational bands

The nuclear superdeformed rotational bands in the $A\ensuremath{\sim}190$ region were systematically analyzed by using the Harris two-parameter formula and the ab expression. Similar to the situation in normally deformed nuclei, there exists obvious and systematic deviation of the Harris formula from the experiment. In contrast, the prediction of the $\mathrm{ab}$ formula is very close to the experiment, and may be conveniently used for the description of both normally deformed and superdeformed bands.

Precision fit of nuclear rotational energies

A five-parameter I(I+1) expansion is presented and is used to analyse systematically the experimentally known level energies of the ground rotational bands in doubly-even deformed nuclei up to very high spin. Better and excellent agreement is obtained with experimental data. The expansion coefficients obtained by a least-squares fitting procedure are compared with those deduced from the previous two-parameter formulae showing closer results to the Holmberg - Lipas formula, as well as the Wu - Zeng expression.

Universal Anharmonic Vibrator Description of Quasiband Structures in Collective Even-Even and Odd-ANuclei

It is found that energy correlations within the bands of the good rotor odd-A nuclei are well described by an anharmonic vibrator formula, which indicates an unexpected constancy of the moment of inertia. Based on a large collection of quasibands in all collective even-even and odd-A nuclei, an empirical recurrence relation is proposed which allows one to calculate the excitation energies of any band starting from its lowest two. This simple two-parameter formula can be reduced approximately to that of a second order anharmonic vibrator. {copyright} {ital 1997} {ital The American Physical Society}