On the Schrödinger equation with a time dependent mass parameter: Nuclear fragmentation theory

We study the dynamics of a system with a coordinate-dependent inertia tensorB _ik (x ₁,...,x _n ), in which one of the coordinates is treated classically,x _j →x _j (t), so that the associated mass parameter becomes time-dependent. As a result, the Hamiltonian must contain a pure imaginary term in order that probability be conserved. The nuclear fragmentation theory is an example of such a system. The inertia parameter associated with the mass fragmentation degree of freedom depends on the relative motion coordinate and, hence, on time. The time dependence of the imaginary term is derived from a fully quantum mechanical treatment, by going to the classical limit in the relative motion variable. A connection is made with the closely related situation which arises if one transforms from an inertial system to one which depends non-linearly on the time.     

The vibration-rotation problem in triatomic molecules allowing for two large-amplitude stretching vibrations

An expression for the Hamiltonian of a vibration-rotating triatomic molecule is derived, using two curvilinear stretching coordinates ?₁ and ?₃ and one rectilinear bending coordinate S₂, in such a way that the Hamiltonian obtained is applicable to any bent triatomic molecule and allows for large displacements along the stretching coordinates. From this, a zeroth-order Hamiltonian H_s⁰ (?₁, ?₃) is obtained, describing the energy levels associated with the two stretching vibrations ν₁ and ν₃. The vibrational energy levels (v₁, v₃^even) of an XY₂ molecule having unequal bond lengths at equilibrium are then calculated. The kinetic energy T₀ (?₁, ?₃) of the Hamiltonian effectively takes into account the two large-amplitude motions in ν₁ and ν₃ together with their interaction. A model calculation is described for a bent XY₂ molecule (SO₂ in its ¹A′ (¹B₂) excited state) in which the ν₃ oscillation occurs in a double-minimum potential. Coupling by kinetic energy terms in the Hamiltonian turns out to be very small in this example.     

Tetravalent vanadium in CaF2 and SrF2 crystals: An EPR study

An EPR study of tetravalent vanadium centers created by room temperature X-irradiation in CaF₂ and SrF₂ is presented. The production efficiency of these centers is enhanced by previous annealing of the samples at 1000 K in air. The symmetry of V⁴⁺ ions is tetragonal and its EPR spectrum can be described by an axial spin Hamiltonian including a Zeeman and hyperfine term with $\text{S =}\text{1}\text{2}$ and $\text{I =}\text{7}\text{2}$ (corresponding to ⁵¹V nuclei). The following values for the spin Hamiltonian parameters are obtained g_∥ = 1.947 ± 0.002, g_⊥ = 1.935 ± 0.005, A_∥ = 500 ± 5 MHz, A_⊥ = 150 ± 10 MHz in the case of SrF₂ and g_∥ = 1.945 ± 0.002, A_∥ = 505 ± 5 MHz and A_⊥ < 200 MHz, in the case of CaF₂. A model for the center including an interstitial O^2? ion is tentatively proposed.     

Structure of A = 3 nuclear systems using realistic Hamiltonians

The structure of A = 3 low-energy scattering states is described using the hyperspherical harmonics method with realistic Hamiltonian models, consisting of two- and three-nucleon interactions. Both coordinate and momentum space two-nucleon potential models are considered.     

Toward automated analysis of dipole-coupled NMR spectra of solutes in liquid crystals

Dipolar-coupled spectra of molecules dissolved in liquid crystalline solvents are analyzed by applying pattern recognition procedures to two-dimensional z-filtered correlation spectra (z-COSY). Transitions that share common common energy levels can be recognized as such by recording spectra with different flip angles. This allows one to obtain a picture of the energy-level diagram without knowledge of the Hamiltonian, although in the current preliminary version of the program, the number of spins and their symmetry must be known prior to analysis. By adapting the programs NMREN and NMRIT, described bySwalen and Reilly (J Chem. Phys. 37, 21 (1962)) and Ferguson and Marquardt (J. Chem. Phys. 41, 2087 (1964)), the energies of the eigenstates derived from the spectra are assigned to the diagonal elements of the Hamiltonian in the product base. These assignments can be made without guesswork, except for states with the same quantum numbers, for which it is necessary to permute assignments by trial and error. The algorithm is iterated for different tentative assignments of the eigenstates to yield the shifts and couplings. The best Hamiltonian (best assignment) is chosen on the grounds of a comparison between simulated and experimental spectra. Examples of entirely automated analysis are shown for A₃B and A₂BC systems.     

Simulation of the thermal fragmentation of fullerene C60

Defect formation and annealing processes in fullerene C₆₀ at T = (4000–6000) K are studied using molecular dynamics with a tight-binding potential. The cluster lifetime until fragmentation, which proceeds, as a rule, through the loss of a C₂ dimer, has been found as a function of temperature. The activation energy and the frequency factor in the Arrhenius equation for the fragmentation rate have been found to be E _a = (9.2 ± 0.4) eV and A = (8 ± 1) × 10¹⁹s^?1. It is shown that fragmentation can occur already after the C₆₀ cluster loses its spherical shape. This fact must be taken into account in theoretical calculations of E _a.     

Two dimensional NQR separation of inhomogeneous and homogeneous lineshapes in disordered solids

A two-dimensional (2D) zero field NQR separation of inhomogeneous and homogeneous lineshapes technique is described. The nuclear spin Hamiltonian for spinsI>1/2 in zero magnetic field consists to a good approximation only of the electric quadrupole term. The 2D separation technique enables a separate spectroscopic observation of a static and a randomly time-fluctuating dynamic part of the quadrupole interaction. The separation is based on the fact that nuclear spin precession under a static quadrupolar Hamiltonian can be time-reversed whereas this can not be achieved under the action of a randomly time-fluctuaing Hamiltonian. The 2D spectrum displays in theω ₂-domain the inhomogeneously broadened lineshape, which is a convolution of the inhomogeneous static frequency distribution function and the homogeneous (adiabatic) lineshape. Theω ₁-domain shows the pure homogeneous lineshape. A deconvolution of the inhomogeneous lineshape with the homogeneous one yields a pure static inhomogeneous frequency distribution function which is characteristic and theoretically known for many different models of disordered solids like glasses and incommensurate systems. This technique is important in studies where both lineshapes have comparable widths. The 2D NQR separation technique has been applied to⁷⁵As in a proton glass Rb_0.98(NH₄)_0.02H₂AsO₄.     

Dynamical calculation of the mass fragmentation in the collision238U-238U

The fragmentation of the nuclear system²³⁸U-²³⁸U is studied by treating the fragmentation coordinate quantum mechanically. The time dependent Schrödinger equation, the Hamiltonian of which is calculated from the microscopic asymmetric two center shell model, is solved by the finite difference method. In order to study the fragmentation mechanism, model calculations have been carried out, by assuming the collective fragmentation potential as an oscillator potential which is adjusted to the realistic potential. Effects arising from the dynamical treatment of the fragmentation coordinate are found to be important in the collision of²³⁸U on²³⁸U. The dependence of the fragmentation on the incident energy is discussed.     

Dynamical calculation of the mass fragmentation in the collision238U-238U

The fragmentation of the nuclear system²³⁸U-²³⁸U is studied by treating the fragmentation coordinate quantum mechanically. The time dependent Schrödinger equation, the Hamiltonian of which is calculated from the microscopic asymmetric two center shell model, is solved by the finite difference method.In order to study the fragmentation mechanism, model calculations have been carried out, by assuming the collective fragmentation potential as an oscillator potential which is adjusted to the realistic potential. Effects arising from the dynamical treatment of the fragmentation coordinate are found to be important in the collision of²³⁸U on²³⁸U. The dependence of the fragmentation on the incident energy is discussed.     

Simultaneous analysis of the ν1, ν4, 2ν2, ν2 + ν5, and 2ν5 infrared bands of 12CH3F

The Fourier-transform spectrum of CH₃F from 2800 to 3100 cm^?1, obtained by Guelachvili in Orsay at a resolution of about 0.003 cm^?1, was analyzed. The effective Hamiltonian used contained all symmetry allowed interactions up to second order in the Amat-Nielsen classification, together with selected third-order terms, amongst the set of nine vibrational basis functions represented by the states ν₁(A₁), ν₄(E), 2ν₂(A₁), ν₂ + ν₅(E), 2ν₅⁰(A₁), and 2ν₅^±2(E). A number of strong Fermi and Coriolis resonances are involved. The vibrational Hamiltonian matrix was not factorized beyond the requirements of symmetry. A total of 59 molecular parameters were refined in a simultaneous least-squares analysis to over 1500 upper-state energy levels for J ≤ 20 with a standard deviation of 0.013 cm^?1. Although the standard deviation remains an order of magnitude greater than the precision of the measurements, this work breaks new ground in the simultaneous analysis of interacting symmetric top vibrational levels, in terms of the number of interacting vibrational states and the number of parameters in the Hamiltonian.     

A rotation-torsion-vibration treatment with three-dimensional internal coordinate approach and additional FTIR spectral assignments for the CH3-bending fundamentals of methanol

A theoretical model has been developed to account for certain features of both newly observed and previously reported CH₃-bending subbands between 1450 and 1570 cm⁻¹ in the high-resolution Fourier transform infrared spectrum of CH₃OH [Can. J. Phys. 79 (2001) 435]. The features include (i) an apparent inversion of the rotationless E-A torsional splitting with respect to the ground state, i.e., the A state located above the E state, (ii) a pronounced upward slope in the K-reduced torsion-vibration energy pattern for the subband origins, and (iii) unexpected A₁/A₂ inversion of the K=2A and K=3AJ-rotational levels that led to ambiguity in identifying the vibrational mode as or . The model is an effective internal coordinate Hamiltonian constructed in G₆ molecular symmetry with the CH₃-bends coupled to each other and to torsion and including a- and γ-type Coriolis coupling. With this model, 33 out of 36 experimental upper-state K-term values for newly assigned , and ν₁₀ subbands plus previous ν₄ subbands have together been fitted successfully, employing 9 adjustable parameters and 17 fixed parameters to give a standard deviation of 0.14 cm⁻¹. The P_γ Coriolis term appears to be the leading cause of the upward shift in the K-reduced energies. When J-dependence is introduced via a rotational Hamiltonian including b- and c-type Coriolis terms in addition to molecular asymmetry, the observed A₁/A₂ inversion of the K=2A and 3A rotational levels can also be reproduced. Predictions using the fitted K-rotation-torsion-vibration Hamiltonian show an interesting Coriolis-induced crossover and mixing of the ν₅ and ν₁₀ torsion-vibration energy patterns. These predictions played a role in identifying two of the new ν₅ subbands in the crossing region, thereby helping to validate the model.     

Fine and hyperfine structure in the X2Σ+ state of gas-phase yttrium monoxide

The A²Π_r-X²Σ⁺ visible spectrum of gas-phase yttrium monoxide has been studied at sub-Doppler resolution to obtain information on the spin-rotation and ⁸⁹Y nuclear magnetic hyperfine interactions. The observations were fitted to an effective Hamiltonian model that neglected the nuclear magnetic hyperfine interactions in the excited electronic state. An interpretation of the determined parameters in terms of possible electronic configurations is presented.     

An intermodulated fluorescence study of the A2Σ−-X2Πi band system of copper monoxide

Several lines in the A²Σ⁻(v = 0)-X²Π_i(v = 0) visible band system of gas phase copper monoxide have been measured at sub-Doppler resolution using the technique of intermodulated fluorescence. The copper nuclear magnetic hyperfine splitting is clearly resolved. The observations were fitted to an effective Hamiltonian model and values of the magnetic hyperfine parameters of b_F = −0.0302 cm⁻¹ for the A state and d = 0.0024 cm⁻¹ for the X state were determined. An interpretation of these values in terms of the proposed electronic configurations of these states is presented.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

Above-threshold ionization of hydrogen and hydrogen-like ions by X-ray pulses

This paper adresses the problem of above-threshold ionization (ATI) of hydrogen interacting with an intense X-ray electromagnetic field. Two approaches have been used. In the first approach, we calculate generalized differential and total cross sections based on second-order perturbation theory for the electron interaction with a monochromatic plane wave, with the A ² and A · P contributions from the nonrelativistic Hamiltonian (including retardation) treated exactly. In the second approach, we solve the time-dependent Schrödinger equation (TDSE) for a pulsed plane wave using a spectral approach with a basis of oneelectron orbitals, calculated with L ²-integrable B-spline functions for the radial coordinate and spherical harmonics Y _lm for the angular part. Retardation effects are included up to O(1/c), they induce extra terms forcing the resolution of the TDSE in a three dimensional space. Relativistic effects [of O (1/c ²)] are fully neglected. The isoelectronic series of hydrogen is explored in the range Z = 1 ? 5 in both TDSE and perturbative approaches. Photoelectron angular distributions are obtained for photon energies of 1 keV and 3 keV for hydrogen, and photon energy of 25 keV for the hydrogenic ion B⁴⁺. Perturbative and TDSE calculations are compared.     

Radiative strength functions in odd-A spherical nuclei

The radiative strength functions for the partial γ-transitions from neutron resonances to the ground and low-lying states of odd-A spherical nuclei are calculated within the quasiparticle-phonon nuclear model. The fragmentation of one-quasiparticle and quasiparticle-plus-phonon states is calculated. This allowed one to calculate γ-transitions between the one-quasi-particle components (valence transitions) and γ-transitions between the quasiparticle-plus-phonon and one-quasiparticle components of the wave functions. The energy dependence of the strength functions $\text{C}$(E1, η) and $\text{C}$(M1, η) is calculated near the neutron binding energy B_n for ⁵⁵Fe and ^{59, 61}Ni. The corresponding experimental data are described qualitatively. The contribution of the valence E1 transitions to the strength function is shown to be from 20% to 90%, and M1 transitions about 1%. The influence of the M1 giant resonance is important for M1 transition probabilities.     

Probabilistic quark-model approach to proton fragmentation

A probabilistic approach is formulated to study the behaviour of the incident hadron quark flavours in the fragmentation process of high energy protons in lowp _T reactions. Analysis of available data onpp collisions, mainly on hyperon and antibaryon multiplicities, leads to estimates of the probabilities for the different ways in which the incident valence quarks recombine into final hadrons. We find that all three incident quarks emerge in one and the same outgoing nucleon (or nucleon resonance) with probabilityA ₃=0.35–0.4, that two of them emerge in one baryon and the third in another hadron (mostly a meson or meson resonance) with probabilityA ₂=0.6–0.5, and that they emerge in three distinct hadrons (mostly mesons or meson resonances) with probabilityA ₁=0.05–0.1. We find good support for a very simple probabilistic picture of the fragmentation process.     

EPR of Mn2+ doping unannealed single crystal of (La2O3)0.95(CeO2)0.05

X-band room temperature EPR spectra have been recorded for Mn²⁺ ion doping unannealed (La₂O₃)_0.95(CeO₂)_0.05 host crystal. The data are analysed using a rigorous least-squares fitting procedure in which a large number of lines characterized by ΔM = ± 1, Δm = 0 transition, obtained for several orientations of the static magnetic field, are simultaneously fitted. Combined with the knowledge of the absolute sign of the hyperfine interaction parameter. A, the hyperfine Hamiltonian parameters A, B, Q as reported in this paper, are given with their correct signs. The information on the linewidth is used to deduce the deviation of the crystal-field axes of different Mn²⁺ ions from the c axis; on the basis of the model proposed here these deviations are found to be between 0 and 10°.     

Effective correlation-free reduced Hamiltonian for the νt(E), νn(A1) Coriolis-interacting states of C3v symmetric-top molecules

The effective correlation-free vibrational-rotational Hamiltonian for the Coriolis-interacting ν_t(E) and ν_n(A₁) states in C_3v molecules has been derived. The Hamiltonian includes the terms describing the x-y Coriolis interaction up to the fourth-order, and several useful reduction schemes for the Hamiltonian are suggested.     

Measurement and analysis of the ν2 band of H2S: Comparison among several reduced forms of the rotational Hamiltonian

The frequencies of 123 lines between 1080 and 1260 cm^?1 in the ν₂ band of H₂S were measured with a tunable diode laser and a grating spectrometer. Ground- and upper-state molecular constants appropriate to Watson's A reduced Hamiltonian in the I^r coordinate representation were obtained from a simultaneous fit of the ν₂ band data and previously measured H₂S rotational microwave frequencies. This fit had a standard deviation comparable to the experimental uncertainties. The A reduced Hamiltonian in the III^r representation, however, was found to fit the data very poorly. The large value of the transformation parameter s₁₁₁ calculated for the A reduced Hamiltonian in the III^r representation confirms that this result is due to the slow convergence of this particular reduced Hamiltonian.