Correlation Of Energy Levels Research Articles

Energy-level correlation function and ac conductivity of a finite disordered system.

The space- and energy-difference- (\ensuremath{\omega}) dependent Wigner-type correlation function for the energy levels is studied for the Anderson disordered tight-binding model, including some of the effects of localization. For small metallic particles it is confirmed that these correlations, in the absence of spin and magnetic effects, are similar to those of the Gaussian orthogonal ensemble. This is in agreement with the conjecture made by Gor'kov and Eliashberg and the analytical results obtained by Efetov. The results are used to obtain the effective conductivity ${\ensuremath{\sigma}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}}}$), relevant for the absorption of low-frequency electromagnetic radiation, in all regimes, including the effects of localization and screening. It is found that in the orthogonal case, ${\ensuremath{\sigma}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}}}$) does not change at low \ensuremath{\omega} when level correlations become important. The conditions to observe changes in the absorption in this regime due to changes in the ensemble symmetry are formulated. In all cases except for the microscopic (critical) one, ${\ensuremath{\sigma}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}})\mathrm{\ensuremath{\propto}}{\ensuremath{\omega}}^{2}}$, and in the latter region, ${\ensuremath{\sigma}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}})\mathrm{\ensuremath{\propto}}{\ensuremath{\omega}}^{5/3}}$. For systems much larger than the localization length, the level correlations decay in space with a length \ensuremath{\xi} ln(${\ensuremath{\Delta}}_{\ensuremath{\xi}}$/\ensuremath{\omega}), as \ensuremath{\omega}\ensuremath{\rightarrow}0 (${\ensuremath{\Delta}}_{\ensuremath{\xi}}$ is the average spacing between levels within a localization volume). The connection with Mott's calculation of the ac conductivity in the insulating phase is made. It is shown that these ideas explain the ``level attraction'' found in one dimension by Gor'kov, Dorokhov, and Prigara. These considerations are generalized to the evaluation of the space-dependent level correlation function in an arbitrary dimension.

Quantum Size Effect in Magnetic Field in Cu Small Particles

In order to study the kinetic energy level correlation of conduction electrons in small metallic particles, we measured the magnetic field dependence of the Knight shift of 63Cu in copper small particles. We observed the suppression of the anomaly in spin paramagnetic susceptibility in the magnetic field so high that the Zeeman energy is larger than the average spacing of the energy levels. This suppression is explained as that the levels of electrons with opposite spins at the Fermi energy is lost in high fields.

Electronic energy level and intensity correlations in the spectra of the trivalent actinide aquo ions. III. Bk3+

The solution absorption spectrum of Bk3+(aquo) was measured and the observed band structure interpreted in terms of a free-ion energy level model. The band intensities were successfully analyzed using the Judd–Ofelt theory for transitions within the f N configuration. Parameters of the theory were then used to compute fluorescence branching ratios from most probable fluorescing states, and an experimental search was successful in yielding evidence for a transition from one excited state to the ground state in D2O solvent. Absorption bands attributed to f → d transitions were observed and an interpretation of the electronic structure is presented. Band intensities were compared to those observed for Tb3+(aquo).

Correlation diagrams for rigid and nonrigid five-body and XY5 six-body systems

Correlation diagrams are constructed to connect the rotation–vibration energy levels of idealized limiting models for penta–atomic molecules. One limit is the completely nonrigid homonuclear five-body cluster. The method of plethysms is applied to classify the states of the nonrigid limit.Several rigid limits are considered: trigonal bipyramidal, tetrahedral, square pyramidal, pentagonal planar, and regular linear chain molecules. The tetrahedral and square pyramidal examples are generalized to allow the atom in the unique central position to differ from the rest giving (four+one)-body systems. A rigid/nonrigid energy level correlation diagram is also constructed for an XY5 molecule having trigonal bipyramidal point symmetry in the rigid limit. The steps are outlined that will be necessary for applying these diagrams to the interpretation of real molecular spectra.

Statistical properties of complex states ofZn3067

High resolution neutron total and capture cross section measurements have been made on the separated isotope /sup 66/Zn in the energy range from a few keV to about 1 MeV using the neutron time-of-flight technique and the pulsed Oak Ridge Electron Linear Accelerator. The nominal resolution of the measurements was 0.07 ns/m. The total cross section data have been analyzed up to 380 keV using a multilevel R-matrix formalism and the capture data up to 130 keV using a least squares fit program. From these analyses we have obtained the values of the resonance parameters E/sub 0/, gGAMMA/sub n/, and GAMMA/sub ..gamma../, and values of average quantities: D/sub l/ = 0 = (4.62 +- 0.55) keV, D/sub l/ = 1 = (1.17 +- 0.14) keV, S/sub 0/ = (2.06 +- 0.36), S/sub 1/ = (0.66 +- 0.06), and S/sub 2/ = (0.50 +- 0.12) in units of 10/sup -4/ eV/sup -1/2/. The values of average radiation widths are (0.39 +- 0.06) eV for s-wave resonances and (0.190 +- 0.013) eV for p-wave resonances. The distributions of neutron reduced widths and of level spacings of s- and p-wave resonances show excellent agreement with the Porter-Thomas and Wigner distributions, respectively. The valuemore » of the ..delta../sub 3/ statistic for the long range correlation of s-wave energy levels is found to be in reasonable agreement with the theoretical prediction of Dyson and Mehta. A value of 0.59 +- 0.10 has been determined for the linear coefficient for the correlation between the neutron reduced widths and the radiation widths of 26 s-wave resonances. This value is highly significant and has a confidence level of 99.7%.« less

Quantitative energy level correlations for linear, bent and internally rotating triatomic molecules

This paper is concerned with the study of the quantitative correlation of the rotation-bending energy levels of a linear or bent triatomic molecule with the energy levels of the molecule when there is free internal rotation of a diatomic fragment [as in, for example, Ar(HCl)]. The LiNC and KCN molecules are used as model systems. A correlation parameter γ r is introduced to quantify the position a molecule occupies in these correlation diagrams; this parameter has the value +1 for an ideal bent molecule, −1 for an ideal linear molecule, and −3 for an ideal free internal rotor triatomic molecule. The rotation-bending energy levels of the HCN-HNC isomerization system, and of the double minimum HO 2 system, are also studied.

Quenching of oxalyl fluoride ( 3Au) molecules

Rate constants for quenching of oxalyl fluoride ( 3 A u) molecules by selected classes of quenchers have been investigated using the technique of laser-induced time-resolved phosphorescence. Various inorganic, alkane, halogenated alkane, alkene, halogenated alkene, diene and alkyne quenchers were studied. NO was found to have a much larger quenching rate constant than O 2 (6.8 × 10 10 and 1.4 × 10 8 l mol −1 s −1 respectively). The alkane data are consistent with a hydrogen atom abstraction mechanism. Both alkene and halogenated alkene rate constants were found to increase as the triplet energy of the particular quencher decreased. Similar correlations of rate constants and triplet energy levels were found for the diene and alkyne compounds studied. The phosphorescence emission spectrum was recorded and the 3 A u state of oxalyl fluoride was found to lie at 85.6 kcal mol −1. A single progression in the ground state ν 1 (CO symmetric stretch) is the dominant feature of the spectrum.

Quenching of oxalyl fluoride ( 3Au) molecules

Rate constants for quenching of oxalyl fluoride ( 3 A u) molecules by selected classes of quenchers have been investigated using the technique of laser-induced time-resolved phosphorescence. Various inorganic, alkane, halogenated alkane, alkene, halogenated alkene, diene and alkyne quenchers were studied. NO was found to have a much larger quenching rate constant than O 2 (6.8 × 10 10 and 1.4 × 10 8 l mol −1 s −1 respectively). The alkane data are consistent with a hydrogen atom abstraction mechanism. Both alkene and halogenated alkene rate constants were found to increase as the triplet energy of the particular quencher decreased. Similar correlations of rate constants and triplet energy levels were found for the diene and alkyne compounds studied. The phosphorescence emission spectrum was recorded and the 3 A u state of oxalyl fluoride was found to lie at 85.6 kcal mol −1. A single progression in the ground state ν 1 (CO symmetric stretch) is the dominant feature of the spectrum.

Cardiovascular trauma from angiographic jets--validation of a theoretic concept in dogs.

The relationship of angiographic injections to cardiovascular trauma was determined. The aorta of dogs was studied by a mathematic model that (a) defines the kinetic energy of contrast-media jets emanating from the exit holes of catheters and (b) enables precise correlation of energy levels to potential trauma. A universal curve independent of the Reynolds number of the jet readily determines how far the jet penetrates intravascular blood. A traumatic histopathologic spectrum was found for those injections predicted by the model to be traumatic. The curve enables the angiographer to calculate the safe operating range of any catheter.

Exactly solvable models for vibronic coupling in molecular spectroscopy. I. Nontotally symmetric harmonic mode

Molecular vibronic coupling is studied on a model consisting of two electronic states coupled by one vibrational mode. If adiabatic potentials corresponding to the two states are available (e.g., from experiment), the model can be solved exactly in terms of a coupling function defined implicitly and self-consistently by means of a chosen or calculated diabatic potential. Alternatively and equivalently, the model can be formulated and solved in terms of one adiabatic and two diabatic potentials. In the harmonic-oscillator approximation, the coupling function reduces to one parameter modulated by the electronic energy separation. Near the limit of weak vibronic coupling, the model reduces to the well-known linear Herzberg–Teller approximation. A formally equivalent but complementary linear perturbation approximation is developed for strong coupling. For intermediate coupling, where both of these perturbation schemes are shown to fail, the model is solved by two complementary numerical procedures of which at least one converges quickly for any given set of physically acceptable parameter values. Detailed results are presented for coupling through a nondegenerate, non-totally-symmetric normal mode. Energy level diagrams as well as absorption and emission spectra are calculated for the full range of coupling parameters and energy gaps compatible with the harmonic approximation. The results show anomalous level spacings, intensities, and isotope effects, especially under intermediate-coupling conditions. The anomalies can be qualitatively understood on the basis of avoided crossings in energy-level correlation diagrams. As an example, the model is used for a qualitative explanation of anomalous intensity variations of the 1Lb transition in aromatic hydrocarbons.

Electronic energy level and intensity correlations in the spectra of the trivalent actinide aquo ions. II. Cm3+

The free-ion energy level scheme of Cm3+ (aquo) to ∼40 000 cm−1 was determined. Energy level assignments were made based on the correlation between observed band intensities and those calculated using the Judd–Ofelt theory. Use of an expanded set of effective operators provided the basis for computation of free-ion energies in good agreement with the experimental assignments.

Hindered asymmetric top states for chemical reactions

A computational method for development of hindered asymmetric top rotational wavefunctions for atom–diatom chemical reactions is presented. The external rotation of the triatomic system is that of a nonrigid asymmetric top that is in no way hindered. Internal rotation (bending) of the system is described by hindered rotor functions of the internal angle. The internal rotation-bending angle is the natural collision coordinage γ; a rotating-axes system located by Euler angles is used to orient the three particle triangle. The basis states are obtained as linear combinations of hindered symmetric top states by perturbatively adding on Coriolis and asymmetry terms in the Hamiltonian while allowing limited configuration interaction among unperturbed levels. The coupling mixes states of different K (projection of the total angular momentum on the rotating Z axis). A single rotational potential minimum is considered in computing symmetric and asymmetric top energy level correlation diagrams. The usefulness of this rotational basis in close-coupling reactive scattering calculations is discussed.

On the nature of color center absorptions

AbstractEvidence is presented for absorptions in halide molecules and halide solutions analogous to color center absorptions in crystals. With similar observations by Hersh, several color center absorptions appear not to be unique to the solid state. Color center absorptions in any one crystal may be correlated in a manner similar to energy level correlations in molecules. Some comments are made on the F absorption band in view of these results.

Electronic energy level and intensity correlations in the spectra of the trivalent actinide aquo ions. I. Es3+

The solution absorption spectrum of Es3+ has been measured in the region 10 000–34 000 cm−1. The observed band structure has been interpreted in terms of free-ion energy levels calculated using parameters that are consistent with those obtained in analyses of the specta of earlier members of the 5fN-series. The correlation between observed and computed band intensities has also been examined and shown to be consistent with the energy level scheme adopted.

Correlation of Energy Levels of Linear and Bent Triatomic Molecules, and the Ultraviolet CS2Absorption Spectrum

A set of vibronic levels for a $^{1}\ensuremath{\Sigma}^{+}_{g}$ and a $^{1}\ensuremath{\Pi}_{g}$ electronic state of linear A${\mathrm{B}}_{2}$ is shown in a diagram, together with gyrovibronic levels of the corresponding electronic states (a $^{1}A_{1}$ and a $^{1}B_{2}$ and a $^{1}A_{2}$) of bent A${\mathrm{B}}_{2}$; and a correlation is set up between the sets of gyrovibronic levels of the two cases. This diagram is then used in a discussion of the near-ultraviolet absorption bands of carbon disulfide, where the molecule is linear in the $^{1}\ensuremath{\Sigma}^{+}_{g}$ ground state. The band types and structures for such a transition are discussed; the structures should be practically identical in character whether the upper state is linear or bent. Vibrational intensities, and their relation to a strong allowed electronic transition at shorter wave-lengths, are also discussed. It is concluded that the near-ultraviolet system is an electronic-allowed transition to a $^{1}\mathrm{B}_{2}$ bent molecule upper state related to a $^{1}\ensuremath{\Pi}_{g}$ linear molecule state.

The ultra-violet spectrum of magnesium hydride. 1.—The band at λ 2430

The system of bands in the visible region of the emission spectrum of magnesium hydride is now well known. The bands with heads at λλ 5622, 5211, 4845 were first measured by Prof. A. Fowler, who arranged many of the strongest lines in empirical series for identification with absorption lines in the spectra of sun-spots. Later, Heurlinger rearranged these series in the now familiar form of P, Q and R branches, and considered them, with the OH group, as typical of doublet systems in his classification of the fine structure of bands. More recently, W. W. Watson and P. Rudnick have remeasured these bands, using the second order of a 21-foot concave grating, and have carried out a further investigation of the fine structure in the light of the present theory of band spectra. Their detection of an isotope effect of the right order of magnitude, considered with the general structure of the system, and the experimental work on the production of the spectrum, seems conclusive in assigning these bands to the diatomic molecule MgH. The ultra-violet spectrum of magnesium hydride is not so well known. The band at λ 2430 and the series of double lines in the region λ 2940 to λ 3100, which were recorded by Prof. Fowler in 1909 as accompanying the group of bands in the visible region, appear to have undergone no further investigation. In view of the important part played by hydride band spectra in the correlation of molecular and atomic electronic energy levels, it was thought that a study of these features might prove of interest in yielding further information on the energy states of the MgH molecule. The present paper deals with observations on the band at λ 2430; details of an investigation of the other features of the ultra-violet spectrum will be given in a later communication.