Mixing Of Wave Functions Research Articles

Far-infrared laser oscillation due to cyclotron emission in p-type germanium

Far-infrared laser oscillation due to cyclotron emission in the light hole band of p-type germanium was observed under crossed electric and magnetic fields. The wavelength is inversely proportional to magnetic fields with a cyclotron mass ofm c =0.048m c . Numerical calculations based on Luttinger Hamiltonian show that mixing of wavefunctions between the light and heavy hole bands causes population inversion betweenn=0 andn=1 light-hole Landau levels. It is also shown that non-equidistant energy spacing of light-hole Landau levels is essential to yield net amplification.

Contrasting behaviour of proton and bands in 175,177,179,181Ir interpreted in an intruder model

Abstract Rotational bands in the odd-proton isotopes 175Ir, 177Ir, 179Ir and 181Ir based on the h 9 2 , h 11/NU> 2 , i 13 2 and d 11/NU> 2 orbitals have been identified, the first three to high spin. In all cases the h 9 2 band shows backbending with a crossing frequency and alignment gain consistent with that expected from the neutron i 2 13 2 alignment. In contrast the h 11/NU> 2 bands (and the fragments of d 11/NU> 2 bands observed) show complex alignment curves. These alignments can be explained using a phenomenological three-band model which incorporates a deformed intruder band, analogous to that proposed in the very light Pt and Os nuclei. Mixed wave functions from the model reproduce the observed B(M1)/B(E2) ratios. The presence of the intruder band in the h 11/NU> 2 and d 11/NU> 2 configurations and its absence in the h 9 2 band is compatible with its proposed association with an ( h 2 9 2 ) 0 pair excitation.

CORRELATIONS IN ODD-MASS NUCLEI

Correlations in odd-mass nuclei are described by expanding the wave functions into one and three quasiparticle configurations. The self-energy operator induces mixing of radial wave functions from different major shells. This allows to study the state-dependence of correlated single particle wave functions entering e.g. into transfer and knock-out processes. Applications to 41Ca are discussed.

Interpretation of intensities of forbidden vibration-rotation transitions in Fourier transform spectra of ammonia

The absolute intensities of forbidden Δ k = ±2 transitions to the ν 4 vibrational state of 14NH 3 have been derived from the Fourier transform spectra measured with the multipass White-type cell. Using expressions derived by the contact transformation method, the intensity data have been analyzed in order to obtain information on the mixing of vibration-rotation wavefunctions with different values of quantum numbers ( k − l t ). The parameter γ 4 of the “2, −1” l-type operator, which is indeterminable from an anlaysis of transition frequencies because of correlations in the reduced Hamiltonian, has been determined to be 0.0563 (16) cm −1.

Reduced probabilities of gamma transitions in even-even deformed nuclei

The reduced probabilities of the E2 intraband and interband transitions in even-even erbium isotopes (A=158–170) andN=94 isotones (A=158–164) are analysed in the frame of the perturbation theory assuming the mixing of the wave functions of ground, beta- and gammavibrational states as well as different values of the intrinsic quadrupole moments of these states.

Vacuum polarization currents in finite nuclei

Self-consistent relativistic Hartree calculations of odd-A nuclei, including vacuum polarization, are presented. The contribution to the baryon current from the deformed Dirac sea is calculated using a derivative expansion. This vacuum current does not affect magnetic moments; however, at intermediate momentum transfers it tends to cancel the current from the deformed occupied core, as expected from the random-phase approximation response in nuclear matter. Wave function mixing for the valence state, which arises in the self-consistent solution but not in random-phase approximations, significantly affects both Dirac and anomalous currents.

Stark-effect studies in xenon autoionizing Rydberg states using a tunable extreme-ultraviolet laser source.

The Stark spectra of autoionizing Rydberg states of Xe converging to the second ionization limit 5${p}^{5}$ $^{2}P_{1/2}$ have been investigated as a function of applied electric field over the range 26--2362 V/cm. Single-photon excitation from the Xe ground state was achieved using coherent radiation near 92.5 nm, generated by frequency tripling the pulsed output of a frequency-doubled dye laser in a free jet of rare-gas atoms. For the lowest electric field only the ns' and nd' series were observed. As the electric field was increased the np' series appeared because of the mixing of wave functions with different l values. Further increase of the electric field caused the appearance of hydrogenlike Stark manifolds. The Stark spectrum was simulated in a two-step procedure with no adjustable parameters. Using a jl-coupling basis set the Hamiltonian matrix was diagonalized to determine the energy levels and corresponding eigenfunctions for a given electric field strength. The intensities and line shapes were then calculated using the formalism developed by Fano for a many-discrete-level--single-continuum configuration interaction. This treatment was sufficient to reproduce the positions, widths, and shapes of most observed features. At field strengths between 1000 and 2000 V/cm, however, an additional modulation structure with a spacing of about 2.7 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ appeared on top of the \ensuremath{\Vert}M\ensuremath{\Vert}=1 autoionizing resonances, which could not be explained by our simple theoretical model.

The statistical correlation between rotation and translation: Implications for fine and hyperfine structure in quantum mechanics

Abstract For matrix isolated small molecules such as hydrogen deuteride (HD) the rotational spectra generated form the Schrodinger equation are perturbed by mixing with quantised translational lines which appear from the periodicity imposed by a solvent cage on the motion of the encaged HD molecule. The effect of the mixing of rotational and translational wave functions is that quantum translation/rotation effects occur in the final spectrum. There are shifts in rotational levels, broadening of rotational levels, and relaxation of rotational selection rules. In this paper it is shown that the existence of various new types of cross-correlation function in the laboratory and moving frames of reference implies the existence of new types of fine and hyperfine structure in the infra red spectrum of HD encaged in solvent molecules such as liquid argon. These new features could lead to specific and observable information on the nature of rotation to translation coupling in the liquid state of matter.

A model for the interpretation of the absorption and circular dichroism spectra of tris(di-imine) complexes of iron(II) and ruthenium(II)

A theoretical model of the electronic structure of the metal-to-ligand charge-transfer (m.l.c.t.), ligand-centred (l.c.) and metal-centred (m.c.) states of tris(di-imine)metal(II), [ML3]2+(M = Fe or Ru), is developed. The model contains only three adjustable parameters describing the first-order mixing of metal and ligand wavefunctions. All the state matrix elements are simplified by tensor operator techniques to reduced one-electron matrix elements, which are calculated numerically using Slater type orbitals. A quantitative prediction of the absorption and circular dichroism spectra, including all spin-allowed m.l.c.t., l.c., and m.c. bands of [ML3]2+ is made, using a reasonable set of mixing parameters, which is in good agreement with the experimental results.

Excitation transfer from 31P, 31D to 33D of neutral helium by atomic collisions in a magnetic field

Laser-induced fluorescence spectroscopy is applied to a helium discharge plasma in a magnetic field: the 31P level is excited with pulsed laser light, and subsequent fluorescence intensities of the 501.5 nm (21S-31P), 667.8 nm (21P-31D) and 587.6 nm (23P-33D) lines are observed. The collisional excitation transfer rates for 31P to 33D and 31D to 33D are strongly dependent on the magnetic field: both rate coefficients obtained at 5.5 T are about 10 times larger than those at 0 T. A qualitative explanation is proposed on the basis of the mixing of spin wavefunctions between the 31D2 and 33D2 levels.

Determination of the band structure parameters of Cd 0.95Mn 0.05Se from magnetoabsorption measurements

Band parameters and the exchange integrals of Cd 0.95Mn 0.05Se are evaluated using magnetoabsorption measurement results. The anisotropy of the exchange interaction for valence band electrons and the magnetic field dependent wave function mixing coefficients are obtained. The use of a quasi-cubic model for describing a CdMnSe crystal in the presence of magnetic field is shown to be unjustified.

Parameterization of 4f-4f transition intensities in Eu 3+-doped lanthanum oxychloride

The Simulation of the line-to-line transition intensities of the 5 D 0, 1 M → 7 F JM transitions in Eu 3+-doped lanthanum oxychloride has been carried out according to the Judd-Ofelt theory. Emphasis is placed upon explaining the high intensities of the “hypersensitive” 5D 0 → 7F 2 and the “forbidden” 5D 0 → 7F 0 transitions. The nine, non-zero B( λkq) parameters reproduce the experimental transition probabilities reasonably well. The parameter set is characterized by a large value of the “linear” B(210) term. This is in accordance with electrostatic point charge calculations which yielded a large A 0 1 value. The high B(210) value, together with the strong mixing of the 7F JM wave functions with the 7F 00 function explains the intense 5D 0 → 7F 0 transition. The intensity of the 5D 0 → 7F 2 transitions is also mainly controlled by the B(210) parameter.

Applications of the two-photon doppler-free method: Hyperfine interactions and isotope shift measurements

The hyperfine structures are generally of the same order of magnitude as the Doppler broadening of optical transitions and so are the isotopic shifts in the case of heavy elements. When these structures are too small, one must use Doppler-free methods. Among these methods, the two-photon spectroscopy has obtained good results in highly excited levels. In our laboratory in Paris, we did measurements on neon and helium by two-photon excitation from metastable levels. The precision of the measurements is of the order of one MHz, which permits a precise comparison with theory. We compare the measurements on neon with the theory by Liberman and we obtain a good fit in the first approximation, but must introduce mixing of wave functions for an exact explanation. In the case of helium, we obtain a good fit with the theoretical values obtained from the wave functions by Accad, Pekeris and Schiff. We also give an example where another technique by polarization measurements permits us to obtain experimentally a hyperfine structuresmaller than the natural width. We also present a short review of the measurements done by the two-photon method in other laboratories on other elements, Pb, Tl, In and alkaline earths Ca, Sr. Ba.

Forbidden rotational transitions in symmetric top molecules

The intensities of the forbidden Δ k = ±6 and ±4 transitions in symmetric top molecules arising from rotational intensity borrowing are reported. A perturbation theoretical approach has been used to study the mixing of wavefunctions produced by the additional splitting term in the rotational Hamiltonian of C 3v or C 4 v -type molecules. Numerical values are presented for PH 3, PD 3, and BrF 5 molecules. The importance of these results in structural analysis and astrophysical studies is discussed.

Field dependence of the residual-resistivity anisotropy in sodium and potassium.

Recent measurements of the low-field, induced torque in sodium and potassium by Elliott and Datars show that the resistivity anisotropy increases with increasing magnetic field. The zero-field resistivity anisotropy, unexpected for cubic symmetry, is explained by the charge-density-wave (CDW) structure. Due to the wave-function mixing caused by the CDW potential, the momentum transfer (by isotropic impurities) is much larger for electrons near the CDW energy gap. This is modeled by an anisotropic relaxation time in k\ensuremath{\rightarrow} space. The Boltzmann transport equation in a magnetic field can then be solved exactly. The computed resistivity anisotropy is higher for ${\ensuremath{\omega}}_{c}$\ensuremath{\tau}>1 compared with its zero-field value. The effect of the magnetic field is to ``stir'' the electron distribution f(k\ensuremath{\rightarrow}); this feeds electrons into the region of rapid relaxation and thereby increases the resistivity anisotropy.

Two-quasineutron states in 168Er populated by the 167Er(d, p) and 167Er(t, d) reactions

The nuclear structure of 168Er has been studied with the 167Er(d, p) 168Er and 167Er(t, d) 168Er reactions, using 12 MeV deuterons and 15 MeV tritons from the McMaster tandem Van de Graaff accelerator. The reaction products were analyzed with an Enge split-pole magnetic spectrograph and detected with photographic emulsions. Angular distributions were obtained for levels up to ∼2.5 MeV excitation, with typical resolutions of ∼9 keV and ∼11 keV (FHWM) for the (d, p) and (t, d) reactions, respectively. Since the I, K π values for all levels up to ∼2 MeV were known from previous ( n, γ) studies, the main contribution of this study was to determine the admixtures of specific two-quasineutron configurations to the various bands. Earlier {7/2 +[633]±1/2 −[521]} assignments for the 1094.0, K π = 4 − and 1541.5 keV, K π = 3 − bands have been confirmed, although the full {7/2 +[633]+1/2 −[521]} strength is not observed in the 1094.0 keV band. The K π = 1 − octupole band at 1358.8 keV has a dominant {7/2 +[633] − 5/2 −[512]} component, as predicted by the Soloviev model. The K π = 6 −, {7/2 +[633] + 5/2 −[512]} configuration has been assigned in this work to the 1773.2 keV level. The K π = 4 − band at 2059.9 keV is found to contain ∼35% of the {7/2 +[633] + 1/2 −[510]} strength, and a ∼25% admixture of the {7/2 +[633] − 1/2 −[510]} configuration is tentatively assigned in the K π = 3 − band at 1828.0 keV. Relative cross sections for members of the 168Er ground state band suggest the presence of a mixed wave function for the 167Er target ground state. The K π = 0 +, 1217.1 keV band exhausts the {7/2 +[633] − 7/2 +[633]} strength which does not go to the ground band, while the K π = 0 +, 1422.0 keV band was not observed. The Soloviev model is the only one which has made quantitative predictions that can be compared with the present results. In general, it is quite successful in explaining the experimental data.

Laser studies of near-resonant state-changing collisions of calcium (4s6s 1S0) with the rare gases.

State-changing collisions of Ca(4s6s ${}^{1}$${S}_{0}$) with the rare gases are studied by pulsed laser excitation and time- and wavelength-resolved detection. The total depletion rates of the 4s6s ${}^{1}$${S}_{0}$ state with different rare gases vary by over a factor 10, with the lighter rare gases being markedly more efficient than the heavier ones. Relative branching ratios to the specific near-resonant, energy-transfer product states, 4p4p ${}^{1}$${D}_{2}$ (\ensuremath{\Delta}E=-29 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$), 3d4p ${}^{1}$${F}_{3}$ (\ensuremath{\Delta}E=152 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$), and 4s6s ${}^{3}$${S}_{1}$ (\ensuremath{\Delta}E=216 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$), are measured for $^{3}\mathrm{He}$, $^{4}\mathrm{He}$, Ne, and Xe. Transfer to the $^{1}\mathrm{D}_{2}$ state is always the major relaxation pathway. The rate constant for $^{3}\mathrm{S}_{1}$ production is immeasurably small, implying a small spin-orbit matrix element for curve crossing that leads to a spin change. The results for 4p4p ${}^{1}$${D}_{2}$ production are compared with simple theoretical calculations based on established models for near-resonant energy transfer between states which are coupled at large internuclear distance. These models successfully explain the dominant transfer to the 4p4p ${}^{1}$${D}_{2}$ state and the trends with rare gas, which are attributed largely to enhancement of the near-resonant-transfer process with increasing velocity. The effects of wave-function mixing in the isolated atom (e.g., spin mixing) on the general outcome of state-changing collisions in two-electron atoms are discussed. Two-electron atoms offer numerous possibilities for further testing of theories of near-resonant energy transfer.

Electronic structure of Ni and Pd alloys. II. X-ray photoelectron core-level spectra

Our measurements of the Ni $2p$ core levels in about 30 alloys show a dependence of the satellite on the chemical environment, which for a given stoichiometry correlates with the electronegativity of the partner. In no case was a complete disappearance of the satellite observed, which is interpreted as evidence for $d$ character in unoccupied states. The general view developed in the preceding paper, that despite the band filling there is no large charge transfer in Ni intermetallic compounds, is confirmed by these observations. The band filling is a result of the mixing of wave functions which are centered on different sites. Studies of the satellite intensity as a function of stoichiometry in alloys with electropositive metals show the satellite to weaken as long as the band is being filled, and then to reach an almost constant intensity at some partner-element concentration. It is not possible to extract similar information from the spectra of Pd compounds because the satellites are weaker and partially obscured by inelastic losses.

Internal Rotation Spectrum in the Ground State of cis Propionyl Fluoride

Abstract The microwave ground state spectrum of cis propionyl fluoride has been investigated expanding, with respect to a previous study, the range of the quantum numbers of the transitions, J up to 40 and K up to 18. At low K the spectrum resembles that of an asymmetric rigid rotor, except in a number of cases where the lines are split into A, E doublets. At intermediate and high K the mixing of the if-doublet rigid rotor wave functions makes forbidden transitions appear. The results of the analysis of the methyl top internal rotation ground state splittings are: V3 = 2350 ± 11 cal/mole, ≮ (i, a) = 32.7° ±2.0°, and Iα = 3.18 ± 0.03 uÅ2.

Wave-function mixing of flavor-degenerate baryons

There are two baryons at the center of an octet weight diagram of S${\mathrm{U}(3)}_{\mathrm{flavor}}$ both of which contain the same quarks. We call these baryons "flavor degenerate." We discuss the symmetry of the wave functions of these baryons in terms of a quark-ordering principle and a mixing angle. We explore the consequences of the mixing on predictions of the masses and magnetic moments of flavor-degenerate baryons, including baryons containing heavy quarks. We obtain a simple approximate expression for the mixing angles in terms of the quark masses. The mixing angles turn out to be rather small, but still change some of the calculated magnetic moments substantially. The effect of mixing on the baryon masses is much smaller.