Fit Of Energy Levels Research Articles

Optical spectra and analysis for Pr3+ in SrAl12O19

Energy levels of the 4f2 electronic configuration of Pr3+ in SrAl12O19 (SAM) have been determined from polarized absorption and fluorescence spectra using crystals, charge-compensated with Mg2+, at temperatures between about 20 and 300 K. Energy level assignments were made initially by comparing the crystal spectra with energy levels calculated by using crystal-field parameters, Bkm, compatible with those previously reported for Nd3+ in SAM. The Bkm were varied to obtain a best fit between experimental and theoretical energies, and the final values: B20=−94, B40=1047, B60=−1397, and B66=−1245 cm−1 give an rms fit of 14.5 cm−1. The Bkm for Pr3+ in SAM were also used along with a result relating the Bkm to crystal-field components, Anm, derived in a lattice sum calculation to obtain Bkm values for all the triply ionized rare-earth ions in SAM. Odd-fold Akm are also given that are required in the intensity and lifetime calculations and yield results in good agreement with the measurements. In addition, the results of our energy level fitting were useful in estimating absorption and emission cross sections, as well as lifetimes and branching ratios, for the various J multiplets. The evidence given here that the room temperature fluorescence quantum efficiency of P03 in lightly doped Pr in SAM is near unity strengthens the case that this is a potentially useful laser material. The theoretical model and method of analysis used is useful for the calculation of energy levels, wave functions, transition probabilities, etc., for any rare-earth RE3+ ion in any host of known crystal structure.

Group-chain scheme analysis of the energy levels and magnetic properties of in the crystal

Based on the analysis of group-chain scheme wavefunctions and the use of the constraint condition set down by the ratios of crystal-field parameters calculated by the point-charge model, the crystal-field-level fitting and the study of Zeeman interaction characterized by the g-factor in have been carried out, in which the ions occupy positions with point symmetry . The RMS of energy-level fitting is less than , and the corresponding wavefunctions of Stark sublevels are useful in the calculation of the g-factors of the ground and excited states. The calculated results agree well with the experiment and confirm Karayianis' partial g-sum rule. The method proposed turned out to be effective in the study of not only the spectroscopic and laser properties but also the magnetic properties of rare-earth ions in crystals.

Correlation-crystal-field analysis of Nd3+(4f3) energy-level structures in various crystal hosts

We have performed an in-depth correlation-crystal-field (CCF) analysis of the energy-level structures in 10 Nd3+(4f3) crystal systems: NdF3, Nd2Te4O11, NdVO4, NdPO4, Nd3+:LiYF4, Nd3+:LaVO4, Nd3+:LaCl3, Nd3+:BaY2F8, Nd3+:YAlO3 and Nd3+:LuA1O3. A model Hamiltonian employing 20 free-ion parameters, appropriate one-electron crystal-field interaction parameters and also selected two-particle CCF interaction parameters was diagonalized within the complete 364 SLJMJ basis set of the 4f3 electronic configuration. Inclusion of the fourth-rank g2(4), g10A(4) and g10B(4) CCF operators in the phenomenological energy-level fits yields an overall improved agreement between calculated and empirical energy levels besides eliminating major discrepancies between calculated and observed crystal-field splittings within the anomalous 2H(2)11/2 multiplet of Nd3+ ion. The fits are also in qualitative agreement with the ab initio calculations of CCF effects for lanthanide ions.

Phenomenological spin-correlated crystal-field analyses of energy levels in Ln 3+:LaCl 3 systems

The best energy level data available in the literature for the Ln 3+:LaCl 3 systems are analyzed in terms of an electronic hamiltonian that includes isotropic (free-ion) and crystal-field interaction parameters for C 3 h site symmetry. The crystal-field hamiltonian is defined to include spin-correlated crystal-field (SCCF) interaction terms, and analyses are carried out with and without consideration of these SCCF terms. The analyses reported here yield improved calculated- vs.-experimental energy level fits (compared to fits reported previously in the literature). For example, for Ho 3+:LaCl 3, the value of sigma is improved from 7.8 to 4.9 cm −1 for 168 crystal-field levels. In each case, inclusion of the SCCF terms improves the calculated- vs.-experimental energy level data fits. Ratios of the SCCF vs. one-electron crystal-field parameters are reported and compared for each system.

Optical spectra and crystal-field analyses of samarium in the centrosymmetric hexakis(antipyrine)samarium(III) tri-iodide and tri-perchlorate systems

Optical absorption and emission measurements are reported for single crystals of Sm(AP)6I3 and Sm(AP)6(ClO4)3 (where AP denotes an antipyrine ligand molecule). Crystal-field energy levels split out of the 4f 5 electronic configuration of Sm3+ are located and assigned from the locations and polarizations of magnetic-dipole origin (no-phonon) lines observed in the optical spectra. The energy levels are analyzed in terms of a parametric hamiltonian that assumes D 3d crystal-field symmetry at the Sm3+ sites. Crystal-field interaction parameters obtained from calculated-versus-experimental energy level fits are reported and compared for the Sm(AP)6I3 and Sm(AP)6(ClO4)3 systems. Good data fits are obtained for both systems, and the rank-two and rank-four crystalfield parameters are reasonably well-determined by these fits. The rank-six parameters are less well-determined. The values determined for the rank-four crystal-field parameters are essentially identical for the tri-iodide and triperchlorate systems; however, the magnitude of the rank-two parameter for Sm(AP)6I3 is twice that for Sm(AP)6(ClO4)3. Magnetic-dipole line strengths are calculated for absorptive transitions originating from crystal-field levels of the 6H5/2 (ground) multiplet and for emissive transitions originating from the 4G5/2 (excited) multiplet.

Optical spectra, energy levels and crystal-field analysis of Sm3+in Na3[Sm(oxydiacetate)3] · 2NaClO4· 6H2O

Locations and assignments of 144 crystal-field energy levels are reported for Sm3+ in the trigonal Na3[Sm(oxydiacetate)3] · 2NaClO4 · 6H2O system. These energy levels span the 0–36 000 cm-1 energy region, and they were located and assigned from optical emission spectra and from axial and orthoaxial (σ- and π-polarized) absorption measurements on single crystals. The principal SL parentages of the assigned levels span at least 20 Russell-Saunders terms within the 4f 5 electronic configuration of Sm3+, and these assigned levels represent nearly half the total number of crystal-field energy levels predicted to occur within the 0–36 000 cm-1 region. The assigned levels are analysed in terms of a 26-parameter electronic hamiltonian in which six of the parameters are defined to represent the 4f-electron/crystal-field interactions for Sm3+ located at sites with trigonal dihedral (D 3) symmetry. Of the 26 parameters, 22 (including the six crystal-field parameters) are used in fitting calculated energy levels to the experimental data, and excellent calculated-versus-experimental energy level fits are achieved. Fitted values obtained in this study for the Sm3+ ‘free-ion’ energy parameters are compared to those reported previously for Sm3+ in LaF3 and LaCl3. Quantitatively determined line strengths are reported for 50 (4f → 4f) crystal-field transitions observed in the single-crystal absorption spectra of Na3[Sm(oxydiacetate)3] · 2NaClO4 · 6H2O at 10 K.

Determination of empirical configuration parameters from experimental data in l4l+1l' electron configurations: application to Ar I 3p54s, 3p54p and 3p53d

It is shown that experimental oscillator strengths or Lande g values of a small number of energy levels may be used to obtain certain of the Slater-Condon parameters of l4l+1l' electron configurations which may be used to predict a large number of the other energy levels. The analysis uses deduced values of eigenvector expansion coefficients for LS-coupled basis functions to transform the diagonal energy level matrix into a numerical LS Hamiltonian matrix. Equating this with the analytical Hamiltonian leads to simultaneous equations in the configuration parameters. The method is tested in three Ar I 3p5nl configurations with satisfactory results. In the case of Ar I 3p54d it is found that the introduction of an additional empirical electrostatic parameter leads to a substantially improved energy level fit. This parameter effectively takes into account the LS dependence of 3F Slater integrals.

Electronic structure of the Si(111) reconstructed surface in the vacancy model. II

The electronic structure of a 2 × 2 superlattice of vacancies on the Si(111) surface is examined by the self-consistent pseudopotential method. The results, together with a previous work on the 3 × 3 structure, show that there is a large dispersion in the surface state energy bands associated with the dangling bonds toward the vacancy. This fact suggests that the surface electronic structure in the vacancy model for the 7 × 7 reconstructed surface is sensitive to the arrangement of vacancies. Next the three-vacancy unit model proposed by Ino for the 7 × 7 surface is investigateFd in the parametrized tight-binding scheme with the parameter values determined from the fitting of energy levels of the periodic 2 × 2 system to the results of the self-consistent calculation.

Zeeman effect and level identification of As i

The Zeeman effect of the neutral arsenic spectrum was observed photographically from 1937 to 11 244 Å. Well-defined Zeeman patterns of 130 As i lines were recorded and yielded 61 Landé g values of which 9 were previously available. The light source was an electrodeless lamp of quartz containing metallic arsenic and helium carrier gas at 3–5 Torr; it was operated in a magnetic field of 24 025 or 10 000 G and excited by a standard microwave generator. Conditions that were favorable for the excitation of the first spectrum of arsenic were noted and characterized. The J and g values obtained from our Zeeman patterns confirmed most of the existing As i analysis but forced reclassification of several terms. There is acceptable agreement between the g values and those calculated from intermediate-coupling wave functions obtained by the least-squares fitting of energy levels for the single configuration approximation to 4s2 4p3 and 4s2 4p2 5s. Also, adequate fits of energy levels could be obtained in the single configuration approximation for each of the 4s2 4p2 5p and 4s2 4p2 6p configurations, but large discrepancies occurred in the g values. Such discrepancies were reduced considerably when configuration interaction was included in the calculations, providing positive evidence of configuration mixing that led to the reclassification of several energy levels and to the addition of some of the missing classifications.

Hartree-Fock calculations for many configurations ofUI

Hartree-Fock calculations with the inclusion of a relativistic correction for more than 30 configurations of UI indicate the existence of many simple relationships between spin-orbit coupling constants and Slater integrals for different configurations. These are discussed in detail and compared with experimental values wherever possible. The use of these relationships in empirical least-squares energy-level fits should allow considerable reduction in the number of parameters required to analyze these complex configurations with open $f$, $p$, and $d$ shells. The calculated integrals also provide some bounds on parameter values and guidelines to further analysis of energy levels of U and other actinides.

Application of the projected Hartree-Fock method to rotational motion in1p-shell odd-Anuclei

The rotational properties of A = 7, 9, 11, and 13 nuclei are studied in a self-consistent way by using the effective two-body interactions derived either from the Sussex relative harmonic-oscillator matrix elements or from energy-level fitting. Special care is taken in choosing the appropriate solution among others corresponding to oblate or prolate deformed orbitals. In particular, a pronounced band mixing in A = 9 and A = 11 nuclei is investigated by diagonalizing the effective Hamiltonian within the orthonormalized basis setup with the Hartree--Fock projected states. Moments of inertia of A = 6, 8, 10, and 12 nuclei are also computed by means of the cranking model. The energy levels and the magnetic dipole moments, as well as the M1 transition rates, are in good agreement with shell-model calculations which use the same effective two-body interactions. (AIP)

Phenomenological Description of High Spin States in Doubly Even Rare Earths

Without recourse to a specific model, assuming only that the global structure of spectra is rotational, a good fit of energy levels in doubly even rare earth nuclei is achieved, including the so-called backbending at high spins.

Calculation of Xe II line strengths and radiative lifetimes in intermediate coupling

Line strengths for 5s2 5p4 6s−5s2 5p4 6p transition in xenon are calculated in intermediate coupling scheme. The parametric potential method is used in calculating the radial part of the wave functions while the method of least squares fit of energy levels is applied in obtaining the angular part of the wave functions. Lifetimes of the 6p states obtained using these methods show a good agreement with data from electron-impact experiments.

Crystal-field calculations for energy levels ofU4+in ZrSiO4

A crystal-field calculation has been made for the energy levels of an ion with an ${f}^{2}$ configuration in a ${D}_{2d}$ crystal field, appropriate to ${\mathrm{U}}^{4+}$ in zircon, ZrSi${\mathrm{O}}_{4}$. A previous analysis is found to be incorrect. The present analysis, using ten parameters, gives an energy-level fit which has a standard deviation of 112 ${\mathrm{cm}}^{\ensuremath{-}1}$ relative to 30 observed levels. This fit is comparable to those obtained by other workers for ${\mathrm{U}}^{4+}$ in octahedral and tetrahedral symmetries. Zeeman experiments and magnetic-circular-dichroism measurements have yielded g values for a number of the $\ensuremath{\sigma}$-polarized absorption lines. The values of the parameters for the best fit are (in units of ${\mathrm{cm}}^{\ensuremath{-}1}$): ${F}_{2}=196.7$, ${F}_{4}=37.0$, ${F}_{6}=4.25$, $\ensuremath{\zeta}=1740$, $\ensuremath{\alpha}=22.8$, ${A}_{2}^{0}〈{\mathcal{r}}^{2}〉=\ensuremath{-}1000$, ${A}_{4}^{0}〈{\mathcal{r}}^{4}〉=250$, ${A}_{4}^{4}〈{\mathcal{r}}^{4}〉=5360$, ${A}_{6}^{0}〈{\mathcal{r}}^{6}〉=\ensuremath{-}362$, and ${A}_{6}^{4}〈{\mathcal{r}}^{6}〉=\ensuremath{-}300$.

Wavelengths, energy levels, and analysis of the second spectrum of chlorine (Cl ii)*

We have observed the spectrum of singly ionized chlorine (CI ii) with grating spectrographs from 500 to 11 000 A using a ring discharge and a pulsed rf-discharge source. The accuracy of the observed wavelengths is about 0.002 A from 500 to 3000 A and 0.01 A from 3000 to 11 000 A. The number of known energy levels has increased from 140 to 270 and the number of classified lines from 600 to over 1100. We have found 25 levels of 3p3ng (n = 5–8) and 59 levels of 3p3nf(n = 4–10), including all 20 levels of 3p3 (2D)4f. Combinations with the latter levels gave new information on the singlets in the 3p3 3d configuration, which led to the rejection of some levels, the establishment of others, and many changed assignments. Ab initio energy-level and spectrum calculations and least-squares energy-level fits showed large interactions between the 3s 3p5, 3s2 3p3 3d, and 3s2 3p3 4d configurations. The ionization limit of Cl+ is 192070 ± 1 cm−1.

Cancellation effects in computed lifetimes of 3s3p n levels for third-row atoms and ions

It is well known that calculated transition probabilities values are often unreliable because of cancellation effects resulting from configuration mixing. A typical example is provided by the transitions 3s 23p m −3s3p m+1 for third-row atoms and ions. In fact, the values of the corresponding transition probabilities which are calculated by means of an independent-particle model (such as Hartree-Fock) are generally of the order of 10 to 1000 times larger than the observed data. For the isoelectronic sequences of P I and S I, we have shown, with the use of a parametrized potential, that the mixture of 3s3p m+1 and 3s 23p m−1 3d configurations is responsible for the most important part of the discrepancy. The general procedure used for calculating transition probabilities has been described already in a previous paper given at the Second International Conference of Beam Foil Spectroscopy. However, while for the noble gases, the angular part of the wave functions was obtained by a least squares fit of energy levels, the wave functions (including configuration mixing) here are determined completely from the central potential. The agreement between observed (especially by the beam-foil technique) and calculated lifetimes for the 3s3p m+1 levels is generally good for the cases where comparisons are possible.

Zeeman Effect of P ii*

Electrodeless-discharge tubes operated in a field of 32 215 G have given Zeeman patterns for 223 P ii spectral lines from 2285 to 7846 Å and yielded 76 Landé g factors. There is good agreement between these g values and those calculated from wave functions obtained by least-squares fitting of energy levels for the single configurations, 3p4p, 3p5p, 3p4f, 3p4s, 3p5s, and 3p6s. However, even when configuration interaction is introduced in the multiconfiguration case, 3s3p3–3p3d–3p4d–3p5d, experimental energy levels will not give good agreement with the corresponding calculated values unless LS-dependent parameters are used in the least-squares fitting. The newly obtained g factors, combined with the predictions of the least-squares energy-level fitting process, have forced a rejection of one even and two odd previously found levels, established nine new levels, reclassified three even and ten odd levels. The new levels belong to the now completed configurations 3p5p and 3p5d. A careful search for transitions to support poorly established levels yielded 44 new lines that confirm newly found levels and fill gaps in the previously known level array.

Transition probabilities and radiative lifetimes for Ne II

Theoretical transition probabilities for 2s 22p 43s−2s 22p 43p and 2s 22p 43p−2s 22p 43d transitions in Ne II have been obtained by using the “exact” intermediate coupling wave function. The angular part of the wave function is obtained be means of a least squares fit of energy levels. Some effective operators are included and the interaction between the 2p 43p and 2p 5 configurations is also taken into account. It is assumed that the radial part of the dipole matrix element for these configurations can be given by the Coulomb approximation to high accuracy.