Nuclear moments and isotope shifts of europium isotopes by hyperfine structure investigations with collinear laser — Ion beam spectroscopy

Hyperfine structures and isotope shifts of the transition 4f⁷(⁸S ₇ ^2/0 )6p_3/2 (7/2,3/2)₄?4f⁷(⁸S ₇ ^2/0 ) 5d⁹D ₄ ⁰ ,λ=604.95 nm, of stable¹⁵¹Eu and¹⁵³Eu and radioactive¹⁵²Eu and¹⁵⁴Eu have been investigated by collinear laser — ion beam Spectroscopy. Nuclear quadrupole moments have been derived.     

Investigations of europium isotopes with neutron numbers between 84 and 93 by collinear laser ion-beam spectroscopy

Hyperfine structures and isotope shifts of the transition Eu II 4f⁷(⁸S ₇ ^2/O )6p_3/2 (7/2,3/2)₄ — 4f⁷ (⁸S ₇ ^2/O )5d⁹D ₄ ^O ,λ=6O4.95 nm, of the radioactive isotopes^{147, 149, 155, 156}Eu have been studied by collinear laser-ion beam spectroscopy (CLIBS). The nuclear moments and the differences of the mean square nuclear charge radii as well as the deformation parameters have been deduced. The experimental data for the odd-A nuclei are in a good agreement with simple Nilsson model calculations.     

Measurement of theg-factor of the 4 1 + -state of202Pb

Theg-factor of the 4 ₁ ⁺ -state of²⁰²Pb was investigated by the IPAC-technique. Inspite of the long half-life, which was measured as:T _1/2(4 ₁ ⁺ -state)=1.97(2)ns and the strong applied external magnetic field of 95.0 kG no rotation of the 787 keV-(422keV)-961 keVγ-γ angular correlation could be observed. Theg-factor must therefore be very small. A computer fit gave the limits:g(4 ₁ ⁺ -state)=+0.002(4). This smallg-factor can be understood by the assumption of an accidental cancellation of a positive contribution by the main [f _5/2 ² ]₄-configuration and several negative contributions by admixtures of all other possible two neutron configurations.     

g-factors of the ground state rotational band of158Dy

Theg-factors of the four lowest states of the ground state rotational band of¹⁵⁸Dy have been determined asg(2 ₁ ⁺ )=+0.362(23),g(4 ₁ ^su+ )=+0.340(20),g(6 ₁ ^su+ )=+0.207(36) andg(8 ₁ ^su+ )=+0.21(11). Theg-factors of the 2⁺ and 4⁺ states were measured by the IPAC method with radioactive samples of 2.4 h¹⁵⁸Er in external magnetic fields. To investigate the higher states, for the first time an on-line γ—γ IPAC experiment was performed with the reaction¹⁵⁶Gd(α, 2n)¹⁵⁸Dy by use of the static hyperfine field of DyGd.     

Gyromagnetic ratios of 4+ and 6+ states in156Gd and158Gd

The followingg-factors have been derived from time integral measurements of γ-γ angular correlations in the static magnetic hyperfine field of magnetized gadolinium metal probes:¹⁵⁶Gd:g(4 ₁ ⁺ )=+0.310(19)g(6 ₁ ⁺ )=+0.25(21)g(4 ₃ ⁺ , 1511 keV)=+0.809(27)¹⁵⁸Gd:g(4 ₁ ⁺ )=+0.409(15). The 5.35d ¹⁵⁶Tb sources were produced by the reaction¹⁵⁶Gd(d, 2n)¹⁵⁶Tb in our cyclotron. A carrier-free 150y ¹⁵⁸Tb source was obtained from ISOLDE/CERN. In comparison with the precisely knowng-factors of the 2 ₁ ⁺ states,g(2 ₁ ^su+ ,¹⁵⁶Gd) =+0.386(4) andg(2 ₁ ⁺ ,¹⁵⁸Gd)=0.381(4), we observe a large reduction for the¹⁵⁶Gd 4 ₁ ⁺ state whereasg increases slightly for¹⁵⁸Gd. The half-life of the 4 ₁ ⁺ state of¹⁵⁸Gd was remeasured as¹⁵⁸Gd:T _1/2(4 ₁ ⁺ )=148(2) ps. A measurement of the rotation in the 4 ₃ ⁺ state of¹⁵⁶Gd in external magnetic fields of various strengths up toB _ext=9.5 T did not confirm the anomalous dependence of the magnetic hyperfine field in gadolinium metal on the external field, which has been reported by Persson et al. [29].     

FastM1 transitions in85Rb

Excited states in⁸⁵Rb have been studied via the⁸²Se(⁷Li, 4n) reaction by using in-beamγ-ray spectroscopy. On the basis of new experimental data on coincidence relations, angular distributions and linear polarizations of theγ-rays the yrast sequence has been extended up to a 33/2⁺ level at 7.1 MeV excitation energy. Using the DSA method mean lifetimes have been determined for 9 levels. Above the 21/2⁺ yrast state 5 fast M1 transitions withB (M 1) ≧0.3 W.u. have been identified. They are interpreted as transitions between recoupled states of the configuration πg_9/2ν(g_9/2) ₈₊ ^?2 or πg_9/2(f ₅ ² /?1p ₃ ^2?1 )₄₊ν(g_9/2) ₈₊ ^?2 .     

High spin shell model excitations in149Gd

The high spin level structure of the three-neutron nucleus¹⁴⁹Gd has been investigated by in-beamγ-ray and electron spectroscopy with (α, xn) reactions. The observed levels are characterized as members of the shell model multipletsνf ₇ ^2/3 ,νh _9/2 f ₇ ^2/2 ,νf ₇ ^2/3 ×3^?,νh _9/2 f ₇ ^2/2 ×3^?, and tentativelyνf ₇ ^2/3 ×(3^?)². The energies of theν f ₇ ^2/3 states agree only moderately with those calculated using empirical two-nucleon interactions taken from¹⁴⁸Gd, which indicates the importance of long range contributions already atN=85.     

Remeasurement of the lifetime of the 4 1 + -state of156Gd and recalculation of itsg-factor

The half-life of the 4 ₁ ⁺ -state of¹⁵⁶Gd has been remeasured by the delayed coincidence technique. The excellent time resolution which can now be achieved by use of small BaF₂ detectors allows a more reliable determination than with previously applied methods. The resultT _1/2(¹⁵⁶Gd,4 ₁ ⁺ )=108(2) ps is smaller than previously published data, but it fits well into the systematics of theB(E2) values of the rotational transitions of this nucleus. A recalculation of the previously measuredg-factor of the same state givesg(¹⁵⁶Gd,4 ₁ ⁺ ) =0.327(19). This value is still smaller than theg-factor of the 2 ₁ ⁺ state, but the magnitude of the reduction can now easier be interpreted by nuclear structure calculations.     

g-factors of the lowest 2+ and 4+ states in198, 200, 202Hg

Theg-factors of the 2 ₁ ⁺ and 4 ₁ ⁺ states in^198,200,202Hg were simultaneously measured by means of the transient-field perturbed angular correlation method in Coulomb excitation using multi-isotopic targets and thin polarized Gd foils as ferromagnetic host. Theg(2 ₁ ⁺ ) andg(4 ₁ ⁺ ) were found identical within errors in^{198, 200}Hg, while in²⁰⁰Hg lowerg-factor values have been determined. The experimentalg-factors were compared with the predictions of the pairing-plus-quadrupole, dynamical deformation and interacting boson models.     

Gyromagnetic ratios of excited states in186W

The gyromagnetic ratios of the 4 ₁ ⁺ , 6 ₁ ⁺ , and 2 ₂ ⁺ states in¹⁸⁶W were measured relative to that of the 2 ₁ ⁺ level by means of the transient field implantation perturbedγ-ray angular distribution technique. The nuclei in the states of interest were Coulomb excited using a beam of 220-MeV⁶³Cu projectiles and recoiled swiftly through a thin, polarized Fe foil. The present measurements yielded ratiosg(4 ₁ ⁺ )/g(2 ₁ ⁺ )=1.04±0.07,g(6 ₁ ⁺ )/g(2 ₁ ⁺ )=1.03 ±0.20 andg(2 ₂ ⁺ )/g(2 ₁ ⁺ )=0.63±0.13. The sizable disparity between the measuredg-factors of the ground- and excited-band is examined within the context of the interacting boson approximation model.     

Embedding theorem for weighted Sobolev classes on a John domain with weights that are functions of the distance to some h-set

Let Ω be a John domain, let Γ ? ?Ω be an h-set, and let g and v be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class W _p,g ^r (Ω) is continuously embedded in the space L _q,v (Ω). Moreover, bounds for the approximation of functions in W _p,g ^r (Ω) by polynomials of degree not exceeding r ? 1 in L _q,v ( $\tilde \Omega $ ) are found, where $\tilde \Omega $ is a subdomain generated by a subtree of the tree T defining the structure of Ω.     

The147Sm(n,α)144Nd Reaction on 2 keV neutrons

Through (p,2n) in-beam γ and e^? measurements the excitation of the 235 μs 9⁺ isomer in¹⁴⁶Eu has been determined to be 666.3 keV. High spin members of the πd ₅ ^2/?1 νf_7/2 and πg ₇ ^2/?1 νf_7/2 particle hole multiplets populated in the isomeric decay were located.     

Yrast states in the π- particle ν- hole nucleus 65 146 Tb81

The nucleus ¹⁴⁶Tb was studied from in beam γγ-and conversion electron measurements. The level scheme was established up to ～5MeV above the (πh_11/2 vd _3/2 ^?1 )_5? β-isomer. In addition to the known (πh_11/2 vh _11/2 ^?1 )₁₀₊ E3-isomer, the 8⁺ and 11⁺ members of this configuration were located. The levels at the yrast line are dominated by the couplings of the πh_11/2 vh _11/2 ^?1 valence nucleons to the collective 3^? octupole state and to the πh_11/2d _5/2 ^?1 and πh_11/2g _7/2 ^?1 particle-hole excitations of ¹⁴⁶Gd.     

g-factors of rotational states in176Hf,177Hf,and180Hf

g-factors of rotational states in¹⁷⁶Hf and¹⁸⁰Hf were measured with the twelve detector IPAC-apparatus of our laboratory [1]. The natural radioactivity 3.78·10¹⁰y¹⁷⁶Lu and the 5.5 h isomer^180mHf were used which populate the ground-state rotational bands of¹⁷⁶Hf and¹⁸⁰Hf. The integral rotations ofγ-γ directional correlations in strong external magnetic fields and in static hyperfine fields of (Lu→Hf)Fe₂ and HfFe₂ were observed. The following results were obtained: $$\begin{array}{l} ^{176} Hf: g\left( {4_1^ + } \right) = + 0.334\left( {38} \right) \\ ^{180} Hf: g\left( {2_1^ + } \right) = + 0.305\left( {14} \right) \\ g\left( {4_1^ + } \right) = + 0.358\left( {43} \right) \\ {{ g\left( {6_1^ + } \right)} \mathord{\left/ {\vphantom {{ g\left( {6_1^ + } \right)} {g\left( {4_1^ + } \right)}}} \right. \kern-\nulldelimiterspace} {g\left( {4_1^ + } \right)}} = + 0.95\left( {12} \right) \\ \end{array}$$ . The hyperfine field in (Lu→Hf)Fe₂ was calibrated by observing the integral rotation of the 9/2^? first excited state of¹⁷⁷Hf populated in the decay of 6.7d¹⁷⁷Lu. Theg-factor of this state was redetermined in an external magnetic field as $$^{177} Hf: g\left( {{9 \mathord{\left/ {\vphantom {9 {2^ - }}} \right. \kern-\nulldelimiterspace} {2^ - }}} \right) = + 0.228\left( 7 \right)$$ . Finally theg-factor of the 2 ₁ ⁺ state of¹⁷⁶Hf was derived from the measuredg(2 ₁ ⁺ ) of¹⁸⁰Hf by use of the precisely known ratiog(2 ₁ ⁺ ,¹⁷⁶Hf)/g(2 ₁ ⁺ ,¹⁸⁰Hf) [2] as $$^{176} Hf: g\left( {2_1^ + } \right) = + 0.315\left( {30} \right)$$ .     

The predissociation of the 2σ u −1 (c 4Σ u − , v states of the oxygen molecular ion

The dynamics of predissociation of the 2σ _u ^?1 (c ⁴Σ _u ^? ), v vibrational states of the O ₂ ⁺ ion was studied theoretically using the method of coupled differential equations. The main equations describing the vibrational motions of nuclei in the adiabatic and diabatic approximations are given. The applicability scope of approximate methods for solving these equations was studied. The predissociation widths for the v = 0 and 1 vibrational levels were found to be Γ₀ = 0.054 meV and Γ₁ = 9.71 meV. This substantiated the results of recent observations of neutral fragments formed after the dissociation of the O₂ molecule. About 99% of the O ₂ ⁺ ions in the 2σ _u ^?1 (c ⁴Σ _u ^? ), v states were found to decompose to the O(¹ D) + O⁺(⁴ S) dissociation products.     

MeanM-subshell fluorescence yields atZ=88, 90, 92, and 94

M x-ray —L x-ray coincidence measurements with high resolution, cooled Si(Li) x-ray detectors were made on transitions following the alpha decays of²²⁸Th,²³²U,²³⁸Pu, and²⁴⁴Cm, in order to determine the meanM-subshell fluorescence yields. The values obtained are:v ₄ ^M =0.032±0.002, andv ₅ ^M =0.024±0.002 atZ=88;v ₁ ^M =0.038±0.003,v ₄ ^M =0.042±0.002, andv ₅ ^M =0.038±0.002 atZ=90;v ₁ ^M =0.047±0.002,v ₄ ^M =0.048±0.002, andv ₅ ^M =0.044±0.002 atZ=92;v ₁ ^M =0.066±0.002,v ₄ ^M =0.062±0.002, andv ₅ ^M =0.063±0.002 atZ=94. The quantityΩ ₁ ^M +f ₁₂ ^MΩ ₂ ^M was measured as (56±10)×10^?4, (62±12)×10^?4, (99±18) ×10^?4, and (93±15)×10^?4 forZ=88, 90, 92, and 94, respectively, which agree well with the calculations of McGuire. The radiativeL ₁-L ₃ transition intensity was measured for the four atomic numbers and found to be consistently less than the calculations of Scofield by about 45 percent.     

Construction of kernels of the nonlinear collision integral in the Boltzmann equation using laplace transformation

It is shown that the relation between kernels L _l (v, v ₁) of the linear collision integral and kernels G _l,0 ^l (v, v ₁, v ₂) of the nonlinear collision integral can be reduced to the Laplace transformation. Analytic expressions for nonlinear kernels G _0,0 ⁺⁰ (v, v ₁, v ₂) and G _1,0 ⁺¹ (v, v ₁, v ₂) are determined for hard spheres and pseudo-Maxwellian molecules.     

Embedding theorem for weighted Sobolev classes with weights that are functions of the distance to some h-set

The paper continues the first part (Russ. J. Math. Phys. 20 (3), 360–373). Let Ω be a John domain, let Γ ? ?Ω be an h-set, and let g and υ be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class W _p,g ^r (Ω) is continuously embedded in the space L _q,v (Ω). Moreover, bounds for the approximation of functions in W _p,g ^r (Ω) by polynomials of degree not exceeding r ? 1 in L _q,v ( $\tilde \Omega $ ) are found, where $\tilde \Omega $ is a subdomain generated by a subtree of the tree T defining the structure of Ω.     

Conserved-vector-current hypothesis and the\bar v_e e^ - \to \pi ^ - \pi ^0 process

Based on the conserved-vector-current (CVC) hypothesis and a four-ρ-resonance unitary and analytic VMD model of the pion electromagnetic form factor, theσ _tot(E _v ^lab ) and dσdE _π ^lab of the weak \(\bar v_e e^ - \to \pi ^ - \pi ^0\) process are predicted theoretically for the first time. Their experimental approval could verify the CVC hypothesis for all energies above the two-pion threshold. Since, unlike the electromagnetic e⁺e^?→π⁺π^? process, there is no isoscalar vector-meson contribution to the weak \(\bar v_e e^ - \to \pi ^ - \pi ^0\) reaction, accurate measurements of theσ _tot(E _v ^lab ) that moreover is strengthened with energyE _v ^lab linearly could solve now a widely discussed problem of the mass specification of the first excited state of theρ(770) meson. As a by-product, an equality \(\sigma _{tot} (\bar v_e e^ - \to \pi ^ - \pi ^0 ) = \sigma _{tot} (e^ + e^ - \to \pi ^ - \pi ^0 )\) is predicted for \(\sqrt s \approx 70 GeV\) .     

Investigation of N 2 + (C 2Σ u + → X 2Σ g + fluorescence excited through auger decay of the 1s −1π* resonance

A theoretical investigation of N ₂ ⁺ (C ²Σ _u ⁺ → X ²Σ _g ⁺ molecular fluorescence excited through the Auger decay of the 1s ^?1π* resonance is carried out. The fluorescence cross sections are calculated with due regard for the dependence of the matrix element of the C → X dipole transition on the internuclear distance, the interference between channels of excitation via different vibrational levels v _r of the 1s ^?1π* resonance, the rotational structure of the fluorescence band, and the predissociation of the N ₂ ⁺ C ²Σ _u ⁺ v′ ≥3) states. The calculated cross sections are in good agreement with the experimental results of recent measurements. The results of the calculations have demonstrated that the observed dependence of the cross section of the (C ²Σ _u ⁺ (v′) → X ²Σ _g ⁺ (v″) fluorescence on the excitation energy and the fluorescence wavelength for a group of bands with equal values of the difference Δv = v′ ? v″ is associated with transitions between the vibrational levels of the electronic states involved in the excitation and subsequent cascade decay of the 1s ^?1π* resonance: N₂ (v ₀ = 0) → N*₂(1s ^?1π*(v _r)) ? N ₂ ⁺ : (C ²Σ _u ⁺ (v′) → X ²Σ _g ⁺ (v″).