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The influence on the pairing correlations and the coriolis force on the E2 ΔK = 1 transitions in deformed odd-mass nuclei
Abstract
The influence of the pairing correlations, of the Coriolis mixing and of corrections to the Coriolis force mainly due to the rotation-vibration interaction on the E2 ΔK = 1 transitions are studied. Because the rotational transitions are as strong as 200 single-particle units and the Nilsson transitions lie between 10−2 and 10−5 units, already a small mixing can enhance the transition probability by several orders of magnitude. The pairing correlations affect the single-particle and the collective part of the transition probability in different ways. Because the single particle part of the Coriolis force is negative under time reversal, the Coriolis mixing is not reduced by the pairing force. However, the single-particle transition probability can be diminished by more than a factor 100 due to the pair correlations. The agreement of the collective values with the experimental results is good, while the Nilsson transition probabilities with pair correlations are up to several orders of magnitude too small.
The experimental results published before August 2020 from the various reaction and decay studies leading to nuclides of Z=55 to Z=72, ¹⁵³Ba, ¹⁵³La, ¹⁵³Ce, ¹⁵³Pr, ¹⁵³Nd, ¹⁵³Pm, ¹⁵³Sm, ¹⁵³Eu, ¹⁵³Gd, ¹⁵³Tb, ¹⁵³Dy, ¹⁵³Ho, ¹⁵³Er, ¹⁵³Tm, ¹⁵³Yb, ¹⁵³Lu, ¹⁵³Hf in the A=153 mass chain have been reviewed. These data are summarized and presented, together with adopted level schemes and properties. This work is intended to supersede the previous evaluation of the A=153 nuclides by R.G. Helmer (2006He06), which was published in Nuclear Data Sheets 107, 507 (2006).
The B(E2) values for transitions between the [402 {5}/{2}] and [404
{7}/{2}] Nilsson states in 171Lu and 173Lu are
determined from half-life and γ-branching measurements. Coriolis
coupling matrix elements are derived from these B(E2) values and
compared with corresponding results in 175-177Lu and
177-181Ta. The experimental matrix elements for the Lu
isotopes are only approximately 10% smaller than the calculated value
whereas the results for the Ta isotopes are attenuated by factors
between two and four. The reduced M1 transitions between the KI =
{5}/{2}{5}/{2}and{7}/{2}{7}/{2} levels are reanalysed.
The experimental results from studies of the radioactive decays and reactions leading to nuclides in the A=161 mass chain, and α decays from it, have been reviewed. These data are summarized and presented, together with adopted level schemes and properties.
The experimental results from the various reaction and decay studies leading to nuclides in the A=158 mass chain have been reviewed. These data are summarized and presented, together with adopted level schemes and properties.
New experimental evidence on the M1 + E2 transitions following the beta decay of 70 d 175Hf is presented. The absolute transition probabilities and the signs of the mixing parameters, which may be measured in directional correlation experiments, are used to derive the reduced Coriolis matrix element AK and the intrinsic inter-band M1 matrix element . The results are in good agreement with calculations based on the Nilsson Model.
An experimental investigation of the 70.5 keV gamma-ray transition from the 5/25/2+ [402] state to the 7/27/2+ [404] ground state in 177Ta was made. The half-life and magnetic moment of the initial state were found to be 73 +/- 5 ns and 4.7 +/- 0.5 muN, respectively. The E2/M1 mixing ratio of the transition was determined to be -0.10 +/- 0.04. The reduced transition probabilities, B(E2) and B(M1), were found to be 2.9 +/- 2.3 e2 . fm4 and (1.1 +/- 0.1) × 10-4mu2N, respectively, and the M1 penetration parameter was determined to be 6 +/- 2. The Coriolis mixing amplitude of the 7/27/2+ [402] state in the 7/27/2+ [404] ground state was determined from the B(E2) value to be -0.011 +/- 0.004. It was found that the experimental value of the Coriolis matrix element exp was one-tenth of that calculated using the Nilsson wave function. The coefficient k2, introduced by Wahlborn et al., was-re-analyzed using the present experimental results and found to be less than 0.8. Present address: Department of Physics, University of Tsukuba, Ibaraki 300-31, Japan.
The M1 and E2 transition rates between the [411 3/2] and [413 5/2] Nilsson states in 153Eu, 155Eu and 159Tb, the [521 3/2] and [523 5/2] Nilsson states in 161Dy, 163Dy and 165Er, and the [404 5/2] and [404 7/2] Nilsson states in 175Lu, 177Ta, 179Ta and 181Ta are studied. The Coriolis coupling matrix elements, derived from the B(E2) values, are approximately a factor of three smaller than the calculated values. The influence of the Coriolis coupling on the Ml transition rates in 153Eu, 161Dy and 173Lu is studied. On leave from the University of Lisbon with a fellowship from the Instituto de Alta Cultura of Portugal.
The couplings between the pair vibrational 2p-2h state in 90Zr and the two-phonon surface oscillations of multipole order 2, 3 and 5 are evaluated with due consideration of non-orthogonality. The resulting distribution of neutron pair correlations is in agreement with two-nucleon transfer data. The disappearence of the pair vibration in isotopes away from the neutron major shell closing is explained by consideration of anharmonic pairing correlations.
Great theoretical interest has recently been shown in the asymptotically forbidden 5/25/2+ [402] --> 7/27/2+ [404] M1 transitions in 181Ta and 175Lu [refs.4,6)]. Half-lives of the 5/25/2+ [402] level in 175Lu have been reported which differ by more than a factor of 10. Using an electron-electron coincidence spectrometer the half-life of this level has been re-measured and a value of (0.26+/-0.02) nsec has been established.
The retarded transitions assigned as 5/2, 5/2+[402]-->7/2, 7/2+[404] in 181Ta (482 keV) and 175Lu (343 keV) are analysed with special regard to the large hindrance of the M1 radiation in 181Ta. The E2 component is dominated by the band mixing which is thus determined by B(E2). For the analysis of B(M1), we exploit the relation between the Coriolis and M1 intrinsic matrix elements. The result shows a marked difference between 181Ta and 175Lu. Calculations with extended models for particle motion indicate possible effects of the detailed deformation structure for the two cases. Furthermore, the predicted tensor term k2[Y2, s]1 in the mu operator gives a significant contribution to B(M1), when k2 is of the order of unity. The present data are consistent with the condition k2 ~ is 0 preferred because of consistency with the model calculations. The effective magnetic moment operator can be written in the form mu ~ (gl0 + deltaglex)l+(gs0+deltagspol+deltagsex)s+(k2pol+k2 ex)[Y2, s]1, where gl0 and gs0 are the unrenormalized nucleon g-factors. The term deltagspol represents the well-known gs renormalization due to spin polarization (deltagsex is small). We have estimated the pion-exchange contributions, giving tau3glex ~ 0.1 and tau3k2ex ~ - 1. When our preferred range for k2 from the present analysis is combined with previous data from other work, we can give the conservative semiempirical estimate 0 < tau3k2pol
The level structure of 169Yb has been studied by radiative capture of thermal neutrons in 168Yb. High resolution measurements of the gamma-ray spectrum have been performed using a Ge(Li) anti-Compton spectrometer in the low-energy region and a Ge(Li) pair spectrometer for the high-energy transitions. The target was Yb2O3 enriched to 19.5% in 168Yb thus corresponding to a cross section contribution of (97.5-2.3+0.8)% for 168Yb. More than 300 gamma lines have been detected in the spectrum. The high accuracy of the data allows the application of Ritz' combination principle to excitation energies up to 1.5 MeV. The results clearly demonstrate the presence of remarkable band-mixing effects. The analysis suggests the following spectroscopic interpretation (bandhead energies and dominant structure): 0 keV, 7/2+(633); 24.25 keV, 1/2-(521) 191.18 keV; 5/2-(512) 569.78 keV, 5/2-(523) 659.61 keV, 3/2-(521)+1/2-(521)+Q2-2 720.02 keV, 7/2+(633)+Q2-2+3/2+(651); 813.46 keV, 5/2-(512)+Q2-2+1/2-(510) 960.28 keV, case7/2-(514) 1033.82 keV, 1/2+(660)+1/2+(660)+Q20(?); 1110.68 keV, 1/2-(521)+Q22 1204.28 keV, 3/2+(651)+7/2(633)+Q2-2 1231.45 keV, 1/2-(521)+Q2-2+3/2-(521) 1319.72 keV, 1/2-(510)+.... The neutron binding energy was found to be 6867.21+/-0.46 keV. Theoretical calculations have been performed which take into account pair correlation, quasiparticle-phonon interaction, rotation-vibration interaction and Coriolis coupling. The energy and structure of individual levels, the ratios of intensities for gamma-ray transitions and multipolarity admixtures are predicted. In general, good agreement is achieved between the theoretical calculations and the experimental results.
The internal conversion electron spectrum of the decay 155Tb --> 155Gd in the energy range 8 keV-600 keV was measured with the aid of a semicircular, beta-ray spectrograph. The activity was produced in the spallation of Ta nuclei with 600 MeV protons. The energies and relative intensities of the conversion lines were measured, and for some transitions their multipolarities and mixing ratios were also determined. The obtained experimental data are presented in the decay scheme of 155Tb. The decay scheme is discussed in terms of the Nilson model by taking into account the Coriolis coupling and the interaction between states differing in oscillator quantum number by DeltaN = 2. From the obtained experimental data, the matrix element of the interaction and the Coriolis coupling parameters of the interaction between the states [651]3/2+ and [642]5/2+ and between the states [660]1/2+ and [651]3/2+ were estimated and the values obtained compared with the theoretical predictions.
A method is described which extends the usual Coriolis coupling band mixing calculations by the inclusion of rotation-vibration interaction and quasiparticle-phonon interaction. It is based on Bohr's collective Hamiltonian and on the rotation-vibration interaction model of Faessler and Greiner. These collective models are described in detail. The success of the method depends on a careful treatment of the pairing correlations which have a striking influence. This is because the operators of the quasiparticle-phonon interaction commute with the time-reversal operator, so that their matrix elements between single-particle states are in many cases considerably reduced by the pairing correlations. Because this interaction mixes high-lying configurations with low-lying ones, it is important to take the blocking effect into account. The influence of the number conservation is investigated.
Energies of nuclear states and their theoretical level structure, allowed and K-forbidden gamma-ray transitions in odd-mass rare earth nuclei are calculated and compared with the experimental values. Good agreement with experimental results is obtained for M 1 transitions; K-forbidden E 1 transitions are about an order of magnitude too fast.
The level structure of 167Er and the de-excitation mechanism m this nucleus have been studied by radiative capture of thermal neutrons in 166Er. High-resolution measurements of the gamma-ray spectrum have been performed using a Ge(Li) anti-Compton spectrometer in the low-energy region and a Ge(Li) pair spectrometer for the high-energy transitions. Coincidence relationships have been obtained from measurements with a Ge(Li)-NaI(T1) coincidence apparatus. The target was Er2O3 enriched to 95.6 % in 166Er thus corresponding to a cross-section contribution of (68.8+5.0−6.4) % for 166Er. More than 350 gamma-ray lines have been detected in the spectrum. Isotope assignment has been achieved by comparison of the spectra with those obtained from a sample of natural erbium. The high accuracy of the data and the coincidence results allow the construction of a transition diagram up to 1.5 MeV excitation energy. A large number of the observed energy levels have been assigned to specific configurations and their superimposed rotational bands. Detailed analysis of the data suggests considerable mixing between Nilsson states and the quadrupole vibrations Q22, Q2−2 and Q20. Theoretical calculations have been performed which take into account pair correlations, quasiparticle-phonon interaction, rotation-vibration interaction and Coriolis coupling. It is shown that neglect of pair correlations leads to unreasonable matrix elements. While quasiparticle-phonon interaction and Coriolis coupling have decisive influence on the level structure and the de-excitation mechanism, the effects due to rotation-vibration interaction are in general small. The model predicts the energy and structure of individual levels, branching ratios for gamma-ray transitions, multipole mixtures and partial gamma-ray half-lives. In most cases, good agreement is achieved between the theoretical calculations and the experimental results. The neutron separation energy was found to be 436.15 ± 0.48 keV.
The available nuclear structure information for all nuclei with mass number A=217 is presented. Various decay and reaction data are evaluated and compared. Adopted data, levels, and spin and parity assignments are given.
The experimental results from the various reaction and decay studies leading to nuclides in the A=154 mass chain, and alpha decays from it, have been reviewed. These data are summarized and presented, together with adopted level schemes and properties.
The γ-ray and conversion electron spectra of 155Eu from the (n, γ) reaction were studied at the Institut Laue-Langevin using the curved crystal spectrometers GAMS 1, GAMS 2/3, the β-spectrometer BILL and the pair spectrometer. The γ-transition multipolarities were determined in the energy region 40–1200 keV. The level scheme of 155Eu was developed with 51 levels up to the excitation energy of 1.7 MeV; these levels were depopulated via 140 transitions out of 360 transitions observed in the γ-ray spectrum. The levels were grouped into 13 rotational bands, and of these −[541],−[550],+[413]+Q20,+[411]+Q20 were entirely new for 155Eu was found Bn = 8151.3(3) keV. It has been determined that the E1 transitions of ΔK = 0 type were enhanced in comparison with these of the ΔK = 1 type. The Nilsson + RPC, cranked shell model and odd RV model calculations have indicated the presence of strong collectivization, even for the 155Eu lowest states.
The relation of the quadrupole deformation with v = ±1 to the rotational motion is discussed and the intrinsic excitation modes with Kπ = 1+ are estimated. The value of the effective charge eeff (E2, v = ±1) is evaluated in order to discuss the magnitude of Coriolis matrix elements quantitatively.
An investigation of the electromagnetic properties of the Kπ = 2− octupole band in 182W populated in the β− decay of 182Ta was undertaken. The multipolarity mixings of the intraband 114 keV and 85 keV transitions were determined from L-subshell measurements. The mixing parameters δ2(E2/M1) are 0.130(7) and 0.124(2) respectively. The directional correlations of the (100–1121) keV, (100–1157) keV and (100–1189) keV cascades were investigated. The E1, M2, E3 mixing of the 1189 keV transition was extracted from the angular correlation coefficients and conversion data taken from the literature. We get a mixing of (57.0 ± 3.7) % E1 + (13.7 ± 1.9) % M2 + (29.3 ± 6.6) % E3. A centroid shift determination of the lifetime of the 3−(1374keV)state gave the result τ = 112 ± 15 ps. From this we obtain the reduced E3 transition probability from the ground state as B(E3; O+ → 3−) = (8.0 ± 1.2) · 104e2 · fm6 = (41 ± 6)SPU. The g-factor of this state was measured using the integral PAC method in the hyperfine field of an iron host lattice yielding g = 0.32(9). The experimental data are discussed in terms of the collective model taking into account Coriolis coupling.
E2 and M1 transition probabilities for odd-mass rare earth nuclei, have been calculated using both the usual BCS wave functions and the strict particle conserving, projected BCS functions. The blocking effect has been exactly and systematically taken into account. The influence of the Coriolis interaction has been studied using the first order perturbation theory. Allowance has been made for the β and γ vibrations. The unphysical effects, due to particle fluctuation in the BCS theory, are not always negligible, but are in most cases, less important than the Coriolis effect. Les probabilités de transitions E2 et M1 dans plusieurs noyaux de la région des terres rares, ont été calculées avec les fonctions BCS habituelles et avec les fonctions BCS projetées sur les états propres de l'opérateur nombre de particules. Il a été tenu compte systématiquement et exactement de l'effet de bloquage. Le couplage de Coriolis a été traité par une méthode perturbative. Les effets des vibrations β et γ ont été approximativement pris en considération. Il s'avère que les erreurs dues à la non-conservation stricte du nombre de nucléons, si elles ne sont pas toujours négligeables, sont cependant moins importantes en moyenne que les effets de Coriolis.
Several examples of the effect of pairing correlations on nuclear gamma-
ray transitions are presented. It is shown that even it collective coupling
exists, gamma-ray transitions of electric multipole type between low-lying
quasiparticle states whose free shell-model orbit energies are symmetrically
related to the Fermi energy may be extremely retirded. The factor (U/sub i/U/sub
f/-V/sub i/V/sub f/) that arises fro m the pairing correlation is directly
estimated from empirical even-odd mass difference data. The agreement of the
calculated transition probabilities with experimental results is good. A shell-
model calculation taking into account the pairing correlation was also made for
transitions of E2 type between d/sub 3/2/ and s/sub 1/2/ orbits in Sn/sup 117/
and Sn/sup 11 8/ and the results are compared with that of the quasiparticle
approximation. It is shown that the quasiparticle approximation is more improved
by using renormalized wave functions that are obtained by projecting out only the
terms having the correct number of particles than by taking into account only the
blocking effect except for special cases. (auth)
The L subshell ratios of a number of M1-E2 transitions have been measured accurately with the Berkeley 50-cm iron free π √2 spectrometer. When the three L subshell conversion lines are resolved, the M1-E2 mixing ratio is overdetermined, because a unique value of δ2 (i.e. E2/M1) is given by any one of the ratios LI/LII, LI/LIII and LII/LIII. (Algebraically only two of these ratios are independent.) Thus, if all three ratios are measured experimentally, the same value of δ2 should in principle be obtained. In fact, this does not always appear to be the case. One of the most pronounced discrepancies is found in the case of the 103 keV transition in Eu153. The measured ratios are LI/LII = 8.22±0.10, LI/LIII = 20.65±0.28 and LII/LIII = 2.51±0.05. With use of carefully interpolated values from Sliv's tables, the E2 admixtures obtained are 2.02±0.08, 1.66±0.3 and 1.44±0.06 percent, respectively. With Rose's values the E2 admixtures are 2.18±0.08, 1.52±0.03 and 1.15±0.06 percent. Similar discrepancies were observed in the 114 keV transition in Lu175 and 42 and 52 keV transitions in Bk249. Relatively good agreement with Sliv's values is found for the 69 keV transition in Eu153 and for the 58 keV transition in Tb159. The agreement is generally better with Sliv's than with Rose's values. The two sets of theoretical L conversion coefficients differ sometimes by as much as 50%.
A model for deformed odd-mass nuclei is described in which the nucleon is coupled to a vibrating core with axially symmetric equilibrium deformation. The Hamiltonian consists of two parts: the Hamiltonian Hc of the rotating and vibrating core, including the rotation-vibration interaction and the Hamiltonian Hp for the motion of a single particle in a rotating and vibrating oscillator potential with 1 · s and 12 terms. A consistent solution of this problem gives the well-known rotation-particle coupling term and the following further terms: a vibration-particle, a rotation vibration and a rotation-vibration-particle coupling term. The energy eigenvalues and eigen-functions are obtained by an exact diagonalization of the Hamiltonian using an appropriate core-particle basis. The model yields the two different γ-bands which have been detected recently. For testing the model all available experimental data for Eu153, Dy161 and Re185 are compared with the adiabatic Bohr-Mottelson-Nilsson theory. The agreement is much better with the model here developed than with the adiabatic model.
The rotation vibration model for deformed nuclei is summarized, various matrix elements are given explicitly and multipole operators are evaluated. The rotation vibration Hamiltonian is diagonalized up to spin 20 and presented in a manner which is independent of the individual nucleus. Energies and wave functions for the various states in the ground state, beta and gamma rotational bands of even deformed nuclei are given in an extensive table. The use of the table is explained in detail. An extensive comparison of the rotation vibration model and the asymmetric rotator model with available experimental data indicates an approximate equivalence of the two theories in explaining the three lowest rotational bands. However, the rotation vibration model appears to have a decided advantage in explaining the few two-phonon vibrations experimentally observed. Small systematic deviations between the rotation vibration model and experiment for the energies of the vibrational bands are interpreted in terms of the Coriolis anti-pairing and blocking effects. These effects lead to lower limits on the pairing correlation energy and the critical spin above which all pairing ceases.
Resonance fluorescence from the 343 keV level of Lu175 was studied with the centrifuge method. Assuming a branching ratio Γ0/Γ = 0.883 to the ground state, a mean gamma life of the 343 keV transition was calculated from the resonance scattering measured at different source velocities. The angular distribution of the 343 keV resonance radiation from a Lu2O3 powder scatterer is isotropic. Combining the lifetime of the 343 keV level with the range of the mixing ratio consistent with conversion studies and the angular distribution, one calculates that the M1 transition from the 343 keV level to the ground state is hindered by a factor of 700 over the single particle prediction. This hindrance is a factor of 4 × 103s less than in a similar M1 transition from the 482 keV level to the ground state in Ta181, despite the fact that both γ-rays result from hindered transitions between the same orbitals.
In order to obtain more evidence on the systematic behavior of nuclear energy levels in the deformed region, a number of neutron-deficient odd-mass activities were studied with internal conversion electron spectrographs. Some scintillation counter measurements were made. Level schemes are postulated for decays leading to the following odd-neutron nuclei: Pm151 --> Sm151, Tb151 --> Gd151, Tb153 --> Gd153, Tb155 --> Gd155, Eu157 --> Gd157, Tm163 --> Er163, Tm165 --> Er165, Ho167 --> Er167, Re183 --> W183, Ir185 --> Os185, Ir187 --> Os187, and Ir189 --> Os189. As a corollary, data are presented for the odd-proton nuclei populated in the decays of Pt189 --> Ir189 and Pt191 --> Ir191. The proposed states are described using the Mottelson and Nilsson predictions. Some conclusions may be drawn concerning such properties of these orbitals as moments of inertia, M1E2 mixing ratios, and positions of the intrinsic excitations. Ratios of gamma-ray transition probabilities from the various states are given. Previously published studies of odd-A nuclei in the region of odd-N numbers 99-107 are incorporated in the energy-level systematics.
Sm155 was found to decay to Eu155 with a half-life of 21.9+/-0.2 minutes. Internal conversion electrons corresponding to the gamma rays in Eu155 were observed in a magnetic spectrometer. The gamma ray spectrum was studied with a well crystal. The spectra of gamma rays coincident with the x ray, 104-kev, 142-kev, and 246-kev gamma rays were observed. A number of new, weak transitions are proposed. Directional correlation measurements were made on the 142-kev-104-kev cascade. Possible spin assignments are discussed.
For deformed even-mass nuclei the model of the axial-symmetric vibrating top with rotation-vibration-interaction is compared to the Davydov-model without and with a-band. TheE2-transition-probabilities are calculated. The coefficientsB
ze andC fromE=A I(I+1) –B
ze
I
2(I+1)2+C I
3(I+1)3 are obtained. For the transition-probabilities from the states of the -band to the states of the ground state band as well as forB andC the Davydov-model yields values only about half of those obtained from the RV-theory. A comparision to the present availuable experimental data is made. In the RV-model the agreement is better than in the Davydov-theory.