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Abstract:In this study quasiparticle random-phase approximation with the translational invariant Hamiltonian using deformed mean field potential has been conducted to describe electric dipole excitations in 136 Xe, 138 Ba, 140 Ce, 142 Nd, 144 Sm and 146 Gd isotones. The distribution of the calculated E1 strength shows a resonance like structure at energies between 6-8 MeV exhausting up to 1% of the isovector electric dipole Energy Weighted Sum Rule and in some aspects nicely confirms the experimental data. It has been shown that the main part of E1 strength, observed below the threshold in these nuclei may be interpreted as main fragments of the Pygmy Dipole resonance. The agreement between calculated mean excitation energies as well as summed B(E1) value of the 1 − excitations and the available experimental data is quite good. The calculations indicate the presence of a few prominent positive parity 1 + States in heavy N = 82 isotones in the energy interval 6-8 MeV which shows not all dipole excitations were of electric character in this energy range.

An approach, called fully renormalized QRPA (FR-QRPA), which takes into account the Pauli principle for the ground state correlations (GSC) and fulfills the Ikeda sum rule has recently been developed. Here a new rotational invariant model of the 1+ states is formulated within the FR-QRPA and deformed nuclei 150Nd, 154Sm and 168Er are studied. Two different Hamiltonians are used to describe the low-lying excited states (the famous scissors mode). One Hamiltonian is constructed to be rotational invariant and the spurious rotational state is separated by moving its energy to zero. The second Hamiltonian mixes the spurious rotational state with different vibrational states having the same quantum numbers. The GSC measured by the number of quasiparticles in the ground state are studied for the Hamiltonians within the QRPA, R-QRPA and FR-QRPA approaches. The present investigation demonstrates an advantage of the FR-QRPA over the other approaches. Within the rotational invariant FR-QRPA the GSC are stronger than within the R-QRPA by about 20%. The spurious rotational state in the rotational invariant case contributes within the FR-QRPA more than 50% to the total number of quasiparticles in the ground state of the nuclei in question.

The resolution of the acceleration vector of a particle moving along a space curve is well known thanks to Siacci [1]. This resolution comprises two special oblique components which lie in the osculating plane of the curve. The jerk is the time derivative of acceleration vector. For the jerk vector of the aforementioned particle, a similar resolution is presented as a new contribution to field [2]. It comprises three special oblique components which lie in the osculating and rectifying planes. In this paper, we have studied the Siacci’s resolution of the acceleration vector and aforementioned resolution of the jerk vector for the space curves which are equipped with the modified orthogonal frame. Moreover, we have given some illustrative examples to show how the our theorems work.

Summary The jerk vector of a moving particle is the third time derivative of the position vector and thus the time derivative of the acceleration vector. A useful resolution of the acceleration vector of a particle traveling along a space curve is well known in the literature due to Siacci's study in 1879. A similar resolution of the jerk vector was given as a new contribution to field by Özen and Tosun in 2019. In the present article, we take into account a particle moving on a space curve which is equipped with the Bishop frame and study the aforesaid resolution of the jerk vector for this particle. Furthermore, we have given an illustrative example to explain how the our result works.