Narrow Results

We present the case of quantal transmission through a smooth, single-piece exponential potential, V(x) = -V₀e^x/a, in contrast to the piecewise continuous potentials, as an intersecting model to demonstrate the analytic extraction of transmission (reflection) coefficient. Several other aspects of quantal transmission are also discussed in this simple and exactly solvable model.

In the paper the closed Friedmann–Robertson–Walker model with quantization in the presence of the positive cosmological constant and radiation is studied. For analysis of tunneling probability for birth of an asymptotically de Sitter, inflationary universe as a function of the radiation energy, a new definition of a "free" wave propagating inside strong fields is proposed. On such a basis, tunneling boundary condition is corrected, penetrability and reflection relative to the barrier are calculated in fully quantum stationary approach. For the first time nonzero interference between the incident and reflected waves has been taken into account which turns out to play important role inside cosmological potentials and could be explained by non-locality of barriers in quantum mechanics. Inside the whole region of radiation energy the tunneling probability for the birth of the inflationary universe is found to be close to its value obtained in the semiclassical approach. The reflection from the barrier is determined for the first time (which differs essentially on 1 at the energy of radiation close to the barrier height). The proposed method could be easily generalized on the cosmological models with barriers of arbitrary shape, as demonstrated for the FRW model with included Chaplygin gas. The result is stable for variations of the studied barriers, accuracy is found to be 11–18 digits for all coefficients and energies below the barrier height.

Starting from the WKB approximation, a new barrier penetration formula is proposed for potential barriers containing a long-range Coulomb interaction. This formula is especially proper for the barrier penetration with penetration energy much lower than the Coulomb barrier. The penetrabilities calculated from the new formula agree well with the results from the WKB method. As a first attempt, this new formula is used to evaluate α-decay half-lives of atomic nuclei and a good agreement with the experiment is obtained.

We study the half-lives of some nuclei via the alpha-decay process in ground-state to ground-state (g.s.→g.s.) transitions. To go through the problem, we consider the parabola and WKB approximations to fin the penetration probability. This new formula is used to evaluate alpha-decay half-lives and a good agreement with the experiment is obtained.

The $\alpha$-decay half-lives for odd $Z$ Pa isotopes were estimated by using the modified generalized liquid drop model (MGLDM) in the neutron number range $120\le N\le 150$. These half-lives are found to be an excellent match with available experimental data certified by the least root mean-square error when compared with the other recently proposed empirical formulas. The applicability of MGLDM is proved on the odd $Z$-odd $N$ (even $N$) nuclei in a successful manner in describing their $\alpha$-decay half-lives which are employed to predict half-lives of few unknown Pa isotopes. Our results are expected to provide more extensive and precise information of half-lives for the upcoming experiments eyeing on the boundaries of the trans-led nuclei.

The new analytical barrier penetration formula proposed by Li et al. [Int. J. Mod. Phys. E 19 (2010) 359] for potential barriers containing a long-range Coulomb interaction is adopted in the empirical coupled-channel (ECC) model for calculating fusion cross-sections. As compared with the Hill–Wheeler (HW) formula based on the parabolic approximation, this formula is more appropriate for the barrier penetration with incident energies much lower than the Coulomb barrier. The calculated results show that the ECC model with the new barrier penetration formula can describe the fusion cross-section data well, especially for light systems at energies much lower than the Coulomb barrier. Then the systematics of the difference between the ECC calculation with the new penetration formula and that with the HW formula is investigated. The results show that the difference between the results with the HW formula and the new penetration formula is less than one order of magnitude at $E_{\mathrm{\text{c.m.}}}>0.89V_{\mathrm{\text{B}}}$.

The Modified Generalized Liquid Drop Model (MGLDM) has been used for calculating the ground-to-ground states of $\alpha$-decay half-lives for various Ac, Th, Pa, U and Np isotopes in the mass number range of $205\le A\le 245$. To account for the nuclear proximity energy, the potential of Blocki et al.⁷⁴ [J. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. Phys. NY 105 (1977) 427] is used. The experimental $Q_{\alpha }$ values are used to evaluate the $\alpha$-decay half-lives. It can be observed that the logarithm of $\alpha$-decay half-life increases with increasing neutron number ($N$), with evident minimum at the magic number 126, where the neutron shell is closed. The calculated results are compared with available experimental data and with the predictions of different empirical formulae such as, ARF, MARF, Royer, AKRE, Akrawy, NRB, VS, MVS, YQZR and MYQZR. It is found that the MGLDM is able to predict the experimental half-lives with a standard deviation of 0.6892 (the lowest one among all the other applied empirical formulae), and is therefore suitable for predicting the $\alpha$-decay half-lives for unknown and new heavy nuclei.