Periodic systems in time: double-well potential

Time analysis of oscillations of a particle between wells in the one-dimensional double-well potential with infinite high outside walls, based on wave packet use and energy spectrum analysis, is presented. For the double-well potential of the form $x^{2}+1/x^{2}$ in the external regions, an exact analytical solution of the energy spectrum is found (by standard QM approach), an analysis of oscillation periodicity is fulfilled, an approach for exact analitical calculation of the oscillation period is proposed (for the first time for the double-well potential).Comment: 11 pages, no figures. LaTeX 2e, class `elsart' (Elsevier Science) is used. Reference correctio

Model of the bremsstrahlung emission accompanying interactions between protons and nuclei from low up to intermediate energies: role of magnetic emission

A new model of the bremsstrahlung emission which accompanies proton decay and collisions of protons off nuclei in the energy region from the lowest up to intermediate, has been developed. This model includes spin formalism, potential approach for description of interaction between protons and nuclei, and operator of emission includes component of the magnetic emission (defined on the basis of Pauli equation). In the problem of the bremsstrahlung during the proton decay in the first time a role of the magnetic emission is studied using such a model. For the studied $^{146}{\rm Tm}$ nucleus it has been studied the following: (1) How much does the magnetic emission change the full bremsstrahlung spectrum? (2) At which angle is the magnetic emission the most intensive relatively electric one? (3) Is there some space region where the magnetic emission increases strongly relatively electric one? (4) How intensive is the magnetic emission in the tunneling region? (5) Which values has the probability at its maximum and at zero energy limit of the emitted photons? It is demonstrated that the model is able to describe enough well experimental data of the bremsstrahlung emission which accompanies collisions of protons off the $^{9}{\rm Be}$, $^{12}{\rm C}$ and $^{208}{\rm Pb}$ nuclei at the incident energy $T_{\rm lab}=140$ MeV (at the photon energy up to 120 MeV).Comment: 25 pages, 23 figures. The following changes are included: (1) comments on why model does not describe well experimental data for 64-Cu (than data for 12-C, 107-Ag obtained by the same group of experimentalists) at 72 MeV; (2) analysis of multipolar components, their role in angular analysis; (3) minor corrections of definitions of photons quantum numbers and bremsstrahlung probabilit

Angular bremsstrahlung during α\alpha-decay, emission from the internal region and unified formula of the bremsstrahlung probability

Model of angular bremsstrahlung of photons emitted during $\alpha$-decay is presented. A special emphasis is given on development of unified formalism of matrix elements in the dipole and multipolar approaches. A probability of the emission of photons calculated on the basis of the multipole model without any normalization on experimental data (i. e. in absolute scale) is found at $90^{\circ}$ of the angle $\vartheta_{\alpha\gamma}$ between directions of motion of the $\alpha$-particle (with its tunneling under barrier) and emission of photons to be in a good agreement with the newest experimental data for the $^{210}{\rm Po}$, $^{214}{\rm Po}$, and $^{226}{\rm Ra}$ nuclei. The spectrum for $^{244}{\rm Cm}$ is found at $\vartheta_{\alpha\gamma} = 25^{\circ}$ to be in satisfactory agreement with high limit of errors of experimental data of Japanese group. A comparative analysis for the spectra calculated for $^{210}{\rm Po}$ by the multipole and dipole approaches in the absolute scale and with normalization on experimental data is performed. The emission of photons from the internal well in the dipole approach is found to be not small while the multipolar approach does not show such a strong dependence. Distribution of the bremsstrahlung probability on the numbers of protons and nucleons of the $\alpha$-decaying nucleus in selected region close to $^{210}{\rm Po}$ is obtained. An unified formula of the bremsstrahlung probability during the $\alpha$-decay of arbitrary nucleus expressed directly through the $Q_{\alpha}$-value and numbers $A_{p}$, $Z_{p}$ of nucleons and protons of this nucleus is proposed.Comment: 23 pages, 9 figures, 2 tabl

Calculations of widths in the problem of decay by proton emission

We develop a new fully quantum method for determination of widths for nuclear decay by proton emission where multiple internal reflections of wave packet describing tunneling process inside proton--nucleus radial barrier are taken into account. Exact solutions for amplitudes of wave function, penetrability $T$ and reflection $R$ are found for $n$-step barrier (at arbitrary $n$) which approximates the realistic barrier. In contrast to semiclassical approach and two-potential approach, we establish by this method essential dependence of the penetrability on the starting point $R_{\rm form}$ in the internal well where proton starts to move outside (for example, for $^{157}_{73}{\rm Ta}$ the penetrability is changed up to 200 times; accuracy is $|T+R-1| < 1.5 \cdot 10^{-15}$). We impose a new condition: in the beginning of the proton decay the proton starts to move outside from minimum of the well. Such a condition provides minimal calculated half-life and gives stable basis for predictions. However, the half-lives calculated by such an approach turn out to be a little closer to experimental data in comparison with the semiclassical half-lives. Estimated influence of the external barrier region is up to 1.5 times for changed penetrability.Comment: 14 pages, 4 figures eps; content is change