Integral Equations for Heat Kernel in Compound Media

By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator $-a^2\Delta$ in the case of compound media. In each of the media the parameter $a^2$ acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the `temperature' and the `heat flows'. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions.Comment: 26 pages, no figures, no tables, REVTeX4; two items are added into the Reference List; a new section is added, a version that will be published in J. Math. Phy

Decay properties of neutron-deficient isotopes 256,257Db^{256, 257}Db, 255Rf^{255}Rf, 252,253Lr^{252, 253}Lr

Isotopes of dubnium (element 105) with mass numbers $A = 256$, 257, and 258 were produced by the reaction $^{209}$Bi($^{50}$Ti,xn)$^{259-{x}}$Db ($x=1,2,3$) at projectile energies of (4.59-5.08) AMeV. Excitation functions were measured for the 1n, 2n and 3n evaporation channels. The same position of the excitation function was observed for the 1n channel as for the previously measured 1n channel of the reaction $^{208}$Pb($^{50}$Ti,1n)$^{257}$Rf. The measured $\alpha$-decay data of $^{257}$Db and its daughter products resulted in the identification of $\alpha$-decaying isomeric states in $^{257}$Db and $^{253}$Lr. Two new isotopes, $^{256}$Db and $^{252}$Lr, were produced at the highest bombarding energies of 4.97 AMeV and 5.08 AMeV. They were identified by delayed $\alpha$-$\alpha$ coincidences. The measured half-lives are $(1.6^{+0.5}_{-0.3})$ s for $^{256}$Db and $(0.36^{+0.11}_{-0.07})$ s for $^{252}$Lr. Besides $\alpha$-decay, a spontaneous fission activity of $T_{1/2}=(2.3^{+1.1}_{-0.6})$ s was observed and attributed to an electron-capture branch of $^{256}$Db, which feeds the fissioning nucleus $^{256}$Rf. A branching ratio of $0.36 \pm 0.12$ was obtained. The isotope $^{255}$Rf was produced by the reaction $^{207}$Pb($^{50}$Ti,2n)$^{255}$Rf. Improved decay data have been obtained by means of $\alpha$- and $\alpha$-$\gamma$ spectroscopy