Non-adiabatic transitions for random processes in a two-level system

Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ?ω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω _e exp (- αx) + ω₀) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω₁τ _c ? 1 where τ _c is the correlation time and ω₁ is the off-diagonal matrix element.     

On the kinetics of a two-level system with a diffusion-modulated interaction

For a two-level system corresponding to a particle of spin ½ in a random field in the Z direction, the relaxation function

has been estimated, the magnitude Ω(t) being the sum of the isotropic interactions of the particle in question with particles j executing diffusional motion. Specifically Ω(t)=ω₀ + Σ ω(Mj, rj), where ω₀ = constant, M_j is a random time-independent parameter, ω(M, r) decreases with r faster than r ^-3 and r _j = r _j (t) is a diffusional, random function. From the expression for <σ⁺(t)>, we establish general features of the relaxation phenomenon for diffusional processes, and calculate the relaxation rate 1/T ₂ and relaxation shift Δω to be 1/T ₂-iΔω = 4π CAv_M λ _M , where C is the concentration of particles and λ _M is the scattering length for an equation of the Schrödinger type with an imaginary potential -iω(M, r) instead of U/?, and diffusion coefficient instead of ?/2m. We also found that for the case of ‘external’ relaxation, the Redfield approach proved valid only under the simultaneous restrictions of low concentration and weak interaction.     

An Elementary Method on a Minimization Problem

一、ＴｈｅＰｒｏｂｌｅｍ .Ｌｅｔｆ(ｘ) =ａｘ2 ｂｘ ｃ ａｘ2 ｐｘ ｑ,ｗｈｅｒｅａ >0 ,ｂ2 - 4ａｃ≤ 0ａｎｄｐ2 - 4ａｑ≤ 0 .ＯｕｒＰｒｏｂｌｅｍｉｓ“Ｗｈｅｔｈｅｒｄｏｅｓｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓ?Ｍｏｒｅｏｖｅｒ ,,ｉｆｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓｔｓ,ｔｈｅｎｍｉｎｘ∈Ｒｆ(ｘ)=?ａｎｄｉｎｔｈｉｓｃａｓｅｘ =?” .Ｎａｔｕｒａｌｌｙ ,ｗｅｋｎｏｗｔｈａｔｍｉｎｘ∈Ｒｆ(ｘ)ｅｘｉｓｔｓｆｒｏｍｔｈｅｋｎｏｗｌｅｄｇｅｏｆｍａｔｈｅｍａｔｉｃａｌａｎａｌｙｓｉｓ.Ａｌｓｏ,ｗｅｃａｎｇｉｖｅｔｈｅ…     

The Synthesis and Properties of New Metal-free and Metallophthalocyanines Containing Four Diloop Macrocyclic Moieties

The synthesis and characterization of new metal-free (9) and metal-containing (Zn, Ni or Cu 10, 11, 12) derivatives of a symmetrically octasubstituted phthalocyanine derived from 21,22-dicyano-2,3,5,6,8,9,11,12,15,17,18,25,26,28-tetradecahydro[1,4,7,12] benzodioxadithiacyclotetradeceno[6,7-b][1,4,7,10,13]benzopentaoxacyclopentadecene (7), which was synthesized in a multi-step reaction sequence, have been described. The novel compouds have been characterized by a combination of elemental analysis, ¹H and ¹³C NMR, IR, UV–vis and MS spectral data.