Beyond Euler angles: exploiting the angle-axis parametrization in a multipole expansion of the rotation operator

Euler angles (alpha,beta,gamma) are cumbersome from a computational point of view, and their link to experimental parameters is oblique. The angle-axis {Phi, n} parametrization, especially in the form of quaternions (or Euler-Rodrigues parameters), has served as the most promising alternative, and they have enjoyed considerable success in rf pulse design and optimization. We focus on the benefits of angle-axis parameters by considering a multipole operator expansion of the rotation operator D(Phi, n), and a Clebsch-Gordan expansion of the rotation matrices D(MM')(J)(Phi, n). Each of the coefficients in the Clebsch-Gordan expansion is proportional to the product of a spherical harmonic of the vector n specifying the axis of rotation, Y(lambdamu)(n), with a fixed function of the rotation angle Phi, a Gegenbauer polynomial C(2J-lambda)(lambda+1)(cosPhi/2). Several application examples demonstrate that this Clebsch-Gordan expansion gives easy and direct access to many of the parameters of experimental interest, including coherence order changes (isolated in the Clebsch-Gordan coefficients), and rotation angle (isolated in the Gegenbauer polynomials).     

Eine Anzahl von Verbesserungsvorschl?gen an Gasdruckregulatoren

Ohne Zusammenfassung     

Alkohol-Messapparat

Ohne Zusammenfassung     

Elektrisehe Messungen

Ohne Zusammenfassung     

Reaction mechanism and rheological properties of polypropylene irradiated under various atmospheres

It is well-known that the melt-strength properties of a polymer increases with molecular weight and with long chain branching due to the increase in the entanglement level. This study is a contribution for the understanding of the following points: — the role of branching, crosslinking and degradation on melt strength properties; — the mechanism and the kinetics of PP irradiation with time of irradiation and the importance of double bond formation.

The results showed that degradation was the major reaction in the initial step of irradiation no matter the atmosphere and or antioxidant. However, double bond formation increased the production of branching and crosslinking reactions. Double bond formation had no effect on the crystallization kinetics, on the other hand, long chain branching had a marked effect on the crystallization temperature.     

Forces between heavy nuclei in the independent-particle model

We use the linear response theory to study the force on a heavy nucleus moving near the symmetry axis of ²³⁸U, in an independent-particle model. We present the first detailed microscopic computation showing relaxation of the single-particle motion due to incoherence, and find a relaxation time of about $\text{0.3}\text{MeV}\text{≈ 2 × 10}{}^{\mathrm{?22}}\text{sec}$. We consider three stages of the reaction. While the nuclei approach each other, the induced force is very weak, and is computed as a function of the position, velocity and acceleration of the other nucleus; it involves excitons of 10 to 20 MeV. The computed force agrees well with phenomenological values. The second stage of the reaction is the formation of a neck, which we argue will begin already when the nuclear half-density surfaces are more than 4 fm from each other. For slow collective motion, thought to occur after the neck forms, we show how to represent the incoherent excitation of low-frequency excitons by a friction force, and present a method suitable for numerical computations of the friction coefficient.