Experimental L-shell x-ray production and ionization cross sections for proton impact

We present methods for the reduction of the complexity of computational problems, both time-dependent and stationary, together with connections to renormalization, scaling, and irreversible statistical mechanics. Most of the methods have been presented before; what is new here is the common framework which relates the several constructions to each other and to methods of theoretical physics, as well as the analysis of the approximate reductions for time-dependent problems. The key conclusions are: (i) in time dependent problems, it is not in general legitimate to average equations without taking into account memory effects and noise; (ii) mathematical tools developed in physics for carrying out renormalization group transformations yield effective block Monte Carlo methods; (iii) the Mori–Zwanzig formalism, which in principle yields exact reduction methods but is often hard to use, can be tamed by approximation; and (iv) more generally, problem reduction is a search for hidden similarities.

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Problem reduction, renormalization, and memory

Publisher Summary This chapter discusses some outcomes from molecular dynamics simulations of the glass transition and of the annealing processes with a view to clarifying the nature of amorphous structures. Some of the properties have been found from the macroscopic and microscopic analyses of the atomic configurations. Melting on heating, crystallization on slow cooling, and a transformation to a glass on rapid cooling are observed in a system composed of Lennard–Jones (LJ) atoms. Relaxation processes in a supercooled liquid have been compared with those in a glass by laying special emphasis on the relationship between fluctuations and relaxations in disordered systems. The mechanism of the glass transition is essentially of a kinetic nature, which is reflected in most of the physical properties. Atoms in an amorphous system are classified according to the local symmetry around each atom. For instance, atoms with icosahedral symmetry are typical of amorphous structures. The chapter discusses that the validity of computer simulations is reinforced by minimizing the effects because of finite sizes and to the periodic boundary condition. The scope of the molecular dynamics methods is widened by extending the classes of subjects to be studied.

Glass Transition and Relaxation of Disordered Structures

The "t-model" for dimensional reduction is applied to the estimation of the rate of decay of solutions of the Burgers equation and of the Euler equations in two and three space dimensions. The model was first derived in a statistical mechanics context, but here we analyze it purely as a numerical tool and prove its convergence. In the Burgers case, the model captures the rate of decay exactly, as was previously shown. For the Euler equations in two space dimensions, the model preserves energy as it should. In three dimensions, we find a power law decay in time and observe a temporal intermittency.

Optimal prediction and the rate of decay for solutions of the Euler equations in two and three dimensions.

The average values of the powers of $r$ for the $4f$ orbital, ${〈{r}^{n}〉}_{4f}$, were calculated using accurate analytical self-consistent-field wave functions. The results were compared with experimental and other theoretical values.

Expectation Values of 〈 r n 〉 4 f for Rare-Earth Ions

The need for an accurate description of the electronic structure of the important laseractive ion ${\mathrm{Nd}}^{3+}$ stimulated this work. We calculated self-consistent-field wave functions by the analytical-expansion method for the ground state of the laser-active ion ${\mathrm{Nd}}^{3+}(Z=60)$, $4{f}^{3}$, $^{4}I$. The results represent the most accurate wave functions for ${\mathrm{Nd}}^{3+}$ published. All the exponents of the basis functions in the analytical expansion were exhaustively optimized. The virial theorem was satisfied to seven significant figures. Values of $〈{\mathcal{r}}^{n}〉$ for the $4f$ orbital are included.

M. Synek

Papers

Accurate Analytical Self-Consistent-Field Wave Functions for Nd 3 +