Eikonal theory

For the Eikonal theory to be valid, the distortions must be suffieiently small that the eoneept of a ray is retained. The criterion is that the radius of curvature of the reflecting planes does not exceed a critical value R. approximately equivalent to an angular rotation of the Bragg planes by half the reflecting curve width in an extinction distance. The critical radius of curvature is thus... 

Byron Jr., F.W. and Joachain, C.J. (1977b). Eikonal theory of electron- and positron-atom collisions. Phys. Rep. 34 233-324. 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

The new fundamental equations of the dynamical theory were written within the Eikonal representation. Such equations are valid for any kind and strength of a regular deformation and, in opposite to the THW equations, they describe only interbranch transitions of electrons. 

In the present section we present a theoretical description for the continuum-distorted-wave eikonal-initial-state (CDW-EIS) model. This model is one of the most advanced and complete perturbative theories of heavy particle collisions which has been formulated to date. The reasons for the success of this model particularly in describing ionization at high energies in the MeV/amu range are that ... 

The characteristic function W plays the same role as the eikonal cf> and a/2m(E — V) serves as the index of refraction. It becomes clear why Huygens wave theory and Newton s formulation were able to account equally well for the phenomena of reflection and refraction. 

Quantum theory was developed primarily to find an explanation for the stability of atomic matter, specifically the planetary model of the hydrogen atom. In the Schrodinger formulation the correct equation was obtained by recognizing the wave-like properties of an electron. The first derivation by Schrodinger [30] was done by analogy with the relationship that was known to exist between wave optics and geometrical optics in the limit where the index of refraction, n does not change appreciably over distances of order A. This condition leads to the eikonal equation (T3.15)... 

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