Plane-wave representation

PL for a beam consisting of photons, each having the energy hv. In this regime we are thus back to the plane-wave representation, here applying to the collective behavior of photons in a beam. Plane EMS waves could also be involved under these conditions, such as in the special process of total reflection described in Section VLB. 

We have argued here and elsewhere [44] that the plane-wave representation of classical electromagnetism is far from complete. In tensor language, this incompleteness means that the antisymmetric electromagnetic field tensor on the 0(3) level must be proportional to an antisymmetric frame tensor of spacetime, 7 , derived from the Riemannian tensor by contraction on two indices ... 

Using a plane wave representation for the electron wavefunction with 163 grid points and approximately 800 independent electronic and molecular configurations from the path integral molecular dynamics trajectories, we have also computed the density of states for the electron under different supercritical conditions of the solvents and the corresponding steady-state optical absorption spectra. The latter were computed within the dipolar approximation from the following expression within the Frank-Condon approximation ... 

Some techniques also involve a well-defined incident electron beam, even though the primary process at some point imparts an arbitrary parallel momentum to the electrons. This happens, for example, with energy loss in HREELS and ILEED, with diffuse scattering in LEED and with Auger emission in ARAES. In these cases the direction of the electrons leaving the surface has an arbitrary relationship to the incident beam direction. Up to the primary process, however, conventional LEED can be applied in the plane-wave representation, at least in the ordered part of the surface, using the finite set of plane waves defined by the direction of incidence. 

Another development of the situation where the plane-wave representation is adequate is the physically-obvious fact that the final state is the time reversal of the initial state. It is necessary to define the T-matrix elements by... 

To carry out the sums appearing in Eqs. (57-60), we use the plane-wave representation (43) of the vacuum modes and work in the spherical representation of the unit orthogonal polarization vectors eki and ek2- Substituting Eq. (43) into Eqs. (57-60), and evaluating the sums over k and s, we obtain... 

With plane-wave representation of wavefuntions the method for calculation of total energy has been described by Wendel and Martin (1978, 1979). Explicit expressions covering also nonlocal (angular momentum dependent) pseudopotentials and forces were given by Ihm, Zunger and Cohen (1979)- The method is described by Martin (this volume). 

Eq. (3.2.1.61) shows that for the selfconsistent calculation of the full interlayer multiple diffraction, we have again to invert a matrix, now in plane-wave representation of the wave field. The dimension of the matrix is correlated to the number of visible beams, which is determined by the request that — (iko — g ) is... 

Apparently, by the method described, the reflection and transmission matrices of a double layer have resulted from those of the single layers. The latter even do not need to be the same as assumed in our formulas for the sake of simpHdty. Of course, one can repeat the procedure using the new matrices as input to calculate the matrices of a quadruple layer and so on. As the total number of layers considered in this way grows as 2 , with n being the number of doubling steps, this layer doubling method (LD) quickly reaches the finite depth probed by the electrons. Yet, for very small interlayer spacings (below about 0.7 A), the method fails, as the plane-wave representation of the wave field between layers is not adequate anymore. 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

Rgure 3 Experimental and calculated results (a) for epitaxial Cu on Ni (001). The solid lines represent experimental data at the Cu coverage indicated and the dashed lines represent single-scattering cluster calculations assuming a plane wave final state for the Cu IMM Auger electron A schematic representation lb) of the Ni (010) plane with 1-5 monolayers of Cu on top. The arrows indicate directions in which forward scattering events should produce diffraction peaks in (a). 

Figure 12.2 Sine wave representation of electromagnetic radiation. It consists of two in-phase waves, with oscillation of the electric field in the xy plane, and the magnetic field perpendicular to it, in the vz plane.

Figure 5.37 Sine-wave representation of stress variation with interatomic separation distance for two atomic planes. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

The boundary conditions can also have a decisive influence on the type of representation. A first example is given by the transmitted wave at a vacuum boundary, as discussed in Section VI.B. Here the incident and reflected plane waves can be matched at the interface by a plane transmitted wave, but hardly by a transmitted beam of axisymmetric photon wavepackets. 

For the purposes of this review it is convenient to focus attention on that class of molecules in which the valence electrons are easily distinguished from the core electrons (e.g., -n electron systems) and which have a large number of vibrational degrees of freedom. There have been several studies of the photoionization of aromatic molecules.206-209 In the earliest calculations either a free electron model, or a molecule-centered expansion in plane waves, or coulomb functions, has been used. Only the recent calculation by Johnson and Rice210 explicitly considered the interference effects which must accompany any process in a system with interatomic spacings and electron wavelength of comparable magnitude. The importance of atomic interference effects in the representation of molecular continuum states has been emphasized by Cohen and Fano,211 but, as far as we know, only the Johnson-Rice calculation incorporates this phenomenon in a detailed analysis. 

It is concluded that the B(3) component in the field interpretation is nonzero in the light-like condition and in the rest frame. The B cyclic theorem is a Lorentz-invariant, and the product B x B<2> is an experimental observable [44], In this representation, B(3> is a phaseless and fundamental field spin, an intrinsic property of the field in the same way that J(3) is an intrinsic property of the photon. It is incorrect to infer from the Lie algebra (796) that Ii(3) must be zero for plane waves. For the latter, we have the particular choice (803) and the algebra (796) reduces to... 

Nano-scale and molecular-scale systems are naturally described by discrete-level models, for example eigenstates of quantum dots, molecular orbitals, or atomic orbitals. But the leads are very large (infinite) and have a continuous energy spectrum. To include the lead effects systematically, it is reasonable to start from the discrete-level representation for the whole system. It can be made by the tight-binding (TB) model, which was proposed to describe quantum systems in which the localized electronic states play an essential role, it is widely used as an alternative to the plane wave description of electrons in solids, and also as a method to calculate the electronic structure of molecules in quantum chemistry. 

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