Spectrum continuum

Application of the exact continuum analysis of dispersion forces requires significant calculations and the knowledge of the frequency spectmm of the material dielectric response over wavelengths X = 2irc/j/ around 10-10 nm. Because of these complications, it is common to assume that a primary absorption peak at one frequency in the ultraviolet, j/uv. dominates the dielectric spectrum of most materials. This leads to an expression for the dielectric response... 

Emission spectrum from a typical continuum source. 

Figure 9.18 shows a typical energy level diagram of a dye molecule including the lowest electronic states Sq, and S2 in the singlet manifold and and T2 in the triplet manifold. Associated with each of these states are vibrational and rotational sub-levels broadened to such an extent in the liquid that they form a continuum. As a result the absorption spectrum, such as that in Figure 9.17, is typical of a liquid phase spectrum showing almost no structure within the band system. 

In an electron-excited X-ray spectrum the discrete X-ray lines are superimposed on a continuous background this is the well-known bremsstrahlung continuum ranging from 0 to the primary energy Eq of the electrons. The reason for this continuum is that because of the fundamental laws of electrodynamics, electrons emit X-rays when they are decelerated in the Coulomb field of an atom. As a result the upper energy limit of X-ray quanta is identical with the primary electron energy. 

Kaplunovsky and Weinstein [kaplu85j develop a field-theoretic formalism that treats the topology and dimension of the spacetime continuum as dynamically generated variables. Dimensionality is introduced out of the characteristic behavior of the energy spectrum of a system of a large number of coupled oscillators. 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

Each spectrum is a continuum that extends over a wide range of wavelengths. The curves are left unfinished to call attention to the difficulty of working with long-wavelength x-rays. 

Plastomers represent a major advancement for polyolefins. Their success allows polyolefins to have a continuum of products from amorphous EPR to thermoplastic PE and iPP. This development coincides with the advent of single-site catalysts these are necessary for copolymers of components of widely different reactivity such as ethylene and octene. Their rapid introduction into the mainstream polymer use indicates that this spectrum of properties and the inherent economy, stability and processibility of polyolefins are finding new applications to enter. 

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