Continuous energy spectrum

An analysis of the rate of release of adsorbed atoms from sites with a continuous energy spectrum for the case of an arbitrary distribution function of initial site populations was given by Carter (32). The rate equation for the t th desorption process with x = 1 and negligible readsorption is... 

For a continuous energy spectrum the sums may be replaced by the integrals... 

The typical way to open a billiard is to attach some reservoirs with continuous energy spectrum, for example, the leads or microwave waveguides, as shown in fig. 3 below. Full information about the scattering properties of the billiard is given by the scattering wave function which is a solution of the Schrodinger equation Hip = Exp with the total Hamiltonian... 

For X-rays from a continuous energy spectrum, p E), passing through a series of materials indexed by i, the formula for attenuation generahzes to... 

There are several complementary aspects to the study of metastable states. First, we seek to show that the distribution function for the energy in a metastable state (recall that this is not a stationary state, hence by the uncertainty principle AE 0) is directly related to the functional form of the rate of decay of that state.17 Suppose that the total Hamiltonian of the system has a continuous energy spectrum. Let this Hamiltonian be... 

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. 

The functions —Q2(0 and —Q2(rj) are qualitatively depicted in Figs. 5.1a-c and Figs. 5.2a-c for the cases m/0 and m = 0, respectively. The physically acceptable wave functions with the -variable correspond to discrete energy eigenvalues, while with the -variable there is associated a continuous energy spectrum, unless one encloses the r -variable in a finite region, as we have done in Chapter 2. 

X-ray machine X-rays 0.05-0.3 Pulsed beam unless constant potential power supply used, continuous energy spectrum. 

In a model with continuous energy spectrum, such as the Heisenberg model, the energy range is divided into discrete bins of width AE and the histograms are constructed for these bins. 

When one solves for the possible energies for the particles in a potential U, the result is that you will have a continuous energy spectrum of free particles above the potential, and a discrete set of energy levels in the potential, see Fig.(1.1). The discrete set is called bound states and represent the energies of the particles, usually electrons, that are bound by the potential. This does not mean that the particles do not move, of course, only that they have a fix energy. 

Beta particles have a continuous energy spectrum extending from zero energy up to maximum kinetic energy (see Sec. 3.7.3). If the transmission experiment shown in Fig. 4.5 is repeated with an incident beam of jS particles, the result will look as shown in Fig. 4.11. The number of betas N(t) transmitted through a thickness t is very closely represented by... 

In a series of papers since 1993-1994, we have demonstrated that it is possible to solve quantitatively a variety of TDMEPs in atoms and small molecules, by expanding the nonstationary in terms of the state-specific wavefunc-tions for the discrete and the continuous energy spectrum of the unperturbed system. This SSEA to the solution of the TDSE bypasses the serious, and at present insurmountable, difficulties that the extensively used "grid" methods have, when it comes to solving problems with arbitrary polyelectronic, ground or excited states. Furthermore, it allows, in a transparent and systematic way, the monitoring and control of the dependence of the final resulfs on the type and number of fhe sfafionary states that enter into the expansion that defines fhe wavepackef 4>(f). 

There is a sharp drop at the end of the continuous energy spectrum of the Compton electrons, the Compton edge, distinctly below the full energy of the original photon. The electron energy reaches its maximum when the photon is totally back scattered 9 - 180°). The energy of such backscattered photons - Ey — is therefore at minimum, but for... 

Both the step function and the continuous function in O Fig. 47.2 had been introduced in the former recommendation (in ICRP 1991). The recently recommended continuous function of radiation weighting factors for neutrons reflects only practical considerations and does not imply the availability of more precise data. Mainly due to the continuous energy spectrum of the neutrons, in calculations the continuous function of the Wr has been used rather than the step function. 

Reactor Neutrons. Nuclear reactors are the largest and most often used neutron sources. Neutrons formed in the reactor have a continuous energy spectrum extending from nearly 0 to 15MeV. Conventionally the neutron spectrum is divided in three components ... 

In the next section, techniques are discussed to compute the bond energy in case one of the fragments has a continuous energy spectrum, as for a metal surface. 

Here p(e) is the density of states. This expression immediately suggests a deeper connection between the pseudostates and the continuum. We have replaced an integration over the continuous energy spectrum by some sort of gaussian integration rule. But what are the points and what are the weights ... 

Prompt fission neutrons are emitted with the continuous energy spectrum shown in Figure 2.7. The spectrum spans an energy range of approximately 10 MeV, from less than 1 10 MeV. The average prompt neutron energy is about 2 MeV. 

We note that the ejected K-shell electrons have a continuous energy spectrum and therefore provide a poor signature for ionization a good signature is provided by the X-ray radiation or by one of the Auger electron lines which follow the ionization. (Since the X-ray is emitted by an electron whose initial state is any of a number of p-states ---- primarily the 2p state ---- with equally populated... 

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