Discrete levels

This expression assumes a system with a discrete level structure for systems with both a discrete and a continuous portion to their spectrum the expression consists of a sum over the discrete states and an integral over the continuous states.) Flere, ifi (v) is a solution of the time-independent Sclirodinger equation,... 

Molecules possess discrete levels of rotational and vibrational energy. Transitions between vibrational levels occur by absorption of photons with frequencies v in the infrared range (wavelength 1-1000 p,m, wavenumbers 10,000-10 cm , energy differences 1240-1.24 meV). The C-0 stretch vibration, for example, is at 2143 cm . For small deviations of the atoms in a vibrating diatomic molecule from their equilibrium positions, the potential energy V(r) can be approximated by that of the harmonic oscillator ... 

Molecules possess discrete levels of rotational and vibrational energy, and transitions between vibrational levels occur by absorption of photons with frequencies in the mid-infrared range. There are four types of vibration ... 

Figure 14-1. Left Relative errors (RELE) in the force as a function of radial distance from the center of the active dynamical region for the VEP-RVM charge-scaling method [80] for the solvated hammerhead ribozyme at different discretization levels [151] of die co surface. Right The projected total electrostatic potential due to the fully solvated hammerhead ribozyme projected onto die YEP surface [80]...

The situation is illustrated in Figure 46-13. The actual voltage is a continuous, linear physical phenomenon. The values represented by the output of the A/D converter, however, can only take discrete levels, as illustrated. The double-headed arrows represent the error introduced by digitizing the continuous physical voltage at various points. The error cannot be greater than 1/2 the difference between representing adjacent levels of the converter if the voltage increases beyond 1/2 the difference between levels, then the conversion will provide the next step s representation of the value. 

Figure 46-13 The actual voltage is a continuous, linear function. The values represented by the output of the A/D converter, however, can only take discrete levels. The double-headed arrows represent the error introduced by digitizing the continuous physical voltage at various points.

Safety integrity level (SIF) Discrete level (one out of a possible four SIL categories) used to specify the probability that a safety instrumented function will perform its required function under all operational states within a specified time. 

Qualitative factors are also frequently considered in a robustness test. " For CE methods, factors such as the batch or manufacturer of the capillary, reagent or solvent can be selected. When evaluating the influence of such qualitative factor, the analyst should be aware that the estimated effect is only valid or representative for the examined discrete levels and not for any other level of that factor, and certainly not for the whole population. For example, when examining two capillaries X and Y, the estimated effect only allows drawing conclusions about these two capillaries and not about other capillaries available on the market. Such approach allows evaluating whether capillary Y is an alternative for capillary X, used, for instance, to develop the method. 

The randomized complete block design has provided a sensitive way of viewing the data from this set of experiments involving two qualitative factors. The linear model using dummy variables ignores much of the variation in the data by again focusing on pairwise differences associated with the different discrete levels of the factors of interest. 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

The second and perhaps most probable explanation is damping and broadening of the resonance, due to size dependent, single electron 5d- 6p,6s interband transitions. Their explanation is that the discrete level structure of the Au 55 cluster acts as an effective decay channel. In reducing the plasmon lifetime, it would also strongly increase the bandwidth of the resonance, washing out the resonance peak. 

In the rough experimental space the distance between discrete levels of the experimental variables is relatively large. After testing three - four catalyst generations different Data Processing methods, such as general statistical approaches or Artificial Neural Networks (ANNs), can be applied to determine the contribution of each variable into the overall performance or establish the Activity - Composition Relationship (ACR). 

Small metal particles reveal a not fully developed valence band (they have a system of discrete levels rather than a quasi-continuous metallic-like band), which effect influences the binding energy as determined by XPS and might be, in principle, important also for chemisorption and catalysis (99, 100). 

Let us consider a solid that contains only one type of electron trap of volume density N at the discrete level ( and a set of occupied deeper electron traps of density M. The experiment is performed in a temperature range in which traps N empty but traps M remain thermally disconnected and act only as an untapped reservoir of trapped electrons. The density of recombination centers is unspecified, but a density/of them is empty. AtT = Tq, a concentration of /o empty recombination centers exists due to excitation. At these conditions, charge neutrality of the sample is of the form... 

Each component may be studied at c different chemical potentials or concentrations. If C, represents the discrete levels of concentrations at which the component is studied in the formulation, then... 

In most semiconductors, there are, in addition to the allowed energy levels for electrons in the conduction and filled bands of the ideal crystal, discrete levels with energies in the forbidden gap which correspond to electrons localized at impurity atoms or imperfections. In zinc oxide, such levels arise when there are excess zinc atoms located interstitially in the lattice. At very low temperatures the interstitial zinc is in the form of neutral atoms. However, the ionization energy of the interstitial atoms in the crystal is small and at room temperature most are singly ionized, their electrons being thermally excited into the conduction band. These electrons give rise to the observed A-type conductivity. 

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