Energy level model

Scientists have known that nuclides which have certain "magic numbers" of protons and neutrons are especially stable. Nuclides with a number of protons or a number of neutrons or a sum of the two equal to 2, 8, 20, 28, 50, 82 or 126 have unusual stability. Examples of this are He, gO, 2oCa, Sr, and 2gfPb. This suggests a shell (energy level) model for the nucleus similar to the shell model of electron configurations. 

Fig. 10 Electrochemical energy level model for orbital mediated tunneling. Ap and Ac are the gas-and crystalline-phase electron affinities, 1/2(SCE) is the electrochemical potential referenced to the saturated calomel electrode, and provides the solution-phase electron affinity. Ev, is the Fermi level of the substrate (Au here). The corresponding positions in the OMT spectrum are shown by Ar and A0 and correspond to the electron affinity and ionization potential of the adsorbate film modified by interaction with the supporting metal, At. The spectrum is that of nickel(II) tetraphenyl-porphyrin on Au (111). (Reprinted with permission from [26])...

Figure 1. Schematic energy-level model for a single species in two valence states. (Reproduced with permission from Ref. 11. Copyright 1988 Optical Society of America.)...

Fig. 6.10. Energy level model used in the thermodynamic calculations described in the text, showing donor, defect and band tail states. Conduction band...

Fig. 25. Energy-level model for the chemical etching of silicon based on mixed-potential theory.

An energy-level model for this mechanism, proposed hy Gerischer et al. [123], is shown in Fig. 30. The energy levels associated with Si - H2 groups at kink sites are assumed to be located just above the valence hand edge and hence sites for hole capture. A hole trapped at the kink site oxidizes one of the Si - H groups to release a... 

The Fluctuating-Energy-Level Model Proposed by Gerischer... 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

II. Discrete Energy-Level Model for Unimolecular Reactions. . 367... 

II. DISCRETE ENERGY-LEVEL MODEL FOR UNIMOLECULAR REACTIONS... 

The steepest descent method is very effective far from the minimum of , but is always much less efficient than the Gauss-Newton method near the minimum of . Marquardt (1963) has proposed a hybrid method that combines the advantages of both Gauss-Newton and steepest descent methods. Mar-quardt s method, combined with the Hellmann-Feynman pseudolinearization of the Hamiltonian energy level model, is the method of choice for most nonlinear molecular spectroscopic problems. 

Table VI. Transition Probabilities for tbe Evenly Spaced Energy Levels Models, (3/tv = 2.0000 and pAv = 3.0000 Cases... 

Table VII. Transition Probabilities for the Evenly Spaced Energy Levels Models, pAv = 5.0000 and p/ v = 6.0000... 

Fig. 2.19 Energy level model for electron transfer from a redox species in the electrolyte to a photogenerated hole in the semiconductor valence band. The density of states Z red depend on the concentration of the reductant species according to = c edlFredfE). A large overlap of the redox DOS with the energy of the hole (left) gives higher currents than a small overlap (right)...

We shall consider next the theory of condition 2 for an extrinsic photoconductor, then the theory of G RA for condition 3, and finally the dependence of t on material parameters. We shall analyze the geometrical model of Fig. 4.8 and assume a simple energy level model of an n-type extrinsic semiconductor consisting of a photoionizable donor level and a compensating acceptor level properties of a corresponding p-type model would be analogous. This material is extrinsic as both a photoconductor and semiconductor. 

Equation 16.32 is the energy as a function of k for a free wave. The function E(k) is a continuous band of energy levels. Models of this type are referred to as band models. From the above, it follows that the Fermi level is located at... 

Comparison of energy-level model parameters (in cra ) used in calculating 3+ lanthanide and actinide free-ion energy levels for f through f configurations. Lanthanide parameters are from Carnall et al. (1989) and actinide parameters are from Carnall (1992), n denotes the number of 4f or 5f electrons. 

Only two studies of transcurium-ion fluorescence in solution have been published. Carnall etal. (1984) measured the absorption spectrum of Bk ", interpreted its energy-level structure in terms of a free-ion energy-level model, analyzed its absorption band intensities in terms of Judd-Ofelt theory, and reported luminescence lifetime data for aquated Bk in DjO. Beitz et al. (1983) carried out LIF studies on Es " in HjO and DjO solutions as well as complexed Es " in an organic phase. No luminescence studies have been reported for actinide elements heavier than Es. 

The diagram shown in Fig. 3.10 is now widely used to describe electron transfer processes at electrodes, and it has the merit that it can be extended readily to the discussion of electron transfer at semiconductor and insulator electrodes [16]. The theoretical basis for the diagram is to be found in the fluctuating energy level model of electron transfer which has been discussed by Marcus [12,13], Gerischer [17-19], Levich [20], and Dogonadze [21]. 

The fluctuating energy level model of electron transfer... 

The application of the fluctuating energy level model to the symmetrical reaction (Equation (3.54)) is of particular interest since it suggests that it should be possible to compare heterogeneous and homogeneous rate constants. The expressions for the rate constant in each case are... 

In this section a rather simple energy-level model is first briefly described, which enables a rough analysis of real-time decay processes obtained by MPI. It has been successfully applied to the investigations of the Nas state. To use the model for the larger alkali clusters, however, it has to be modified. Both models can be regarded as a first approximation to estimate the time constants of the induced photodissociation process. It has to be stated that neither of the two models takes into account the dynamics of wave packets prepared on the repulsive PES. 

Simple Energy-Level Model. Figure 2.35 sketches a simplified case of TPI spectroscopy of a dissociative electronic state. Fragmentation with a probability l/rfrag is regarded as the only relaxation channel. In a real-time TPI experiment, first an ultrashort pump pulse transfers an ensemble of e.g. molecules or clusters to an excited state (2.18a). Then, either the excited... 

This simple energy-level model has been successfully tested by analyzing real-time TPI experiments with picosecond time resolution carried out in... 

Extended Energy-Level Model. The real-time spectra - performed with femtosecond time resolution - of the Naa system, as well as of the larger sodium and potassium clusters (see Sects. 4.2, 4.3 and 4.4), reveal a nonexponential decay which cannot be explained within the simple energy-level model introduced above. It seems reasonable that this different behavior is caused by clusters in the beam which are larger than those of interest. Therefore, the simple model has to be slightly extended. This extended energy-level model has the following features. 

To describe the decay of the real-time spectra, one might think of using the simple energy-level model described in Sect. 2.2.2. However, the experimental data (ignoring the fast oscillation) do not at all fit to a convolution of the overall system response with a single exponential decay. Therefore, the extended energy-level model developed in Sect. 2.2.2 was applied. The real-time spectra were fitted with the convolution function ... 

Thus, the real-time spectra of Naa excited to the C state can be explained within the simple energy-level model (see Sect. 2.2.2) shown in the inset of Fig. 4.5. The pump pulse will generate an initial population No of excited trimers within its pulse width. Owing to radiative transitions with lifetime Trad and fragmentation with lifetime Tfrag, this population will decrease with the progress of time t. The decay is characterized by the lifetime r of the excited ensemble of trimers given by (see 4.1)... 

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