Discrete energy levels

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 ... 

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). 

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). 

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. 

Discrete-Level Descriptions of Nuclei at Low Excitation Energies... 

Table 3. The two methods of deducing the energy-dependent spin cut-off parameter, < (E) = (6/ 2MlkA2/3K [ - 6])1/2, from discrete-level sets.

Figure 4. (a,b) The 89Y spin cut-off parameter, a1, vs E, and the spin distribution, P(J), obtained from 60 discrete levels and from calculations with k values of 0.146 and 0.24. At energies below the pairing energy, the value of a2 is arbitrarily held constant and equal to (6/tt2) (A A2/3). (c,d) The MY spin cutoff parameter and the spin distribution obtained from 25 discrete levels and from calculations with k values of 0.146 and 0.29. 

We see that many problems still need to be solved in order to obtain accurate results in Hauser-Feshbach calculations. Some examples are the energy dependence of rotational enhancement of levels in deformed nuclei, the energy and mass dependence of Ml gamma-ray transitions, the importance of E2 transitions, and better estimates of fission barriers. Work in each of these areas will benefit greatly from a better understanding of the discrete levels, particularly in nuclei away from stability. 

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. 

Broad maxima in Cm versus T (the so-called Schottky anomalies) frequently indicate partially populated discrete levels that are separated by an energy difference AE in the range of knT. For a simple two-level system, the maximum occurs at B niax 0.42A/i. Phase transitions, for example, transitions between a long-range ordered ferromagnetic phase and a paramagnetic phase, produce a characteristic peak in Cm versus T graphs. 

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