Crystal transition probabilities

For crystal growth from the vapor phase, one better chooses the transition probability appropriate to the physical situation. The adsorption occurs ballistically with its rate dependent only on the chemical potential difference Aj.1, while the desorption rate contains all the information of local conformation on the surface [35,48]. As long as the system is close to equilibrium, the specific choice of the transition probability is not of crucial importance. 

In the first topic, we will briefly describe a semi-empirical method that is commonly nsed to estimate the radiative transition probabilities from energy levels of (RE) + ions in crystals. This is certainly very nsefnl in order to determine the efficiency of a (RE) + based system as a luminescent or laser material. In the previous chapter (Section 5.7), we have described a method for determining the qnantnm efficiency of a Inminescent system. However, the application of that method is limited to certain... 

The crystal field model may also provide a calciflation scheme for the transition probabilities between levels perturbed by the crystal field. It is so called weak crystal field approximation. In this case the crystal field has little effect on the total Hamiltonian and it is regarded as a perturbation of the energy levels of the free ion. Judd and Ofelt, who showed that the odd terms in the crystal field expansion might connect the 4/ configuration with the 5d and 5g configurations, made such calculations. The result of the calculation for the oscillator strength, due to a forced electric dipole transition between the two states makes it possible to calculate the intensities of the lines due to forced electric dipole transitions. 

He points out that the variation of lifetime with glass matrix is due to at least two causes, the first being the changes in refractive index. If the wave functions of the ion remain essentially the same from host to host, the spontaneous-transition probability will increase with increasing refractive index because of the increase in density of final states. The second cause is configuration mixing of 4/ and 5d states, which must reflect the size and symmetry of the crystal field produced at the ion by the surroundings. 

Pollack also observes that the transition probabilities in his erbium-doped crystal change with concentration. He ascribes the decrease in emission intensity of certain transitions to reductions in radiative-transition... 

A possible modification of this expression is presented elsewhere (82). The value of t, can be related to a diffusion coefficient (e.g., tj = l2/6D, where / is the jump distance), thereby making the Ar expressions qualitatively similar for continuous and jump diffusion. A point of major contrast, however, is the inclusion of anisotropic effects in the jump diffusion model (85). That is, jumps perpendicular to the y-ray direction do not broaden the y-ray resonance. This diffusive anisotropy will be reflected in the Mossbauer effect in a manner analogous to that for the anisotropic recoil-free fraction, i.e., for single-crystal systems and for randomly oriented samples through the angular dependence of the nuclear transition probabilities (78). In this case, the various components of the Mossbauer spectrum are broadened to different extents, while for an anisotropic recoil-free fraction the relative intensities of these peaks were affected. 

The EPR line shapes produced by powder samples was calculated in many papers1,7,8. The spectrum of powder sample is considered as the superposition of spectra of large number small single crystals with chaotic orientation with respect to direction of external magnetic field. The sum intensity in each point of absorption band is determined by both the number of particles with given orientation and transition probability for this orientation. 

As stated above, expression (9) for the rate constant of transition in Einstein s crystal was first calculated analytically by the method of the straight search in the pioneer works of Pekar [1] and Kun and Rhys [2]. Their analytic expression remains till now the unique exact expression for multi-phonon transition probability in the time unit. Then, there appeared different methods that permit to derive the integral expressions for the rate constant in the general case of the phonon frequencies dispersion the operator calculation method [5], the method of generating polynomial [6], and the method of density matrix [7]. The detailed consideration of these methods was made in the Perlin s review [9],... 

The possible explanation [233] for the environmental sensitivity of the transitions (nonhypersensitive) may be due to an increase in 7) parameters which results in lowering of the excited states of the ions since the surrounding crystal field is influenced by the symmetric part of the crystal field. The observed changes in 7 and the small nephelauxetic effect show that the nonsymmetric part of the crystal field affects 7 values. The modifier ions distort the oxygen bonded to the lanthanide and thus lowers the site symmetry and increases the f-f transition probability. 

The time-sequence of any kinetic process is evidently determined if the particle population of certain valleys, i, in the crystal is known for die initial state, and if the elementary transition probabilities wy. (transitions per unit time) into a valley k, neighboring i> can be given. Retaining the assumption that defect interactions are to be excluded, we shall limit ourselves here to i — k transitions of elementary particles located at a sufficient distance from all other defects. In formulating the theoretical expression for w we can make use of die treatment by Jost (10) and that by Schottky (ref. 10, p. 155). It is found that Wjj. can be represented as the product of a frequency... 

The result shown in Fig. 6.8 that the two species of crystallites respond to the mechanical field in sequence - first the homo-PBT crystallites and later those arising from PEE, means that the homo-PBT crystals are probably dispersed within PEE in such way that they experience the mechanical field from the very beginning of loading. Moreover, one can assume that in the blend some internal stress and/or strain pre-exists since the strain-induced polymorphic transition starts even at lower... 

Experimental results (12) showed a transition to a lamellar liquid crystal for 14 added water molecules. Our calculations (to be reported at a later occasion) showed no discontinuity or any other indication of instability of the soap/acid water complex for the subsequent water molecules added in excess of 14. It appears reasonable to assume that the isotropic liquid/liquid crystal transition does not depend on the energy levels of the polar group interactions. The phase transition probably depends on the hydrophobic/hydrophilic volume ratio and estimations according to Israelachvili/Ninham (15) approach may offer a better potential for an understanding. 

The Transition Probability Anomalies of Crystal Field Theory.47... 

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