Avalanching regime

Finally, it is worth mentioning an up-conversion avalanche laser in LaClsiPr " single crystals (Koch et al. 1990). A CW emission in a 4-level scheme is obtained at 644 nm through an up-conversion avalanche process (sect. 3.4.3), under 677 imi pumping which corresponds to ESA pumping for the second step and probably to multiphonon absorption for the first one, as recently observed for ASE emission at 850 nm in an ZBLAN Er fiber in an avalanche regime (Chen and Auzel 1994a,b). 

For moderately doped substrates the crossover from tunneling to avalanche breakdown occurs at pore diameters of about 500 nm, corresponding to a bias in excess of 10 V. Above doping densities of 1017 cm-3 breakdown is always dominated by tunneling. Tunneling is therefore expected to dominate all pore formation in the mesoporous regime and extends well into the lower macropore regime, while avalanche breakdown is expected to produce structures of macropor-ous size. 

For n-type doping densities below 1017 cm-3 and an anodization bias above 10 V, avalanche breakdown becomes relevant. The interface morphology generated in this regime is very complex and shows large etch pits, macropores and mesopores. The formation of this structure is not understood in detail. A hypothetical model will be discussed in Section 8.5. 

In contrast to p-type electrodes, an n-type electrode is under reverse conditions in the anodic regime. This has several consequences for pore formation. Significant currents in a reverse biased Schottky diode are expected under breakdown conditions or if injected or photogenerated minority carriers can be collected. Breakdown at the pore tip due to tunneling generates mainly mesopores, while avalanche breakdown forms larger etch pits. Both cases are discussed in Chapter 8. Macropore formation by collection of minority carriers is understood in detail and a quantitative description is possible [Le9], which is in contrast to the pore formation mechanisms discussed so far. 

In the early years of the regime, the death of inmates caused problems for the Nazi leaders. An avalanche of deaths was unacceptable for their policies which had to take account of public opinion. The stability [of the number of camp inmates] was therefore chiefly attributable to the number of released inmates, as well as the arrival of new inmates, which maintained the stability of the total camp population. "... 

Equation (4.2) includes translational and collisional effects but not static effects. However, most engineering applications (e.g., chute flows) and other natural flow situations (mudflows, snow avalanches, debris flow, etc.) appear to fit into a regime for which the total stress must be represented by a linear combination of a rate-independent static component plus the rate-dependent viscous component just described (Savage, 1989). The flow patterns observed for material flow in rotary kilns appear to fit these descriptions and the constitutive equations for granular flow may apply within the relevant boundary conditions. 

Under the assumptions that the number of incoming links per node A is small (A N) and that the overall avalanche is small V N), it can be proven, as it will be shown in Sect. 3, that the distribution of avalanches depends only upon the same Derrida parameter that determines the dynamical regime of the network. The assumptions made here amount to suppose that an avalanche never interferes with itself. Precisely an affected node B is defined to be the parent of another affected node C if the first deviation of C from the unperturbed value is due to the influence of B. The noninterference condition amounts to assuming that every node C in the avalanche is not affected by any other affected node different from B (neither at a later stage nor at the same time). Therefore, under these assumptions the topology of a spreading avalanche is that of a tree, where each node has a single parent. 

J. Rajchenbach. Flow in powders From discrete avalanches to continuous regime. Physical Review Letters, 65 2221-2224, October 1990. 

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