Diffusion lengths

T v o physical parameters required in every application of the one-velocity model are the diffusion coefficient D and the diffusion length L. So far, our interest in the one-velocity model has been confined to descriptions of the spatial distribution of thermal neutrons, and these parameters have therefore been defined in terms of the thermal cross sections. We will show later that similar quantities can be defined in the treatment of multivelocity systems by the energy-group method, or for that matter, in any system wherein it is convenient to describe the diffusion properties of a particular group of neutrons in terms of a single speed. 

The diffusion coefficient and the diffusion length are fundamental macroscopic properties of a material which are useful in the one-velocity formulation. Both quantities can be measured directly in the laboratory by suitable experiments. The direct measurement of the diffusion coefficient, however, entails the use of a pulsed beam of neutrons. Inasmuch as this experiment involves a time-dependent phenomenon, a discussion of the experiment will be deferred until after a suitable model has been developed for the analysis of nonstationary problems in neutron diffusion. An experiment for the direct determination of the diffusion length, however, is based on a steady-state phenomenon, and the important features of this experiment can be displayed by means of the models and concepts already developed. Because of the close relationship between the parameters L and D, it would be desirable to examine the techniques for their measurement simultaneously. This is not possible because of the complications mentioned above thus for the present we confine our attention to the study of the diffusion length and an experiment for its measurement. 

It should be noted that a separate measurement of both D and L is not ahvays required. Because of the relationship = D/2a, it is clear that a knowledge of one parameter yields the other, provided of course that 2b is known. This is precisely the situation encountered in the L experiment thus, in order to determine D we require a separate measurement of 2 . The diffusion-coefficient experiment based on the pulsed-neutron- 

For this calculation we use the problem of neutron diffusion from a point source in an infinite medium. Similar calculations can be made for the plane and line source, but the basic relationship between L and the crow-flight distance will prove to be the same in all cases. Consider then the diffusion of neutrons in an infinite medium from a point source of strength go neutrons per unit time. For convenience, we place the source at the brigin of a suitable coordinate system. As previously noted, this situation has spherical symmetry, and the neutron distribution may be described completely by the radial coordinate r alone. Let us compute now the quantity r for this system. We define as the average square 

The frequency function can be computed from our knowledge of the distribution of absorption reactions throughout the entire medium. Now qo is the total number of absorptions per unit time in this system and Sa (r) dr the absorptions per unit time in dr at r. Thus the probability of a neutron being absorbed in dr at r, namely, the function /(r) dr, is simply 

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Lp = D r ) is the minority carrier diffusion length for electrons in the -region, (0) is the minority carrier concentration at the boundary between the depletion layer and the neutral region. The sign of this equation indicates that electron injection into the -region results in a positive current flow from p to n a.s shown in Figure 7. 

Urp Pi 1 — / If) where Wis the base width, is the distance between emitter and collector junctions and is the minority carrier diffusion length ia the... 

Global AMI.5 sun illumination of intensity 100 mW/cm ). The DOS (or defect) is found to be low with a dangling bond (DB) density, as measured by electron spin resonance (esr) of - 10 cm . The inherent disorder possessed by these materials manifests itself as band tails which emanate from the conduction and valence bands and are characterized by exponential tails with an energy of 25 and 45 meV, respectively the broader tail from the valence band provides for dispersive transport (shallow defect controlled) for holes with alow drift mobiUty of 10 cm /(s-V), whereas electrons exhibit nondispersive transport behavior with a higher mobiUty of - 1 cm /(s-V). Hence the material exhibits poor minority (hole) carrier transport with a diffusion length <0.5 //m, which puts a design limitation on electronic devices such as solar cells. 

Assuming that the average diffusion length of particles to the cavity from the shell surface is equal to tire thickness of the shell, and conversely tlrat there is a counter-cutTent of vacancies from tire cavity to the shell surface, the rate of densification dp/dt is given by... 

Obtaining information on a material s electronic band structure (related to the fundamental band gap) and analysis of luminescence centers Measurements of the dopant concentration and of the minority carrier diffusion length and lifetime... 

Many inorganic solids lend themselves to study by PL, to probe their intrinsic properties and to look at impurities and defects. Such materials include alkali-halides, semiconductors, crystalline ceramics, and glasses. In opaque materials PL is particularly surface sensitive, being restricted by the optical penetration depth and carrier diffusion length to a region of 0.05 to several pm beneath the surface. 

If a laser pulse of sub-picosecond duration is used, deposition of the laser energy to the sample is so rapid that the thermal diffusion length is determined by the diffusion of hot electrons before they transfer the energy to the lattice of the solid sample. 

FIG. 2 Growth rates as a function of the driving force A//. Comparison of theory and computer simulation for different values of the diffusion length and at temperatures above and below the roughening temperature. The spinodal value corresponds to the metastability limit A//, of the mean-field theory [49]. The Wilson-Frenkel rate WF is the upper limit of the growth rate. 

If spiral growth occurs due to the existence of screw dislocations, the results depend upon whether the diffusion length ijy is smaller or larger than the typical separation of the spiral arms i. In the first case the situation hardly changes from the purely kinetic situation without diffusion, but in the second case interaction between steps comes into effect [90] and phenomena such as step bunching [91] take place. We can estimate qualitatively the... 

Furthermore, assuming a constant deposition rate J (particles per area and time) during MBE, we can define a further length scale, namely the free diffusion length or the capture length... 

This length is apparently related to the capture time by the relation Pi J Tc and il A physical meaning of the free diffusion length 4 is that the maximum size of a stable adsorbed two-dimensional nucleus on a facet cannot essentially exceed this free diffusion length. If the nucleus is smaller, all atoms depositing on the surface can still find the path to the boundary of a nucleus in order to be incorporated there. If the nucleus is larger, a new nucleus can develop on its surface. 

A normal diffusion process, however, runs at a finite concentration of particles different from zero. In this situation it was found [101] that a fractal character (73) of the resulting structure is restricted to an interval a < R < if), where d is the diffusion length (67). Larger clusters have a constant density on a length scale larger than They are no longer fractal there. These observations have various consequences for crystal growth, and will be discussed in the next section. 

A serious point is the neglect of surface tension and anisotropy in these derivations. In the experiments analyzed so far the relation VX const, seems to hold approximately, but what happens when the capillary anisotropy e goes to zero Numerically, tip-splitting occurs at lower velocities for smaller e. Most likely in a system with anisotropy e = 0 (and zero kinetic coefficient) the structures show seaweed patterns at velocities where the diffusion length is smaller than the short wavelength hmit of the neutral stability curve, as discussed in Sec. V B. 

Here, D is the diffusion constant for heat or material and the kinematic viscosity of the liquid. A consequence of the existence of such a diffusive surface barrier is that the diffusion length = D/F is to be replaced by in all formulas, as soon as growth rate V the more important become the hydrodynamic convection effects. 

The Debye length of the electrode material can be determined from the constant B, and the sensitivity factor S from C, provided the diffusion length and the diffusion constant for minority carriers are known. 

For an electrode with high interfacial rate constants, for example, relation (28) can be plotted, which yields the flatband potential. It allows determination of the constant C, from which the sensitivity factor S can be calculated when the diffusion constant D, the absorption coefficient a, the diffusion length L, and the incident photon density I0 (corrected for reflection) are known ... 

The diffusion length can thus be calculated since a is typically known, or since =(t D)1/2, the bulk lifetime provided the diffusion coefficient... 

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