Random impurity distribution

Jackie J. (1980), Potential fluctuations in doped semiconductors with random impurity distribution , Phil. Mag. B 41,681-687. 

Inspection of Fig. 1(c) reveals that there are a few pairs of atoms with a preferred distance. Analysis of many such images in terms of site occupation probabilities as a function of adatom distances revealed significant deviations from a random distance distribution, and the existence of adsorbate interactions which indeed oscillate with a wave vector of 2kp [16]. The decay followed the l/r2-prediction only for large distances, while significant deviations were observed at distances below 20 A and interpreted as a shortcoming of theory [16]. However, an independent study, carried out in parallel, focused on two body interactions only, i.e., the authors counted only those distances r from a selected atom to a nearby atom where no third scatterer (adatom or impurity) was closer than r [17]. This way, many body interactions were eliminated and the interaction energy E(r) yielded perfect... 

The calculation of distribution function of random field, created by different independent sources, has been carried out in the statistical theory framework [87]. The similar calculations had been performed earlier for bulk incipient ferroelectrics with off-center impurities [84] and bulk relaxors [88]. The first calculation of random field distribution function in the films of relaxor ferroelectrics has been performed in Ref. [89]. 

As a final remark it must be mentioned that theoretical and experimental works have been dedicated to investigating the effect of the finite size of the chains [65]. In fact, as grows exponentially, at low temperatures it can become comparable with the distance between two consecutive defects (e.g. impurities and vacancies) which are always present in real systems and hardly separated by more than 103 -104 elementary units. In case of Z < , the nucleation of the DW is energetically favoured if occurring at the boundaries, because the energy cost is halved. However the probability to have a boundary spin is inversely proportional to L thus the pre-exponential factor becomes linearly dependent on L, as experimentally found in doped SCMs. As doping occurs at random positions on the chain, a distribution of lengths is observed in a real system. However, as the relaxation time is only linearly dependent on L, a relatively narrow distribution is expected. 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

In LSC measurements precautions are required to avoid impurities which may cause scintillation quenching. Since radioactive decay is random and is described with the Poisson distribution, the standard deviation for a given count, C, is equal to C1/2. 

One application of these formulae in 4his book is to impurity bands in doped silicon or germanium. Here the centres are distributed at random the appropriate formulae for this case are discussed in Section 7 and Chapter 6. 

As it is known [5], the intensity of the scattered light gives us an information about the system s disorder, e.g., presence therein of pores, impurities etc. Since macroscopically liquid is homogeneous, critical opalescence arises due to local microscopic inhomogeneities - an appearance of small domains with different local densities. In other words, liquid is ordered inside these domains but still disorded on the whole since domains are randomly distributed in size and space, they appear and disappear by chance. Fluctuations of the order parameter have large amplitude and involve a wide spectrum of the wavelengths (which results in the milk colour of the scattered light). 

Most of theories of spin dynamics in semiconductors, as discussed above, consider the SO coupling as uniform, that is electron-coordinate independent. The reality is, however, different the coupling produced by randomly distributed charged impurities is spatially random as well. 

Since the enantiomeric distribution, i.e., the incidence of generating the R or S product, did not appear clearly as random but seemed to be correlated to the specific set-up used and to the order in which the experiments were carried out, the authors interpreted these results as possibly caused by chiral impurities in the starting mixture [34]. However, attempts to detect these impurities failed. In essence, the experiments of Singleton and Vo showed that... 

Subsequent attempts by Singleton and Vo supported these results [36]. Also in this case, a clear random distribution of the R and S enantiomers in a number of the 54 experiments was observed, indicating that a systematic effect coming from chiral impurities may be excluded. Each experimental rim afforded a clearly detectable prevalence of either the R or the S enantiomer, giving rise to a total of 27 events in which the R enantiomer dominated and 27 events in which the S enantiomer was found in excess (no quantification of the particular ee was given). 

Obviously, the equivalence (randomness) of the spatial distribution of the acceptor impurities (V4+ ions) in polycrystalline lattice, or its heterogeneity, should strongly affect electrophysical, photoelectrochemical and catalytic properties of nano-... 

Thus, one may summarize the physical picture of the relaxation dynamics in KTN crystal-doped with Cu+ ions in the following way In the paraelectric phase, as the ferroelectric phase transition is approached, the Nb5+ ions form dipolar clusters around the randomly distributed Cu+ impurity ions. The interaction between these clusters gives rise to a cooperative behavior according to the AG theory of glass-forming liquids. At the ferroelectric phase transition the cooperative relaxation of the Cu+ ions is effectively frozen. ... 

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