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The curve in Fig. 4.7 illustrates the change in the electron energy corresponding to the diffusion process of Fig. 4.6. The magnitude of the variation is substantial and easily detectable by experiment [78-80,90-92]. For a theoretical interpretation of such experiments, including the change in exciton binding energy, see [77]. 4.6 CONCENTRATION DEPENDENT DIFFUSION COEFFICIENT There are many examples for which the diffusion coefficient of a species is a function of the concentration of that species itself [83,93,94]. The work by Tuck is perhaps the most interesting as it makes a detailed comparison with experiment. Tuck considered the diffusant of a dopant, Zn in bulk GaAs, a well studied system in which the 'substitutional-interstitial' mechanism is thought to dominate [95]. In this system the dopant (Zn) sits at two different positions within the crystal lattice, i.e. as the name implies both substitutionally and at interstitial sites. Atoms on the substitutional sites, which are in the majority, diffuse very slowly; however, the minority of interstitial atoms diffuse so readily that they dominate the evolution of the dopant profile with time. The diffusion coefficient describing the process is concentration (c) dependent, and is given mathematically as:

Exercise 11.2. What is the value of ( k: k = 4: k ) in the case that the type of k is (a) a natural number, (b) an integer D

hence the diffusion equation becomes:

[Fahlman 1989] SE Fahlman, it Fast Learning Variations on Back-Propagation: An Empirical Study, in DS Touretzky, GE Hinton, TJ Sejnowski (eds) Proceedings of the 1988 Connectionist Summer School, Morgan Kaufmann, 1988, pp 38 51..

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In the next section, we formulate general properties of summation. Several of these rules have so-called side conditions that prevent improper use of the rule. The side conditions are, primarily, syntactic and not semantic. This means they are conditions on the way expressions are written (their syntax) rather than conditions on the values of the expressions (their semantics). So, the conditions may apply to one expression but not to an equal expression. For example, the condition 'the symbol '0' occurs in the expression' is a syntactic condition on expressions, which is true of '0' but not true of '1-1', even though 0 and 1-1 are equal. A consequence of the syntactic nature of the side conditions is that they are cumbersome to state even though they are, in fact, quite straightforward. In order to understand them, we need to have a clear understanding of the notions of 'free' and 'bound' variables in an expression. (These notions provide the semantic justification for the syntactic side conditions.) Note that although all the examples given in this section are of summations, the discussion applies equally well to all quantifiers. Recall that a dummy in a summation is said to be bound. For example, all occurrences of 'k' in (Zk : 1 < k < 3 : k) are bound to the S quantifier. Variables that have occurrences in an expression that are not bound to a quantifier are called free variables. For example, n is free in 2n, and m and n are free in (Zk: 0 ^ k < ra: k n ). Free and bound variables have different roles. The value of an expression depends on the value of its free variables. For example, the value of 2" depends

The constant k was deduced from experiment, and the index n, for this system, was found to be 2. Tuck considered several situations and implemented solutions which agreed very well with experiment. In this work, the numerical simulation can be implemented directly.

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