Tablas De Totalización De Puntajes De Las Herramientas 62

A detailed comparison of the EPA and the percolation theory approach to d.c. conductivity shows that, in all

must be constant and determines its value in an unambiguous way, independently of fitting any numerical conductivity data. This is accomplished by demanding that our approximate equations reproduce known percolation properties. To be explicit, we substitute equation (3.4.5), i.e.

g = g^0(s — srt) , into equation (6.2.5) and find the value

mn 0 0

of sA when Y just vanishes. By definition this is s and is

0 p

a function of n; we fix ic by requiring that this s^ be identical to that predicted by percolation theory. This procedure leads to accurate d.c. conductivity exponents because the factor exp(-s > is either explicitly contained

P

in the solutions to the EPA equations or inferred from numerical studies. In section 3.4 we considered energy independent (R-hopping> and energy dependent models; here we determine k for each case considered therein.

6

•6.3.1 R- Hopping in d Dimensions

When we put s = 2ar and substitute u = 0 and p(€) = n s S(f> into equation (6.2.5) we immediately find that the EPA predicts

cases, the correction factor k * introduced in section 6 . 2

(6.3.la) in three dimensions and

- [

]

s (6.3.lb)

P n n

percolation thresholds (3.4.4) if r = N . We therefore set it = 2.7 in three dimensions and it = 4.5 in two dimensions. Moreover, considering now the d.c. R-hopping problem with the original exponential conductances g = grtexp(-2ar )f

mn u mn

the methods given in appendix A readily yield the result Y = g^exp(-s ) when n -* 0. Since Y is a factor in d(0) we

0 p s

have complete agreement with the predictions of percolation theory.

¿>.3.2 Energy Dependent Model in Three Dimensions

We shall discuss the model defined in section 3.4.3 where, we recall

a percolation theory approach, that leads to good agreement with computer simulation data, is given by

Solution of the percolation prob lem by our equations is complicated by the fact that the admittance Y(f> now has an energy dependence determined by a n o n - linear integral

equation. However, we proceed by first reducing equation (6.2.5 ) to a non-dimensional form; we find, at low

temperatures, that Y(f> = g^y(x) where x = f/k^Ts^ and y(x) P 2 c»r ♦ (If I + If I ♦ If - f„|)/2k T m n m n o (6.3.2a) mn e(f > If I < W/2 (6.3.2b) 0 otherwi se P

[

y(x) = | j1 0CP(x,x') -

n c i

We seek the critical value of % for which the

solution of the non-linear equation (6.3.4) approaches zero so that, from equation (6.2.4), d(0) = 0. The corresponding value of s is the critical value of s. and is obtained by

P °

solving equation (6.3.6) with % = . We find that

, r c°- I

np(p*)k„T

(6.3.7)

If k i s a dimensionless number then so too i s 1 . therefore

c

c h o o s i n g k a s s u c h i s s u f f i c i e n t t o r e p r o d u c e t h e c o r r e c t

f o r m o f s . To d e t e r m i n e $ we n e g l e c t y(x') in t h e

P C Jf

denominator of the integrand in equation (6.3.7) since we

a r e seeking a solution which vanishes as t approaches

Then the equation becomes a linear eigenvalue problem which can be solved by standard methods.

We write equation (6.3.^) in the form

y(x) = 1 j K ( x , x ' > y( x ' )dx' (6.3.8)

-1

w h e r e

- Q> K (x , x ' > 0(Q - 1) (1

It is clear that is the minimum value of % for which there exists a non-trivial solution to equation (6.3.8). By inspection we see that y(x) vanishes when x = +1. Therefore we expand y(x) in orthonormal functions with that property, specifically we put

The quantities X = l/< are the eigenvalues of the matrix K whose elements are given by

- 1

and we seek the maximum value of X, X • We h ave max

performed this calcuation numerically and found that taking the first 5 x 5 block of K is sufficient to giv e

convergence in Xmax to three significant figures. We find X = 0.315 and hence 5 = 3.175. Equation (6.3.7) therefore

max c

becomes identical to equation (6.3.3) if we cho ose k ■ 4.4. It is also nessessar y to solve a non-linear integral equation to determine Y(f) for energy dependent hopping with the original exponential conductances. At low

a good approximation to Y(0). It follows from equation (6.2.4) that this factor also appears in d(0> as we confirm by the numerical results to be presented in section 6.5. Thus we again have complete agreement with the predictions of percolation theory but the energy dependence of Y(f)

y (x) = Z a U (x) n n (6.3.9) n and choose U (x) = cosC(2n-l> nx/23 n (6.3.10) (6.3.11) necasAOLr-tj

prevents us from exhibiting this fact analytically.