5 Results

[190.1.1] The results for $\omega=0$ will be expressed via the so called correlation factor $f$ . [190.1.2] It is a measure which is often used to characterize the amount by which the self diffusion coefficient of a tracer particle in a lattice gas differs from its mean field value given by the vacancy concentration. [190.1.3] All results will be presented and discussed with the conventions $\omega=1$ , $p^{{\prime\prime}}(0)=1$ , and assuming a unit lattice constant. [190.1.4] In these dimensionless units $f$ is defined by the equation $D(0)=f(1-p)$ or, more generally,

$f=\frac{D(0)}{D(\infty)}$

(5.1)

because for our hopping models $(1-p)=D(\infty)$ . [190.1.5] In the limit $p\to 1$ exact results for the correlation factor are known for the case $\tau=1$ (and $b=1$ )[11, 13, 14]. [page 191, §0] [191.0.1] For the hexagonal lattice one has $f=1/3$ , while $f=0.781..$ for the fcc lattice. [191.0.2] These values will be used below to determine $c$ in eq. (4.2).

[191.1.1] We now have to solve eq. (3.10) in conjunction with eqs. (3.13) and (4.4) for the cases of interest, the hexagonal and face centered cubic lattice. [191.1.2] The lattice Greens functions for these situations are well known and can be expressed in terms of the complete elliptic integral of the first kind

$K(m)=\int^{{\pi/2}}_{0}(1-m\sin^{2}\phi)^{{-1/2}}\mathrm{d}\phi.$

(5.2)

[191.1.3] For the hexagonal lattice we have

$G(x)=-\frac{2(3+x)}{\pi(2+x)^{{3/2}}(6+x)^{{1/2}}}K\left(\frac{16(x+3)}{(x+2)^{3}(x+6)}\right).$

(5.3)

For the fcc lattice the Greens function is given by

$G(x)=-\frac{1}{\pi^{2}}(4+x)^{{-1}}K(x_{+})K(x_{-})$

(5.4)

where

$x^{2}_{\pm}=\left(4+\frac{x}{4}\right)^{{-2}}\left\{\frac{1}{16}\left[\left(4+\frac{x}{4}\right)^{{1/2}}-\left(\frac{x}{4}\right)^{{1/2}}\right]^{4}+\left[\left(4+\frac{x}{4}\right)^{{1/2}}\pm\left(3+\frac{x}{4}\right)^{{1/2}}\right]\right\}.$

(5.5)

[191.1.4] We now solve eq. (3.10) iteratively on the computer. [191.1.5] We stop the iteration when the maximal relative error between two consecutive solutions falls below $10^{8}$ . [191.1.6] The resulting $A^{0}(u)$ is then used to calculate $D(u)$ according to eqs. (3.13) and (4.4). [191.1.7] The results are displayed in Figures 2 through 10.

[191.2.1] First we determine the proportionality constant $c$ in eq. (4.2). [191.2.2] This is achieved by requiring that the calculated correlation factor reproduces the known exact results for the limit $p=1,\,\tau=1,\, b=1$ . [191.2.3] We find $c_{\mathrm{hcx}}\approx 1.00$ for the hexagonal lattice, and $c_{\mathrm{fcc}}\approx 0.16$ for the fcc lattice. [191.2.4] These numbers are difficult to determine numerically, and we estimate the error to be roughly $0.03$ . [191.2.5] In all subsequent calculations we then use these values for $c$ .

Figure 2: Correlation factor $f$ as function of blocker concentration $p$ for the face centered cubic lattice ( $b=1$ , $c=0.16$ ). The crosses are the simulation results of Ref. [7] for the case $\tau=1$ . The circles are simulation results from Ref. [10] for the case $\tau=\infty$ . The dashed line is obtained by using the exact value $p_{c}=0.198...$ from Ref. [8] instead of the effective medium value $p_{c}=1/6$ .

Figure 3: Correlation factor $f$ vs. blocker concentration $p$ and ratio of hopping rates $\tau$ for the hexagonal lattice ( $b=1$ , $c=1$ ). The crosses are simulation results from Ref. [12] for the case $\tau=1$ .

[page 193, §1] [193.1.1] In Figure 2 we have extracted the correlation factor from $D(0)$ and plotted it versus blocker concentration $p$ . [193.1.2] All curves are for the uncorrelated case, i. e. $b=1$ , on the fcc lattice. [193.1.3] We give results for $\tau=0.1,~1,~10,~100,~1000$ and $\tau=\infty$ . [193.1.4] The crosses are the results of the Monte Carlo simulation for the case $\tau=1$ taken from Ref. [7]. [193.1.5] The circles are MC-results for $\tau=\infty$ and were taken from Ref. [10]. [193.1.6] Clearly there will be a discrepancy for this case because our results are approximate and for bond percolation while the simulation is exact and for site percolation. [193.1.7] An immediate problem is the value of $p_{c}$ for which the effective medium theory gives $p_{c}=1/6$ while the exact value is $p_{c}=0.198..$ [8]. [193.1.8] If we simply use the exact value for $p_{c}$ in our calculation we obtain the dashed line displayed in Fig. 2 which is found to be in good agreement. [193.1.9] In Fig. 3 we plot $f$ vs. $p$ for the hexagonal lattice. [193.1.10] Here the simulations have been taken from Ref. 10. [193.1.11] Keeping in mind that there are no free parameters (remember $b=1$ ) we find very good agreement for both lattices. [193.1.12] However, additional simulation data especially for $\tau$ in the range $1<\tau<\infty$ , and a more accurate determination of $c$ are required to fully evaluate the quality of the theoretical results. [page 194, §0] [194.0.1] We now turn to the results for our primary objective, the frequency dependent diffusion coefficient.

[194.1.1] We consider first the uncorrelated case $(b=1)$ on the fcc-lattice. [194.1.2] In Figure 4 and Figure 5 we plot $\operatorname{Re}D(\omega)$ over ten decades in frequency on a log-log plot. [194.1.3] Figure 4 corresponds to a blocker concentration $p=0.9$ which is below the percolation threshold for vacancies, and shows the results for $\tau=1,~10^{3},~10^{6},~10^{9}$ and $\infty$ . [194.1.4] Figure 5 has $p=0.8$ and $\tau=1,~10,~100,~1000$ . [194.1.5] From Figure 4 we see immediately that below the percolation threshold $D(\omega)$ vanishes quadratically with frequency for $\tau=\infty$ . [194.1.6] This behaviour is well known from the analysis of the EM theory for the frozen case. [194.1.7] For $\tau<\infty$ we find a crossover to a constant proportional to $1/\tau$ . [194.1.8] This could have been expected because the blocker motion now allows the Aâparticle to get through the network although the vacancy concentration at each instant is below $p_{c}$ . [194.1.9] The mobility of the A-particles will be completely determined by the mobility of the blockers. [page 197, §0] [197.0.1] The crossover frequency is seen to vary as $\omega _{\tau}\sim\tau^{{-1/2}}$ . [197.0.2] This will be discussed further in the next section. [197.0.3] On the other hand above the vacancy threshold Figure 5 shows that the effect of the blocker rearrangement is only noticeable for $\tau$ values smaller than roughly $10^{4}$ . [197.0.4] Indeed one expects that the effect of blocker motion will become negligible if $1/\tau$ is much smaler than the d. c. conductivity in the frozen case which is proportional to $1-p-p_{c}$ .

Figure 4: Real part of the genaralized diffusion coefficient $\operatorname{Re}D(\omega)$ for the fcc-lattice as a function of frequency $\omega$ in a logarithmic plot for several values of $\tau$ and $p=0.9$ , $b=1$ ( $C=0.16$ ).

Figure 5: Real part of the genaralized diffusion coefficient $\operatorname{Re}D(\omega)$ for the fcc-lattice as a function of frequency $\omega$ in a logarithmic plot for several values of $\tau$ and $p=0.8$ , $b=1$ $(c=0.16)$ .

[197.1.1] In Figures 6 and 7 we now turn to the correlated case, i. e. $b\neq 1$ . [197.1.2] Again we consider the fcc-lattice and plot the real (Fig. 6) and imaginary (Fig. 7) part of $D(\omega)$ for the two concentrations $p=0.8$ and $p=0.9$ with fixed $\tau=100$ but variable $b$ . [197.1.3] We have chosen $b=0.1,\, 0.5,\, 1,\, 2,\, 10$ for the correlation factor. [197.1.4] The case $b=1$ is included as a reference and has been distinguished graphically by a dashed line. [197.1.5] As before the real part approaches a constant as $\omega\to 0$ irrespective of $p$ because $\tau$ is finite. [197.1.6] A new phenomenon however is the appearance of nonmonotonous behaviour for $b=0.1$ . [197.1.7] In this case $\operatorname{Re}D(\omega)$ is found to increase at low frequencies, and to decrease at high frequencies thereby exhibiting a maximum at a finite frequency. [197.1.8] In general $\operatorname{Re}D(\omega)$ is found to decrease as $b\to 0$ at high frequencies, and to increase at low frequencies. [197.1.9] The reverse is seen for $b\to\infty$ . [197.1.10] This will also be discussed in the next section in more detail. [197.1.11] For the imaginary part of $D(\omega)$ we find a change of sign for sufficiently small $b<1$ . [197.1.12] See for example the case $p=0.8$ , $b=0.1$ . [197.1.13] On the other hand for $p=0.9$ , $b=0.1$ there is no change of sign in the imaginary part while the real part still shows a maximum.

[197.2.1] The same calculations have been performed for the hexagonal lattice. [197.2.2] The results are displayed in Figures 8 and 9. [197.2.3] The only difference lies in the parameter values. [197.2.4] We have chosen different concentrations, $p=0.5,\, 0.7$ , $b=0.1$ and fixed $\tau$ at $\tau=10$ . [197.2.5] The results show qualitatively the same behaviour as for the fcc-lattice.

Figure 6: Real part of $D(\omega)$ for the fcc-lattice as a function of frequency in a linear plot for fixed $\tau=100$ , and values for $b$ and $p$ as indicated. The dashed line corresponds to the uncorrelated case $b=1$ .

Figure 7: Imaginary part of $D(\omega)$ for the fcc-lattice as a function of frequency in a linear plot for fixed $\tau=100$ , and values for $b$ and $p$ as indicated. The dashed line is for the uncorrelated case $b=1$ .

Figure 8: Real part of $D(\omega)$ for the hexagonal lattice as a function of frequency in a linear plot for fixed $\tau=10$ , and values for $b$ and $p$ as indicated. The dashed line indicates the uncorrelated case $b=1$ .

Figure 9: Imaginary part of $D(\omega)$ for the hexagonal lattice as a function of frequency in a linear plot for fixed $\tau=10$ , and values for $b$ and $p$ as indicated. The dashed line corresponds to the uncorrelated case $b=1$ .

[197.3.1] In Figure 10 we have plotted some results for the correlated case ( $b\neq 1$ ) in a log-log plot. [197.3.2] We show $\operatorname{Re}D(\omega)$ for $p=0.7$ , $\tau=100$ and $b=1,2,10$ on the hexagonal lattice. [197.3.3] We note that as a consequence of the correlations the crossover into the constant high frequency limit is smeared out and resembles a power law over more than a decade in frequency. [197.3.4] This is particularly apparent for the case $b=2$ .

Figure 10: Real part of $D(\omega)$ for the hexagonal lattice plotted logarithmically against frequency for $p=0.7$ , $\tau=100$ and $b=1,2,10$ . The slope of the straight line is roughly $0.5$ .

[page 198, §0] [198.1.1] For reference we have included a straight line into the graph whose slope is found to be roughly $0.5$ . [198.1.2] We remark that such a power law behaviour for the frequency dependent conductivity is often found experimentally in disordered systems. [198.1.3] As a particular example we mention $\mathrm{Na}$ - $\beta$ -alumina where the ionic transport is also known to be highly correlated[34].

Author:	R. Hilfer and R. Orbach
originally published in:	Dynamical Processes in Condensed Molecular Systems, J. Klafter et al. (eds.), World Scientific, Singapore, 1989, pages 175-202, ISBN 981-02-3457-0
originally published:	1989