Appendix A

According to Eq. (2.37)

$$\omega (r)=\frac{1}{N+1}\sum_{\alpha }\sum_{\beta }\Omega _{\alpha \beta }(r)$$ (2.39)

From the discussion at the end of section 2.4 it follows

$$\hat{\Omega}_{\alpha \beta }(k) \int d^{3}re^{i\vec{k}\cdot \vec{r} }\Omega _{\alpha \beta }(r)$$ =

$$\delta _{\alpha \beta }+(1-\delta _{\alpha \beta })\int d^{3}re^{i\vec{k }\cdot \vec{r}}\Omega _{\alpha \beta }(r)$$ =

$$\delta _{\alpha \beta }+(1-\delta _{\alpha \beta })2\pi \int_{0}^... ...rr^{2}\Omega _{\alpha \beta }(r)\int_{-1}^{1}d(\cos \theta )e^{ikr\cos \theta }$$ =

$$\delta _{\alpha \beta }+(1-\delta _{\alpha \beta })4\pi \int_{0}^{\infty }drr^{2}\Omega _{\alpha \beta }(r)\frac{\sin kr}{kr}$$ =

$$\hat{\Omega}_{\alpha \beta }(k)=\delta _{\alpha \beta }+(1-\d... ...le \frac{\sin kr_{\alpha \beta }}{kr_{\alpha \beta }}\rangle .$$ (2.40)

The average may be calculated using the methods of Chapter 1. Usually it is a good approximation to describe $\hat{\Omega}_{\alpha \beta }(k)$ with a damped sine function

$$\hat{\Omega}_{\alpha \beta }(k) \frac{\sin (B_{\alpha \beta }k)}{ B_{\alpha \beta }k}\exp \{-A_{\alpha \beta }^{2}k^{2}\}$$ =

$$A_{\alpha \beta }^{2} \frac{1}{6}\langle r_{\alpha \beta }^{2}\rangle (1-C_{\alpha \beta })$$ =

$$B_{\alpha \beta }^{2} \langle r_{\alpha \beta }^{2}\rangle C_{\alpha \beta }$$ =

$$C_{\alpha \beta }^{2} \frac{1}{2}\left\{ 5-3\frac{\langle r_{\alpha \beta }^{4}\rangle }{\langle r_{\alpha \beta }^{2}\rangle ^{2}}\right\} .$$ = (2.41)

From Fig. (1.3) we conclude that $c_{\alpha \beta }=0$ for $\vert\alpha -\beta \vert\geq 1000$, in which case $\hat{\Omega}_{\alpha \beta }(k)$is a simple Gaussian.