Fraunhofer diffraction from the correlation sphere

Neighbour-shell radii can be defined as the distances where dN(r) = (G(r)+4$\pi$r $N_0$) rdr has maxima. We will also employ the reduced pair-correlation function, G(r), or even the pair-correlation function, g(r) = 1 + G(r)/(4$\pi$r$N_0$), instead of dN(r), despite of slightly smaller shell radii due to the latter ones. The neighbour-shell structure describes the residual order if observations with reference to atoms are averaged.

In common diffraction experiments the diffracted intensity is proportinal to the structure factor,

$$S(q) = 1 + \int dr G(r) \frac{sin(qr)}{q}.$$ (7)

As a consequence of the Fraunhofer-type observation, mainly the asymptotic shell spacing, $\Delta$, is resolved. The diffraction peak at $K_p$ = 2$\pi$/$\Delta$ appears particularly pronounced if the shell radii, $R_n$, are adapted to the uniformly spaced standard, $\sin(K_pr)$, according to

$$R_n = \Delta \left(\frac{5}{4} + (n-1)\right).$$ (8)

This is a two-fold condition for most-constructive interference in the Fraunhofer diffraction. It demands (i) a constant shell-spacing of $\Delta$ down to the first shell and (ii) the first neighbour shell precisely at the distance (5/4) $\Delta$.

We have demonstrated in section 4 that the empirical $K_p-a_0$ relation (6) is really the Nagel-Tauc condition, $K_p = 2 k_F$, provided that the $k_F-a_0$ relation (3) is inserted. At least the asymptotic neighbour-shell sequence must thus be uniformly spaced with the Friedel wave length $\lambda_{FR} = 2\pi/(2k_F)$. Experimental results [4] indicate that the shell radii may even approximate the sequence (8) at $\overline{Z} \approx 1.8$. In the latter case we insert (3) into (8) and obtain

$$R_n = \frac{5}{6} a_0\; \; (\frac{5}{4} + (n-1)) \approx \left(1.042 + 0.833 (n-1)\right) N_0^{-1/3}.$$ (9)

This is just the sequence of shell radii which guarantees that electronic transitions at the Fermi surface as well as the Fraunhofer diffraction at 2$k_F$ are both accompanied by highly constructive interference.