Fresnel diffraction in the electronic density of states

We deal with the local electronic density of states of atomic spheres (ASDOS) within the multiple-scattering approach. A model is adopted where the same representative muffin-tin (MT) potential is put into each atomic sphere. $E=k^2$, $L\equiv(l,m)$, $\delta_l$ and $f_l = e^{i\delta_l} sin(\delta_l)/k$ designate the electron energy, the angular momentum, the MT scattering phase shifts and the partial scattering amplitudes, respectively. The partial ASDOS of the embedded atomic sphere number $s$,

$$D_{sl}= D^{(0)}_l \; (1 + {\rm Re}T_{sl}),$$ (10)

is obtained in terms of both the bare ASDOS, $D^{(0)}_l$, and complex amplitudes, $T_{sl}$, which describe the inward back scattering of MT waves from the environment of the considered atomic sphere. On passing from MT waves to the multiple scattering of vacuum waves one obtains [8]

$$T_{sl} = e^{i2\delta_{l}} \; \frac{1}{2l+1}\sum_m \langle sL\vert(I-C)^{-1}C\vert sL \rangle \; (ik f_l)^{-1}.$$ (11)

The matrix C comprises all links between the atomic angular-momentum channels via scattering followed by vacuum-wave propagation. The representation (11) has a physically appealing interpretation. The atomic sphere $number$ $s$ performs Fresnel-type diffraction experiments: Vacuum L-waves are emitted (cf. $\vert sL \rangle$) and propagate towards the environment (cf. $C/ikf_l$). After multiple scattering which includes the emitting atomic sphere (cf. $(I-C)^{-1}$), the amplitude of the back-scattered vacuum L-wave is projected out (cf. $\langle sL\vert$). The final step transforms from vacuum waves to MT waves (cf. $e^{i2\delta_l}$).

We want to know the radial sequence of neighbour shells which provides low ASDOS at the Fermi energy. Suppose an approximation to the ensemble average of the environment correction (11) where the pair correlation function $g(r)$ is compared with certain standards of the radial behaviour similar to the $\sin(qr)$ in (7). In that case the neighbour-shell radii and the radii where the standards have extreme values have to be correlated in just the same way as demonstrated in connection with (7) and (8). This kind of approximation to (11) is obtained upon confining to single-scattering in the environment ($(I-C)^{-1}$ replaced by $C$). The ensemble average can be performed and we arrive at

$$\overline{T_l} = e^{i2\delta_l} \sum_{l_1}(2l_1+1) i k f_... ... dr \; 4\pi r^2 N_0 g(r) \; \left(h_{l_2}^{(1)}(kr))\right)^2.$$ (12)

$P_l(x)$ are the Legendre polynomials. The spherical Hankel functions, $h_l^{(1)}(z)$, are products of the asymptotically important factors, $i^{-l} \; exp(iz)/(iz)$, and certain l-th order close-range polynomials of 1/z.

The following items are to be noted: (i) Similar to (7) one finds in (12) the pair correlation function $g(r)$ correlated with certain standards of the radial behaviour, $r^2 \; (h_l^{(1)}(kr))^2$. At $l > 0$ the atomic spheres resolve the close-range behaviour of the neighbour-shell sequence $g(r)$ via Fresnel-type diffraction. This is beyond the information extracted by $\sin(qr)$ in (7). (ii) Hybridization effects are represented by scattering-path loops which include at least three scattering events with two different angular-momentum channels of the considered atomic sphere. Hybridization is thus disregarded in (12). Nevertheless, the hybridizing elements of the linking matrix $C$ which are non-diagonal in the angular momentum contribute in (12).

The Nagel-Tauc stability concept demands a minimum of the ASDOS at the Fermi energy. According to (10) ASDOS minima arise from strongly negative values of Re $\overline{T_l}$. In the following, employing the single-scattering approximation (12), the trends are inferred which follows the neighbour-shell sequence in order to achieve low ASDOS at $E_F$.