Most-constructive interference

The Fermi sphere, the interference cube and $\overline{Z}$ are connected by the well-known free-electron relation

$$k_F = (3\pi^2 \overline{Z} N_0)^{1/3}.$$ (1)

Suppose the Fermi sphere approaches the interference cube from inside. Up to $\overline{Z}$ = $\pi$/3 = 1.05, this is the situation where the sphere and the cube touch each other ($k_F$ = $\pi/a_0$), only "umklapp scattering" is amplified by constructive interference. The next remarkable geometry is reached at $\overline{Z} = \pi/3 \; 2^{3/2}$ = 2.96 where the Fermi sphere touches just the edges of the interference cube ($k_F$ = $\pi \sqrt{2}$/$a_0$). Obviously, at $\overline{Z} \approx 1.8$ we will find six caps of the Fermi sphere outside the interference cube and the rest inside. We ask if there is an optimal size of the Fermi sphere which makes most transitions between points at the Fermi surface accompanied by at least one almost fulfilled interference condition.

In order to answer this question we associate the six quadratic surface planes of the interference cube with corresponding parts of the Fermi surface: As seen from the common center of the two bodies, each quadratic face fills a solid angle of 4$\pi$/6=2$\pi$/3. We conclude that each quadratic face is "responsible" for just the part of the Fermi surface which belongs to its solid angle (fig.1b). Suppose a cap (solid angle $\Omega$) of each $2\pi$/3 Fermi surface lies outside the interference cube and the rest, 2$\pi$/3 - $\Omega$, lies inside. The following statement is fundamental for this approach: Transitions which start at any point of any cap outside the interference cube and terminate on the opposit side at any point inside the interference cube have one component of the momentum transfer approximately equal to the side length of the interference cube. Hence, they are accompanied by constructive interference. Just the same holds for the inside-outside transitions.

Now, the condition for most-constructive interference among scattered waves can be set up: The total number of transitions, $N_{tot}$, is proportional to $(4\pi)^2$. The part $N_{con}$ of them which is accompanied by almost constructive interference will be proportional to

$$6(\Omega \; (\frac{2\pi}{3} - \Omega) + (\frac{2\pi}{3} - \Omega) \; \Omega).$$ (2)

The maximum of $N_{con}/N_{tot}$ is 1/12 and it occurs at $\Omega$=$\pi$/3. Hence, we have to find the radius $k_F$ of a sphere which supplies a cap of $\pi$/3 solid angle if cut by a plane at the distance $\pi/a_0$ from the center of the sphere (fig.1c) The polar angle, $\theta$, fulfils cos($\theta$)=5/6 which provides a second equation for $k_F$.

$$k_F = \frac{1}{cos(\theta)} \frac{\pi}{a_0} = \frac{6}{5} \frac{\pi}{a_0}.$$ (3)

With (1) and (3) one obtains

$$\overline{Z} = \frac{\pi}{3} (\frac{6}{5})^3 = 1.80956.$$ (4)

This is the critical electron concentration to be obtained.

The present treatment proceeds as follows: (i) A model is postulated which describes the behaviour of the electrons at the Fermi surface. (ii) Employing this model, the conditions (3) and (4) are obtained which ensure that the electrons respond most efficiently to the ions. Former estimates of the critical electron concentration have accounted for the perturbing influence of the core electrons. The results scatter from 1.7 [3] to values between 1.7 and 2.5 [6]. Haeussler [4] estimated values below 2.0 in order to achieve a space-filling arrangement of Fermi spheres.

Our final relation (3) can be contrasted with empirical results for certain metallic alloys where the diffraction behaviour is reasonably approximated by packings of almost equally-sized spheres. Blétry [7] reported for such alloys that both the packing fraction, $\eta$, and the position of the first diffraction peak, $K_p$, are well represented by

$$(\eta)_{emp} = \frac{\pi}{6}\; \left(\frac{d}{a_0}\right)^3 \approx 0.535, \; \; (K_p)_{emp} \approx 7.64 \; \frac{1}{d}$$ (5)

where $d$ means the diameter of the averages sphere in the alloy. Upon eliminating the diameter $d$ one obtains

$$(K_p)_{emp} \approx 7.64 \left(\frac{\pi/6}{0.535}\right)^{... ...c{\pi}{a_0} \right\} = 2 \left\{1.207 \pi N_0^{1/3} \right\}.$$ (6)

This is just the demand of the Nagel-Tauc criterion if one adopts $k_F \approx 1.207 \; \pi/a_0$ which is very close to our result (3). We conclude that our $k_F$-$a_0$ relation (3) meets the situation of certain metallic glasses which exist obviously close to the conditions of most-constructive electronic interference.