## 5 Diffraction of Electrons by Linear Chains of Nanomagnets  
Consider the scattering of electrons from a linear chain of equally spaced nanomagnets with lattice constant \(d\). We assume that all nanomagnets are identical. The magnetic moments of the nanomagnets are directed along the \(z\)-axis (see Figure 6). This linear chain of nanomagnets can be treated as a diffraction grating for 2D electron beams. According to the Huygens-Fresnel principle, every individual nanomagnet can be treated as the source of the cylindrical waves. In the case of 2D spin-dependent scattering by nanomagnets, the amplitude of the scattered wave depends on the scattering angle. As it was illustrated in the previous section, there are some scattering angles (for particular energy of incident electrons) where the unpolarized incident beams of electrons is practically completely polarized. Particularly, according to Figure 3(a), the scattered particles with \(ka=0.5\) are completely polarized at \(\varphi=1.5\) rad. Similarly, it follows from Figure 3(b) that complete polarization is observed for scattered electrons with \(ka=1\) at \(\varphi=1\) rad.  
Consider the case when all path differences \(\Delta\) (\(\Delta=d\sin\varphi\)) between the neighboring scattered beams are the same. This condition allows us to find the distance between nanomagnets when all the scattering amplitudes added coherently (if they put together by focusing the electrons beams far from the chain of the nanomagnets). This condition gives the following relation for the distance between the nanomagnets  
\[d=\frac{2\pi a}{ka\sin\varphi_{d}} \tag{42}\]  
where \(a\) is the radius of individual nanomagnets and \(\varphi_{d}\) is the scattering angle corresponding to almost the complete polarization. Let us apply this relation to the case \(ka=0.5\) and \(\varphi_{d}\)=1.5 rad=87\({}^{\circ}\). Substituting this parameters in Equation (42), we get \(d=4\pi a\gg a\). The last condition is favorable for not taking into account the multiple scattering by the neighboring nanomagnets. The typical size of the nanomagnet \(a\) must satisfy inequality Equation (13). For large nanomagnets, the magnetization vectors of the scatterers in the chain will wander and destroy constructive interference.  
It is clear that the resultant scattering amplitude at the scattering angle \(\varphi_{d}\) is proportional to \(N\) (\(N\) is the number of the scatterers). The total scattering length corresponding to \(N\) scatterers is \(\sim N^{2}\tilde{L}_{\uparrow}^{\parallel}\simeq 2.5\times 10^{-2}N^{2}\). Even setting \(N=10\), we get considerable increment in the scattering lengths.  
We considered the chain of scatterers with magnetic moment parallel to \(z\)-axis (Figure 6). In the same manner, one can consider the chain of nanomagnets with the magnetic moments transverse to \(z\)-axis. In this case, the largest scattering length of complete polarization \(\tilde{L}_{\uparrow}^{\perp}\simeq 0.034\) is obtained for \(ka=0.5\) at \(\varphi=2\) rad (Figure 5(a)). It is similar to the chain with \(\vec{\mu}\) parallel to \(z\)-axis. However, it is necessary to note that the scattering angle \(\varphi=2\) rad relates to the back scattering that may complicate the experimental verification of the theory.