## 3 Scattering Amplitudes and Scattering Lengths
### Magnetic Moment of Nanomagnet Perpendicular to Velocity of Incident Electron
The ratio of numbers of the scattered electrons with spin up and spin down depends on \(\varphi\). Figure 2(b) illustrates the scattering lengths for \(ka=0.2\). At \(\varphi\simeq\pi/2\), the ratio of \(\tilde{L}_{\uparrow}^{\parallel}\) to \(\tilde{L}_{\downarrow}^{\parallel}\) is about 4. This is to mean that around the scattering angle \(\varphi=\pi/2\) rad, about 80% of the scattered electrons have spin up and 20% spin down. However, the values of the corresponding scattering lengths are very small and become smaller with further increase in \(ka\). Figure 3(a) and 3(b) illustrates the scattering of unpolarized beams of electron by nanomagnets for \(ka=0.5\) and \(ka=1\), respectively. As one can see, the scattering lengths become smaller as the energy of the incident electron increases. The polarization of the beams considerably increases, and for particular values of \(\varphi\) one can get almost complete polarization. Figure 3(a) reveals that around \(\varphi=\pi/2\) rad practically all scattered electrons have spin up and around \(\varphi=3\pi/2\) rad they have spin down. The complete polarization can also be seen in Figure 3(b) at \(\varphi=1\) and \(\varphi=4\) rad (spin up) and at \(\varphi=2.3\) and \(\varphi=5.3\) rad (spin down). The inspections of Figure 3(a) and 3(b) shows that in the range \(0.5<ka<1\), the scattering length roughly behaves like \(1/ka\). Let us consider the scattering of unpolarized beams of electrons with initial velocity perpendicular to \(\vec{\mu}\). Figure 4(a) and 4(b) presents the corresponding scattering lengths of electrons for \(ka=0.01\) and \(ka=0.2\), respectively. The properties of this scattering are practically the same compared to the above described Figure 2. Particularly, for \(ka=0.01\), the scattering is anisotropically weak, and for \(ka=0.2\), ratio of the scattering lengths for electrons with spin up and spin down around \(\varphi=\pi/2\) rad is close to \(4\), just as the case of \(\vec{\mu}\) parallel to \(\vec{v}\). **\[f_{\downarrow}^{\perp}(\varphi)=\sqrt{L_{0}}\big{[}\alpha\sqrt{\tilde{L}_{2}} -\beta\sqrt{\tilde{L}_{3}}\big{]} \tag{34}\]0**\(\tilde{L}_{\uparrow}^{\perp}\) (curve I) and \(\tilde{L}_{\downarrow}^{\perp}\) (curve II) _versus_\(\varphi\) (in radian). **(a)** for \(ka=0.01\), and **(b)** for \(ka=0.2\) according to Equations (40) and (41). Figure 5(a) and 5(b) indicates the scattering lengths \(\tilde{L}_{\uparrow}^{\perp}\) (curve I) and \(\tilde{L}_{\downarrow}^{\perp}\) (curve II) _versus_\(\varphi\), respectively. Just as the case of \(\vec{\mu}\) parallel to \(\vec{v}\), we have complete polarization of the electron beams for particular values of \(\varphi\). For example, for \(ka=0.5\) around \(\varphi=1\) and \(2\) rad practically all electrons have spin up and around \(\varphi=4.3\) and \(5.3\) rad almost all electrons have spin down. Figure 5(b) shows that around \(\varphi=0.5\) and \(2.5\) rad practically all spins of electrons pointed up and for \(\varphi=3.8\) and \(5.8\) rad their spins are down. The dimensionless differential scattering lengths behave like \(1/ka\) in the range of \(0.5\leq ka\leq 1\). From the above graphs it is possible to conclude that at particular scattering angles in the range of \(0.5\leq ka\leq 1\), the initially unpolarized beam of electrons becomes practically completely polarized. However, the corresponding scattering lengths are relatively small. The scattering lengthes can considerably increase through interference effects of periodical structures of nanomagnets.