## 3 Scattering Amplitudes and Scattering Lengths
### Magnetic Moment of Nanomagnet Perpendicular to Velocity of Incident Electron  
Now we consider the case when the magnetic moment \(\vec{\mu}\) is transverse to \(z\)-axis and the velocity \(\vec{v}\) of electrons along the \(z\)-axis. Let \(\vec{\mu}\) be along positive \(x\)-axis. Acting as in the pervious subsection and taking in to account that the components of the magnetic moment of the nanomagnet in Equation (17) are \(\mu_{x^{\prime}}=\mu\cos\frac{\varphi}{2}\) and \(\mu_{z^{\prime}}=\sin\frac{\varphi}{2}\), we obtain the following scattering amplitudes  
\[\vec{f}(\varphi)=f_{\uparrow}^{\perp}(\varphi)\left( \begin{array}{c}1\\ 0\end{array} \right)+f_{\downarrow}^{\perp}(\varphi)\left( \begin{array}{c}0 \\ 1\end{array} \right) \tag{32}\]  
where \(f_{\uparrow}^{\perp}(\varphi)\) and \(f_{\downarrow}^{\perp}(\varphi)\) represent the spin-dependent 2D scattering amplitudes of electron with initial velocity perpendicular (\(\perp\)) to \(\vec{\mu}\) with the spin up and down after scattering, respectively. The scattering amplitudes \(f_{\uparrow}^{\perp}(\varphi)\) and \(f_{\downarrow}^{\perp}(\varphi)\) may be presented as  
\[f_{\uparrow}^{\perp}(\varphi)=\sqrt{L_{0}}\big{[}\beta\sqrt{\tilde{L}_{2}}+ \alpha\sqrt{\tilde{L}_{3}}\big{]} \tag{33}\]  
\[f_{\downarrow}^{\perp}(\varphi)=\sqrt{L_{0}}\big{[}\alpha\sqrt{\tilde{L}_{2}} -\beta\sqrt{\tilde{L}_{3}}\big{]} \tag{34}\]  
where we used Equations (27) and (29) together with  
\[\tilde{L}_{2}=\frac{1}{ka}\big{[}I_{1}\cos^{2}\frac{\varphi}{2}+I_{2}\sin^{2} \frac{\varphi}{2}\big{]}^{2} \tag{35}\]  
The differential scattering lengths can be presented as  
\[|f_{\uparrow}^{\perp}(ka,\varphi)|^{2}=L_{0}\bigg{[}\beta\sqrt{ \tilde{L}_{2}}+\alpha\sqrt{\tilde{L}_{3}}\bigg{]}^{2} \tag{36}\] \[|f_{\downarrow}^{\perp}(ka,\varphi)|^{2}=L_{0}\bigg{[}\alpha \sqrt{\tilde{L}_{2}}-\beta\sqrt{\tilde{L}_{3}}\bigg{]}^{2} \tag{37}\]  
The next section discusses the general properties of the differential scattering lengths (30, 31) and (36, 37) as functions of the scattering angle \(\varphi\) and the dimensionless parameter \(ka\), which characterizes the energy of unpolarized beams of incident electrons \(\alpha=\beta=1/\sqrt{2}\). 4 Graphical Presentations of the Scattering Lengths of Unpolarized Beams of Electrons by Nanomagnets  
The above formulas help us to specify peculiarities of scattering of unpolarized beams of electrons. Consider the scattering of electrons initially moving parallel to \(\vec{\mu}\) (Equations (30) and (31)) and perpendicular to \(\vec{\mu}\) (Equations (36) and (37)). Thus, we present the graphs of the following dimensionless differential scattering lengths  
\[\tilde{L}_{\uparrow}^{\parallel}=\frac{1}{L_{0}}\frac{dL_{\uparrow}^{ \parallel}}{d\varphi}=\frac{1}{L_{0}}\big{[}f_{\uparrow}^{\parallel}(ka, \varphi)\big{]}^{2} \tag{38}\]  
\[\tilde{L}_{\downarrow}^{\parallel}=\frac{1}{L_{0}}\frac{dL_{\downarrow}^{ \parallel}}{d\varphi}=\frac{1}{L_{0}}\big{[}f_{\downarrow}^{\parallel}(ka, \varphi)\big{]}^{2} \tag{39}\]  
\[\tilde{L}_{\uparrow}^{\perp}=\frac{1}{L_{0}}\frac{dL_{\uparrow}^{\perp}}{d \varphi}=\frac{1}{L_{0}}\big{[}f_{\uparrow}^{\perp}(ka,\varphi)\big{]}^{2} \tag{40}\]\[\tilde{L}_{\downarrow}^{\perp}=\frac{1}{L_{0}}\frac{dL_{\downarrow}^{\perp}}{d \varphi}=\frac{1}{L_{0}}\big{[}f_{\downarrow}^{\perp}(ka,\varphi)\big{]}^{2} \tag{41}\]  
In [10], we obtained that 2D spin-dependent scattering of polarized beams of electrons by nanomagnets is practically isotropic for very slow particles \(ka\ll 1\). It is true for the scattering of unpolarized beams of slow electrons as well. Let us consider the scattering of electrons with the initial velocity \(\vec{v}\) parallel to \(\vec{\mu}\). Figure 2(a) shows the scattering lengths for spin up (curve I) and spin down (curve II). The maximum difference between these scattering lengths is 8% for \(\varphi=2\) rad. It allows us to claim that the scattering is anisotropically weak. With increasing in \(ka\), the scattering becomes anisotropic.