## 3 Scattering Amplitudes and Scattering Lengths  
### Magnetic Moment of Nanoparticle Parallel to the Velocity of Incident Electron  
The scattering amplitude Equation (11) may be presented as  
\[\vec{f}(k,\varphi)=-\hat{I}\vec{\chi}(S_{i}) \tag{14}\]  
where the operator \(\hat{I}\) is given by  
\[\hat{I}=\frac{m}{\hbar^{2}\sqrt{2\pi k}}\int e^{-i\vec{\mu}\vec{\rho}^{\prime }}\hat{V}(\vec{\rho^{\prime}})d^{2}\rho^{\prime} \tag{15}\]  
To evaluate integral Equation (15) one has to choose appropriate coordinate system. Figure 1 shows two coordinate systems. In the laboratory coordinate system \((x,z)\), the magnetic moment of nanoparticle \(\vec{\mu}\) is along the \(z\)-axis and the velocity of the incident electron is parallel to \(\vec{\mu}\). The calculation of Equation (15) is, however, convenient to carry out in other coordinate system \((x^{\prime},z^{\prime})\) with \(x^{\prime}\)-axis directed along \(\vec{q}\), [10]. In this coordinate system, the dot product in the exponent Equation (15) has the simplest form \(\vec{q}\cdot\vec{\rho^{\prime}}=q\rho^{\prime}\cos\varphi^{\prime}\) (\(\rho^{\prime}\) and \(\varphi^{\prime}\) are variables of integration), and Equation (15) can be written as  
\[\hat{I}=\frac{m}{\hbar^{2}\sqrt{2\pi k}}\int_{0}^{2\pi}\int_{a}^{\infty}e^{-iq \rho^{\prime}\cos\varphi^{\prime}}\hat{V}(\vec{\rho^{\prime}})\rho^{\prime}d \rho^{\prime}d\varphi^{\prime} \tag{16}\]  
where \(a\) is the radius of the nanomagnet, and the operator of potential energy Equation (3) is given by  
\[\hat{V}=\frac{\mu_{B}}{\rho^{\prime 3}}\bigg{\{}\bigg{[}\mu_{x^{\prime}} \big{(}3\cos^{2}\varphi^{\prime}-1\big{)}+\frac{3}{2}\mu_{z^{\prime}}\sin 2 \varphi^{\prime}\bigg{]}\hat{\sigma}_{x}+\bigg{[}\mu_{z^{\prime}}\big{(}3\sin ^{2}\varphi^{\prime}-1\big{)}+\frac{3}{2}\mu_{x^{\prime}}\sin 2\varphi^{ \prime}\bigg{]}\hat{\sigma}_{z}\bigg{\}} \tag{17}\]  
In the prime coordinate system, the magnetic moment \(\mu\) has two components \(\mu_{z^{\prime}}=\mu\cos\frac{\varphi}{2}\), and \(\mu_{x^{\prime}}=-\mu\sin\frac{\varphi}{2}\). Inserting Equation (17) into Equation (16) and carrying out integration respect to \(\varphi^{\prime}\) we obtain  
\[\hat{I}^{\prime} =\frac{6\pi\gamma}{a}\bigg{(}-I_{1}\sin\frac{\varphi}{2}\hat{ \sigma}_{x}+I_{2}\cos\frac{\varphi}{2}\hat{\sigma}_{z}\bigg{)} \tag{18}\] \[I_{1} =qa\int_{qa}^{\infty}\frac{1}{x^{2}}\big{[}\frac{2}{3}J_{0}(x)- \frac{1}{x}J_{1}(x)\big{]}dx\] (19) \[I_{2} =qa\int_{qa}^{\infty}\frac{1}{x^{2}}\big{[}\frac{1}{x}J_{1}(x)- \frac{1}{3}J_{0}(x)\big{]}dx \tag{20}\]  
where \(\gamma=-\frac{m\mu_{B}}{\hbar^{2}\sqrt{2\pi k}}\), \(J_{0}(x)\), and \(J_{1}(x)\) are the first kind Bessel functions of the zeroth, and first order, respectively. Details of calculations of the integrals \(I_{1}\) and \(I_{2}\) are given in Appendixes \((A1-A8)\). In Equations (19) and (20), we introduced new variable of integration \(x=q\rho^{\prime}\). It is necessary to remember that result Equation (18) is obtained in the \(x^{\prime}-z^{\prime}\) coordinate system. To get the correct expression of the scattering amplitudes in the laboratory coordinate system, we can proceed in the following way: transform the spinor \(\vec{\chi}(S_{i})\) to \(x^{\prime}-z^{\prime}\) coordinate system, find the result \(\hat{I}^{\prime}\vec{\chi}(S_{i})\), and transform this result back to the laboratory coordinate system. Let us consider the unpolarized beam of electrons, which is in the \(x-z\) coordinate given by the linear combination of the spinors  
\[\vec{\chi}(S_{i})=\alpha\left| \begin{array}{c}1\\ 0\end{array} \right|+\beta\left| \begin{array}{c}0\\ 1\end{array} \right| \tag{21}\]  
with \(\alpha^{2}+\beta^{2}=1\), \(\alpha^{2}\) is the fraction of electrons whose spin is parallel to their initial velocity (along \(z\)-axis) and \(\beta^{2}\) is the fraction of electrons with opposite spin. By carrying out the above specified operations, we obtain  
\[\vec{f}(\varphi)=f^{\parallel}_{\uparrow}(\varphi)\left( \begin{array}{c}1 \\ 0\end{array} \right)+f^{\parallel}_{\downarrow}(\varphi)\left(