## 2 Probabilities of Spin-Dependent Scattering by Nanomagnets
Integration in Equation (6) with respect to \(p_{f}\) with account of the relation \(dp_{f}^{2}=1/2d(p_{f}^{2})d\varphi\) gives the following expression of the differential scattering length  
\[dL(k,\varphi;S_{f},S_{i})=\frac{m^{2}}{2\pi\hbar^{4}k}\bigg{|}\langle\vec{ \chi}(S_{f})|\int e^{-i\vec{q}\cdot\vec{\rho}}\hat{V}(\vec{\rho})d^{2}\rho| \vec{\chi}(S_{i})\rangle\bigg{|}^{2}d\varphi \tag{9}\]  
where \(\vec{q}=\vec{k}-\vec{k}_{0}\), is the transferred momentum, \(\vec{k}_{0}=\vec{p}_{i}/\hbar\), \(\vec{k}=\vec{p}_{f}/\hbar\), \(k_{0}^{2}=k^{2}=\frac{2 mE}{\hbar^{2}}\), \(q=2k\sin\frac{\varphi}{2}\), \(\varphi\) is the scattering angle. Integration in Equation (6) with account of the delta function means that we consider only elastic collisions. Because of the spin-dependent character of the scattering, the differential scattering length Equation (9) includes the process with spin flipping and non-spin flipping. To distinguish them, it is better to use the spin-dependent scattering amplitudes introduced in [10]. Comparison of Equation (9) with corresponding result of [10] allows one to write down the following expression  
\[dL(k,\varphi;S_{f},S_{i})=\big{|}\langle\vec{\chi}(S_{f})|\vec{f}(k,\varphi) \rangle\big{|}^{2} \tag{10}\]where the scattering amplitude \(\vec{f}(k,\varphi)\) is the two component spinor given by the relation  
\[\hat{H}=\frac{1}{2m}\hat{p}_{2}^{2}+\hat{V}(\vec{\rho}) \tag{2}\]0  
It is worth noting that from Equation (10) one can not obtain the minus sign in Equation (11). It is obvious that the usage of the perturbation theory does not allow to obtain the phase factor [11]. The scattering amplitude Equation (11) was obtained in [10] with the help of the Born approximation. Its applicability coincides with the applicability of the perturbation theory \(|\hat{V}|\ll\hbar^{2}/(m\bar{\rho}^{2})\), where \(\bar{\rho}\) is the range of action of the scattering potential \(\hat{V}(\rho)\), which is given by Equation (3). For evaluation, we set \(|V|\sim\frac{\mu\mu_{B}}{\rho^{2}}\). Due to the fast decay of the magnetic dipole-dipole interaction with distance, we take \(\bar{\rho}=2a\). Keeping this in mind, we obtain that the Born approximation is applicable if the following inequality is true [10]  
\[\hat{H}=\frac{1}{2m}\hat{p}_{2}^{2}+\hat{V}(\vec{\rho}) \tag{2}\]1  
For the magnetic moment of nanoparticle, we used the formula \(\mu=\frac{4\pi}{3}\
u\mu_{B}n_{a}a^{3}\), \(\
u\) is a number of Bohr magnetons carried by the ferromagnetic atom, and \(n_{a}\) is the density of atoms of the nanomagnet. Substituting the numerical values of universal constants and \(n_{a}=10^{22}cm^{-3}\) we obtain the following restriction on the size of the nanomagnet  
\[\hat{H}=\frac{1}{2m}\hat{p}_{2}^{2}+\hat{V}(\vec{\rho}) \tag{2}\]2  
For the typical \(\
u\) Equation (13) gives \(a\) of the order of \(10\ nm\) that allows us to consider only one domain nanomagnets. The perturbation Equation (3) is non-spherically symmetric and depends on the spin degrees of freedom of the electron. This creates some problems while evaluating integral Equation (11). Basically the scattering amplitude is a two component spinor that depends on spin polarization of incident electron.