Quantum states of neutrons in magnetic thin films

We have studied experimentally and theoretically the interaction of polarized neutrons with magnetic thin films and magnetic multilayers. In particular, we have analyzed the behavior of the critical edges for total external reflection in both cases. For a single film we have observed experimentally and theoretically a simple behavior: the critical edges remain fixed and the intensity varies according to the angle between the polarization axis and the magnetization vector inside the film. For the multilayer case we find that the critical edges for spin-up and spin-down polarized neutrons move toward each other as a function of the angle between the magnetization vectors in adjacent ferromagnetic films. Although the results for multilayers and single thick layers appear to be different, in fact, the same spinor method explains both results. An interpretation of the critical edges behavior for the multilyers as a superposition of ferromagnetic and antifferomagnetic states is given.

Figure 1

Bottom: Hysteresis loop of the polycrystalline $\mathrm{Fe}∕\mathrm{Si}$ sample measured by MOKE. Top: The behavior of the remanent magnetization as a function of the rotation angle extracted from hysteresis loops.

Figure 2

(Color online). Top: Polarized neutron reflectivity curves ${\mathrm{R}}^{+}$ (solid black circles) and ${\mathrm{R}}^{-}$ (open black circles) of the $\mathrm{Fe}∕\mathrm{Si}$ sample. The black line is the simulated ${\mathrm{R}}^{+}$ reflectivity and the red (light-gray) line is the simulated ${\mathrm{R}}^{-}$ reflectivity. The applied magnetic field was 2000 Oe. In the inset, the magnetic profile obtained from fitting the data is shown. Bottom: Experimental (open black symbols) and simulated (black line) spin asymmetry $[(R^{+}-R^{-})∕(R^{+}+R^{-})]$ are plotted for the same sample in saturation. All lines in the top and bottom panels are fits to the data points using the GMM (for more details see text). The abscissa is the wave vector transfer: $Q=4\pi \sin \left(\alpha \right)∕\lambda$.

Figure 3

(Color online). Experimental (symbols) and simulated (lines) reflectivity curves (${\mathrm{R}}^{+}$ and ${\mathrm{R}}^{-}$) from $\mathrm{Fe}\left(1000\mathrm{\AA }\right)∕\mathrm{Si}$ sample. The abscissa is the wave-vector transfer. The two sets of $R^{+}$ (solid black symbols) and $R^{-}$ (open black symbols) reflectivity curves were measured for four different $\theta$ angles between the neutron polarization along ${\mathit{B}}_{\mathbf{0}}$ and the direction of the magnetic induction $\left(\mathit{B}\right)$, which lies in the sample plane. The guiding field is $B_0\approx 10\mathrm{Oe}$. The blue (thick dark-gray) lines are the simulated $\mathrm{R}+$ reflectivities and the red (thin light-gray) lines are the simulated $\mathrm{R}-$ reflectivities. In the right side the experimental geometry is shown. The figure shows that the critical edges $Q_{+}^c$ and $Q_{-}^c$ are not sensitive to the $\theta$ angle.

Figure 4

(Color online). Solid black symbols: The experimental normalized spin asymmetries $[nSA\left(\theta \right)=SA\left(\theta \right)∕\left(SA\left(0\right)\right)]$, plotted as a function of the wave-vector transfer $Q$. Lines: The three lines are the cosines of the corresponding angles set during the experiment. From top: cosine of 48° [thin gray (red) line], 68° [thick gray (blue) line], and 88° (thin black line), respectively. The angles are the experimental rotation angles $\theta$ used during the experiment shown in Fig. 3. The experimental normalized spin asymmetries are assembled from the $R^{+}$ and $R^{-}$ reflectivities shown in Fig 3. This figure shows that the equation $nSA\left(\theta \right)=\cos \left(\theta \right)$ is valid over the whole $Q$ range for a single magnetic layer. It can be used to extract the angle $\theta$ directly from the experimental reflectivities.

Figure 5

(Color online). Top: Simulation of polarized neutron reflectivities [${\mathrm{R}}^{+}$ (dashed lines, shifting from right to left), ${\mathrm{R}}^{-}$ (solid lines, shifting left to right)] for a $\mathrm{Fe}∕\mathrm{Cr}$ multilayer as a function of coupling angle $\left(\gamma \right)$ between the magnetization vectors of adjacent Fe layers. Bottom: Simulations of ${\mathrm{R}}^{+}$ (solid lines, shifting from top down) and ${\mathrm{R}}^{-}$ (dashed lines, shifting from bottom up) as a function of rotation angle $\theta$ for $\gamma =90°$.