Magnetic small-angle neutron scattering on bulk metallic glasses: A feasibility study for imaging displacement fields

Magnetic-field-dependent small-angle neutron scattering (SANS) has been utilized to study the magnetic microstructure of bulk metallic glasses (BMGs). In particular, the magnetic scattering from soft magnetic ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ and hard magnetic ${\left({\mathrm{Nd}}_{60}{\mathrm{Fe}}_{30}{\mathrm{Al}}_{10}\right)}_{92}{\mathrm{Ni}}_8$ alloys in the as-prepared, aged, and mechanically deformed state is compared. While the soft magnetic BMGs exhibit a large field-dependent SANS response with perturbations originating predominantly from spatially varying magnetic anisotropy fields, the SANS cross sections of the hard magnetic BMGs are only weakly dependent on the field, and their angular anisotropy indicates the presence of scattering contributions due to spatially dependent saturation magnetization. Moreover, we observe an unusual increase in the magnetization of the rare-earth-based alloy after deformation. Analysis of the SANS cross sections in terms of the correlation function of the spin misalignment reveals the existence of field-dependent anisotropic long-wavelength magnetization fluctuations on a scale of a few tens of nanometers. We also give a detailed account of how the SANS technique relates to unraveling displacement fields on a mesoscopic length scale in disordered magnetic materials.

Figure 1

Sketch of the perpendicular scattering SANS geometry, where the applied magnetic field ${\mathbf{H}}_0$ is perpendicular to the wave vector ${\mathbf{k}}_0$ of the incoming neutron beam. The scattering vector $\mathbf{q}$ is defined as the difference between the wave vectors of the scattered $\mathbf{k}$ and incident ${\mathbf{k}}_0$ neutrons. The mean wavelength $\lambda$ of the neutrons is selected by the velocity selector; $2\psi$ is the scattering angle, and $\theta$ is the angle between ${\mathbf{H}}_0$ and $\mathbf{q}$ in the plane of the detector.

Figure 2

Room-temperature magnetization curves of as-cast, aged, and deformed ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ (only the upper right quadrant is shown). The inset shows the field dependence of the neutron count rate of the samples normalized by the respective sample thickness. The measured $q$-range for the count rate is $0.035\lesssim q\lesssim 0.3{\mathrm{nm}}^{-1}$.

Figure 3

Two-dimensional total (nuclear and magnetic) unpolarized SANS cross sections $d\mathrm{\Sigma }/d\mathrm{\Omega }$ of as-cast, aged, and deformed BMG alloy ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ at selected applied magnetic fields (logarithmic color scale). ${\mathbf{H}}_0$ is horizontal in the plane. Note the relatively pronounced and sharp maximum (“spike”) of the $d\mathrm{\Sigma }/d\mathrm{\Omega }$ of the deformed sample at $70\mathrm{mT}$ and low $q$.

Figure 4

Comparison of the angular $\theta$-dependence of $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of as-cast, aged, and deformed ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ at ${\mu }_0{H}_0=70\mathrm{mT}$. Red corresponds to low $q$-values around the beamstop ($q\cong 0.035{\mathrm{nm}}^{-1}$), while blue represents high $q$-values from around the borders of the detector ($q\cong 0.2{\mathrm{nm}}^{-1}$).

Figure 5

Azimuthally averaged spin-misalignment SANS cross sections $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of as-cast, aged, and deformed ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ at selected applied magnetic fields (see the insets, log-log scale). The respective scattering at saturation ($1496\mathrm{mT}$) has been subtracted in each data set.

Figure 6

Comparison of $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of as-cast, aged, and deformed ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ at selected applied magnetic fields (see the insets, log-log scale).

Figure 7

Field dependence of the two-dimensional (normalized) correlation functions $c(y,z)$ [computed according to Eq. (14)] of as-cast, aged, and deformed ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ at selected applied magnetic fields (see the insets). ${\mathbf{H}}_0$ is horizontal in the plane. Dashed contour lines: $c(y,z)=exp(-1)$.

Figure 8

Field dependence of the correlation length $l_C$ of as-cast, aged, and deformed ${\mathrm{Fe}}_{70}{\mathrm{Mo}}_5{\mathrm{Ni}}_5{\mathrm{P}}_{12.5}{\mathrm{B}}_{2.5}{\mathrm{C}}_5$ (lines are guides to the eyes). Note that the $l_C$ were obtained using the correlation functions computed according to Eq. (15).

Figure 9

Room-temperature magnetization curves of as-cast, aged, and deformed ${\left({\mathrm{Nd}}_{60}{\mathrm{Fe}}_{30}{\mathrm{Al}}_{10}\right)}_{92}{\mathrm{Ni}}_8$ (only the upper quadrants are shown). The inset shows the field dependence of the neutron count rate of the samples normalized by the respective sample thickness. The measured $q$-range for the count rate is $0.035\lesssim q\lesssim 0.3{\mathrm{nm}}^{-1}$.

Figure 10

Comparison of the two-dimensional spin-misalignment SANS cross sections $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of as-cast, aged, and deformed ${\left({\mathrm{Nd}}_{60}{\mathrm{Fe}}_{30}{\mathrm{Al}}_{10}\right)}_{92}{\mathrm{Ni}}_8$ at selected applied magnetic fields (logarithmic color scale). $d\mathrm{\Sigma }/d\mathrm{\Omega }$ at $8\mathrm{T}$ has been subtracted from each dataset. ${\mathbf{H}}_0$ is horizontal in the plane.

Figure 11

Azimuthally averaged correlation function $c\left(r\right)$ of the deformed ${\left({\mathrm{Nd}}_{60}{\mathrm{Fe}}_{30}{\mathrm{Al}}_{10}\right)}_{92}{\mathrm{Ni}}_8$ sample at selected applied magnetic fields (see the insets). Dashed horizontal line: $c\left(r\right)=exp(-1)$. Note that the $c\left(r\right)$ were obtained by azimuthally averaging the two-dimensional correlation functions [computed according to Eq. (14)].

Figure 12

Field dependence of the correlation length $l_C$ of as-cast, aged, and deformed ${\left({\mathrm{Nd}}_{60}{\mathrm{Fe}}_{30}{\mathrm{Al}}_{10}\right)}_{92}{\mathrm{Ni}}_8$ (lines are guides to the eyes). The $l_C$ were computed using the $c\left(r\right)$ data displayed in Fig. 11.

Figure 13

(a) Computed spin misalignment (at ${\mu }_0{H}_0=0.6\mathrm{T}$) around a spherical pore (diameter: $12\mathrm{nm}$) in a ferromagnetic iron matrix (2D cut out of a 3D simulation). Shown is the magnetization component ${\mathbf{M}}_{\perp }\left(\mathbf{r}\right)$ perpendicular to ${\mathbf{H}}_0$; the thickness of the arrows is proportional to the magnitude of ${\mathbf{M}}_{\perp }$. Solid gray lines: magnetodipolar field distribution. (b) Comparison between experimental SANS data [38] and micromagnetic simulations for the spin-misalignment SANS cross section of nanoporous iron. Images taken from Ref. [39].