Magnetic small-angle neutron scattering

Small-angle neutron scattering (SANS) is one of the most important techniques for microstructure determination, being utilized in a wide range of scientific disciplines, such as materials science, physics, chemistry, and biology. The reason for its great significance is that conventional SANS is probably the only method capable of probing structural inhomogeneities in the bulk of materials on a mesoscopic real-space length scale from roughly 1 to 300 nm. Moreover, the exploitation of the spin degree of freedom of the neutron provides SANS with a unique sensitivity to study magnetism and magnetic materials at the nanoscale. As such, magnetic SANS ideally complements more real-space and surface-sensitive magnetic imaging techniques, e.g., Lorentz transmission electron microscopy, electron holography, magnetic force microscopy, Kerr microscopy, or spin-polarized scanning tunneling microscopy. This review summarizes the recent applications of the SANS method to study magnetism and magnetic materials. This includes a wide range of materials classes from nanomagnetic systems such as soft magnetic Fe-based nanocomposites, hard magnetic Nd-Fe-B-based permanent magnets, magnetic steels, ferrofluids, nanoparticles, and magnetic oxides to more fundamental open issues in contemporary condensed matter physics such as skyrmion crystals, noncollinear magnetic structures in noncentrosymmetric compounds, magnetic or electronic phase separation, and vortex lattices in type-II superconductors. Special attention is paid not only to the vast variety of magnetic materials and problems where SANS has provided direct insight, but also to the enormous progress made regarding the micromagnetic simulation of magnetic neutron scattering.

Figure 1

Schematic of the SANS setup and the two commonly employed scattering geometries in magnetic SANS experiments. (a) ${\mathbf{k}}_0\perp {\mathbf{H}}_0$; (b) ${\mathbf{k}}_0\parallel {\mathbf{H}}_0$. The scattering vector $\mathbf{q}$ is defined as $\mathbf{q}={\mathbf{k}}_1-{\mathbf{k}}_0$, where ${\mathbf{k}}_0$ and ${\mathbf{k}}_1$ are the wave vectors of the incident and scattered neutrons; $q=|\mathbf{q}|=(4\pi /\lambda )\sin (\psi /2)$ depends on the mean wavelength $\lambda$ of the neutrons and on the scattering angle $\psi$. The symbols “P,” “F,” and “A” denote, respectively, the polarizer, spin flipper, and analyzer, which are optional neutron optical devices. $SD=\text{sample}\text{−}\text{to}\text{−}\text{detector}\text{distance}$; $r=\text{radial distance on the detector}$ (measured from the beam center). SANS usually assumes elastic scattering ($k_0=k_1=2\pi /\lambda$), and the component of $\mathbf{q}$ along the incident neutron beam [i.e., $q_x$ in (a) and $q_z$ in (b)] is neglected. The azimuthal angle $\theta$ describes the angular anisotropy of the recorded scattering pattern on a two-dimensional position-sensitive detector. The applied magnetic field ${\mathbf{H}}_0$ is always taken parallel to ${\mathbf{e}}_z$, in this way defining the longitudinal magnetization.

Figure 2

(a)–(d) Contour plots of normalized $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ at applied magnetic fields as indicated (${\mathbf{k}}_0\perp {\mathbf{H}}_0$; ${\mathbf{H}}_0$ is horizontal); the scattering of the saturated state has been subtracted. (e)–(h) Corresponding two-dimensional correlation functions $c(y,z)$ [$=2\mathrm{D}$ Fourier transforms of $d{\mathrm{\Sigma }}_M(q_y,q_z)/d\mathrm{\Omega }$]. The DMI has not been taken into account. From [431].

Figure 3

(a) Spike anisotropy observed in the total unpolarized $d\mathrm{\Sigma }/d\mathrm{\Omega }$ of a sintered Nd-Fe-B-based permanent magnet in the remanent state (${\mathbf{k}}_0\perp {\mathbf{H}}_0$). (b) $d\mathrm{\Sigma }/d\mathrm{\Omega }$ vs azimuthal angle $\theta$ at $q=0.10\pm 0.02{\mathrm{nm}}^{-1}$. From [506].

Figure 4

Contour plots of the spin-flip difference cross section $-2i\chi (\mathbf{q})$ at selected applied magnetic fields (${\mathbf{k}}_0\perp {\mathbf{H}}_0$). From [443].

Figure 5

(a) Azimuthally averaged $d\mathrm{\Sigma }/d\mathrm{\Omega }$ of the two-phase alloy $({\mathrm{Fe}}_{0.985}{\mathrm{Co}}_{0.015}{)}_{90}{\mathrm{Zr}}_7{\mathrm{B}}_3$ at selected applied magnetic fields (log-log scale). Field values (in mT) from bottom to top: 1270, 312, 103, 61, and 33. Solid lines: fits to the micromagnetic theory. The inset depicts the mean-square deviation between experiment and fit, ${\chi }^2/\nu$, as a function of the exchange-stiffness constant $A$. (b) Best-fit results for the scattering function of the anisotropy field $S_H\propto |{\tilde{H}}_p(q){|}^2$ and for the scattering function of the longitudinal magnetization $S_M\propto |{\tilde{M}}_z(q){|}^2$. $d{\mathrm{\Sigma }}_{\mathrm{nuc}}/d\mathrm{\Omega }$ denotes the nuclear SANS (log-log scale). From [269].

Figure 6

The correlation length $l_C$ of the spin misalignment. (Left) Computed spin misalignment (at ${\mu }_0{H}_0=0.6\mathrm{T}$) around a spherical pore ($2R=12\mathrm{nm}$) in a ferromagnetic iron matrix (two-dimensional cut out of a three-dimensional simulation). Shown is the magnetization component ${\mathbf{M}}_{\perp }(\mathbf{r})$ perpendicular to ${\mathbf{H}}_0\parallel {\mathbf{e}}_z$; the thickness of the arrows is proportional to the magnitude of ${\mathbf{M}}_{\perp }$. Solid gray lines: magnetodipolar field distribution. The correlation length $l_C$ is a measure for the size of the inhomogeneously magnetized region around the defect; $l_C$ consists of a field-independent contribution $R$, which specifies the structural size of the defect, and of a field-dependent exchange length $l_H$, which transmits the perturbation at the pore-matrix interface into the surrounding crystal lattice. (Right) Corresponding magnetization Fourier components $|{\tilde{M}}_x(\mathbf{q}){|}^2$ and $|{\tilde{M}}_y(\mathbf{q}){|}^2$ projected into the plane $q_x=0$. The bright colors correspond to “high” values and the dark colors to “low” values of the Fourier components. Pixels in the corners of the images have $q\cong 0.4{\mathrm{nm}}^{-1}$. A logarithmic color scale is used. From [168].

Figure 7

(a) Azimuthally averaged total SANS cross section $d\mathrm{\Sigma }/d\mathrm{\Omega }$ of ${\mathrm{Nd}}_2{{\mathrm{Fe}}_{14}\mathrm{B}/\mathrm{Fe}}_3\mathrm{B}$ as a function of momentum transfer $q$ and applied magnetic field $H_0$ ($T=300\mathrm{K}$) (${\mathbf{k}}_0\perp {\mathbf{H}}_0$) (log-log scale). Solid circles: applied-field values (in Tesla) decrease from bottom to top: 10, 6, 1, $-0.25$, and $-0.55$; squares: $-1\mathrm{T}$; triangles: $-3\mathrm{T}$. Inset: room-temperature magnetization curve of ${\mathrm{Nd}}_2{{\mathrm{Fe}}_{14}\mathrm{B}/\mathrm{Fe}}_3\mathrm{B}$. (b) Applied-field dependence of the spin-misalignment SANS cross section $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ of nanocrystalline ${\mathrm{Nd}}_2{{\mathrm{Fe}}_{14}\mathrm{B}/\mathrm{Fe}}_3\mathrm{B}$. Solid circles: field values (in Tesla) decrease from bottom to top: 6, 1, $-0.25$, and $-0.55$; squares: $-1\mathrm{T}$; triangles: $-3\mathrm{T}$. The $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ data displayed in (b) were obtained by subtracting the 10 T data shown in (a) from the $d\mathrm{\Sigma }/d\mathrm{\Omega }$ at lower fields. Dashed line: $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }\propto q^{-5.5}$. From [61].

Figure 8

Field dependence of the correlation function $C(r)$ of the spin misalignment of nanocrystalline ${\mathrm{Nd}}_2{{\mathrm{Fe}}_{14}\mathrm{B}/\mathrm{Fe}}_3\mathrm{B}$ (log-linear scale). The field values follow the course of a hysteresis loop, starting from a large positive field and then reducing the field to negative values (see insets). Dotted line (extrapolating the 6 T data to $r=0$): $C(r)=4.58-0.043r^2$. From [61].

Figure 9

(a) Applied-field dependence of the correlation length $l_C$ of the spin misalignment of nanocrystalline ${\mathrm{Nd}}_2{{\mathrm{Fe}}_{14}\mathrm{B}/\mathrm{Fe}}_3\mathrm{B}$. Solid line: fit of the data to Eq. (32), where $R=10.9\mathrm{nm}$ and ${\mu }_0{H}^{\star }=+0.60\mathrm{T}$ are treated as adjustable parameters, and the quantities $A=12.5\mathrm{pJ}/\mathrm{m}$ and ${\mu }_0{M}_s=1.6\mathrm{T}$ are held fixed. In addition to $l_C(H_0)$ data obtained at the instrument Quokka (ANSTO), results obtained at the SANS instruments KWS 1 [Jülich Center for Neutron Science (JCNS)] and D11 [Institut Laue-Langevin (ILL)] are also shown. Dashed horizontal line: average radius of the ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$ particles ($R=11\mathrm{nm}$). Dotted vertical line: coercive field ${\mu }_0{H}_c=-0.55\mathrm{T}$. From [61]. (b) Normalized correlation function $C(r)$ of the spin misalignment for a textured (hot-deformed) and isotropic ${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}/\alpha \text{−}\mathrm{Fe}$ nanocomposite in the remanent state. The $C(r)$ of the textured sample is computed by using $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ averaged along the vertical and horizontal directions ($\pm 7.5°$ sector averages) as well as using the full circular ($2\pi$) average of $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$; the $C(r)$ of the isotropic sample is computed using the corresponding $2\pi$-averaged $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ (see inset). Solid horizontal line: $C(r)=e^{-1}$. Data from [448].

Figure 10

The effect of the grain-boundary diffusion process (GBDP) on the field dependence of the correlation length $l_C$ of the spin misalignment of isotropic sintered Nd-Fe-B. The inset shows the hysteresis loops of the as-received (AR) and GBDP samples. Data from [507] and [508].

Figure 11

Trial-and-error procedure to specify the composition $c_{\mathrm{prec}}$ of precipitates for the case of nonmagnetic precipitates in ferromagnetic steels. The blue color indicates known or directly derived quantities. The red color indicates trials to be checked by way of comparison of both theoretical with experimental $A$ ratio and calculated with experimental bulk composition. The numbers refer to the sequence of steps (see the main text).

Figure 12

Effect of neutron flux (for equal neutron exposure of $0.032\mathrm{dpa}$) on the size distribution of irradiation-induced CRPs for an RPV weld (0.22 wt % Cu, 1.1 wt % Mn, 1.1 wt % Ni). The measured magnetic scattering is shown in the inset. Low and high fluxes correspond to $0.087\times {10}^{-9}$ and $3.05\times {10}^{-9}\mathrm{dpa}/\mathrm{s}$ (11.6 years of irradiation), respectively. Data from [52], nonmagnetic scatterers assumed. The size of CRPs decreases, and the volume fraction is roughly constant.

Figure 13

The effect of neutron exposure on $A(q)$ for neutron-irradiated ${\mathrm{Fe}}_{1-x}{\mathrm{Cr}}_x$ ($x=0.125$). The measured magnetic scattering is shown in the inset. The unirradiated reference was subtracted before calculating $A(q)$. Theoretical $A$ ratios are indicated.

Figure 14

(a) Experimental SANS data of a Co-based ferrofluid. The black lines are the results of a simultaneous fit of Eq. (46) for different external fields. The inset shows the core-shell particle with confidence intervals for the core radius (green dashed line) and shell thickness (pink dashed line). (b) Fits of the cross term lead to dashed curves within the full flipper on and off data. The missing intensity originates from nonmagnetic contributions and can be fitted separately. From [252].

Figure 15

Field dependence of the magnetic SLD ${\eta }_m$ in the nanoparticle core of maghemite nanospheres (top) and nanocubes (bottom) with fits of the Langevin behavior. Inset: spatial magnetization distribution of the nanospheres (SANS) compared to the macroscopic [vibrating sample magnetometry (VSM)] and the theoretical bulk maghemite moments (dashed line: 300 K). From [153].

Figure 16

SANS patterns of iron oxide nanoparticles in solution, exposed to external magnetic fields aligned vertically. (f) The calculated reflections of a face-centered cubic superstructure are shown as white circles. From [195].

Figure 17

(a) Schematics of the two scattering geometries and the microscopic structure of the nanocomposite sample with simulation volume $V=250\times 600\times 600{\mathrm{nm}}^3$. Blue polyhedrons: Fe particles; yellow, orange, and red polyhedrons: the matrix phase. From [441]. (b) Examples of the spatial distribution of hard magnetic crystallites (soft crystallites not shown) with different aspect ratios of corresponding ellipsoids of revolution. From [165]. (c) Core-shell microstructure used in the micromagnetic simulations of a Nd-Fe-B nanocomposite and example of a polyhedron mesh-element distribution in a single grain; blue (yellow) elements correspond to the core (shell), typical mesh-element size is 2 nm.

Figure 18

(a) Results of a micromagnetic simulation for the spin distribution around two selected Fe nanoparticles (blue circles), which are assumed to be in a single-domain state. The external magnetic field $\mathbf{H}$ is applied horizontally in the plane. The magnetization component ${\mathbf{M}}_{\perp }$ perpendicular to $\mathbf{H}$ is shown by red arrows; the thickness of the arrows is proportional to the magnitude of ${\mathbf{M}}_{\perp }$. Blue lines visualize the magnetodipolar field. From [441]. (b) Dependence of the spin structure of a nanoparticle on the particle size represented by selected magnetization configurations of Fe particles with diameters of $D=20$ and 40 nm at an external field of $-30\mathrm{Oe}$. (c) Simulated upper part of the hysteresis loop (green line) and magnetization distribution (two-dimensional cuts out of three-dimensional distributions) at selected points on the hysteresis curve (approach to saturation, remanence, coercivity) obtained in the core-shell model of a Nd-Fe-B nanocomposite. Contributions to the hysteresis loop only from cores (blue dashed line) and shells (red dashed line) are also shown as lower parts of this loop.

Figure 19

Results of micromagnetic simulations for the Fourier components of the magnetization. The images represent projections of the respective functions into the detector plane $q_x=0$ for ${\mathbf{k}}_0\perp \mathbf{H}$. Columns from left to right: $|{\tilde{M}}_x{|}^2$, $|{\tilde{M}}_y{|}^2$, $|{\tilde{M}}_z{|}^2$, $CT=-({\tilde{M}}_y{\tilde{M}}_z^{*}+{\tilde{M}}_y^{*}{\tilde{M}}_z)$, and $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$. Pixels in the image corners correspond to $q\cong 1.8{\mathrm{nm}}^{-1}$. A logarithmic color scale is used (color bars are in arbitrary units).

Figure 20

(a), (b) Two-dimensional cuts (shifted for a better visibility) of $|{\tilde{M}}_x{|}^2(q_x,q_y,q_z)$. (c) Radially averaged individual scattering contributions to $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ as a function of the scattering vector $q$ (${\mathbf{k}}_0\perp \mathbf{H}$) (log-log scale): total $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ (black); $|{\tilde{M}}_z{|}^2{\sin }^2\theta$ (blue); $CT$ (magenta); $|{\tilde{M}}_y{|}^2{\cos }^2\theta$ (green); $|{\tilde{M}}_x{|}^2$ (red). (d) Radially averaged total magnetic SANS cross section $d{\mathrm{\Sigma }}_M/d\mathrm{\Omega }$ as a function of the scattering vector $q$ and for several applied magnetic fields $H$ (see the inset) (${\mathbf{k}}_0\perp \mathbf{H}$) (log-log scale). Materials parameters for NANOPERM were used.

Figure 21

Comparison between simulated (upper row) and experimental (lower row) data for the difference cross section $\propto (|{\tilde{M}}_x{|}^2+|{\tilde{M}}_y{|}^2{\cos }^2\theta +CT\sin \theta \cos \theta )$ for various external fields as indicated (${\mathbf{k}}_0\perp \mathbf{H}$). Pixels in the image corners correspond to $q\cong 0.64{\mathrm{nm}}^{-1}$. A logarithmic color scale is used. Since experimental data were not obtained in absolute units, their values are scaled by a constant factor for comparison with simulated data. $\mathbf{H}$ is horizontal in the figure plane. From [441].

Figure 22

Comparison between the normalized autocorrelation function of the spin misalignment $C_{SM}$ (solid lines) and the normalized correlation function of the spin-misalignment SANS cross section $C_M$ (dashed lines) along different directions in the $y\text{−}z$ detector plane (${\mu }_{0}H=0.6\mathrm{T}$ along $z$). The right images show the corresponding combination of Fourier components, projected into the detector plane: (autocorrelation) $|{\tilde{M}}_x{|}^2+|{\tilde{M}}_y{|}^2$; (SANS) $|{\tilde{M}}_x{|}^2+|{\tilde{M}}_y{|}^2{\cos }^2\theta -({\tilde{M}}_y{\tilde{M}}_z^{*}+{\tilde{M}}_y^{*}{\tilde{M}}_z)\sin \theta \cos \theta$. Pixels in the image corners correspond to $q\cong 0.4{\mathrm{nm}}^{-1}$. A logarithmic color scale is used. From [168].

Figure 23

Images of electronic and magnetic inhomogeneities in complex oxides. (a) Dark-field transmission electron microscope image of ${\mathrm{La}}_{0.25}{\mathrm{Pr}}_{0.38}{\mathrm{Ca}}_{0.38}{\mathrm{MnO}}_3$ at 20 K using a charge-ordered superlattice peak for contrast. The upper dark region is a charge-disordered ferromagnetic metallic region, while the lower textured region is a charge-ordered insulating region. From [109]. (b) Scanning tunneling spectroscopy images of a ${\mathrm{La}}_{0.7}{\mathrm{Ca}}_{0.3}{\mathrm{MnO}}_3$ film just below its Curie temperature in 0 and 9 T. The color scale depicts local conductance, black being metallic, white insulating. Note the percolating ferromagnetic metallic region in the 9 T image. From [176]. (c) Scanning tunneling spectroscopy of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ in the normal state at 93 K. The color scale depicts local conductance. Note the nanoscale heterogeneity. From [496]. (d) Finite temperature Monte Carlo simulation from a random-field Ising model designed to describe manganites. $100\times 100$ sites are shown, black and white corresponding, crudely, to ferromagnetic metal and nonferromagnetic insulator, respectively. From [422].

Figure 24

Temperature dependence of (a) the $q=0.012{\AA }^{-1}$ SANS intensity, (b) the $q=0.046{\AA }^{-1}$ SANS intensity, and (c) the magnetic correlation length from polycrystalline ${\mathrm{Tl}}_2{\mathrm{Mn}}_2{\mathrm{O}}_7$. From [397]. (d) The temperature dependence (normalized to the Curie temperature) of the magnetic SANS intensity in polycrystalline ${\mathrm{La}}_{2/3}{\mathrm{Ca}}_{1/3}{\mathrm{MnO}}_3$ (LCMO) in three magnetic fields. From [147].

Figure 25

$q$ dependence of the elastic magnetic SANS cross section from polycrystalline ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{MnO}}_3$ at (a) $x=0.02$, (b) $x=0.09$, and (c) $x=0.95$. The magnetic contribution was isolated by subtracting high-temperature data, as illustrated in the insets. Solid lines are fits to a model for a liquidlike distribution of nanoscopic magnetic clusters. From [385].

Figure 26

(a), (b) Magnetic field dependence of the 30 K magnetization and resistance of a ${\mathrm{Pr}}_{0.7}{\mathrm{Ca}}_{0.3}{\mathrm{MnO}}_3$ single crystal. From [566]. (c), (d) Schematic of the zero-field and in-field magnetic phase separations in the same compound. Gray signifies ferromagnetic metallic, while white is antiferromagnetic insulating. Adapted from [425]. (e) Magnetic filaments obtained from Monte Carlo simulations for the same compound; arrows illustrate magnetization directions. From [641].

Figure 27

$q$ dependence of the SANS cross section from single crystal ${\mathrm{LaCoO}}_3$ at multiple temperatures. Inset: Magnetic cross section, isolated by subtracting the 300 K data; solid lines are Guinier fits. From [161].

Figure 28

Temperature dependence of the SANS cross section at (a) $q=0.007{\AA }^{-1}$ and (b) $q=0.049{\AA }^{-1}$ along with the extracted magnetic correlation length (c), in single crystal ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x{\mathrm{CoO}}_3$ at various $x$. (d), (e) The 10 K $x$ dependence of the $q=0.007$ and $0.049{\AA }^{-1}$ cross sections. Dashed lines are guides to the eye, and the right axis in (e) shows the normalized intensity of the incommensurate satellite peak from [517]. From [248].

Figure 29

Temperature dependence of the $q=0.1{\AA }^{-1}$ SANS cross section (a)–(c) and magnetic correlation length (d)–(f) of polycrystalline ${\mathrm{La}}_{1-x}{\mathrm{Sr}}_x{\mathrm{CoO}}_3$ at $x=0.40$, 0.30, and 0.20. The insets in (d)–(f) are expanded views of the low correlation length region, the arrows marking $T^{*}$, where correlations turn on. From [249]. (g) Temperature dependence of the ferromagnetic Bragg intensity (left axis), and $q=0.1{\AA }^{-1}$ magnetic SANS intensity (right axis) of polycrystalline ${\mathrm{Tb}}_5{\mathrm{Si}}_2{\mathrm{Ge}}_2$ at various magnetic fields. Curie and Griffiths temperatures are marked with vertical dashed lines. From [399]).

Figure 30

(a) Phase diagram of ${\mathrm{Ni}}_{50-x}{\mathrm{Co}}_x{\mathrm{Mn}}_{40}{\mathrm{Sn}}_{10}$. From [58]. $T_M$, $T_C$, $T_B^{SP}$, and $T_B^{EB}$ denote the martensitic phase transformation temperature, Curie temperature, and superparamagnetic and exchange bias blocking temperatures, respectively. P, F, AF, and SP denote paramagnetic, ferromagnetic, antiferromagnetic, and superparamagnetic; “Aust.” and “Mart.” refer to austenite and martensite. The top axis shows the valence electron per atom ratio. (b), (c) The temperature dependence of the SANS cross section from ${\mathrm{Ni}}_{44}{\mathrm{Co}}_6{\mathrm{Mn}}_{40}{\mathrm{Sn}}_{10}$ at $q=0.005$ and $0.1{\AA }^{-1}$, taken on cooling and heating. The inset shows a close-up at low temperature. From [57].

Figure 31

(a) Schematic depiction of a spin helix derived from a FM (left) or AF (right) spin structure. (b1)–(b3) The transformation from real space to reciprocal space for a smooth texture (b1), a discrete atomic lattice with lattice spacing $a$ (b2), and a smooth incommensurate modulation on top of a discrete atomic lattice (b3) where the width of the points corresponds to the color code of (b1). (c) SDW of multiferroic ${\mathrm{BiFeO}}_3$ (thin arrows) that is caused by a slight canting of an otherwise AF cycloidal spin spiral (thick arrows show the staggered moment of the AF cycloidal structure). The canting leads to a small FM SDW visible by polarized SANS (d). (c), (d) From [537].

Figure 32

(a) Concentration-temperature phase diagram of the isostructural series ${\mathrm{Fe}}_{1-x}{\mathrm{Mn}}_x\mathrm{Si}$ to ${\mathrm{Fe}}_{1-y}{\mathrm{Co}}_x\mathrm{Si}$. The abbreviations HMM, PMM, and PMI denote helimagnetic metal, PM metal, and PM insulator, respectively ([409]) (b) Generic B20 phase diagram with an illustration of the characteristic main phases. From Markus Garst.

Figure 33

(a) Evolution of sharp diffraction peaks associated with the helical domains, observed below $T_C$, to a diffuse ring, representative for a cut through a sphere for temperatures above $T_C$. (b) Characteristic temperature of the inverse correlation length $\kappa$ as inferred from SANS and measurements of the susceptibility. From [294].

Figure 34

(a), (b) Schematic depictions of a Néel-type and a Bloch-type single skyrmion. A cut through a single skyrmion yields the spin structure of a Néel and a Bloch domain wall, respectively. From Markus Garst. (c) Schematic depiction of a skyrmion lattice with the hexagonal arrangement of skyrmion lines through the sample. From [450].

Figure 35

(a) The phase diagram of MnSi calculated by means of a Ginzburg-Landau ansatz based on a triple-$k$ state. The inset shows the effect of Gaussian fluctuations on lowering the free energy of the SkL phase. (b), (c)  SANS data of the SkL in MnSi for a magnetic field parallel to (b) (110) and (c) a random direction of the magnetic field ([457]). (d) A SANS pattern of the SkL in ferroelectric ${\mathrm{Cu}}_2{\mathrm{OSeO}}_3$ with magnetic field along the (100) axis ([658]). (e) SANS data of the SkL in a single-crystalline sample of ${\mathrm{Co}}_8{\mathrm{Zn}}_8{\mathrm{Mn}}_4$ with magnetic field along (110) ([625]).

Figure 36

SANS diffraction patterns showing (a) a square VL in ${\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}$ ([145]) and (b) a rhombic VL in ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ ([152]).

Figure 37

Density driven VL symmetry transition for $\mathbf{H}\parallel \mathbf{c}$ in superconductors with a fourfold basal plane anisotropy. (a) Equilibrium VL opening angle predicted from nonlocal corrections to the London model ([340]). (b) Measured opening angle in ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ for different temperatures ([152]). (c) Field dependence of the opening angle (2 K) for the VL structures in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ ([656]). Gray shading in (b) and (c) indicate where first-order VL reorientation transitions are observed.

Figure 38

VL phases in niobium. (a) Phase diagram for applied fields along the fourfold $[001]$ axis ([374]). (b) VL structures observed as a function of field direction ([371]). Isosceles half-unit cells are indicated by filled triangles, and equilateral triangles are colored in red. Thick lines show sudden changes in the VL orientation. (c) Intermediate mixed phase indicated by a constant VL scattering vector below 120 mT at two different positions within the same crystal ([541]). Gray lines correspond to a vortex density $\propto H$, Eq. (64).

Figure 39

Field dependence of the VL form factor in (a) ${\mathrm{LuNi}}_2{\mathrm{B}}_2\mathrm{C}$ ([145]) and (b) ${\mathrm{CeCoIn}}_5$ ([655]). The inset in (a) shows the real-space field reconstruction.

Figure 40

Temperature dependence of the VL peak intensity in ${\mathrm{KFe}}_2{\mathrm{As}}_2$ ([313]). (b)–(d) Different fits to the original data in (a). The difference between a full and nodal gap is illustrated in (b).

Figure 41

Probing complex superconducting states. (a) Diffraction patterns in ${\mathrm{UPt}}_3$ for $\mathbf{H}\parallel \mathbf{c}$, “grown” at 190 mT and $T_Q$ and quenched to the measurement temperature (100–150 mK) ([276]). (b) VL anisotropy as a function of magnetic fields applied close to the basal plane in ${\mathrm{KFe}}_2{\mathrm{As}}_2$ ([363]). The inset shows a schematic of VL Bragg reflections lying on an ellipse with major-to-minor axis ratio ${\mathrm{\Gamma }}_{\mathrm{VL}}$. (c) VL anisotropy vs field angle in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ consistent with a superconducting anisotropy ${\mathrm{\Gamma }}_{ac}=58.5\pm 2.3$ ([538]).

Figure 42

Structural properties of the VL. (a) Positional correlation ($R_{az}$) length for $(\mathrm{K},\mathrm{Ba})\mathrm{Bi}{\mathrm{O}}_3$ and BiSrCaCuO ([332]). The horizontal dotted line is where $R_{az}$ becomes larger than the experimental resolution. (b) VL positions in a reverse Monte Carlo ensemble based on SANS measurements in niobium ([372]). Colors denote the magnitude (lightness) and direction (hue) of the in-plane displacement.

Figure 43

Peak effect in ${\mathrm{NbSe}}_2$ ([415]). (a) Linear ac susceptibility in the vicinity of the PE for different field histories. (b) Rocking curve after a field cooling procedure. (c) RCs after “shaking” the VL at two different temperatures below the PE and with widths that are resolution limited.

Figure 44

Metastable VL phases in ${\mathrm{MgB}}_2$ ([135]). (a), (d) Cooling or heating across the F-L phase transition leaves the VL in a metastable configuration. (b), (c) The ground state VL is obtained by applying a damped field oscillation.