Third-order effect in magnetic small-angle neutron scattering by a spatially inhomogeneous medium
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Third-order effect in magnetic small-angle neutron scattering by a spatiallyinhomogeneous medium
Konstantin L. Metlov
Donetsk Institute for Physics and Technology NAS, Donetsk, Ukraine 83114 ∗ Andreas Michels
Physics and Material Science Research Unit, University of Luxembourg,162A Avenue de la Fa¨ıencerie, L-1511 Luxembourg, Grand Duchy of Luxembourg (Dated: June 11, 2018)Magnetic small-angle neutron scattering (SANS) is a powerful tool for investigating nonuniformmagnetization structures inside magnetic materials. Here, considering a ferromagnetic medium withweakly inhomogeneous uniaxial magnetic anisotropy, saturation magnetization, and exchange stiff-ness, we derive the second-order (in the amplitude of the inhomogeneities) micromagnetic solutionsfor the equilibrium magnetization textures and compute the corresponding magnetic SANS crosssections up to the next, third order. We find that in the case of perpendicular scattering (the inci-dent neutron beam is perpendicular to the applied magnetic field) if twice the cross section alongthe direction orthogonal to both the field and the neutron beam is subtracted from the cross sectionalong the field direction, the result has only a third-order contribution (the lower-order terms arecanceled). This difference does not depend on the amplitude of the exchange inhomogeneities andprovides a separate gateway for a deeper analysis of the sample’s magnetic structure. We deriveand analyze analytical expressions for the dependence of this combination on the scattering-vectormagnitude for the case of spherical Gaussian inhomogeneities.
PACS numbers: 61.05.fg, 75.60.-d, 75.25.-jKeywords: micromagnetics, neutron scattering, magnetic small-angle neutron scattering
I. INTRODUCTION
Magnetic small-angle neutron scattering (SANS) is animportant tool for the analysis of magnetic structureson the nanoscale. Traditional scalar magnetometry, forexample, only measures the sample’s total magnetic mo-ment and has no spatial resolution. Magnetic force mi-croscopy is sensitive to the spatial features of the magne-tization only in the near-vicinity of the sample’s surfaceand is also prone to disturbing the magnetic structureduring the measurement. Optical magnetometry is eitheralso only surface-sensitive (such as in Kerr microscopy) oris applicable only to optically transparent magnets (suchas in Faraday microscopy). Magnetic SANS complementsthese techniques by permitting analysis of the sample’smagnetic structure throughout the volume, even in non-transparent materials, while also being sensitive to thespatial arrangement of the magnetization.The analysis of magnetic SANS cross sections isclosely interwoven with the continuum theory ofmicromagnetics. This is because, unlike nonmagneticnuclear SANS (which is sensitive to nanoscale densityand compositional fluctuations), magnetic SANS crosssection images are formed by the distribution of themagnetic moments within the sample. These magneticmoments are influenced by magnetic material inhomo-geneities, but, due to their mutual interaction, do notfollow the inhomogeneities exactly. Thus, in order to un-derstand magnetic SANS cross sections, one must alsounderstand the process of magnetic-structure formationand its dependence on the external magnetic field, which is the subject of micromagnetic theory.Currently, the interpretation of magnetic SANS crosssections of heterogeneous multiphase magnets with smallinhomogeneities of the saturation magnetization and themagnetic anisotropy is based on a second-order (in theinhomogeneities amplitude) theory, which has its originin the theory of the approach to magnetic saturation. The latter stems from the works of Schl¨omann, which,in turn, is a follow up on the work by N´eel. The motivation for the present study is to probe thelimits of the second-order magnetic SANS theory bylooking for prominent third-order effects. Specifically,those, which are not masked by the second-order ones.One such effect—a central result of this work—is pin-pointed at the end of the paper (Sec. VI). An attempt wasmade to include all the interactions which are commonin micromagnetics. In particular, our solution for themicromagnetic problem of weakly inhomogeneous mag-nets includes inhomogeneous exchange interaction, whichis irrelevant for the problem of the approach to mag-netic saturation and for the second-order SANS theory.Also, the present theory explicitly includes a weak, fluc-tuating random-axis uniaxial anisotropy and full three-dimensional anisotropy-direction averaging.The paper is organized as follows. Sec. II introducesthe magnetic SANS cross sections, Sec. III details the so-lution of the micromagnetic problem in Fourier space,Sec. IV provides a discussion of the defect distribu-tion and averaging procedures, the second and higher-order SANS cross sections are, respectively, discussed inSecs. V and VI, and Sec. VII summarizes the main resultsof this study. II. MAGNETIC SANS CROSS SECTIONS
The current theoretical and experimental understand-ing of magnetic SANS of bulk ferromagnets has recentlybeen summarized. The quantity of interest—the differ-ential SANS cross section dΣdΩ —is related to the Fouriertransform of the Cartesian components of the magneti-zation vector field f M = { f M X , f M Y , f M Z } . In particular,the total unpolarized nuclear and magnetic SANS crosssection is dΣ ⊥ dΩ = 8 π V b " | e N | b + | f M X | + | f M Y | cos α + | f M Z | sin α − f M Y f M Z ) sin α cos α i , (1)dΣ k dΩ = 8 π V b " | e N | b + | f M X | sin β + | f M Y | cos β + | f M Z | − f M Y f M X ) sin β cos β i , (2)where the first expression refers to the perpendicularscattering geometry, for which q ⊥ = q { , sin α, cos α } ,and the second equation relates to the parallel geometry,with q k = q { cos β, sin β, } ; V is the scattering volume, b H = 2 . × A − m − , e N ( q ) is the nuclear scatteringamplitude, Re stands for taking the real part of a com-plex number and over-bar for its complex conjugate. Thecross section is generally measured in units of cm − sr − .Fourier transforms (distinguished by tildes above thesymbol) are defined for a representative cube of the ma-terial with dimensions L × L × L . Most of the time weshall work with discrete transforms (the continuous onescorrespond to the limit L → ∞ ): e X ( q ) = 1 V ZZZ V X ( r ) e − ı qr d r , (3) X ( r ) = X q e X ( q ) e ı qr , (4)where ı = √− V = L is the representative cube vol-ume and X is a quantity which is defined inside the vol-ume. The components of q = { q X , q Y , q Z } take on all val-ues which are integer multiples of 2 π/L . Note that unlikeRef. 8 our definition of the forward Fourier transform car-ries a prefactor of 1 /V , so that the Fourier transform hasthe same dimension as the transformed quantity. This,however, renders the prefactor in the expressions for thecross sections, Eqs. (1) and (2), also slightly different.In order to get rid of the nuclear scattering | e N | , thecross sections are typically split into the residual and themagnetic parts, Σ = Σ res + Σ M , where the residual partcorresponds to the magnet in the fully saturated state.Assuming that a large saturating external magnetic field is directed along the Z -axis,dΣ ⊥ res dΩ = 8 π V b " | e N | b + | f M S | sin α , (5)dΣ k res dΩ = 8 π V b " | e N | b + | f M S | , (6)where f M S is the Fourier transform of the inhomogeneoussaturation magnetization of the magnet, which is definedin the next section. The residual part can then be mea-sured independently and subtracted from the measuredtotal cross section at a lower field to yield the magneticpart:dΣ ⊥ M dΩ = 8 π V b h | f M X | + | f M Y | cos α +( | f M Z | − | f M S | ) sin α − f M Y f M Z ) sin α cos α i , (7)dΣ k M dΩ = 8 π V b h | f M X | sin β + | f M Y | cos β +( | f M Z | − | f M S | ) − f M Y f M X ) sin β cos β i . (8)Thus, in order to compute the magnetic SANS crosssection one needs to know the Fourier components ofthe magnetization vector field inside the material. Theirderivation is the main subject of the two following sec-tions. III. MAGNETIZATION DISTRIBUTION IN AWEAKLY INHOMOGENEOUS MAGNET
Consider an infinite magnet, whose saturation magne-tization depends explicitly on the position vector r , M S ( r ) = M [1 + I m ( r )] , (9)where the magnitude of I m ( r ) is a small quantity. Wealso assume that the spatial average h I m ( r ) i = 0, so that M = h M S ( r ) i is the average saturation magnetizationof the magnet.If the representative volume contains a magnetic ma-terial, the equilibrium distribution of the magnetizationvector M ( r ) is the solution of Brown’s equations at eachpoint r in the volume,[ H eff , M ] = 0 , (10)where the square brackets denote the vectorial crossproduct. The effective field H eff ( r ) is defined as thefunctional derivative of the ferromagnet’s energy-densityfunctional e over the magnetization vector field, H eff ( r ) = − µ δeδ M = − µ ∂e∂ M − ∂∂ r ∂e ∂ M ∂ r ! . (11)From the magnetic-units standpoint, followingAharoni, we use the defining relation for the magneticinduction B = µ ( H + γ B M ) , (12)which can be made valid in all systems of magnetic unitsby appropriately choosing the constants µ and γ B . Forexample, in the SI system µ is the permeability of vac-uum and γ B = 1, in the CGS system µ = 1 and γ B = 4 π .The energy density e represents our knowledge of theinteractions in the magnetic material. Here, we includethe effects of exchange, random uniaxial anisotropy, mag-netostatic interaction, and the influence of the uniformexternal magnetic field, so that the total energy densityof the magnet can be written as a sum, e = e EX + e A + e MS + e Z . (13)Different interactions enter both the energy density andthe effective field additively, so that H eff = H EX + H A + H MS + H Z , (14)where H Z is simply the external field.The exchange interaction is deemed inessential in thetheory of the approach to magnetic saturation, butSANS is sensitive to small spatial variations of the mag-netization vector field, despite their negligible contribu-tion to the total magnetization of the sample. That iswhy we have included the exchange interaction into con-sideration. Its energy density in a material with varyingsaturation magnetization is conventionally defined as e EX = C ( r )2 X i = X,Y,Z (cid:18) ∇ (cid:20) M i ( r ) M S ( r ) (cid:21)(cid:19) , (15)where C ( r ) is the exchange stiffness; X , Y , Z are thelabels of the Cartesian coordinate-system axes; ∇ = { ∂/∂X, ∂/∂Y , ∂/∂Z } is the gradient operator. Usingvector-calculus identities, it can be transformed into e EX = µ γ B L ( r )2 − [ ∇ M S ( r )] + X i = X,Y,Z [ ∇ M i ( r )] , (16)with the exchange length L EX ( r ) = p C ( r ) / [ µ γ B M S ( r )] . (17)The first term in Eq. (16) vanishes under the variation of M and gives no contribution to the effective field. Thisis a manifestation of the fact that the exchange energydepends only on relative angles of the magnetization vec-tors and not on their magnitude. Thus, H EX = γ B L ( r )∆ M ( r ) + γ B ∇ L ( r ) ∇ M ( r ) , (18)where ∆ is the Laplace operator and ∇ M ( r ) is a matrix,whose rows are the gradients of the components of the vector M ( r ). Similarly to Eq. (9) we will now assumethat the squared exchange length is weakly inhomoge-neous L ( r ) = L [1 + I e ( r )] , (19)where I e is a small position-dependent quantity ofthe same order as I m and L is an average position-independent exchange length. In real materials, sincethe values of both C and M S are determined by thesame quantum exchange interaction (and both grow asexchange becomes stronger), the value of L displays lit-tle variation across a wide range of magnetic materialsand is of the order of 5 −
10 nm for most of them. Nev-ertheless, we shall keep track of the weak spatial depen-dence of L EX in this calculation.The presence of uniaxial anisotropy creates the follow-ing energy density e A = − k U ( r ) [ d ( r ) · M ( r )] M ( r ) , (20)where k U ( r ) is the spatially inhomogeneous anisotropyconstant, and d ( r ) is a unit vector along the local direc-tion of the anisotropy axis. The corresponding effectivefield is H A = γ B Q ( r )[ d ( r ) · M ( r )] d ( r ) , (21)where the dimensionless quality factor Q ( r ) = 2 k U ( r ) / [ µ γ B M S ( r )] = I k ( r ) (22)is assumed to be small and of the same order as I m .The magnetostatic energy density is e MS = − µ ( H D · M ) , (23)where H D is the magnetostatic (or demagnetizing) field.The expression for the latter in the static case with nomacroscopic currents is simplest in Fourier representa-tion. It follows from the expression of the magneticinduction (12) with the internal field H = H Z + f H D and Maxwell’s equations ∇ × H D = 0 and ∇ · B = 0 that f H D = − γ B q ( q · f M ) q for q = 0 . (24)The average demagnetizing field f H D (0) is anti-parallel tothe external field H Z . Thus, we can add these fields asscalars H = H Z − | f H D (0) | . All the following results willbe computed as functions of this internal field H (which,in particular, contains the information about the shapeof the sample) and not directly of the external field H Z .Having a simple expression for the demagnetizing field,Eq. (24), suggests trying to solve Brown’s equations,Eqs. (10), directly in Fourier space. There is, however,a complication, since products of functions in real spacebecome their convolutions in Fourier space. Thus, tosimplify the expressions, let us introduce a shorthandnotation for convolutions: X ⊗ Y ( q ) = X q ′ X ( q ′ ) Y ( q − q ′ ) , (25)where the argument q on the left hand side (which some-times will be omitted in the following text) is the argu-ment of the whole convolution (not just of Y ) and sum-mation is carried out over all the values of q ′ . The alge-bra of convolutions is commutative ( X ⊗ Y = Y ⊗ X ),distributive ( X ⊗ ( Y + Z ) = X ⊗ Y + X ⊗ Z ), and as-sociative with respect to multiplication of a constant, a ( X ⊗ Y ) = ( aX ) ⊗ Y = X ⊗ ( aY ), where a is a constant.It also has an identity element (a product of Kroneckerdeltas in the discrete case or Dirac’s delta functions in thecontinuous case), which we will denote as δ ( q ), so that δ ⊗ X = X . We will also sometimes specify functions in-line by underlining them, so that qZ ⊗ Y is convolutionof the function X ( q ) = qZ ( q ) with the function Y ( q ).Using this notation, we can now express Fourier rep-resentations of the effective-field terms: f H EX = − γ B g L ⊗ q f M − γ B q g L ⊗ q × f M , (26) f H A = γ B e Q ⊗ (cid:16) e d X ⊗ f M X + e d Y ⊗ f M Y + e d Z ⊗ f M Z (cid:17) ⊗ e d , (27) f H MS + f H Z = H δ − γ B q [ q · ( f M − δ f M )] q , (28)where the cross ( × ) denotes a direct product of two vec-tors (forming a matrix, having the products of the leftvector by each element of the right vector in the rows)and convolution of a vector with a matrix is like theirnormal product but with convolutions instead of multi-plications. The final subtraction, together with the con-dition that f M − δ f M → q → q = 0. This limiting condition is fulfilled,if h I m ( r ) i = e I m (0) = 0, which is assumed from the startof this computation.In completely homogeneous infinite isotropic magnets,the magnetization will always be uniform, saturated, andaligned parallel to the external (and the internal) field(however small it is). In our weakly-inhomogeneous andweakly anisotropic case, there will be a small deviationfrom uniformity. Let us choose the coordinate system insuch a way that the direction of the external field coin-cides with Z -axis, so that H = { , , H } and, using themagnitude of I m as a small parameter, represent thisweakly inhomogeneous magnetization via Taylor-seriesexpansion f M = { , , M } δ + f M (1) + f M (2) + . . . , (29)where M ( i ) contains the terms of the order i in I m . Due to the constraint M ( r ) = M ( r ) there are onlytwo independent components of M . Considering M X = M (1)X + M (2)X and M Y = M (1)Y + M (2)Y independent andsmall, the expansion of the constraint up to the secondorder allows us to express the remaining component of M in real space as M Z = M + M I m − ( M (1)X ) + ( M (1)Y ) M . (30)Rendering products as convolutions in Fourier space andintroducing the dimensionless magnetization vector m = M /M we get e m Z = δ + e I m − F Z , (31)where F Z = e m (1)X ⊗ e m (1)X + e m (1)Y ⊗ e m (1)Y . (32)Brown’s equations (of which only two are independent)in Fourier space also contain convolutions. For example,the first one of them reads e H effZ ⊗ f M Y = e H effY ⊗ f M Z , (33)while the second independent one can be obtained byreplacing the subscript Y by X everywhere. After substi-tuting the expression for the effective field and the Taylorexpansion of the magnetization components in powers of I m , Brown’s equations become Taylor series themselves.By collecting the terms of the same order in I m , we geta chain of coupled equations for Taylor-expansion coeffi-cients m (1)X / Y , m (2)X / Y , etc. For example, the equations inthe first order read( h + L q + y q ) e m (1)Y + x q y q e m (1)X = e A Y − y q z q e I m , ( h + L q + x q ) e m (1)X + x q y q e m (1)Y = e A X − x q z q e I m , where h = H/ ( γ B M ) is the dimensionless field, e A X = e d X ⊗ e d Z ⊗ e I k , e A Y = e d Y ⊗ e d Z ⊗ e I k , and the dimensionlesscomponents of the q direction vector are { x q , y q , z q } = q /q . Quantities L and q are still carrying dimensions,but their product is dimensionless. These linear equa-tions are solved by e m (1)X = e A X ( h q + y q ) − x q ( e A Y y q + h q z q e I m ) h q ( h q + x q + y q ) , (34) e m (1)Y = e A Y ( h q + x q ) − y q ( e A X x q + h q z q e I m ) h q ( h q + x q + y q ) , (35)where h q = h + L q . When both the exchange interac-tion and the anisotropy are neglected ( e A X = 0, e A Y = 0, h q = h ), these expressions coincide with the first-ordersolution by Schl¨omann. Otherwise they coincide withthe first order solution, which is extensively used atpresent as a basis for magnetic SANS, except that nowwe have an explicit expression for e A X and e A Y via themagnitude and the direction fields of the local uniaxialanisotropy.In the second order (as well as higher orders), the equa-tions are also linear and differ from the first-order onesonly by their right hand side, which now contains sumsover the lower-order solutions:( h + L q + y q ) e m (2)Y + x q y q e m (2)X = F Y + y q z q F Z , ( h + L q + x q ) e m (2)X + x q y q e m (2)Y = F X + x q z q F Z . Their solutions are also similar, e m (2)X = F X ( h q + y q ) − x q ( F Y y q − h q z q F Z ) h q ( h q + x q + y q ) , (36) e m (2)Y = F Y ( h q + x q ) − y q ( F X x q − h q z q F Z ) h q ( h q + x q + y q ) . (37)The special functions are F X = e d X ⊗ e I k ⊗ (2 e d Z ⊗ e I m + e d X ⊗ e m (1)X + e d Y ⊗ e m (1)Y ) − qL x q e I e ⊗ qL x q e m (1)X − qL y q e I e ⊗ qL y q e m (1)X − qL z q e I e ⊗ qL z q e m (1)X − e d Z ⊗ e d Z ⊗ e I k ⊗ e m (1)X − e I e ⊗ L q e m (1)X − e I m ⊗ L q e m (1)X + x q ( z q e I m + x q e m (1)X + y q e m (1)Y ) + e m (1)X ⊗ L q e I m + z q ( z q e I m + x q e m (1)X + y q e m (1)Y ) , and a similar expression is obtained for F Y with the X and Y subscripts as well as the functions x q and y q inter-changed. The function F Z = − e m (2)Z is defined by Eq. (32).Just to give a simpler example: if the effects of in-homogeneous anisotropy and exchange are neglected (byputting e I k = 0 and e I e = 0) and the expressions for e m (1)X and e m (1)Y are substituted, the special functions are F X / Y / Z ( q ) = X q ′ e I ( q ′ ) e I ( q − q ′ ) u q u q − q ′ f X / Y / Z ( q ′ , q − q ′ )with f X = − hx q ′ z q ′ u q − q ′ − ( z q ′ ( h + L q ′ ) + u q ′ L q ′ ) x q − q ′ z q − q ′ ,f Y = − hy q ′ z q ′ u q − q ′ − ( z q ′ ( h + L q ′ ) + u q ′ L q ′ ) y q − q ′ z q − q ′ ,f Z = 12 z q ′ z q − q ′ ( x q ′ x q − q ′ + y q ′ y q − q ′ ) , (38)where u q = h q + x q + y q . Expression (38) for f Z isvalid even if inhomogeneous exchange and anisotropy arepresent. If we further neglect the effects of exchange (byputting L = 0), the solutions for f m (2) coincide exactlywith those obtained by Schl¨omann. These analytical calculations complete the second-order solution of the micromagnetic problem of a weakly inhomogeneous magnetic material under the influence ofan externally applied magnetic field. Let us now proceedwith the evaluation of the ensuing magnetic SANS crosssections.
IV. MODEL FOR DEFECTS AND THEIRAVERAGING
The theory of magnetic SANS relates to experiment ina similar way as the theory of the approach to magneticsaturation, being a microscopic theory for a macroscopicmeasurement. The micromagnetic analysis of the previ-ous section allows us to express the magnetization Fourierimage at a specified magnetic field via those of the in-homogeneous saturation magnetization, anisotropy, andexchange. The latter, however, are usually unknown fora specific piece of magnetic material. In fact, it is realis-tic to assume that the inhomogeneity functions ( e I m , e I k ,and e I e ) are random processes, having specific realizationsin each representative volume into which a macroscopicmagnet is subdivided. Then, the magnetic SANS crosssection, resulting from the scattering of the neutron beamoff the macroscopic magnet comprising many representa-tive volumes, can be expressed as an average (both overthe random process realization and over the orientation,since the defect realizations in representative volumes arealso randomly oriented).Also, the inhomogeneities of different material param-eters are usually not independent. The underlying phys-ical reasons behind their formation (such as nanocrys-tallization) imply that the material consists of two ormore phases, each having a specific set of magnetic pa-rameters, separated by transition regions (such as grainboundaries). That is why later on we will assume thatthe inhomogeneity functions are proportional to a uni-versal inhomogeneity function e I , describing the materialmicrostructure: e I m = e I , e I k = κ e I , e I e = ǫ e I , where κ, ǫ . I ( r ) = X n f n ( r − p n ) , (39)where p n are uniformly-distributed random vectors andthe summation is carried out over all the inhomogeneitiesin the representative volume. The Fourier transform ofthis function is e I ( q ) = X n e − ı qp n e f n ( q ) , (40)where e f n ( q ) is the Fourier transform of f n ( r ).We further assume that the inhomogeneities have aGaussian profile, f n ( r ) = a n e − r ⊺ Ar , (41)where a n denotes their (random) amplitude, the sub-script ⊺ indicates transposition, and the bold capital sym-bol denotes a square matrix. Moreover, the matrix A is assumed to be positive definite and its elements haveunits of inverse squared length (so that the argumentof the exponential is dimensionless). Assuming that theinhomogeneities are much smaller than the representa-tive volume, we can extend the integration limits in theFourier transform up to infinity to get a simple represen-tation for e f n ( q ), e f n ( q ) = a n vV e − q ⊺ A − q , (42)where v = (2 π ) / / √ D is the volume of a single inhomo-geneity, D = det A is the determinant of A , and A − isits inverse.Since we are going to perform the directional averag-ing over all the possible inhomogeneity orientations, it issufficient, without loss of generality, to specify the pos-itive definite matrix A in diagonal form. Specifically,to consider spheroidal inhomogeneities, we can write thematrix A as A = τ /s τ /s
00 0 1 / ( τ s ) , (43)where s is a real number with units of length specifyingthe defect size, and τ is a dimensionless quantity, speci-fying their shape. The case τ = 1 corresponds to spher-ical inhomogeneities, τ ≪ τ → ∞ toneedle-like elongated defects. The above parametrizationis chosen in such a way that the volume v = (2 π ) / s ofa single inhomogeneity is independent of τ .Now we can explicitly include a rotation matrix O intothe description of the inhomogeneities. For example, wecan use a matrix which is parametrized via the sphericalangles ϕ R ∈ [0 , π ] and θ R ∈ [0 , π ]: O = c ϕ c θ + s ϕ c ϕ s ϕ ( c θ − c ϕ s θ c ϕ s ϕ ( c θ − c ϕ + s ϕ c θ s ϕ s θ − c ϕ s θ − s ϕ s θ c θ , (44)where c ϕ = cos ϕ R , s ϕ = sin ϕ R , c θ = cos θ R , and s θ =sin θ R ; it rotates the direction vector { , , } towardsthe unit-vector v = { c ϕ s θ , s ϕ s θ , s θ } and, consequently,has the property O − v = { , , } .The quadratic-form matrix A in the rotated coordinatesystem can be represented as OAO − = OAO ⊺ , since forthe rotation matrix O − = O ⊺ . Similarly, ( OAO ⊺ ) − = OA − O ⊺ and q ⊺ p n in the rotated coordinate system shouldbe replaced by q ⊺ Op n . Thus, for the Fourier image of therotated inhomogeneity function we have e I ( q ) = vV e J ( q ) e − q ⊺ OA − O ⊺ q , (45) e J ( q ) = X n a n e − ı q ⊺ Op n . (46)Besides the averaging over the full range of the rotationangles ϕ R and θ R , the expressions containing e I ( q ) will need to be averaged over the random defect positions p n .The function e I depends on these positions only via thefactor J . In order to learn how to compute the configura-tional average of this function, consider its mean-squaredvalue: D | e J ( q ) | E = *X n X n ′ a n a n ′ e ı q ′ ⊺ O ( p n − p n ′ ) + = N h a n i , where N is the number of defects in the representativevolume. This is because the averaging of the exponent inthe last expression yields a Kronecker delta. The sum-mation can then be easily performed. Similarly, it is pos-sible to show that the various m -products of e J , such asthe triple product | e J ( q ) | Re J ( q ) or the quadruple prod-uct | e J ( q ) | , average over various defect configurations to N h a mn i , which is independent of q .Finally, we will assume that the direction of the localanisotropy axis is independent of the particle shape. Thismeans that all the expressions for the SANS cross sec-tions will need to be averaged over the random anisotropydirection as well. V. SECOND-ORDER MAGNETIC SANS CROSSSECTIONS
As we have seen in Sec. II, the magnetic SANS crosssections depend on the squared magnetization Fouriercomponents. Since the magnetization components startwith the first order in e I , the lowest order terms in thecross sections will be of second order. Let us computethese terms.For simplicity, we assume that the magnitude of theanisotropy inhomogeneities is related to the magnitude ofthe saturation-magnetization inhomogeneities by a fac-tor e I k = κ e I , and also that the anisotropy direction isconstant inside each inclusion (but randomly orientedin different ones), so that e d X = δ cos ϕ A sin θ A , e d Y = δ sin ϕ A sin θ A , and e d Z = δ cos θ A . Then, substituting themagnetization components, Eqs. (34) and (35), into theexpressions for the parallel [Eq. (8)] and perpendicular[Eq. (7)] magnetic SANS cross sections, and averagingover the anisotropy directions, h F i A = 14 π Z π Z π F sin θ A d ϕ A d θ A , (47)we getdΣ k M dΩ = 8 π V b M h e I i κ h q , (48)dΣ ⊥ M dΩ = 8 π V b M h e I i (cid:20) κ cos α h q + sin α ) + κ h q +(3 + 4 h q − cos 2 α ) sin α h q + sin α ) (cid:21) , (49) τ /(1+ τ )00.20.40.60.81 Υ ( µ , δ ) µ =0 µ =0.1 µ =0.2 µ =0.3 FIG. 1. Dependence of the mean-squared inhomogeneity func-tion h e I i = N h a n i ( v/V ) Υ( qs, τ ) on the inclusion shape τ fordifferent values of µ = qs in the range from 0 to 2 in equalsteps of 0 . h e I i has an extremum at τ = 1, correspondingto spherical defects. The left side of the plot corresponds toplanar defects, while the right side to needle-like ones. where we have introduced an angle in the plane ofthe detector ( q X = 0) for the perpendicular cross sec-tion q Z = q cos α , q Y = q sin α and angular bracketsstand for the directional averaging over the representa-tive volume orientations ( ϕ R , θ R ). For Gaussian defects,Eqs. (41) − (43), this averaging can be performed analyt-ically, yielding h e I i = N h a n i ( v/V ) Υ( qs, τ ) withΥ( µ, τ ) = e − µ τ √ π µ q τ − τ Erfi( µ q − τ τ ) τ < e − µ τ = 1 e − µ τ √ π µ q ττ − Erf( µ q τ − τ ) τ > , (50)where Erf( z ) = (2 / √ π ) R z e − t d t denotes the error func-tion, and Erfi( z ) = Erf( ız ) /ı is its imaginary counterpart.Dependence of Υ( µ, τ ) on particle shape at different val-ues of µ = qs is plotted in Fig. 1.The parallel cross section in the second order, Eq. (48),is fully isotropic in the detector plane ( q Z = 0), while theperpendicular one, Eq. (49), besides the isotropic term κ h q , contains two terms, which depend on α . One ofthese terms is due to the effect of magnetic anisotropy,while the other is of purely magnetostatic origin. Theyare plotted in Fig. 2.Remember that h q = h + L q , which means that h q takes on values starting with the external field h > h q is small (small h and small q with re-spect to the inverse exchange length squared 1 /L ) andwhen h q is large (either when h is large, or when q islarge for small h ). In the former regime, the angular c o s ( α ) / ( h q + s i n ( α )) h q = 0 h q = 0.1 h q =2 h q = 0.2 a) α / π ( + h q - c o s ( α ) s i n ( α ) / ( ( h q + s i n ( α )) ) h q =0 h q =0.1 h q =0.2 h q =2 b) FIG. 2. Angular dependence of the anisotropic terms in theperpendicular magnetic SANS cross section at different valuesof h q . (a) displays the first and (b) the third term in Eq. (49). dependence of both anisotropic terms displays a similartwo-fold angular dependence with sharp maxima along α = 0 , π . Together with the isotropic halo, described bythe second term in Eq. (49), this gives rise to the recentlyobserved UFO-like shape of the SANS image, shownin Fig. 3.At large h q , the angular dependence of the first andthe third term in Eq. (49) is different. The formeris two-fold, while the latter tends asymptotically to(1 − cos 4 α ) / (4 h q ), which has fourfold symmetry. Thisopens the possibility of separating the anisotropy andthe magnetostatic contributions by performing a Fourieranalysis of the cross sections at large h q (either at mod-erately large h or at the outskirts of the cross section,measured at small h ).Regarding the asymptotic q -dependence, it is readilyverified that both magnetic SANS cross sections vary (forspherical inclusions with τ = 1) as ∼ e − s q ( sq ) − , where s denotes the defect size. Other assumptions about theprofile of the inclusions, e.g. , a sharp interface, may resultin different asymptotic dependencies. VI. HIGH-ORDER TERMS IN MAGNETICSANS CROSS SECTIONS
The structure of the second-order solutions for themagnetization components, Eqs. (36) and (37), is similarto that of the first-order ones, but now magnetostatic ef- -0.3-0.2-0.1 0 0.1 0.2 0.3-0.3 -0.2 -0.1 0 0.1 0.2 0.3 q Y q Z FIG. 3. UFO-like magnetic SANS cross section shape at small h = 0 .
01 and q in a sample with spherical ( τ = 1) Gaussianinclusions. The other parameters for this plot are: κ = 1, s = 1, L = 1. The outer contour corresponds to a value of15, which increases inwards in steps of 100. There is a verysharp maximum at the center. fects also contribute to F X and F Y . These functions playthe same role in the second-order solutions as the func-tions A X and A Y do in the first-order ones, except thatthey have an additional dependence on the magnitudeand the direction of the q -vector.The main problem with this (and any other) high-ordercontribution to physical properties is that it is usuallyvery small and, if lower-order effects are present at thesame time, is completely masked by them. On the otherhand, analysis of the higher-order effects allows one toextract independently additional information about thesystem, which the lower-order effects do not provide.Therefore, it is desirable to establish the experimentalconditions when the lower-order effects are canceled outand only the higher-order terms contribute, thus, en-abling their analysis.In the present problem this can be achieved by con-sidering the following combination of SANS cross-sectionvalues: ∆Σ ⊥ M = dΣ ⊥ M dΩ (cid:12)(cid:12)(cid:12)(cid:12) α =0 − ⊥ M dΩ (cid:12)(cid:12)(cid:12)(cid:12) α = π/ . (51)As can be readily checked from Eq. (49), this combina-tion is exactly zero in second order. This is true bothin the presence of anisotropy inhomogeneities κ > h e I i alwaysdepends only on the magnitude of q due to the direc-tional averaging. It is also independent of the assump-tion that the anisotropy inhomogeneities are related tothe saturation-magnetization inhomogeneities by a fac-tor of e I k = κ e I . In other words, the cancellation of thesecond-order terms in ∆Σ ⊥ M is a universal property of the g A h=0.1h=0.2h=0.3 λ =0.70 0.5 1 1.5 2 2.5 3 µ g A h=0.4 λ =0.2 λ =0.4 λ =0.6 λ =0.8 FIG. 4. The functions g A ( µ, h, λ ) (solid lines) and their ap-proximation by decaying exponentials (dotted lines) for dif-ferent values of h at fixed λ (upper plot) and for differentvalues of λ at fixed h (lower plot). SANS cross sections, which is independent of the specificmodel.In next significant order (which is the third one), thecontributions of F X and F Y are also canceled and ∆Σ ⊥ M takes on an especially simple form,∆Σ ⊥ M = 32 π V b M h F Z e I i (cid:12)(cid:12)(cid:12) q Z =0 , (52)where q = q Y , F Z is defined by Eq. (32), and the angularbrackets denote a triple (configurational, directional, andanisotropy direction) average.To make the following expressions simpler, let us as-sume a spherical particle shape ( τ = 1), which obvi-ates the directional averaging, and, again, assume that e I k = κ e I . Then, averaging is easy to perform,∆Σ ⊥ M = 32 π b M ρ v h a n i [ κ g A ( qs ) + g MS ( qs )] , (53)where ρ = N/V is the defect density. The dimensionlessfunctions g A ( µ ) and g MS ( µ ), which also depend on h and λ = L /s , are described in the Appendix and plotted inFigs. 4 and 5. The remaining integrals in these functionsare due to the convolution embedded in the definition ofthe function F Z .The dependence of the third-order perpendicular mag-netic difference SANS cross section, Eq. (53), on µ forthe considered spherical Gaussian defect model is mostly g M S h=0.1h=0.2h=0.3 λ =0.70 0.5 1 1.5 2 2.5 3 µ g M S h=0.4 λ =0.2 λ =0.4 λ =0.6 λ =0.8 FIG. 5. The functions g MS ( µ, h, λ ). The curves correspond tothe same values of parameters as those in Fig. 4. a featureless decaying exponential. Only for small val-ues of the externally applied magnetic field h does thisdependence become sharper at small values of µ . In thecase of a very small amplitude of the anisotropy inhomo-geneities κ , such that the cross section is dominated bythe function g MS , it is possible to have negative values of∆Σ ⊥ M for µ ∼ = 1 .
5. This does not, of course, imply thatthe total cross section is negative.
VII. SUMMARY AND CONCLUSIONS
We have presented an analytical solution of the micro-magnetic problem of a weakly inhomogeneous magneticmaterial in an applied magnetic field up to the second or-der in the amplitude of inhomogeneities. On the basis ofthis solution, we have computed the second-order mag-netic SANS cross sections, which, at sufficiently smallvalues of the applied magnetic field h , inevitably displaya prominent UFO-like shape. It is shown that under verygeneral assumptions in a magnet with arbitrary smallinhomogeneities of exchange, anisotropy, and saturationmagnetization, a specific combination of the perpendicu-lar SANS cross-section values, Eq. (51), is exactly zero inthe second order. The next significant third-order con-tribution to this combination is also computed here andis non-zero. Detection and analysis of its q -dependence should provide a deeper insight into the magnet’s mi-crostructure. So far there is no experimental confirma-tion of this newly predicted effect. ACKNOWLEDGMENTS
Financial support by the National Research Fund ofLuxembourg (Project No. FNR/A09/01) is gratefully ac-knowledged.
Appendix A: The functions g A and g MS The functions g A and g MS appear as the result of com-puting the average h F Z e I i over random defect positions,anisotropy direction, and the orientation of the repre-sentative volume (if the inclusions are not of sphericalshape) with F Z defined by Eq. (32) and e I by Eq. (40). F Z , however, contains convolutions of the first-order so-lutions for the magnetization vector field, Eqs. (34) and(35), which, in turn, are proportional to e I . For comput-ing these convolutions, it is easiest to approximate thetriple summation by a triple integration, according to X q . . . = V (2 π ) ZZZ . . . d q , (A1)and integrate over the whole q space in a spherical co-ordinate system. Nevertheless, even in the simplest caseof spherical defects (which obviates the directional aver-ages) the full expressions are too complex to be presentedhere; they are given in the attached Mathematica file and plotted in Figs. 4 and 5 (solid lines).A relatively simple formula can be written for the val-ues of g A and g MS at µ = 0, which reads g | µ =0 = ∞ Z √ h e − ( p − − h ) /λ u ( p ) p p − − h √ πλ d p, (A2)where the functions u ( p ) are given by u A = p (3 p − p − + coth − p p , (A3) u MS = − p + (3 p −
1) coth − p. (A4)Also, a simple closed-form asymptotic expression for g A at large h can be obtained, g A ( h ≫
1) = e − µ / √ h . (A5)0 ∗ [email protected] A. Michels, Journal of Physics: Condensed Matter ,383201 (2014) H. Kronm¨uller, A. Seeger, and M. Wilkens,Zeitschrift f¨ur Physik , 291 (1963), ISSN 0044-3328, http://dx.doi.org/10.1007/BF01379357 W. Brown,
Micromagnetics , Interscience tracts onphysics and astronomy (Interscience Publishers, 1963) http://books.google.com.ua/books?id=v9rvAAAAMAAJ D. Honecker and A. Michels, Phys. Rev. B , 224426(2013) H. Kronm¨uller and M. F¨ahnle,
Micromagnetismand the Microstructure of Ferromagnetic Solids ,Cambridge studies in magnetism (CambridgeUniversity Press, 2003) ISBN 9780521331357, http://books.google.com.ua/books?id=h6nKtwcYyNEC E. Schl¨omann, J. Appl. Phys. , 5798 (Dec. 1971) L. N´eel, Compt. Rend. , 738 (1945) A. Michels and J. Weissm¨uller, Reportson Progress in Physics , 066501 (2008), http://stacks.iop.org/0034-4885/71/i=6/a=066501 J. W. F. Brown,
Micromagnetics (New York: Wiley, 1963) A. Aharoni,
Introduction to the theory of ferromagnetism (Oxford University Press, Oxford, 1996) ISBN 0198517912 C. Herring and C. Kittel, Phys. Rev. , 869 (1951) ´E. P´erigo, E. P. Gilbert, G. C. Hadjipanayis, K. L. Metlov,and A. Michels, “Experimental observation of magneticpoles inside the bulk magnets via q = 0 Fourier modesof magnetostatic field,” (2014), accepted to New Journalof Physics J. Weissm¨uller, R. McMichael, A. Michels, and R. Shull,J. Research NIST , 261 (1999)14