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| Copyright (c) 2009-2022, SasView Developers | ||
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| Copyright (c) 2009-2024, SasView Developers | ||
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| All rights reserved. | ||
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| //These functions are required for magnetic analysis models. They are copies | ||
| //from sasmodels/kernel_iq.c, to enables magnetic parameters for 1D and 2D models. | ||
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| static double langevin( | ||
| double x) { | ||
| // cotanh(x)-1\x | ||
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| if (x < 0.00001) { | ||
| // avoid dividing by zero | ||
| return 1.0/3.0*x-1.0/45.0 * pow(x, 3) + 2.0/945.0 * pow(x, 5) - 1.0/4725.0 * pow(x, 7); | ||
| } else { | ||
| return 1.0/tanh(x)-1/x; | ||
| } | ||
| } | ||
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| static double langevinoverx( | ||
| double x) | ||
| { | ||
| // cotanh(x)-1\x | ||
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| if (x < 0.00001) { | ||
| // avoid dividing by zero | ||
| return 1.0/3.0-1.0/45.0 * pow(x, 2) + 2.0/945.0 * pow(x, 4) - 1.0/4725.0 * pow(x, 6); | ||
| } else { | ||
| return langevin(x)/x; | ||
| } | ||
| } | ||
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| //weighting of spin resolved cross sections to reconstruct partially polarised beam with imperfect optics using up_i/up_f. | ||
| static void set_weights(double in_spin, double out_spin, double weight[8]) //from kernel_iq.c | ||
| { | ||
| double norm=out_spin; | ||
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| in_spin = clip(sqrt(square(in_spin)), 0.0, 1.0);//opencl has ambiguities for abs() | ||
| out_spin = clip(sqrt(square(out_spin)), 0.0, 1.0); | ||
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| if (out_spin < 0.5){norm=1-out_spin;} | ||
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| // The norm is needed to make sure that the scattering cross sections are | ||
| //correctly weighted, such that the sum of spin-resolved measurements adds up to | ||
| // the unpolarised or half-polarised scattering cross section. No intensity weighting | ||
| // needed on the incoming polariser side assuming that a user has normalised | ||
| // to the incoming flux with polariser in for SANSPOl and unpolarised beam, respectively. | ||
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| weight[0] = (1.0-in_spin) * (1.0-out_spin) / norm; // dd.real | ||
| weight[1] = weight[0]; // dd.imag | ||
| weight[2] = in_spin * out_spin / norm; // uu.real | ||
| weight[3] = weight[2]; // uu.imag | ||
| weight[4] = (1.0-in_spin) * out_spin / norm; // du.real | ||
| weight[5] = weight[4]; // du.imag | ||
| weight[6] = in_spin * (1.0-out_spin) / norm; // ud.real | ||
| weight[7] = weight[6]; // ud.imag | ||
| } | ||
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| //transforms scattering vector q in polarisation/magnetisation coordinate system | ||
| static void set_scatvec(double *qrot, double q, | ||
| double cos_theta, double sin_theta, | ||
| double alpha, double beta) | ||
| { | ||
| double cos_alpha, sin_alpha; | ||
| double cos_beta, sin_beta; | ||
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| SINCOS(alpha*M_PI/180, sin_alpha, cos_alpha); | ||
| SINCOS(beta*M_PI/180, sin_beta, cos_beta); | ||
| //field is defined along (0,0,1), orientation of detector | ||
| //is precessing in a cone around B with an inclination of theta | ||
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| qrot[0] = q*(cos_alpha * cos_theta); | ||
| qrot[1] = q*(cos_theta * sin_alpha*sin_beta + | ||
| cos_beta * sin_theta); | ||
| qrot[2] = q*(-cos_beta * cos_theta* sin_alpha + | ||
| sin_beta * sin_theta); | ||
| } | ||
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| //Evaluating the magnetic scattering vector (Halpern Johnson vector) for general orientation of q and collecting terms for the spin-resolved (POLARIS) cross sections. Mz is along the applied magnetic field direction, which is also the polarisation direction. | ||
| static void mag_sld( | ||
| // 0=dd.real, 1=dd.imag, 2=uu.real, 3=uu.imag, 4=du.real, 5=du.imag, 6=ud.real, 7=ud.imag | ||
| double qx, double qy, double qz, | ||
| double mxreal, double mximag, double myreal, double myimag, double mzreal,double mzimag, double nuc, double sld[8]) | ||
| { | ||
| double vector[3]; | ||
| //The (transversal) magnetisation and hence the magnetic scattering sector is here a complex quantity. The spin-flip (magnetic) scattering amplitude is given with | ||
| // MperpPperpQ \pm i MperpP (Moon-Riste-Koehler Phys Rev 181, 920, 1969) with Mperp and MperpPperpQ the | ||
| //magnetisation scattering vector components perpendicular to the polarisation/field direction. Collecting terms in SF that are real (MperpPperpQreal + SCALAR_VEC(MperpPimag,qvector) ) and imaginary (MperpPperpQimag \pm SCALAR_VEC(MperpPreal,qvector) ) | ||
| double Mvectorreal[3]; | ||
| double Mvectorimag[3]; | ||
| double Pvector[3]; | ||
| double perpy[3]; | ||
| double perpx[3]; | ||
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| double Mperpreal[3]; | ||
| double Mperpimag[3]; | ||
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| const double q = sqrt(qx*qx + qy*qy + qz*qz); | ||
| SET_VEC(vector, qx/q, qy/q, qz/q); | ||
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| //Moon-Riste-Koehler notation choose z as pointing along field/polarisation axis | ||
| //totally different to what is used in SASview (historical reasons) | ||
| SET_VEC(Mvectorreal, mxreal, myreal, mzreal); | ||
| SET_VEC(Mvectorimag, mximag, myimag, mzimag); | ||
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| SET_VEC(Pvector, 0, 0, 1); | ||
| SET_VEC(perpy, 0, 1, 0); | ||
| SET_VEC(perpx, 1, 0, 0); | ||
| //Magnetic scattering vector Mperp could be simplified like in Moon-Riste-Koehler | ||
| //leave the generic computation just to check | ||
| ORTH_VEC(Mperpreal, Mvectorreal, vector); | ||
| ORTH_VEC(Mperpimag, Mvectorimag, vector); | ||
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| sld[0] = nuc - SCALAR_VEC(Pvector,Mperpreal); // dd.real => sld - D Pvector \cdot Mperp | ||
| sld[1] = +SCALAR_VEC(Pvector,Mperpimag); //dd.imag = nuc_img - SCALAR_VEC(Pvector,Mperpimg); nuc_img only exist for noncentrosymmetric nuclear structures; | ||
| sld[2] = nuc + SCALAR_VEC(Pvector,Mperpreal); // uu => sld + D Pvector \cdot Mperp | ||
| sld[3] = -SCALAR_VEC(Pvector,Mperpimag); //uu.imag | ||
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| sld[4] = SCALAR_VEC(perpy,Mperpreal)+SCALAR_VEC(perpx,Mperpimag); // du.real => real part along y + imaginary part along x | ||
| sld[5] = SCALAR_VEC(perpy,Mperpimag)-SCALAR_VEC(perpx,Mperpreal); // du.imag => imaginary component along y - i *real part along x | ||
| sld[6] = SCALAR_VEC(perpy,Mperpreal)-SCALAR_VEC(perpx,Mperpimag); // ud.real => real part along y - imaginary part along x | ||
| sld[7] = SCALAR_VEC(perpy,Mperpimag)+SCALAR_VEC(perpx,Mperpreal); // ud.imag => imaginary component along y + i * real part along x | ||
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| } |
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| //Core-shell form factor for anisotropy field (Hkx, Hky and Hkz), Nuc and lonitudinal magnetization Mz | ||
| static double | ||
| form_volume(double radius, double thickness) | ||
| { | ||
| if( radius + thickness > radius + thickness) | ||
| return M_4PI_3 * cube(radius + thickness); | ||
| else | ||
| return M_4PI_3 * cube(radius + thickness); | ||
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| } | ||
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| static double fq(double q, double radius, | ||
| double thickness, double core_sld, double shell_sld, double solvent_sld) | ||
| { | ||
| const double form = core_shell_fq(q, | ||
| radius, | ||
| thickness, | ||
| core_sld, | ||
| shell_sld, | ||
| solvent_sld); | ||
| return form; | ||
| } | ||
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| static double reduced_field(double q, double Ms, double Hi, | ||
| double A) | ||
| { | ||
| // q in 10e10 m-1, A in 10e-12 J/m, mu0 in 1e-7 | ||
| return Ms / (fmax(Hi, 1.0e-6) + 2.0 * A * 4.0 * M_PI / Ms * q * q * 10.0); | ||
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| } | ||
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| static double DMI_length(double Ms, double D, double qval) | ||
| { | ||
| return 2.0 * D * 4.0 * M_PI / Ms / Ms * qval ; //q in 10e10 m-1, A in 10e-3 J/m^2, mu0 in 4 M_PI 1e-7 | ||
| } | ||
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| //Mz is defined as the longitudinal magnetisation component along the magnetic field. | ||
| //In the approach to saturation this component is (almost) constant with magnetic | ||
| //field and simplfy reflects the nanoscale variations in the saturation magnetisation | ||
| //value in the sample. The misalignment of the magnetisation due to perturbing | ||
| //magnetic anisotropy or dipolar magnetic fields in the sample enter Mx and My, | ||
| //the two transversal magnetisation components, reacting to a magnetic field. | ||
| //The micromagnetic solution for the magnetisation are from Michels et al. PRB 94, 054424 (2016). | ||
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| static double fqMxreal( double qx, double qy, double qz, double Mz, double Hkx, double Hky, double Hi, double Ms, double A, double D) | ||
| { | ||
| double qsq = qx * qx + qy * qy + qz * qz; | ||
| double q = sqrt(qsq); | ||
| double Hr = reduced_field(q, Ms, Hi, A); | ||
| double DMI = DMI_length(Ms, D,q); | ||
| double DMIz = DMI_length(Ms, D,qz); | ||
| double denominator = (qsq + Hr * (qx * qx + qy * qy) - square(Hr * DMIz * q)) / Hr; | ||
| double f = (Hkx * (qsq + Hr * qy * qy) - Ms * Mz * qx * qz * (1.0 + Hr * square(DMI)) - Hky * Hr * qx * qy) / denominator; | ||
| return f; | ||
| } | ||
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| static double fqMximag(double qx, double qy, double qz, double Mz, double Hkx, double Hky, double Hi, double Ms, double A, double D) | ||
| { | ||
| double qsq = qx * qx + qy * qy + qz * qz; | ||
| double q = sqrt(qsq); | ||
| double Hr = reduced_field(q, Ms, Hi, A); | ||
| double DMI = DMI_length(Ms, D,q); | ||
| double DMIz = DMI_length(Ms, D,qz); | ||
| double DMIy = DMI_length(Ms, D,qy); | ||
| double denominator = (qsq + Hr * (qx * qx + qy * qy) - square(Hr * DMIz * q)) / Hr; | ||
| double f = - qsq * (Ms * Mz * (1.0 + Hr) * DMIy + Hky * Hr * DMIz) / denominator; | ||
| return f; | ||
| } | ||
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| static double fqMyreal( double qx, double qy, double qz, double Mz, double Hkx, double Hky, double Hi, double Ms, double A, double D) | ||
| { | ||
| double qsq = qx * qx + qy * qy + qz * qz; | ||
| double q = sqrt(qsq); | ||
| double Hr = reduced_field(q, Ms, Hi, A); | ||
| double DMI = DMI_length(Ms, D,q); | ||
| double DMIz = DMI_length(Ms, D,qz); | ||
| double denominator = (qsq + Hr * (qx * qx + qy * qy) - square(Hr * DMIz * q))/Hr; | ||
| double f = (Hky * (qsq + Hr * qx * qx) - Ms * Mz * qy * qz * (1.0 + Hr * square(DMI)) - Hkx * Hr * qx * qy)/denominator; | ||
| return f; | ||
| } | ||
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| static double fqMyimag( double qx, double qy, double qz, double Mz, double Hkx, double Hky, double Hi, double Ms, double A, double D) | ||
| { | ||
| double qsq = qx * qx + qy * qy + qz * qz; | ||
| double q = sqrt(qsq); | ||
| double Hr = reduced_field(q, Ms, Hi, A); | ||
| double DMI = DMI_length(Ms, D,q); | ||
| double DMIx = DMI_length(Ms, D,qx); | ||
| double DMIz = DMI_length(Ms, D,qz); | ||
| double denominator = (qsq + Hr * (qx * qx + qy * qy) - square(Hr * DMIz * q))/Hr; | ||
| double f = qsq * (Ms * Mz * (1.0 + Hr) * DMIx - Hkx * Hr * DMIz)/denominator; | ||
| return f; | ||
| } | ||
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| static double | ||
| Calculate_Scattering(double q, double cos_theta, double sin_theta, double radius, double thickness, double nuc_sld_core, double nuc_sld_shell, double nuc_sld_solvent, double mag_sld_core, double mag_sld_shell, double mag_sld_solvent, double hk_sld_core, double Hi, double Ms, double A, double D, double up_i, double up_f, double alpha, double beta) | ||
| { | ||
| double qrot[3]; | ||
| set_scatvec(qrot,q,cos_theta, sin_theta, alpha, beta); | ||
| // 0=dd.real, 1=dd.imag, 2=uu.real, 3=uu.imag, 4=du.real, 6=du.imag, 7=ud.real, 5=ud.imag | ||
| double weights[8]; | ||
| set_weights(up_i, up_f, weights); | ||
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| double mz = fq(q, radius, thickness, mag_sld_core, mag_sld_shell, mag_sld_solvent); | ||
| double nuc = fq(q, radius, thickness, nuc_sld_core, nuc_sld_shell, nuc_sld_solvent); | ||
| double hk = fq(q, radius, 0, hk_sld_core, 0, 0); | ||
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| double cos_gamma, sin_gamma; | ||
| double sld[8]; | ||
| //loop over random anisotropy axis with isotropic orientation gamma for Hkx and Hky | ||
| //To be modified for textured material see also Weissmueller et al. PRB 63, 214414 (2001) | ||
| double total_F2 = 0.0; | ||
| for (int i = 0; i<GAUSS_N ;i++) { | ||
| const double gamma = M_PI * (GAUSS_Z[i] + 1.0); // 0 .. 2 pi | ||
| SINCOS(gamma, sin_gamma, cos_gamma); | ||
| //Only the core of the defect/particle in the matrix has an effective | ||
| //anisotropy (for simplicity), for the effect of different, more complex | ||
| // spatial profile of the anisotropy see Michels PRB 82, 024433 (2010) | ||
| double Hkx = hk * sin_gamma; | ||
| double Hky = hk * cos_gamma; | ||
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| double mxreal = fqMxreal(qrot[0], qrot[1], qrot[2], mz, Hkx, Hky, Hi, Ms, A, D); | ||
| double mximag = fqMximag(qrot[0], qrot[1], qrot[2], mz, Hkx, Hky, Hi, Ms, A, D); | ||
| double myreal = fqMyreal(qrot[0], qrot[1], qrot[2], mz, Hkx, Hky, Hi, Ms, A, D); | ||
| double myimag = fqMyimag(qrot[0], qrot[1], qrot[2], mz, Hkx, Hky, Hi, Ms, A, D); | ||
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| mag_sld(qrot[0], qrot[1], qrot[2], mxreal, mximag, myreal, myimag, mz, 0, nuc, sld); | ||
| double form = 0.0; | ||
| for (unsigned int xs = 0; xs<8; xs++) { | ||
| if (weights[xs] > 1.0e-8) { | ||
| // Since the cross section weight is significant, set the slds | ||
| // to the effective slds for this cross section, call the | ||
| // kernel, and add according to weight. | ||
| // loop over uu, ud real, du real, dd, ud imag, du imag | ||
| form += weights[xs] * sld[xs] * sld[xs]; | ||
| } | ||
| } | ||
| total_F2 += GAUSS_W[i] * form ; | ||
| } | ||
| return total_F2; | ||
| } | ||
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| static double | ||
| Iqxy(double qx, double qy, double radius, double thickness, double nuc_sld_core, double nuc_sld_shell, double nuc_sld_solvent, double mag_sld_core, double mag_sld_shell, double mag_sld_solvent, double hk_sld_core, double Hi, double Ms, double A, double D, double up_i, double up_f, double alpha, double beta) | ||
| { | ||
| double sin_theta, cos_theta; | ||
| const double q = sqrt(qx * qx + qy * qy); | ||
| if (q > 1.0e-16 ) { | ||
| cos_theta = qx/q; | ||
| sin_theta = qy/q; | ||
| } else { | ||
| cos_theta = 0.0; | ||
| sin_theta = 0.0; | ||
| } | ||
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| double total_F2 = Calculate_Scattering(q, cos_theta, sin_theta, radius, thickness, nuc_sld_core, nuc_sld_shell, nuc_sld_solvent, mag_sld_core, mag_sld_shell, mag_sld_solvent, hk_sld_core, Hi, Ms, A, D, up_i, up_f, alpha, beta); | ||
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| return 0.5 * 1.0e-4 * total_F2; | ||
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| } | ||
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| static double | ||
| Iq(double q, double radius, double thickness, double nuc_sld_core, double nuc_sld_shell, double nuc_sld_solvent, double mag_sld_core, double mag_sld_shell, double mag_sld_solvent, double hk_sld_core, double Hi, double Ms, double A, double D, double up_i, double up_f, double alpha, double beta) | ||
| { | ||
| // slots to hold sincos function output of the orientation on the detector plane | ||
| double sin_theta, cos_theta; | ||
| double total_F1D = 0.0; | ||
| for (int j = 0; j<GAUSS_N ;j++) { | ||
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| const double theta = M_PI * (GAUSS_Z[j] + 1.0); // 0 .. 2 pi | ||
| SINCOS(theta, sin_theta, cos_theta); | ||
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| double total_F2 = Calculate_Scattering(q, cos_theta, sin_theta, radius, thickness, nuc_sld_core, nuc_sld_shell, nuc_sld_solvent, mag_sld_core, mag_sld_shell, mag_sld_solvent, hk_sld_core, Hi, Ms, A, D, up_i, up_f, alpha, beta); | ||
| total_F1D += GAUSS_W[j] * total_F2 ; | ||
| } | ||
| //convert from [1e-12 A-1] to [cm-1] | ||
| return 0.25 * 1.0e-4 * total_F1D; | ||
| } | ||
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| r""" | ||
| Definition | ||
| ---------- | ||
| This model is a micromagnetic approach to analyse the SANS that arises from | ||
| nanoscale variations in the magnitude and orientation of the magnetization in | ||
| bulk ferromagnets in the approach to magnetic saturation (single domain state). | ||
| Typical materials are cold-worked elemental magnets, hard and soft magnetic | ||
| nanocomposites, amorphous alloys and precipitates in magnetic steel [#Michels2014]_. | ||
| The magnetic SANS depends on the magnetic interactions, the magnetic microstructure | ||
| (defect/particle size, magnetocrystalline anisotropy, saturation magnetisation) | ||
| and on the applied magnetic field. As shown in [#Michels2016]_ near magnetic | ||
| saturation the scattering cross-section can be evaluated by means of micromagnetic theory | ||
| .. math:: | ||
| I(\mathbf{Q}) = I_{nuc} + I_{mag}(\mathbf{Q},H), | ||
| with the field-independent nuclear and magnetic SANS cross section (due | ||
| to nanoscale spatial variations of the magnetisation). | ||
| .. math:: | ||
| I_{mag}(\mathbf{Q},H)= S_K(Q) R_K(\mathbf{Q}, H_i) + S_M(Q) R_M(\mathbf{Q}, H_i), | ||
| with $H_i$ the internal field, i.e. the external magnetic field corrected for | ||
| demagnetizing effects and the influence of the magnetodipolar field and of the | ||
| magnetic anisotropy [#Bick2013]_. This magnetic field dependence of the scattering | ||
| reflects the increasing magnetisation misalignment with decreasing | ||
| externally applied magnetic field with a contribution $S_K \times R_K$ due to | ||
| perturbations around magnetic anisotropy fields and a term $S_M \times R_M$ | ||
| related to magnetostatic fields. The magnetic moments decorate perturbations in the | ||
| microstructure (precipitates, grain boundaries etc). | ||
| The anisotropy-field function $S_K$ depends on the Fourier transform of the magnetic | ||
| anisotropy distribution (strength and orientation) in the material, and the | ||
| scattering function of the longitudinal magnetisation $S_M$ reflects the | ||
| variations of the saturation magnetisation, e.g. jumps at the particle-matrix | ||
| interface. $R_K$ and $R_M$ denote the micromagnetic response functions that | ||
| describe the magnetisation distribution around a perturbation in magnetic | ||
| anisotropy and flucutations in the saturation magnetisation value. | ||
| .. figure:: img/micromagnetic_FF.png | ||
| Magnetisation distribution around (left) a particle with magnetic easy axis | ||
| in the vertical direction and (right) a precipitation with a magnetisation | ||
| that is higher than the matrix phase. | ||
| The micromagnetic response functions depend on magnetic material parameters $M_S$: | ||
| average saturation magnetisation of the material, $H_i$: the internal magnetic | ||
| field, $A$ the average exchange-stiffness constant. In the vicinity of lattice | ||
| imperfection in ferromagnetic materials, antisymmetric Dzyaloshinskii–Moriya | ||
| interaction (DMI) can occur due to the local structural inversion symmetry | ||
| breaking [#Arrott1963]_. DMI with strength $D$ can give rise to nonuniform spin | ||
| textures resulting in a polarization-dependent asymmetric scattering term for | ||
| polycrystalline ferromagnetic with a centrosymmetric crystal structure [#Michels2016]_. | ||
| We assume (for simplicity) an isotropic microstructure (for $S_M$) and random | ||
| orientation of magnetic easy axes (additionally for $S_K$) such that the | ||
| contributions of the magnetic microstructure only depend on the magnitude of $q$. | ||
| Considerations for a microstructure with a prefered orientation (texture) can be | ||
| found in [#Weissmueller2001]_. In the code the averaging procedure over the random | ||
| anisotropy is explicitely performed. A specific orientation distribution can be | ||
| implemented by rewriting the model. | ||
| The magnetic field is oriented with an inclination of $\alpha$ to the neutron beam | ||
| and rotated by $\beta$. The model for the nuclear scattering amplitude, saturation | ||
| magnetisation is based on spherical particles with a core shell structure. For | ||
| simplicity, only the core has an effective anisotropy, that is varying randomly | ||
| in direction from particle to particle. The effect of different, more complex | ||
| spatial profiles of the anisotropy can be seen in [#Michels2010]_. | ||
| The magnetic scattering length density (SLD) is defined as | ||
| $\rho_{\mathrm{mag}}=b_H M_S$, where $b_H= 2.91*10^{8}A^{-1}m^{-1}$ and $M_S$ | ||
| is the saturation magnetisation (in $A/m$). | ||
| The fraction of "upward" neutrons before ('up_frac_i') and after the sample | ||
| ('up_frac_f') must range between 0 to 1, with 0.5 denoting an unpolarised beam. | ||
| Note that a fit may result in a negative magnetic SLD, and hence magnetisation, | ||
| when the polarisation state is inverted, i.e. if you have analysed for a $I_{00}$ | ||
| state wheras your data are $I_{11}$. The model allows to construct the 4 | ||
| spin-resolved cross sections (non-spin-flip $I_{00}$, $I_{11}$ and spin-flip, here | ||
| $I_{01}=I_{10}$), half-polarised SANS (SANSpol, incoming polarised beam $I_0$ and | ||
| $I_1$, no analysis after sample 'up_frac_f'$=0.5$), and unpolarised beam | ||
| ('up_frac_i'$=$'up_frac_f'$=0.5$). Differences and other combinations between | ||
| polarised scattering cross section, e.g. to obtain the nuclear-magnetic | ||
| interference scattering, or subtraction of the residual scattering of the high | ||
| field reference state can be constructed with a custom model (Fitting> | ||
| Add/Multiply Model) and using approbriate scales. For dense systems, special | ||
| care has to be taken as the nculear structure factor (arrangement of particles) | ||
| does not need to be identical with the magnetic microstructure e.g. local | ||
| textures and correlations between easy axes (see [#Honecker2020]_ for further | ||
| details). The use of structure model is therefore strongly discouraged. Better | ||
| $I_{nuc}$, $S_K$ and $S_M$ are fit independent from each other in a model-free way. | ||
| References | ||
| ---------- | ||
| .. [#Arrott1963] A. Arrott, J. Appl. Phys. 34, 1108 (1963). | ||
| .. [#Weissmueller2001] J. Weissmueller et al., *Phys. Rev. B* 63, 214414 (2001). | ||
| .. [#Bick2013] J.-P. Bick et al., *Appl. Phys. Lett.* 102, 022415 (2013). | ||
| .. [#Michels2010] A. Michels et al., *Phys. Rev. B* 82, 024433 (2010). | ||
| .. [#Michels2014] A. Michels, *J. Phys.: Condens. Matter* 26, 383201 (2014). | ||
| .. [#Michels2016] A. Michels et al., *Phys. Rev. B* 94, 054424 (2016). | ||
| .. [#Honecker2020] D. Honecker, L. Fernandez Barguin, and P. Bender, *Phys. Rev. B* 101, 134401 (2020). | ||
| Authorship and Verification | ||
| ---------------------------- | ||
| * **Author:** Dirk Honecker **Date:** January 14, 2021 | ||
| * **Last Modified by:** Dirk Honecker **Date:** September 23, 2024 | ||
| * **Last Reviewed by:** | ||
| """ | ||
|
|
||
| import numpy as np | ||
| from numpy import pi, inf | ||
|
|
||
| name = "micromagnetic_FF_3D" | ||
| title = "Field-dependent magnetic microstructure around imperfections in bulk ferromagnets" | ||
| description = """ | ||
| I(q) = A (F_N^2(q)+ C F_N F_M + D F_M^2) +B(H) I_mag(q,H) | ||
| A: weighting function =1 for unpolarised beam and non-neutron-spin-flip scattering, zero for spin-flip scattering. The terms in the bracket are the residual scattering at perfectly saturating magnetic field. | ||
| B(H): weighting function for purely magnetic scattering I_mag(q,H) due to misaligned magnetic moments, different for the various possible spin-resolved scattering cross sections | ||
| C: weighting function for nuclear-magnetic interference scattering | ||
| F_N: nuclear form factor | ||
| F_M: magnetic form factor | ||
| The underlying defect can have a core-shell structure. | ||
| """ | ||
| category = "shape:sphere" | ||
|
|
||
| # pylint: disable=bad-whitespace, line-too-long | ||
| # ["name", "units", default, [lower, upper], "type","description"], | ||
| parameters = [["radius", "Ang", 50., [0, inf], "volume", "Structural radius of the core"], | ||
| ["thickness", "Ang", 40., [0, inf], "volume", "Structural thickness of shell"], | ||
| ["nuc_sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "", "Core scattering length density"], | ||
| ["nuc_sld_shell", "1e-6/Ang^2", 1.7, [-inf, inf], "", "Scattering length density of shell"], | ||
| ["nuc_sld_solvent", "1e-6/Ang^2", 6.4, [-inf, inf], "", "Solvent scattering length density"], | ||
| ["mag_sld_core", "1e-6/Ang^2", 1.0, [-inf, inf], "", "Magnetic scattering length density of core"], | ||
| ["mag_sld_shell", "1e-6/Ang^2", 1.7, [-inf, inf], "", "Magnetic scattering length density of shell"], | ||
| ["mag_sld_solvent", "1e-6/Ang^2", 3.0, [-inf, inf], "", "Magnetic scattering length density of solvent"], | ||
| ["hk_sld_core", "1e-6/Ang^2", 1.0, [0, inf], "", "Anisotropy field of defect"], | ||
| ["Hi", "T", 2.0, [0, inf], "", "Effective field inside the material"], | ||
| ["Ms", "T", 1.0, [0, inf], "", "Volume averaged saturation magnetisation"], | ||
| ["A", "pJ/m", 10.0, [0, inf], "", "Average exchange stiffness constant"], | ||
| ["D", "mJ/m^2", 0.0, [0, inf], "", "Average DMI constant"], | ||
| ["up_i", "None", 0.5, [0, 1], "", "Polarisation incoming beam"], | ||
| ["up_f", "None", 0.5, [0, 1], "", "Polarisation outgoing beam"], | ||
| ["alpha", "None", 90, [0, 180], "", "Inclination of field to neutron beam"], | ||
| ["beta", "None", 0, [0, 360], "", "Rotation of field around neutron beam"], | ||
| ] | ||
| # pylint: enable=bad-whitespace, line-too-long | ||
|
|
||
|
|
||
|
|
||
|
|
||
| source = ["lib/sas_3j1x_x.c", "lib/core_shell.c", "lib/gauss76.c", "lib/magnetic_functions.c", "micromagnetic_FF_3D.c"] | ||
| structure_factor = False | ||
| have_Fq = False | ||
| single=False | ||
| opencl = False | ||
|
|
||
|
|
||
| def random(): | ||
| """Return a random parameter set for the model.""" | ||
| outer_radius = 10**np.random.uniform(1.3, 4.3) | ||
| # Use a distribution with a preference for thin shell or thin core | ||
| # Avoid core,shell radii < 1 | ||
| radius = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1 | ||
| thickness = outer_radius - radius | ||
| pars = dict( | ||
| radius=radius, | ||
| thickness=thickness, | ||
|
|
||
| ) | ||
| return pars | ||
|
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|
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|
|
||
| tests = [ | ||
| [{},1.002266990452620e-03, 7.461046163627724e+03], | ||
| [{},(0.0688124, -0.0261013), 22.024], | ||
| ] |
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