SANS of bulk ferromagnets -

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_res+ I_mag(\mathbf{Q},H),

with the field-independent nuclear and magnetic residual SANS cross section (due to nanoscale spatial variations of the saturation magnetisation) measured at complete magnetic saturation. Commonly, a measurement at a high magnetic field taken as reference and subtracted from the scattering cross-section at lower field to obtain the so-called spin-misalignment scattering cross-section 

.. 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 purely magnetic scattering reflects the sensitive response of the transversal magnetisation components with 
to an 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 alignment of the magnetic moments along the magnetic field is disturbed by 
perturbations in the microstructure. 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. 

The magnetic field is oriented with an inclination of $alpha$ to the neutron beam ans 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 Michels PRB 82, 024433 (2010). 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. 


The model needs testing and validation. Your feedback is kindly requested.


.. [#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).
.. [#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:**
* **Last Reviewed by:**


import numpy as np
from numpy import pi, inf

name = "spin_structure_ferromagnets_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", "Radius of the core"],
              ["thickness", "Ang", 40., [0, inf], "volume", "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"],
              ["magnetic_sld_core",    "1e-6/Ang^2", 1.0,  [-inf, inf], "",    "Magnetic scattering length density of core"],
              ["magnetic_sld_shell", "1e-6/Ang^2", 1.7, [-inf, inf], "", "Magnetic scattering length density of shell"],
              ["magnetic_sld_solvent", "1e-6/Ang^2", 3.0,  [-inf, inf], "",  "Magnetic scattering length density of solvent"],
              ["hk_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", "micromagnetic_FF_3D.c"]
structure_factor = False
have_Fq = False

effective_radius_type = ["outer radius", "core radius"]

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(
    return pars

# tests = [
    # [{'radius': 20.0, 'thickness': 10.0}, 0.1, None, None, 30.0, 4.*pi/3*30**3, 1.0],

    # # The SasView test result was 0.00169, with a background of 0.001
    # [{'radius': 60.0, 'thickness': 10.0, 'sld_core': 1.0, 'sld_shell': 2.0,
      # 'sld_solvent': 3.0, 'background': 0.0}, 0.4, 0.000698838],
# ]

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