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#!/usr/bin/env python3
# /// script
# requires-python = ">=3.10"
# dependencies = ["bornagain>=25,<26", "lmfit"]
# ///
"""
Fit of polarized reflectivities of a magnetic spinel film.
Jointly fits the polarized non-spin-flip reflectivities R++ and R-- of a
magnesium aluminum ferrite (MAFO, from Mg-Al-Fe-O; MgAl0.5Fe1.5O4) layer
on a magnesium aluminate (MAO, MgAl2O4) substrate to data measured at NIST
(https://www.nist.gov/ncnr/magnetically-dead-layers-spinel-films).
The spin asymmetry S = (R++ - R--)/(R++ + R--) is a derived diagnostic:
it is plotted, but not fitted separately. The initial model has zero
magnetic SLD and therefore no intrinsic spin contrast; the fit turns the
magnetization on. The companion example SpinAsymmetry simulates the fitted
model.
"""
import os
import numpy as np
import bornagain as ba
ba.require_versions("bornagain>=25,<26")
from bornagain import nm, ba_fitmonitor, ba_io, ba_plot as bp, R3
import lmfit
datadir = ba_io.data_dir()
fname_stem = os.path.join(datadir, "specular/MAFO_Saturated_")
MAO_SLD = (5.377e-06, 0) # SLD in Angstrom^-2
# Magnetic SLD in Angstrom^-2 per magnetization in A/m.
MAGNETIC_SLD_PER_MAGNETIZATION = 2.910429812376859e-12
def total_q_resolution(q_axis, dq_pointwise, sample_broadening):
"""
Combines pointwise instrument resolution with sample broadening.
The pointwise dQ is a standard deviation in 1/nm. The sample broadening is
an angular FWHM in degrees and is added linearly to the reconstructed
instrument angular FWHM, as in Refl1D. The wavelength and its resolution
are fixed by the MAFO experiment.
"""
wavelength = 0.475*nm
wavelength_resolution = 0.003*nm
fwhm_scale = np.sqrt(8*np.log(2))
theta = np.arcsin(q_axis*wavelength/(4*np.pi))
dq_spectral = q_axis*wavelength_resolution/wavelength
dq_angular = np.sqrt(dq_pointwise**2 - dq_spectral**2)
angular_fwhm = (
dq_angular*wavelength*fwhm_scale/(4*np.pi*np.cos(theta)))
angular_fwhm += np.deg2rad(sample_broadening)
dq_angular_broadened = (
(4*np.pi/wavelength)*np.cos(theta)*angular_fwhm/fwhm_scale)
return np.hypot(dq_spectral, dq_angular_broadened)
def get_sample(parameters):
"""
Magnesium aluminum ferrite (MAFO, from Mg-Al-Fe-O; MgAl0.5Fe1.5O4)
layer on a magnesium aluminate (MAO, MgAl2O4) substrate.
"""
magnetic_sld = parameters["mafo_magnetic_sld"]*1e-6
magnetization = R3(0, magnetic_sld/MAGNETIC_SLD_PER_MAGNETIZATION, 0)
vacuum = ba.Vacuum()
film_color = (0.45, 0.32, 0.80)
film_sld = parameters["mafo_sld"]*1e-6
film_material = ba.SLDMaterial(
"MgAl0.5Fe1.5O4", film_color, film_sld, 0, magnetization)
substrate_color = (0.28, 0.57, 0.82)
substrate_material = ba.SLDMaterial("MgAl2O4", substrate_color, *MAO_SLD)
film_autocorr = ba.SelfAffineFractalModel(
parameters["mafo_roughness"]*nm, 0.7, 25*nm)
substrate_autocorr = ba.SelfAffineFractalModel(
parameters["mao_roughness"]*nm, 0.7, 25*nm)
transient = ba.TanhTransient()
film_roughness = ba.Roughness(film_autocorr, transient)
substrate_roughness = ba.Roughness(substrate_autocorr, transient)
ambient_layer = ba.Layer(vacuum)
film_layer = ba.Layer(
film_material, parameters["mafo_thickness"]*nm, film_roughness)
substrate_layer = ba.Layer(substrate_material, substrate_roughness)
sample = ba.Sample()
sample.addLayer(ambient_layer)
sample.addLayer(film_layer)
sample.addLayer(substrate_layer)
return sample
def get_simulation(q_axis, q_resolution, parameters, spin_sign):
"""
Polarized specular simulation of one non-spin-flip channel.
q_axis and q_resolution contain one value per point, both in 1/nm.
spin_sign is +1 for the ++ channel, -1 for the -- channel.
"""
resolution_profile = ba.DistributionGaussian(0., 1., 25, 4.)
scan = ba.QzScan(q_axis)
scan.setOffset(parameters["q_offset"]/nm)
q_resolution = total_q_resolution(
q_axis, q_resolution, parameters["sample_broadening"])
scan.setVectorResolution(resolution_profile, q_resolution)
channel = R3(0, spin_sign, 0)
scan.setPolarization(channel)
scan.setAnalyzer(channel)
return ba.SpecularSimulation(scan, get_sample(parameters))
def load_data(fname):
"""
Reads dimensionless reflectivity, its uncertainty, and q resolution.
Interprets q and its pointwise standard deviation as 1/nm.
"""
q, reflectivity, uncertainty, q_resolution = ba_io.read_columns(
fname, usecols=(0, 1, 2, 3))
return q/nm, reflectivity, uncertainty, q_resolution/nm
def qz_datafield(q, values, errors=()):
"""
Wraps dimensionless values and errors on a q_z axis given in 1/nm.
"""
return ba.Datafield(ba.Frame(ba.ListScan("q_z (1/nm)", list(q))),
np.asarray(values, dtype=float).tolist(),
np.asarray(errors, dtype=float).tolist())
def spin_asymmetry(r_pp, r_mm):
"""
Spin asymmetry S = (R++ - R--)/(R++ + R--); NaN where undefined.
"""
denominator = r_pp + r_mm
with np.errstate(divide='ignore', invalid='ignore'):
return np.where(denominator != 0, (r_pp - r_mm)/denominator, np.nan)
def spin_asymmetry_error(r_pp, r_mm, sigma_pp, sigma_mm):
"""
Uncertainty of the spin asymmetry, assuming independent channels.
"""
denominator = (r_pp + r_mm)**2
with np.errstate(divide='ignore', invalid='ignore'):
return np.where(denominator != 0,
2*np.sqrt(r_mm**2*sigma_pp**2
+ r_pp**2*sigma_mm**2)/denominator,
np.nan)
if __name__ == '__main__':
q_pp, r_pp, sigma_pp, q_res_pp = load_data(fname_stem + "pp.tab")
q_mm, r_mm, sigma_mm, q_res_mm = load_data(fname_stem + "mm.tab")
if not np.array_equal(q_pp, q_mm):
raise ValueError(
"Spin asymmetry requires both channels on the same q grid")
qz_data = q_pp
def residuals(fit_parameters):
"""
Concatenated sigma-weighted residuals of both channels,
so that their sum of squares is the chi-square objective.
"""
values = fit_parameters.valuesdict()
sim_pp = get_simulation(qz_data, q_res_pp, values, +1).simulate()
sim_mm = get_simulation(qz_data, q_res_mm, values, -1).simulate()
return np.concatenate([
(r_pp - sim_pp.intensities())/sigma_pp,
(r_mm - sim_mm.intensities())/sigma_mm
])
parameters = lmfit.Parameters()
parameters.add("sample_broadening", value=0.026, min=0, max=0.1) # deg
parameters.add("q_offset", value=0, min=-0.002, max=0.002) # (1/nm)
parameters.add("mafo_sld", value=6.3649, min=2, max=7) # (1e-6 Å⁻²)
parameters.add("mafo_magnetic_sld", value=0, min=0, max=2) # (1e-6 Å⁻²)
parameters.add("mafo_thickness", value=15, min=6, max=18) # (nm)
parameters.add("mao_roughness", value=0.1, min=0, max=1.2) # (nm)
parameters.add("mafo_roughness", value=0.1, min=0, max=1.2) # (nm)
initial_parameters = parameters.valuesdict()
fit_result = lmfit.minimize(
residuals, parameters, method="leastsq",
iter_cb=ba_fitmonitor.Printer(every_nth=10))
print(lmfit.fit_report(fit_result))
fitted_parameters = fit_result.params.valuesdict()
# Evaluate smooth model curves on a denser q grid than the measured data.
qmin, qmax = 0.05997/nm, 1.96/nm
scan_size = 1500
qz_plot = np.linspace(qmin, qmax, scan_size)
q_res_plot_pp = np.interp(qz_plot, q_pp, q_res_pp)
q_res_plot_mm = np.interp(qz_plot, q_mm, q_res_mm)
# Both initial channels coincide because the magnetic SLD is zero.
initial_result = get_simulation(
qz_plot, q_res_plot_pp, initial_parameters, +1).simulate()
fitted_result_pp = get_simulation(
qz_plot, q_res_plot_pp, fitted_parameters, +1).simulate()
fitted_result_mm = get_simulation(
qz_plot, q_res_plot_mm, fitted_parameters, -1).simulate()
initial_intensity = initial_result.intensities()
initial_sa = spin_asymmetry(initial_intensity, initial_intensity)
fitted_sa = spin_asymmetry(fitted_result_pp.intensities(),
fitted_result_mm.intensities())
measured_sa = spin_asymmetry(r_pp, r_mm)
measured_sa_error = spin_asymmetry_error(r_pp, r_mm, sigma_pp, sigma_mm)
measured_result_pp = qz_datafield(qz_data, r_pp, sigma_pp)
measured_result_mm = qz_datafield(qz_data, r_mm, sigma_mm)
measured_sa_result = qz_datafield(qz_data, measured_sa, measured_sa_error)
initial_sa_result = qz_datafield(qz_plot, initial_sa)
fitted_sa_result = qz_datafield(qz_plot, fitted_sa)
# Plot the measured data and initial model before fitting.
initial_figure, (ax_initial_r, ax_initial_sa) = bp.plt.subplots(
2, 1, figsize=(8, 8))
initial_figure.suptitle("Before fitting")
bp.plot_specular_curves(
[("$R^{++}$ data", measured_result_pp, None),
("$R^{--}$ data", measured_result_mm, None)],
ax=ax_initial_r, ylabel="$R$")
bp.plot_specular_curves(
[("initial model (both channels)", None, initial_result)],
ax=ax_initial_r, ylabel="$R$", color='black')
ax_initial_r.legend()
bp.plot_specular_curves(
[("measured", measured_sa_result, None)],
ax=ax_initial_sa, yscale='linear', ylim=(-0.3, 0.5),
ylabel="Spin asymmetry", color='C0')
bp.plot_specular_curves(
[("initial model", None, initial_sa_result)],
ax=ax_initial_sa, yscale='linear', ylim=(-0.3, 0.5),
ylabel="Spin asymmetry", color='black')
ax_initial_sa.legend()
# Plot the measured data and fitted model after fitting.
fitted_figure, (ax_fitted_r, ax_fitted_sa) = bp.plt.subplots(
2, 1, figsize=(8, 8))
fitted_figure.suptitle("After fitting")
bp.plot_specular_curves(
[("$R^{++}$", measured_result_pp, fitted_result_pp),
("$R^{--}$", measured_result_mm, fitted_result_mm)],
ax=ax_fitted_r, ylabel="$R$")
ax_fitted_r.legend()
bp.plot_specular_curves(
[("measured", measured_sa_result, None)],
ax=ax_fitted_sa, yscale='linear', ylim=(-0.3, 0.5),
ylabel="Spin asymmetry", color='C0')
bp.plot_specular_curves(
[("fitted model", None, fitted_sa_result)],
ax=ax_fitted_sa, yscale='linear', ylim=(-0.3, 0.5),
ylabel="Spin asymmetry", color='C0')
ax_fitted_sa.legend()
initial_figure.tight_layout(rect=(0, 0, 1, 0.96))
fitted_figure.tight_layout(rect=(0, 0, 1, 0.96))
bp.plt.show()
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