Fitting the spin-asymmetry example from NIST

This example shows how to fit the parameters in the spin-asymmetry example.

For this demonstration, we choose initial parameters that are not too far from the fitting results. In particular, the magnetization is initially set to zero, such that the spin asymmetry identically vanishes.

With the initial parameters, we obtain the following reflectivity and spin-asymmetry curves:

Reflectivity

Spin Asymmetry

Setup of the Fit

For fitting of reflectometry data covering several orders of magnitude we use the $\chi^2$ metric

$$\chi^2 = \sum_{i = 1}^N \frac{\left( d_i - s_i \right)^2}{\sigma_i^2}$$

Here $d_i$ is the $i$-thexperimental data point, $\sigma_i$ is its uncertainty and $s_i$ is the corresponding simulation result.

This is supported in BornAgain by setting

fit_objective.setObjectiveMetric("chi2")

It must be noted, that in order to obtain good results, one needs to provide the uncertainties of the reflectivity. If no uncertainties are available, using the relative difference fit_objective.setObjectiveMetric("reldiff") yields better results. If the relative difference is selected and uncertainties are provided, BornAgain automatically falls back to the above $\chi^2$ metric.

The fitting of polarized reflectometry data proceeds similar to the lines presented in the tutorial on multiple datasets. We need to add the reflectivity curves for the up-up and down-down channel to the fit objective:

fit_objective.addSimulationAndData( SimulationFunctionPlusplus,
                                    r_data_pp, r_uncertainty_pp, 1.0)
fit_objective.addSimulationAndData( SimulationFunctionMinusMinus,
                                    r_data_mm, r_uncertainty_mm, 1.0)

SimulationFunctionPlusplus and SimulationFunctionMinusMinus are two function objects that return a simulation result for the up-up and down-down channels, respectively.

The fit parameters are defined in the dictionary startParams, where they are defined as a triple of values (start, min, max). If no fit is performed the values obtained from our own fit are stored in fixedParams and are subsequently used to simulate the system.

We want to fit the following parameters:

  • q_res: Relative $Q$-resolution
  • q_offset: Shift of the $Q$-axis.
  • t_Mafo: The thickness of the layer
  • rho_Mafo: The SLD of the layer
  • rhoM_Mafo: The magnetic SLD of the layer
  • r_Mao: The roughness on top of the substrate
  • r_Mafo: The roughness on top of the magnetic layer

Fit Result

After running the fit using

python3 PolarizedSpinAsymmetryFit.py fit

we get the result

Reflectivity

Spin Asymmetry

This result was already presented in the spin-asymmetry tutorial and can also be obtained by runnning the example without the fit option:

python3 PolarizedSpinAsymmetryFit.py

Here is the complete example:

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"""
This fitting and simulation example demonstrates how to replicate
the fitting example "Magnetically Dead Layers in Spinel Films"
given at the Nist website:
https://www.nist.gov/ncnr/magnetically-dead-layers-spinel-films

For simplicity, here we only reproduce the first part of that
demonstration without the magnetically dead layer.
"""

# import boranagain
import bornagain as ba
from bornagain import deg, angstrom, nm

import numpy
import matplotlib.pyplot as plt

# import more libs needed for data processing
from re import match, DOTALL
from sys import argv
from io import BytesIO
from urllib.request import urlopen
from zipfile import ZipFile
from os.path import isfile

# q-range on which the simulation and fitting are to be performed
qmin = 0.05997
qmax = 1.96

# number of points on which the computed result is plotted
scan_size = 1500

# The SLD of the substrate is kept constant
sldMao = (5.377e-06, 0)

# constant to convert between magnetization and magnetic SLD
RhoMconst = 2.910429812376859e-12

####################################################################
#                  Create Sample and Simulation                    #
####################################################################


def get_sample(params):
    """
    construct the sample with the given parameters
    """
    magnetizationMagnitude = params["rhoM_Mafo"]*1e-6/RhoMconst
    angle = 0
    magnetizationVector = ba.kvector_t(
        magnetizationMagnitude*numpy.sin(angle*deg),
        magnetizationMagnitude*numpy.cos(angle*deg), 0)

    mat_vacuum = ba.MaterialBySLD("Vacuum", 0.0, 0.0)
    mat_layer = ba.MaterialBySLD("(Mg,Al,Fe)3O4", params["rho_Mafo"]*1e-6, 0,
                                 magnetizationVector)
    mat_substrate = ba.MaterialBySLD("MgAl2O4", *sldMao)

    ambient_layer = ba.Layer(mat_vacuum)
    layer = ba.Layer(mat_layer, params["t_Mafo"]*angstrom)
    substrate_layer = ba.Layer(mat_substrate)

    r_Mafo = ba.LayerRoughness()
    r_Mafo.setSigma(params["r_Mafo"]*angstrom)

    r_substrate = ba.LayerRoughness()
    r_substrate.setSigma(params["r_Mao"]*angstrom)

    multi_layer = ba.MultiLayer()
    multi_layer.addLayer(ambient_layer)
    multi_layer.addLayerWithTopRoughness(layer, r_Mafo)
    multi_layer.addLayerWithTopRoughness(substrate_layer, r_substrate)

    return multi_layer


def get_simulation(q_axis, parameters, polarization, analyzer):
    """
    Returns a simulation object.
    Polarization, analyzer and resolution are set
    from given parameters
    """
    simulation = ba.SpecularSimulation()
    q_axis = q_axis + parameters["q_offset"]
    scan = ba.QSpecScan(q_axis)

    dq = parameters["q_res"]*q_axis
    n_sig = 4.0
    n_samples = 25

    distr = ba.RangedDistributionGaussian(n_samples, n_sig)
    scan.setAbsoluteQResolution(distr, parameters["q_res"])

    simulation.beam().setPolarization(polarization)
    simulation.setAnalyzerProperties(analyzer, 1.0, 0.5)

    simulation.setScan(scan)
    return simulation


def run_simulation(q_axis, fitParams, *, polarization, analyzer):
    """
    Run a simulation on the given q-axis, where the sample is constructed
    with the given parameters.
    Vectors for polarization and analyzer need to be provided
    """
    parameters = dict(fitParams, **fixedParams)

    sample = get_sample(parameters)
    simulation = get_simulation(q_axis, parameters, polarization, analyzer)

    simulation.setSample(sample)
    simulation.runSimulation()
    return simulation


def qr(result):
    """
    Returns two arrays that hold the q-values as well as the
    reflectivity from a given simulation result
    """
    q = numpy.array(result.result().axis(ba.Axes.QSPACE))
    r = numpy.array(result.result().array(ba.Axes.QSPACE))

    return q, r


####################################################################
#                         Plot Handling                            #
####################################################################


def plot(qs, rs, exps, labels, filename):
    """
    Plot the simulated result together with the experimental data
    """
    fig = plt.figure()
    ax = fig.add_subplot(111)

    for q, r, exp, l in zip(qs, rs, exps, labels):

        ax.errorbar(exp[0],
                    exp[1],
                    xerr=exp[3],
                    yerr=exp[2],
                    fmt='.',
                    markersize=0.75,
                    linewidth=0.5)

        ax.plot(q, r, label=l)

    ax.set_yscale('log')
    plt.legend()

    plt.xlabel("Q [nm${}^{-1}$]")
    plt.ylabel("R")

    plt.tight_layout()
    plt.savefig(filename)


def plotSpinAsymmetry(data_pp, data_mm, q, r_pp, r_mm, filename):
    """
    Plot the simulated spin asymmetry as well its
    experimental counterpart with errorbars
    """

    # compute the errorbars of the spin asymmetry
    delta = numpy.sqrt(4 * (data_pp[1]**2 * data_mm[2]**2 + \
            data_mm[1]**2 * data_pp[2]**2 ) /
                ( data_pp[1] + data_mm[1] )**4 )

    fig = plt.figure()
    ax = fig.add_subplot(111)

    ax.errorbar(data_pp[0], (data_pp[1] - data_mm[1])/(data_pp[1] + data_mm[1]),
                xerr=data_pp[3],
                yerr=delta,
                fmt='.',
                markersize=0.75,
                linewidth=0.5)

    ax.plot(q, (r_pp - r_mm)/(r_pp + r_mm))

    plt.gca().set_ylim((-0.3, 0.5))

    plt.xlabel("Q [nm${}^{-1}$]")
    plt.ylabel("Spin asymmetry")

    plt.tight_layout()
    plt.savefig(filename)


####################################################################
#                          Data Handling                           #
####################################################################


def normalizeData(data):
    """
    Removes duplicate q values from the input data,
    normalizes it such that the maximum of the reflectivity is
    unity and rescales the q-axis to inverse nm
    """

    r0 = numpy.where(data[0] - numpy.roll(data[0], 1) == 0)
    data = numpy.delete(data, r0, 1)

    data[0] = data[0]/angstrom
    data[3] = data[3]/angstrom

    norm = numpy.max(data[1])
    data[1] = data[1]/norm
    data[2] = data[2]/norm

    so = numpy.argsort(data[0])
    data = data[:, so]

    return data


def filterData(data, qmin, qmax):
    minIndex = numpy.argmin(numpy.abs(data[0] - qmin))
    maxIndex = numpy.argmin(numpy.abs(data[0] - qmax))

    return data[:, minIndex:maxIndex + 1]


def get_Experimental_data(qmin, qmax):
    input_Data = downloadAndExtractData()
    data_pp = normalizeData(input_Data[0])
    data_mm = normalizeData(input_Data[1])

    return (filterData(data_pp, qmin, qmax), filterData(data_mm, qmin, qmax))


def downloadAndExtractData():
    url = "https://www.nist.gov/document/spinelfilmzip"

    if not isfile("spinelfilm.zip"):
        downloadfile = urlopen(url)
        with open("spinelfilm.zip", 'wb') as outfile:
            outfile.write(downloadfile.read())

    zipfile = ZipFile("spinelfilm.zip")

    rawdata = zipfile.open("MAFO_Saturated.refl").read().decode("utf-8")

    table_pp = match(
        r'.*# "polarization": "\+\+"\n#.*?\n# "units".*?\n(.*?)#.*', rawdata,
        DOTALL).group(1)
    table_mm = match(
        r'.*# "polarization": "\-\-"\n#.*?\n# "units".*?\n(.*?)#.*', rawdata,
        DOTALL).group(1)

    data_pp = numpy.genfromtxt(BytesIO(table_pp.encode()), unpack=True)
    data_mm = numpy.genfromtxt(BytesIO(table_mm.encode()), unpack=True)

    return (data_pp, data_mm)


####################################################################
#                          Fit Function                            #
####################################################################


def run_fit_ba(q_axis, r_data, r_uncertainty, simulationFactory, startParams):

    fit_objective = ba.FitObjective()
    fit_objective.setObjectiveMetric("chi2")

    fit_objective.addSimulationAndData(
        lambda params: simulationFactory[0](q_axis[0], params), r_data[0],
        r_uncertainty[0], 1.0)
    fit_objective.addSimulationAndData(
        lambda params: simulationFactory[1](q_axis[1], params), r_data[1],
        r_uncertainty[1], 1.0)

    fit_objective.initPrint(10)

    params = ba.Parameters()
    for name, p in startParams.items():
        params.add(name, p[0], min=p[1], max=p[2])

    minimizer = ba.Minimizer()

    result = minimizer.minimize(fit_objective.evaluate, params)
    fit_objective.finalize(result)

    return {r.name(): r.value for r in result.parameters()}


####################################################################
#                          Main Function                           #
####################################################################

if __name__ == '__main__':

    if len(argv) > 1 and argv[1] == "fit":
        fixedParams = {
            # parameters can be moved here to keep them fixed
        }
        fixedParams = {d: v[0] for d, v in fixedParams.items()}

        startParams = {
            # own starting values
            "q_res": (0.0, 0, 0.1),
            "q_offset": (0, -0.002, 0.002),
            "rho_Mafo": (6.3649, 2, 7),
            "rhoM_Mafo": (0, 0, 2),
            "t_Mafo": (150, 60, 180),
            "r_Mao": (1, 0, 12),
            "r_Mafo": (1, 0, 12),
        }
        fit = True

    else:
        startParams = {}
        fixedParams = {
            # parameters from our own fit run
            'q_res': 0.010542945012551425,
            'q_offset': 7.971243487467318e-05,
            'rho_Mafo': 6.370140108715461,
            'rhoM_Mafo': 0.27399566816062926,
            't_Mafo': 137.46913056084736,
            'r_Mao': 8.60487712674644,
            'r_Mafo': 3.7844265311293483
        }

        fit = False

    paramsInitial = {d: v[0] for d, v in startParams.items()}

    def run_Simulation_pp(qzs, params):
        return run_simulation(qzs,
                              params,
                              polarization=ba.kvector_t(0, 1, 0),
                              analyzer=ba.kvector_t(0, 1, 0))

    def run_Simulation_mm(qzs, params):
        return run_simulation(qzs,
                              params,
                              polarization=ba.kvector_t(0, -1, 0),
                              analyzer=ba.kvector_t(0, -1, 0))

    qzs = numpy.linspace(qmin, qmax, scan_size)
    q_pp, r_pp = qr(run_Simulation_pp(qzs, paramsInitial))
    q_mm, r_mm = qr(run_Simulation_mm(qzs, paramsInitial))

    data_pp, data_mm = get_Experimental_data(qmin, qmax)

    plot([q_pp, q_mm], [r_pp, r_mm], [data_pp, data_mm], ["$++$", "$--$"],
         f'MAFO_Saturated_initial.pdf')

    plotSpinAsymmetry(data_pp, data_mm, qzs, r_pp, r_mm,
                      "MAFO_Saturated_spin_asymmetry_initial.pdf")

    if fit:
        fitResult = run_fit_ba([data_pp[0], data_mm[0]],
                               [data_pp[1], data_mm[1]],
                               [data_pp[2], data_mm[2]],
                               [run_Simulation_pp, run_Simulation_mm],
                               startParams)
        print("Fit Result:")
        print(fitResult)

        q_pp, r_pp = qr(run_Simulation_pp(qzs, fitResult))
        q_mm, r_mm = qr(run_Simulation_mm(qzs, fitResult))

        plot([q_pp, q_mm], [r_pp, r_mm], [data_pp, data_mm], ["$++$", "$--$"],
             f'MAFO_Saturated_fit.pdf')

        plotSpinAsymmetry(data_pp, data_mm, qzs, r_pp, r_mm,
                          "MAFO_Saturated_spin_asymmetry_fit.pdf")
PolarizedSpinAsymmetryFit.py