Fitting reflectometry data

In this example we will fit reflectometry data obtained from GenX for the sample previously used in reflectometry simulation tutorial.

The only fitting parameter of the simulation considered here is the thickness of the Ti layers. The reference data was obtained under the following assumptions:

  • All Ti layers have the same thickness
  • Thickness value was $3 \, nm$

Fit window

The fit view produced by running the fitting script is shown in the picture. The right-hand part of the view contains information about the current iteration of the fitting process, the maximum relative difference $d_{r, max}$ between the reference and the simulated data, and the current values of the fitting parameters.

One should note that in the current example the BornAgain built-in fitting engine and default minimizer (namely, Minuit) was used to fit the data.

The minimizer can be selected by the setMinimizer command:

minimizer = ba.Minimizer()
minimizer.setMinimizer("Genetic", "", "MaxIterations=30")

This code snippet replaces the default Minuit minimizer with the Genetic one, which is recommended to use for complicated multi-dimensional fitting tasks.

Further topics

A much more sophisticated example of fitting experimental reflectometry data with BornAgain and an external minimizer can be found in Examples/python/fitting/ex03_ExtendedExamples/specular/RealLifeReflectometryFitting.py in the BornAgain directory.

Complete script and data

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"""
Example demonstrates how to fit specular data.
Our sample represents twenty interchanging layers of Ti and Ni. We will fit
thicknesses of all Ti layers, assuming them being equal.

Reference data was generated with GENX for ti layers' thicknesses equal to 3 nm
"""

import numpy as np
import bornagain as ba
from matplotlib import pyplot as plt
from os import path


def get_sample(params):
    """
    Creates a sample and returns it
    :param params: a dictionary of optimization parameters
    :return: the sample defined
    """

    # substrate (Si)
    si_sld_real = 2.0704e-06  # \AA^{-2}
    density_si = 0.0499 / ba.angstrom ** 3  # Si atomic number density

    # layers' parameters
    n_repetitions = 10
    # Ni
    ni_sld_real = 9.4245e-06  # \AA^{-2}
    ni_thickness = 70 * ba.angstrom
    # Ti
    ti_sld_real = -1.9493e-06  # \AA^{-2}
    ti_thickness = params["ti_thickness"]

    # defining materials
    m_air = ba.MaterialBySLD()
    m_ni = ba.MaterialBySLD("Ni", ni_sld_real, 0.0)
    m_ti = ba.MaterialBySLD("Ti", ti_sld_real, 0.0)
    m_substrate = ba.MaterialBySLD("SiSubstrate", si_sld_real, 0.0)

    # air layer and substrate form multi layer
    air_layer = ba.Layer(m_air)
    ni_layer = ba.Layer(m_ni, ni_thickness)
    ti_layer = ba.Layer(m_ti, ti_thickness)
    substrate_layer = ba.Layer(m_substrate)
    multi_layer = ba.MultiLayer()
    multi_layer.addLayer(air_layer)
    for i in range(n_repetitions):
        multi_layer.addLayer(ti_layer)
        multi_layer.addLayer(ni_layer)
    multi_layer.addLayer(substrate_layer)
    return multi_layer


def get_real_data():
    """
    Loading data from genx_interchanging_layers.dat
    Returns a Nx2 array (N - the number of experimental data entries)
    with first column being coordinates,
    second one being values.
    """
    if not hasattr(get_real_data, "data"):
        filename = "genx_interchanging_layers.dat.gz"
        filepath = path.join(path.dirname(path.realpath(__file__)), filename)
        real_data = np.loadtxt(filepath, usecols=(0, 1), skiprows=3)

        # translating axis values from double incident angle (degrees)
        # to incident angle (radians)
        real_data[:, 0] *= np.pi / 360
        get_real_data.data = real_data
    return get_real_data.data.copy()


def get_real_data_axis():
    """
    Get axis coordinates of the experimental data
    :return: 1D array with axis coordinates
    """
    return get_real_data()[:, 0]


def get_real_data_values():
    """
    Get experimental data values as a 1D array
    :return: 1D array with experimental data values
    """
    return get_real_data()[:, 1]


def get_simulation(params):
    """
    Create and return specular simulation with its instrument defined
    """
    wavelength = 1.54 * ba.angstrom  # beam wavelength

    simulation = ba.SpecularSimulation()
    scan = ba.AngularSpecScan(wavelength, get_real_data_axis())
    simulation.setScan(scan)
    simulation.setSample(get_sample(params))
    return simulation


def run_fitting():
    """
    Setup simulation and fit
    """

    real_data = get_real_data_values()

    fit_objective = ba.FitObjective()
    fit_objective.addSimulationAndData(get_simulation, real_data, 1.0)

    plot_observer = ba.PlotterSpecular()
    fit_objective.initPrint(10)
    fit_objective.initPlot(10, plot_observer)

    params = ba.Parameters()
    params.add("ti_thickness", 50 * ba.angstrom,
               min=10 * ba.angstrom, max=60 * ba.angstrom)

    minimizer = ba.Minimizer()
    result = minimizer.minimize(fit_objective.evaluate, params)

    fit_objective.finalize(result)


if __name__ == '__main__':
    run_fitting()
    plt.show()
FitSpecularBasics.py

Reference data