Spin-asymmetry example from NIST

This example shows how to simulate the magnetic layer described on the NIST homepage in the Magnetically Dead Layers in Spinel Films example. In particular, we want to show how to use BornAgain in order to simulate the spin asymmetry. The sample simulated in this example is very similar to the previous examples introduced in this section. It just consists of a magnetic layer on top of a substrate. During these tutorials, we neglect the magnetically dead layer that forms below the magnetic layer, as there is currently no API in BornAgain to support such a scenario out of the box.

In this first example, we utilize parameters that are deduced from a fit to the data provided on the NIST homepage. How to perform the fit is described in the extended example.

Spin asymmetry

The spin asymmetry is defined as

$$S = \frac{R^{++} - R^{- -}}{R^{++} + R^{- -}}$$

Therefore, we only need to perform a normal polarized simulation for the up-up and down-down channels and then compute the spin asymmetry.

Given the experimental data, the measured spin asymmetry is calculated in the same way. In addition, the error is computed by:

$$\Delta S = \frac{\sqrt{ 4 {R^{++}}^2 \left( \Delta {R^{- -}} \right)^2 + 4 {R^{- -}}^2 \left( \Delta {R^{++}}\right)^2 }}{ \left( R^{++} + R^{- -}\right)^2 } $$

This is performed in the function plotSpinAsymmetry.

Further corrections

We also apply a resolution correction, as described in the ToF - Resolution effects example.

Furthermore, we introduce an offest in the $Q$-axis, in order to accomodate for experimental uncertainties in the measurement of $\theta$. For this purpose, the provided $Q$-axis is shifted in the function get_simulation:

q_axis = q_axis + parameters["q_offset"]

Data processing

After loading the experimental data, we scale the q-axis in order to obtain inverse nm as they are the default units in BornAgain. Furthermore, the reflectivity data is scaled such that its maximum is unity.

Simulation result

Reflectivity

Spin Asymmetry

Here is the complete example:

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#!/usr/bin/env python3
"""
This 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 os
import numpy
import matplotlib.pyplot as plt
import bornagain as ba
from bornagain import angstrom, ba_plot as bp, deg

# 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.R3(
        magnetizationMagnitude*numpy.sin(angle*deg),
        magnetizationMagnitude*numpy.cos(angle*deg), 0)

    mat_vacuum = ba.MaterialBySLD("Vacuum", 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(params["r_Mafo"]*angstrom)
    r_substrate = ba.LayerRoughness(params["r_Mao"]*angstrom)

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

    return sample


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

    # unused, why was this define: 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"])

    scan.setPolarization(polarizer_dir)
    scan.setAnalyzer(analyzer_dir, 1, 0.5)

    return ba.SpecularSimulation(scan, sample)


def run_simulation(q_axis, fitParams, *, polarizer_dir, analyzer_dir):
    """
    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(sample, q_axis, parameters, polarizer_dir,
                                analyzer_dir)

    return simulation.simulate()


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.convertedBinCenters(ba.Coords_QSPACE))
    r = numpy.array(result.array(ba.Coords_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 load_exp(fname):
    dat = numpy.loadtxt(fname)
    return numpy.transpose(dat)


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]


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

if __name__ == '__main__':
    bp.parse_args()

    datadir = os.getenv('BA_EXAMPLE_DATA_DIR', '')
    fname_stem = os.path.join(datadir, "MAFO_Saturated_")

    expdata_pp = load_exp(fname_stem + "pp.tab")
    expdata_mm = load_exp(fname_stem + "mm.tab")

    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
    }

    def run_Simulation_pp(qzs, params):
        return run_simulation(qzs,
                              params,
                              polarizer_dir=ba.R3(0, 1, 0),
                              analyzer_dir=ba.R3(0, 1, 0))

    def run_Simulation_mm(qzs, params):
        return run_simulation(qzs,
                              params,
                              polarizer_dir=ba.R3(0, -1, 0),
                              analyzer_dir=ba.R3(0, -1, 0))

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

    data_pp = filterData(expdata_pp, qmin, qmax)
    data_mm = filterData(expdata_mm, qmin, qmax)

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

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

    bp.show_or_export()
Examples/specular/PolarizedSpinAsymmetry.py