### Reflectometry: Fit Pt layer

In this example, we want to demonstrate how to fit experimental reflectivity data that was obtained by a time-of-flight experiment with unpolarized neutrons. Experimental data is available for a sample of a roughly 50 nm thick platinum layer on top of a silicon substrate that is published in this repository.

The mesaurements were made by Timothy Charlton, Haile Ambaye and Michael Fitzsimmons (ORNL) on a sample provided by Eric Fullerton (UCSD).

#### Fit model

We describe the above experiment by a three-layer model, where as usual the top layer is the vacuum and the substrate layer is the silicon substrate. On top of the silicon substrate, we place the platinum layer. The materials of both layers are described by their SLD, where we use literature values for both silicon as well as platinum and keep them constant throughout the fitting procedure.

The main parameters of the sample are stored in dictionary, where they are defined by a unique name and the following six parameters are utilized:

• Beam intensity: intensity

We explicitly fit the beam intensity, in order to compensate for possible experimental errors and to circumvent problems with the rather large variance in the reflectivity data at low $Q$-values.

• Roughness on top of the Pt layer: r_pt/nm

• Roughness on top of the Si substrate: r_si/nm

• Thickness of the Pt layer: t_pt/nm

• The relative $Q$-resolution: q_res/q (c.f.)

• A $Q$-offset: q_offset

This global offset is introduced to account for uncertainties in the angle at which the measurement is performed.

Due to saturation of the detector it is possible that the intensity at low $Q$-values (i.e. at high count rates) is underestimated. Furthermore, there is a rather large variance in the data that also leads to a rather bad fit in this region. Therefore, we neglect the data in the low $Q$-region by choosing a cutoff at $Q_{\text{min}} =$ 0.18. This value is selected by hand after performing several fits and visually selecting a good result.

##### $Q$-offset

Currently, BornAgain does not have an API support for an offset of the $Q$-axis. Therefore, we need to shift the $Q$-axis before performing a simulation

q_axis = q_axis + parameters["q_offset"]

This shift then needs to be counter-transformed when returning the results in the qr(result) function

q = numpy.array(result.result().axis(ba.Axes.QSPACE)) - q_offset

##### Initial parameters

In order to successfully fit this example, we chose some sane starting values and the example code, that is fully given below, can be run with the following command:

python3 Pt_layer_fit.py fit
This performs a simulation with the initial parameters and yields the following result:

Immediately afterwards the fit is performed.

#### Fit result

In order to run the fitting procedure, the following command can be issued:

python3 Pt_layer_fit.py fit

We need to allow a few seconds computational time and BornAgain should compute the following result

If the fit keyword is omitted from the command line

python3 Pt_layer_fit.py
a simulation is performed with our fit results and one should obtain the result shown above.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252  #!/usr/bin/env python3 """ This example demonstrates how to fit actual experimental data by M. Fitzsimmons et al. that is published in https://doi.org/10.5281/zenodo.4072376 """ import numpy import matplotlib.pyplot as plt from sys import argv import bornagain as ba from bornagain import angstrom # filename of the experimental data to be loaded filename = 'RvsQ_36563_36662.txt.gz' # restrict the Q-range of the data used for fitting qmin = 0.18 qmax = 2.4 # number of points on which the computed result is plotted scan_size = 1500 # Use fixed values for the SLD of the substrate and Pt layer sldPt = (6.3568e-06, 1.8967e-09) sldSi = (2.0728e-06, 2.3747e-11) #################################################################### # Create Sample and Simulation # #################################################################### def get_sample(params): mat_ambient = ba.MaterialBySLD("Ambient", 0, 0) mat_layer = ba.MaterialBySLD("Pt", *sldPt) mat_substrate = ba.MaterialBySLD("Si", *sldSi) ambient_layer = ba.Layer(mat_ambient) layer = ba.Layer(mat_layer, params["t_pt/nm"]) substrate_layer = ba.Layer(mat_substrate) r_si = ba.LayerRoughness() r_si.setSigma(params["r_si/nm"]) r_pt = ba.LayerRoughness() r_pt.setSigma(params["r_pt/nm"]) multi_layer = ba.MultiLayer() multi_layer.addLayer(ambient_layer) multi_layer.addLayerWithTopRoughness(layer, r_pt) multi_layer.addLayerWithTopRoughness(substrate_layer, r_si) return multi_layer def get_simulation(q_axis, parameters): scan = ba.QSpecScan(q_axis) scan.setOffset(parameters["q_offset"]) n_sig = 4.0 n_samples = 25 distr = ba.RangedDistributionGaussian(n_samples, n_sig) scan.setAbsoluteQResolution(distr, parameters["q_res/q"]) simulation = ba.SpecularSimulation() simulation.beam().setIntensity(parameters["intensity"]) simulation.setScan(scan) return simulation def run_simulation(q_axis, fitParams): parameters = dict(fitParams, **fixedParams) sample = get_sample(parameters) simulation = get_simulation(q_axis, parameters) simulation.setSample(sample) simulation.runSimulation() return simulation #.result() def qr(result): """ helper function to return the q axis and reflectivity from 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(q, r, exp, filename, params=None): """ helper function to plot a result """ fig = plt.figure() ax = fig.add_subplot(111) ax.errorbar(exp[0], exp[1], xerr=exp[3], yerr=exp[2], label="R", fmt='.', markersize=1., linewidth=0.6, color='r') ax.plot(q, r, label="Simulation", color='C0', linewidth=0.5) ax.set_yscale('log') ax.set_xlabel("Q [nm$^{^-1}$]") ax.set_ylabel("R") y = 0.5 if params is not None: for n, v in params.items(): plt.text(0.7, y, f"{n} = {v:.3g}", transform=ax.transAxes) y += 0.05 plt.tight_layout() plt.savefig(filename) #################################################################### # Data Handling # #################################################################### def get_Experimental_data(qmin, qmax): """ read experimental data, remove some duplicate q-values recalculate q axis to inverse nm """ data = numpy.genfromtxt(filename, unpack=True) 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 data[1] = data[1] data[2] = data[2] so = numpy.argsort(data[0]) data = data[:, so] minIndex = numpy.argmin(numpy.abs(data[0] - qmin)) maxIndex = numpy.argmin(numpy.abs(data[0] - qmax)) return data[:, minIndex:maxIndex + 1] #################################################################### # 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(q_axis, params), r_data, r_uncertainty, 1) 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_offset": (0, -0.02, 0.02), "q_res/q": (0, 0, 0.02), "t_pt/nm": (53, 40, 60), "r_si/nm": (1.22, 0, 5), "r_pt/nm": (0.25, 0, 5), "intensity": (1, 0, 2), } fit = True else: startParams = {} fixedParams = { # parameters from our own fit run 'q_offset': 0.015085985992837999, 'q_res/q': 0.010156450689003465, 't_pt/nm': 48.564838355355405, 'r_si/nm': 1.2857515425763575, 'r_pt/nm': 0.2868252673771518, 'intensity': 1.3156374978332654 } fit = False paramsInitial = {d: v[0] for d, v in startParams.items()} qzs = numpy.linspace(qmin, qmax, scan_size) q, r = qr(run_simulation(qzs, paramsInitial)) data = get_Experimental_data(qmin, qmax) plot(q, r, data, f'PtLayerFit_initial.pdf', dict(paramsInitial, **fixedParams)) if fit: fitResult = run_fit_ba(data[0], data[1], data[2], run_simulation, startParams) print("Fit Result:") print(fitResult) q, r = qr(run_simulation(qzs, fitParams=fitResult)) plot(q, r, data, f'PtLayerFit_fit.pdf', dict(fitResult, **fixedParams)) 
Pt_layer_fit.py Reference data