pnr - Jupyter Notebook

Menu Python 3 Trusted File Edit View Insert Cell Kernel WidgetsHelp In [1]: import numpy as npimport escape as escfrom escape.utils import showimport escape.utils.mdb as mdbimport matplotlib.pyplot as plt. . .Polarized neutron reflectivity¶ Polarized neutron reflectometry is a scattering method, which is well-suited for determining the structural parameters of magnetic thin films and multilayers. The specular reflection of polarized neutrons can be analyzed to yield the in-plane average of the vector magnetization depth profile of a magnetic sample along the surface normal. For the analysis of the polarization state of neutrons the PNR experimental setup has a polarizer, analyzer and two filppers after the polarizer and before anaylzer. The polarizer, analyzer and flippers are characterized by their efficiency, which one can define in ESCAPE as model parameters. If one is not familiar with polarized neutron relfectivity, there is a good introduction to this method and its theory [https://www.ncnr.nist.gov/staff/hammouda/distance_learning/pnrchapti.pdf] Below we demonstrate how to create a model for a single magnetic Fe layer on Si substrate. Magnetization of this layer has an angle of 45 degrees towards the direction of external magnetic field. In this example for the parametrization of the nuclear scattering we use scattering length density. The strength of magnetic scattering is described by magnetic scattering length density. In [2]: #SLDsFe_sld=esc.par("Fe SLD", 8.024, scale=1e-4, units="1/nm^2")Fe_sldm=esc.par("Fe SLDM", 5.00, scale=1e-4, units="1/nm^2")Si_sld=esc.par("Si SLD", 2.074, scale=1e-4, units="1/nm^2")thknFe = esc.par("Fe Thkn", 10, units="nm", userlim=[5, 15])bphiFe=esc.par("Fe1 Bphi", 45, units="deg", userlim=[-90, 90]). . .Next we create materials objects, layers and sample. In [3]: Fe = esc.amorphous("Fe", Fe_sld, 0.0, sldm=Fe_sldm)Si = esc.amorphous("Si", Si_sld, 0.0)Fe_layer = esc.magnetic_layer("Layer: Fe", Fe, bphiFe, 90, thknFe, 0.0)sub = esc.substrate("Substrate: Si", Si, 0.0) # roughness is zerosample = esc.multilayer("Fe/Si/Fe", esc.air("Air"), sub)sample.add(Fe_layer)#let's have a look on the profileshow(sample, xlabel="Z[nm]", ylabel="SLDRe[1/nm^2]", yaxis="sld0re")Fe SLDx0.0001[1/nm^2]:8.024Fe SLDMx0.0001[1/nm^2]:5.000Fe1 Bphi[deg]:45.00Fe Thkn[nm]:10.00Si SLDx0.0001[1/nm^2]:2.074Figure . . .Now we create calculation kernel for polarized neutron reflectivity. We use four parameters for efficiency of neutrons polarizers and flippers. The sign of the final efficiency parameters define the direction of neutron's polarization state: up (+) or down (-). In [4]: Qz=esc.var("qz")poleffi=esc.par("Polarization Effi", 1, userlim=[0, 1])polefff=esc.par("Polarization Efff", 1, userlim=[0, 1])flipeffi=esc.par("Flip Effi", 1, userlim=[0, 1])flipefff=esc.par("Flip Efff", 1, userlim=[0, 1])Rpp = esc.pnrspec("Specrefl ++", Qz, sample, poleffi, polefff)Rpm = esc.pnrspec("Specrefl +-", Qz, sample, poleffi, -polefff*flipefff)Rmp = esc.pnrspec("Specrefl -+", Qz, sample, -poleffi*flipeffi, polefff)Rmm = esc.pnrspec("Specrefl --", Qz, sample, -poleffi*flipeffi, -polefff*flipefff). . .We read the data from the file, provided by Dr. Yury Khaydukov. The data is generated in the software package and normalized. The errors are undefined. In order to obtain realistic fit errors for our fit parameters, we multiply the data by I0, apply poisson noise and find errors for every point folllowed by normalizing the data again. In general case, the I0 can be also a parameter. In [5]: qz, rpp, rpm, rmp, rmm = np.loadtxt("data/pnr/fe-si.dat", unpack=True)qz=qz*10 #1/A to 1/nmI0 = 1e8rpp = np.random.poisson(rpp*I0)/I0err_pp = np.sqrt(rpp)/np.sqrt(I0)rmm = np.random.poisson(rmm*I0)/I0err_mm = np.sqrt(rmm)/np.sqrt(I0)rpm = np.random.poisson(rpm*I0)/I0err_pm = np.sqrt(rpm)/np.sqrt(I0)rmp = np.random.poisson(rmp*I0)/I0err_mp = np.sqrt(rmp)/np.sqrt(I0)dobj_pp = esc.data("++", qz, rpp, err_pp, copy=True)dobj_pm = esc.data("+-", qz, rpm, err_pm, copy=True)dobj_mp = esc.data("-+", qz, rmp, err_mp, copy=True)dobj_mm = esc.data("--", qz, rmm, err_mm, copy=True). . .Next we create four models object, each for every polarization-flipping state. In [7]:x mobj_pp = esc.model("R++", Rpp, dobj_pp, residuals_scale="none", weight_type="data")mobj_mm = esc.model("R--", Rmm, dobj_mm, residuals_scale="none", weight_type="data")mobj_pm = esc.model("R+-", Rpm, dobj_pm, residuals_scale="none", weight_type="data")mobj_mp = esc.model("R-+", Rmp, dobj_mp, residuals_scale="none", weight_type="data")show([mobj_pp, mobj_mm, mobj_pm, mobj_mp], ylog=True, xlog=False, xlabel="Q[1/nm]", ylabel="Intensity")R++R--R+-R-+Fe SLDx0.0001[1/nm^2]:8.024Fe SLDMx0.0001[1/nm^2]:5.000Fe1 Bphi[deg]:45.00Fe Thkn[nm]:10.00Si SLDx0.0001[1/nm^2]:2.074Polarization Effi[arbitr.]:1.000Polarization Efff[arbitr.]:1.000Figure . . .Optimizing now with Levenberg-Marquardt algorithm In [9]: opt = esc.levmar("LM", [mobj_pp, mobj_mm, mobj_pm, mobj_mp], maxiter=1000, xtol=1e-15, ftol=1e-15, gtol=1e-15, nupdate=1)show(opt, ylog=True, xlog=False, xlabel="Q[1/nm]", ylabel="|R|^2")RunStopResetShakeBest cost: 1.742 Status: Convergence in parameter value Figure R++R--R+-R-+Fe SLDx0.0001[1/nm^2]:8.051Fe SLDMx0.0001[1/nm^2]:5.007Fe1 Bphi[deg]:45.00Fe Thkn[nm]:10.00Si SLDx0.0001[1/nm^2]:2.082Polarization Effi[arbitr.]:1.000Polarization Efff[arbitr.]:1.000Figure . . .In [10]: #Let's print parameters and their errors after fitoptOut[10]:Name: LM Parameters number: 9 Parameter Value +- Error Units Fixed Fe SLD 8.0509 +- 0.00013167 x0.0001 1/nm^2 0 Fe SLDM 5.007 +- 0.00011096 x0.0001 1/nm^2 0 Fe1 Bphi 44.998 +- 0.00097677 deg 0 Fe Thkn 10.003 +- 0.00019472 nm 0 Si SLD 2.0822 +- 3.7521e-05 x0.0001 1/nm^2 0 Polarization Effi 1 +- 0 arbitr. 0 Polarization Efff 1 +- 4.0483e-06 arbitr. 0 Flip Effi 0.99999 +- 5.6177e-06 arbitr. 0 Flip Efff 1 +- 0 arbitr. 0. . .In [ ]: . . .