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"# Mono Calibration and EXAFS Results\n",
"\n",
"This notebook explores the effect of energy miscalibration on EXAFS results.\n",
"\n",
"EXAFS measurements need a well-calibrated energy, typically coming from a double-cystal monochromator. While EXAFS can be measured in modes without a double-crystal monochromator, that is sort of unusual, and we are not discussing that situation here. A double-cystal monochromator uses Bragg's Law to relate the monochromator angle $\\theta$ to X-ray energy $E$ with\n",
"\n",
" $\\lambda = \\frac{hc}{E} = 2d\\sin(\\theta) $\n",
"\n",
"With $h$ being Planck's constant, $c$ the speed of light, and $d$ the lattice spacing of the monochromator crystal. With $d$ in Angstroms and $E$ in eV, $hc \\approx 12398.419$ eV A. The crystal $d$ values for Si, Ge, and C (the most common crystals used0 are typically very well known, but there may be thermally-induced changes. The values of $d$ and an angular offset ($\\theta_0$ may need to be adjusted to calibrate a monochromator.\n",
"This is typically done by scanning across a known (or at least reproducible) energies, say the $K$-edges of metal foils, and adjusting $d$ or $\\theta_0$ until the energy is correct. \n",
"\n",
"\n",
"Most commonly, the XANES of some metal foil is measured and the maximumn of the first derivative is set to the tabulated edge energy for the metal. This method will give a bit of a variation as \n",
"the energy resolution changes. These days, most good XAFS beamlines have high enough energy \n",
"resolution that this is not much of a problem, but it can be a problem for older data. It should \n",
"also be said that those tabulated values are not always accurate to better than 1 or 2 eV. In fact, \n",
"rather than use the tabulated values such as at https://xraydb.xrayabsorption.org/ the values from Kraft et al Review of Scientific Instruments 67, p681 (1996): (https://doi.org/10.1063/1.1146657) should be used, as they were carefully measured with a single, very high-resolution and consistently calibrated monochromator.\n",
"\n",
"\n",
"Ideally, a well-calibrated monochromator would have a single d-spacing and angular offset that \n",
"stays calibrated across the edges of many edges. That is, if the energy needs to be recalibrated\n",
"at every edge, the energy will be drifting between those edges, and the reason for any need to\n",
"recalibrate should be investigated. If the beamline is well collimated and the angle of the beam \n",
"incident on the monochomator is stable, it is certainly possible to have a monochromator set up t\n",
"that stays calibrated for weeks and over an energy range of 10 keV or more. \n",
"\n",
"\n",
"For any XAS analysis, the energy scale is critical. Because of the variations above in calibrating \n",
"energy, small energy shifts between beamlines or even runs at the same beamline but different months \n",
"are not uncommon. If the shift is relatively small, say 1 to 5 eV at 10 keV, simply adding a constant \n",
"energy offset is satisfactory.\n",
" \n",
"You may hear people say that poor calibration leads to large errors in XAFS results or even leads \n",
"\"the wrong results\". You should know that when people say things are Wrong is describing scientific\n",
"measurements and analysis, they are almost certainly wrong. \n",
"\n",
"Here, we'll look at these effects\n"
]
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"# we start with some imports of python functions we will need to read, \n",
"# modify, and fit xafs data:\n",
"\n",
"import numpy as np\n",
"import scipy.constants as consts\n",
"\n",
"\n",
"from larch.io import read_ascii\n",
"from larch import Group\n",
"from larch.fitting import param_group, param, guess\n",
"from larch.xafs import (autobk, feffpath, feffit_transform,\n",
" feffit_dataset, feffit, feffit_report)\n",
"\n",
"from larch.plot.plotly_xafsplots import plot, multi_plot\n",
"\n",
"PLANCK_HC = 1.e10 * consts.Planck * consts.c / consts.e\n",
"\n",
"RAD2DEG = 180.0/np.pi\n",
"DEG2RAD = np.pi/180.0\n",
"\n",
"# then we will read in some XAFS data (Cu metal) at the Cu K edge, \n",
"# measured with a Si(111) monochromator\n",
"cudat = read_ascii('../xafsdata/cu_10k.xmu', labels='energy, mu')\n",
"\n",
"# this data file set says that hc_2d = 1977.260\n",
"hc_2d = 1977.260\n",
"\n",
"# use the current value of HC to get the dspace they would have used\n",
"dspace = PLANCK_HC / (2 * hc_2d)\n",
"print(dspace)\n",
"\n",
"# now let's calculate the mono angles in degrees and plot that\n",
"cudat.theta = RAD2DEG * np.arcsin(PLANCK_HC / (2.0*dspace*cudat.energy))\n",
"\n",
"\n",
"f = plot(cudat.energy, cudat.theta, xlabel='Energy (eV)', ylabel=r\"$\\theta$\", \n",
" color='red', title='Si(111) at Cu K edge', label='angle', linewidth=2)\n"
]
},
{
"cell_type": "markdown",
"id": "772eaaa4",
"metadata": {},
"source": [
"note from the plot above that the angle is around 10 to 12 degrees, and that Energy and Angle are not linear."
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "9a444e18",
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"name": "stdout",
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"text": [
"E0 = 8977.580 eV\n"
]
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"source": [
"# let's plot the normalized XAFS data around the edge:\n",
"\n",
"autobk(cudat.energy, cudat.mu, group=cudat, rbkg=1.0, kw=2)\n",
"print(f\"E0 = {cudat.e0:.3f} eV\")\n",
"plot(cudat.energy, cudat.norm, xmin=8900, xmax=9500, label='norm',\n",
" xlabel='Energy (eV)', ylabel='$\\mu(E)$', delay_draw=True)\n",
" "
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"text": [
"Result: E0 = 8977.580 eV, R= 2.5494 +/- 0.0017 Ang\n",
"=================== FEFFIT RESULTS ====================\n",
"[[Statistics]]\n",
" nvarys, npts = 4, 104\n",
" n_independent = 13.732\n",
" chi_square = 80.5162009\n",
" reduced chi_square = 8.27300959\n",
" r-factor = 0.00122781\n",
" Akaike info crit = 32.2884981\n",
" Bayesian info crit = 34.7675287\n",
" \n",
"[[Data]]\n",
" fit space = 'r'\n",
" r-range = 1.400, 3.000\n",
" k-range = 2.500, 15.000\n",
" k window, dk = 'kaiser', 4.000\n",
" paths used in fit = ['../feffit/feffcu01.dat']\n",
" k-weight = 2\n",
" epsilon_k = Array(mean=0.00135189, std=9.20931e-4)\n",
" epsilon_r = 0.03323044\n",
" n_independent = 13.732\n",
" \n",
"[[Variables]]\n",
" amp = 0.92213823 +/- 0.02713764 (init= 1.00000000)\n",
" del_e0 = 5.67728180 +/- 0.37592135 (init= 0.00000000)\n",
" del_r = 0.00155256 +/- 0.00168163 (init= 0.00000000)\n",
" sig2 = 0.00354962 +/- 1.85415e-4 (init= 0.00000000)\n",
" \n",
"[[Correlations]] (unreported correlations are < 0.100)\n",
" amp, sig2 = 0.924\n",
" del_e0, del_r = 0.914\n",
" del_r, sig2 = 0.135\n",
" amp, del_r = 0.131\n",
" \n",
"[[Paths]]\n",
" = Path 'p3tf6mewgr' = Cu K Edge\n",
" feffdat file = ../feffit/feffcu01.dat, from feff run 'feffit'\n",
" geometry atom x y z ipot\n",
" Cu 0.0000, 0.0000, 0.0000 0 (absorber)\n",
" Cu 0.0000, -1.8016, 1.8016 1\n",
" reff = 2.54780000\n",
" degen = 12.0000000\n",
" n*s02 = 0.92213823 +/- 0.02713764 := 'amp'\n",
" e0 = 5.67728180 +/- 0.37592135 := 'del_e0'\n",
" r = 2.54935256 +/- 0.00168163 := 'reff + del_r'\n",
" deltar = 0.00155256 +/- 0.00168163 := 'del_r'\n",
" sigma2 = 0.00354962 +/- 1.85415e-4 := 'sig2'\n",
"\n",
"=======================================================\n"
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"# let's now write a function to do a first shell fit to the \n",
"# XAFS data from a dataset (Group)\n",
"def fit_cu_1stshell(dat, full_report=False, with_plot=False):\n",
" \n",
" # extract the EXAFS chi(k)\n",
" autobk(dat.energy, dat.mu, group=dat, rbkg=1.0, kw=2)\n",
"\n",
" # define fitting parameter group\n",
" pars = param_group(amp = param(1.0, vary=True),\n",
" del_e0 = param(0.0, vary=True),\n",
" sig2 = param(0.0, vary=True),\n",
" del_r = guess(0.0, vary=True) )\n",
"\n",
" # define a Feff Path, give expressions for Path Parameters\n",
" path1 = feffpath('../feffit/feffcu01.dat',\n",
" s02 = 'amp',\n",
" e0 = 'del_e0',\n",
" sigma2 = 'sig2',\n",
" deltar = 'del_r')\n",
"\n",
" # do the fit\n",
" trans = feffit_transform(kmin=2.5, kmax=15, kw=2, dk=4, window='kaiser',\n",
" rmin=1.4, rmax=3.0)\n",
" dset = feffit_dataset(data=dat, pathlist=[path1], transform=trans)\n",
" out = feffit(pars, dset)\n",
" \n",
" r1_fit = out.params['del_r'].value + path1.reff\n",
" r1_err = out.params['del_r'].stderr\n",
" print(f\"Result: E0 = {dat.e0:.3f} eV, R= {r1_fit:.4f} +/- {r1_err:.4f} Ang\")\n",
" if full_report:\n",
" print(feffit_report(out))\n",
" if with_plot:\n",
" multi_plot([dict(xdata=dset.data.r, ydata=dset.data.chir_mag, label='data',\n",
" xlabel='R (Ang)', ylabel='$|\\chi(R)|$'),\n",
" dict(xdata=dset.model.r, ydata=dset.model.chir_mag, label='fit')])\n",
" return out\n",
" \n",
"out = fit_cu_1stshell(cudat, full_report=True, with_plot=True)\n",
" "
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "fa9fd5d2",
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"name": "stdout",
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"text": [
"theta_off=0.0200, dspace_off=0.0000: E0= 8977.580 to 8991.481\n"
]
}
],
"source": [
"# OK, so we found E0=8977 eV, and can do a fit with this dataset and get \n",
"# R = 2.5494(0017) Ang.\n",
"\n",
"# If we take the angles for this energy array and change the offset, \n",
"# what do we get? \n",
"\n",
"# we define two functions to shift X-ray energy either by changing the \n",
"# angular offset or the d-spacing of the monochromator\n",
"\n",
"def energy_shift(energy, dspace, theta_off=0, dspace_off=0):\n",
" theta = RAD2DEG * np.arcsin(PLANCK_HC / (2.0*dspace*energy))\n",
" return PLANCK_HC/(2*(dspace-dspace_off)*np.sin(DEG2RAD*(theta-theta_off)))\n",
"\n",
"def change_calib(dgroup, dspace, theta_off=0, dspace_off=0):\n",
" \"make a new data group with an offset in theta and/or dspacing\"\n",
" enew = energy_shift(dgroup.energy, dspace, theta_off=theta_off, \n",
" dspace_off=dspace_off)\n",
"\n",
" dat = Group(energy=enew, mu=dgroup.mu*1.0)\n",
" autobk(dat.energy, dat.mu, group=dat, rbkg=1.0, kw=2)\n",
" print(f\"theta_off={theta_off:.4f}, dspace_off={dspace_off:.4f}: E0= {dgroup.e0:.3f} to {dat.e0:.3f}\")\n",
" return dat\n",
"\n",
"# and let's see how big a shift of 0.02 degrees is:\n",
"new_dat = change_calib(cudat, dspace, theta_off=0.02)"
]
},
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"source": [
"# So, that gave an energy shift of 14 eV. Let's plot those XANES spectra together\n",
"\n",
"multi_plot([dict(xdata=cudat.energy, ydata=cudat.norm, label='E0=8977.6'),\n",
" dict(xdata=new_dat.energy, ydata=new_dat.norm, label='E0=8991.5',\n",
" xmin=8950, xmax=9100, xlabel='Energy (eV)', ylabel='$\\mu(E)$')])\n",
"\n",
"# that is a enormous energy shift"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "638b0a19",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Result: E0 = 8991.481 eV, R= 2.5455 +/- 0.0017 Ang\n"
]
}
],
"source": [
"# and now we can fit that shifted data:\n",
"\n",
"out = fit_cu_1stshell(new_dat, full_report=False, with_plot=False)\n",
" \n",
"# and we see that the 1st shell R changed from R= 2.5494 to 2.5455 (+/- 0.0017 Ang).\n",
"# a shift in R of 0.004 Ang. \n",
"# Yes, that shift is outside of the error bars for this very good dataset. \n",
"# Typical uncertainties and expected accuracies of EXAFS results are 0.01 Ang. "
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "6e00298d",
"metadata": {},
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{
"name": "stdout",
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"text": [
"3000.0 -> 3001.2 eV\n",
"5000.0 -> 5004.1 eV\n",
"7000.0 -> 7008.3 eV\n",
"9000.0 -> 9014.0 eV\n",
"11000.0 -> 11021.1 eV\n",
"20000.0 -> 20070.5 eV\n"
]
}
],
"source": [
"# If we had an angular offset that big, would we notice? Or, how would we know?\n",
"\n",
"# you would change the monochromator energy to another edge. \n",
"# Let's run our `energy_shift` function with different energies:\n",
"\n",
"etest = np.array([3000, 5000, 7000, 9000, 11000, 20000])\n",
"enew = energy_shift(etest, dspace, theta_off=0.02)\n",
"for e_in, e_out in zip(etest, enew):\n",
" print(f\"{e_in:.1f} -> {e_out:.1f} eV\")\n",
" "
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "aa39c4f4",
"metadata": {},
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"text": [
"3000.0 -> 3009.6 eV\n",
"5000.0 -> 5016.0 eV\n",
"7000.0 -> 7022.4 eV\n",
"9000.0 -> 9028.8 eV\n",
"11000.0 -> 11035.2 eV\n",
"20000.0 -> 20064.0 eV\n"
]
}
],
"source": [
"# if we thought a shift of 14 eV at Cu K was big (it is!), then a shift \n",
"# of 70 eV at Mo K edge is enormous.\n",
"# for completeness, what if we had a bad value for $d$?\n",
"enew = energy_shift(etest, dspace, dspace_off=0.01)\n",
"for e_in, e_out in zip(etest, enew):\n",
" print(f\"{e_in:.1f} -> {e_out:.1f} eV\")"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "6ac03624",
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"name": "stdout",
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"text": [
"theta_off=0.0000, dspace_off=0.0000: E0= 8977.580 to 8977.580\n",
"Result: E0 = 8977.580 eV, R= 2.5494 +/- 0.0017 Ang\n",
"theta_off=0.0010, dspace_off=0.0000: E0= 8977.580 to 8978.274\n",
"Result: E0 = 8978.274 eV, R= 2.5492 +/- 0.0017 Ang\n",
"theta_off=0.0020, dspace_off=0.0000: E0= 8977.580 to 8978.968\n",
"Result: E0 = 8978.968 eV, R= 2.5490 +/- 0.0017 Ang\n",
"theta_off=0.0050, dspace_off=0.0000: E0= 8977.580 to 8981.051\n",
"Result: E0 = 8981.051 eV, R= 2.5484 +/- 0.0017 Ang\n",
"theta_off=0.0100, dspace_off=0.0000: E0= 8977.580 to 8984.525\n",
"Result: E0 = 8984.525 eV, R= 2.5474 +/- 0.0017 Ang\n",
"theta_off=0.0200, dspace_off=0.0000: E0= 8977.580 to 8991.481\n",
"Result: E0 = 8991.481 eV, R= 2.5455 +/- 0.0017 Ang\n",
"theta_off=0.0300, dspace_off=0.0000: E0= 8977.580 to 8998.448\n",
"Result: E0 = 8998.448 eV, R= 2.5436 +/- 0.0017 Ang\n",
"theta_off=0.0400, dspace_off=0.0000: E0= 8977.580 to 9005.427\n",
"Result: E0 = 9005.427 eV, R= 2.5417 +/- 0.0016 Ang\n",
"theta_off=0.0500, dspace_off=0.0000: E0= 8977.580 to 9012.416\n",
"Result: E0 = 9012.416 eV, R= 2.5397 +/- 0.0016 Ang\n",
"theta_off=0.0600, dspace_off=0.0000: E0= 8977.580 to 9019.417\n",
"Result: E0 = 9019.417 eV, R= 2.5378 +/- 0.0016 Ang\n",
"theta_off=0.0700, dspace_off=0.0000: E0= 8977.580 to 9026.428\n",
"Result: E0 = 9026.428 eV, R= 2.5359 +/- 0.0016 Ang\n",
"theta_off=0.0800, dspace_off=0.0000: E0= 8977.580 to 9033.451\n",
"Result: E0 = 9033.451 eV, R= 2.5340 +/- 0.0016 Ang\n",
"theta_off=0.0900, dspace_off=0.0000: E0= 8977.580 to 9040.485\n",
"Result: E0 = 9040.485 eV, R= 2.5320 +/- 0.0016 Ang\n",
"theta_off=0.1000, dspace_off=0.0000: E0= 8977.580 to 9047.531\n",
"Result: E0 = 9047.531 eV, R= 2.5301 +/- 0.0016 Ang\n"
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"# you can see that an error is angular offset is more pronounced at high energy, \n",
"# which is lower angle. On the other hand, a bad value for d-spacing gives a \n",
"# slightly less energy-dependent error (\"chromatic aberration\"). \n",
"#\n",
"# When calibrating a monochromator, it is important to use a wide range of \n",
"# energies to set the d-spacing and angular offset.\n",
"\n",
"# But: how far off do you have to be to see real errors in EXAFS results?\n",
"\n",
"e0val = []\n",
"delr1 = []\n",
"\n",
"for theta_off in (0.000, 0.001, 0.002, 0.005, 0.010, 0.020, 0.03, \n",
" 0.04, 0.050, 0.06, 0.07, 0.08, 0.09, 0.10):\n",
" new_dat = change_calib(cudat, dspace, theta_off=theta_off)\n",
"\n",
" out = fit_cu_1stshell(new_dat, full_report=False, with_plot=False)\n",
" \n",
" e0val.append(new_dat.e0-cudat.e0) \n",
" delr1.append(out.params['del_r'].value -0.00155)\n",
" \n",
"f = plot(e0val, delr1, marker='o', linewidth=0.0002, xlabel='E0 (eV)', ylabel='$\\Delta R1$ (Ang)', label='deltaR')"
]
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"# To get an error in R of 0.01 Ang, E0 needs to shift by about 35 eV at Cu K edge. \n",
"\n",
"# If you are expecting to get absolute measures that are accurate to 0.005 Ang,\n",
"# then making sure your energy calibration is good to 10 eV is important. \n",
"# Claiming such absolute accuracies is rare in the EXAFS literature, and requires\n",
"# a very careful analysis.\n",
"\n",
"# Errors or drifts in energy of 5 eV in the 5 to 15 keV range will cause very\n",
"# significant problems for XANES work and would need to be addressed.\n",
"#\n",
"# Errors of that magnitude will not impact EXAFS results at all.\n",
"\n"
]
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