¹⁷O 2D DAS NMR of Coesite

Coesite is a high-pressure (2-3 GPa) and high-temperature (700°C) polymorph of silicon dioxide \(\text{SiO}_2\). Coesite has five crystallographic \(^{17}\text{O}\) sites. The experimental dataset used in this example is published in Grandinetti et al. 1

import numpy as np
import csdmpy as cp
import matplotlib.pyplot as plt
from lmfit import Minimizer, report_fit

from mrsimulator import Simulator
from mrsimulator.methods import Method2D
from mrsimulator import signal_processing as sp
from mrsimulator.utils import spectral_fitting as sf
from mrsimulator.utils import get_spectral_dimensions
from mrsimulator.utils.collection import single_site_system_generator

Import the dataset

filename = "https://sandbox.zenodo.org/record/814455/files/DASCoesite.csdf"
experiment = cp.load(filename)

# standard deviation of noise from the dataset
sigma = 921.6698

# For spectral fitting, we only focus on the real part of the complex dataset
experiment = experiment.real

# Convert the coordinates along each dimension from Hz to ppm.
_ = [item.to("ppm", "nmr_frequency_ratio") for item in experiment.dimensions]

# plot of the dataset.
max_amp = experiment.max()
levels = (np.arange(14) + 1) * max_amp / 15  # contours are drawn at these levels.
options = dict(levels=levels, alpha=0.75, linewidths=0.5)  # plot options

plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.contour(experiment, colors="k", **options)
ax.invert_xaxis()
ax.set_ylim(30, -30)
plt.grid()
plt.tight_layout()
plt.show()

Create a fitting model

Guess model

Create a guess list of spin systems.

shifts = [29, 39, 54.8, 51, 56]  # in ppm
Cq = [6.1e6, 5.4e6, 5.5e6, 5.5e6, 5.1e6]  # in  Hz
eta = [0.1, 0.2, 0.15, 0.15, 0.3]
abundance_ratio = [1, 1, 2, 2, 2]
abundance = np.asarray(abundance_ratio) / 8 * 100  # in %

spin_systems = single_site_system_generator(
    isotopes="17O",
    isotropic_chemical_shifts=shifts,
    quadrupolar={"Cq": Cq, "eta": eta},
    abundance=abundance,
)

Method

Create the DAS method.

# Get the spectral dimension parameters from the experiment.
spectral_dims = get_spectral_dimensions(experiment)

DAS = Method2D(
    channels=["17O"],
    magnetic_flux_density=11.744,  # in T
    spectral_dimensions=[
        {
            **spectral_dims[0],
            "events": [
                {
                    "fraction": 0.5,
                    "rotor_angle": 37.38 * 3.14159 / 180,
                    "transition_query": {"P": [-1], "D": [0]},
                },
                {
                    "fraction": 0.5,
                    "rotor_angle": 79.19 * 3.14159 / 180,
                    "transition_query": {"P": [-1], "D": [0]},
                },
            ],
        },
        # The last spectral dimension block is the direct-dimension
        {
            **spectral_dims[1],
            "events": [
                {
                    "rotor_angle": 54.735 * 3.14159 / 180,
                    "transition_query": {"P": [-1], "D": [0]},
                }
            ],
        },
    ],
    experiment=experiment,  # also add the measurement to the method.
)

# Optimize the script by pre-setting the transition pathways for each spin system from
# the das method.
for sys in spin_systems:
    sys.transition_pathways = DAS.get_transition_pathways(sys)

Guess Spectrum

# Simulation
# ----------
sim = Simulator(spin_systems=spin_systems, methods=[DAS])
sim.config.number_of_sidebands = 1  # no sidebands are required for this dataset.
sim.run()

# Post Simulation Processing
# --------------------------
processor = sp.SignalProcessor(
    operations=[
        # Gaussian convolution along both dimensions.
        sp.IFFT(dim_index=(0, 1)),
        sp.apodization.Gaussian(FWHM="0.15 kHz", dim_index=0),
        sp.apodization.Gaussian(FWHM="0.1 kHz", dim_index=1),
        sp.FFT(dim_index=(0, 1)),
        sp.Scale(factor=4e7),
    ]
)
processed_data = processor.apply_operations(data=sim.methods[0].simulation).real

# Plot of the guess Spectrum
# --------------------------
plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.contour(experiment, colors="k", **options)
ax.contour(processed_data, colors="r", linestyles="--", **options)
ax.invert_xaxis()
ax.set_ylim(30, -30)
plt.grid()
plt.tight_layout()
plt.show()

Least-squares minimization with LMFIT

Use the make_LMFIT_params() for a quick setup of the fitting parameters.

params = sf.make_LMFIT_params(sim, processor)
print(params.pretty_print(columns=["value", "min", "max", "vary", "expr"]))

Out:

Name                                      Value      Min      Max     Vary     Expr
SP_0_operation_1_Gaussian_FWHM             0.15     -inf      inf     True     None
SP_0_operation_2_Gaussian_FWHM              0.1     -inf      inf     True     None
SP_0_operation_4_Scale_factor             4e+07     -inf      inf     True     None
sys_0_abundance                            12.5        0      100     True     None
sys_0_site_0_isotropic_chemical_shift        29     -inf      inf     True     None
sys_0_site_0_quadrupolar_Cq             6.1e+06     -inf      inf     True     None
sys_0_site_0_quadrupolar_eta                0.1        0        1     True     None
sys_1_abundance                            12.5        0      100     True     None
sys_1_site_0_isotropic_chemical_shift        39     -inf      inf     True     None
sys_1_site_0_quadrupolar_Cq             5.4e+06     -inf      inf     True     None
sys_1_site_0_quadrupolar_eta                0.2        0        1     True     None
sys_2_abundance                              25        0      100     True     None
sys_2_site_0_isotropic_chemical_shift      54.8     -inf      inf     True     None
sys_2_site_0_quadrupolar_Cq             5.5e+06     -inf      inf     True     None
sys_2_site_0_quadrupolar_eta               0.15        0        1     True     None
sys_3_abundance                              25        0      100     True     None
sys_3_site_0_isotropic_chemical_shift        51     -inf      inf     True     None
sys_3_site_0_quadrupolar_Cq             5.5e+06     -inf      inf     True     None
sys_3_site_0_quadrupolar_eta               0.15        0        1     True     None
sys_4_abundance                              25        0      100    False 100-sys_0_abundance-sys_1_abundance-sys_2_abundance-sys_3_abundance
sys_4_site_0_isotropic_chemical_shift        56     -inf      inf     True     None
sys_4_site_0_quadrupolar_Cq             5.1e+06     -inf      inf     True     None
sys_4_site_0_quadrupolar_eta                0.3        0        1     True     None
None

Solve the minimizer using LMFIT

minner = Minimizer(sf.LMFIT_min_function, params, fcn_args=(sim, processor, sigma))
result = minner.minimize(method="powell")
report_fit(result)

Out:

[[Fit Statistics]]
    # fitting method   = Powell
    # function evals   = 2194
    # data points      = 131072
    # variables        = 22
    chi-square         = 186495.645
    reduced chi-square = 1.42308771
    Akaike info crit   = 46267.9953
    Bayesian info crit = 46483.2324
##  Warning: uncertainties could not be estimated:
    this fitting method does not natively calculate uncertainties
    and numdifftools is not installed for lmfit to do this. Use
    `pip install numdifftools` for lmfit to estimate uncertainties
    with this fitting method.
[[Variables]]
    sys_0_site_0_isotropic_chemical_shift:  27.8024407 (init = 29)
    sys_0_site_0_quadrupolar_Cq:            6099675.46 (init = 6100000)
    sys_0_site_0_quadrupolar_eta:           0.01078074 (init = 0.1)
    sys_0_abundance:                        11.1052088 (init = 12.5)
    sys_1_site_0_isotropic_chemical_shift:  38.5408329 (init = 39)
    sys_1_site_0_quadrupolar_Cq:            5399536.41 (init = 5400000)
    sys_1_site_0_quadrupolar_eta:           0.20001097 (init = 0.2)
    sys_1_abundance:                        13.8849100 (init = 12.5)
    sys_2_site_0_isotropic_chemical_shift:  55.9052118 (init = 54.8)
    sys_2_site_0_quadrupolar_Cq:            5493792.38 (init = 5500000)
    sys_2_site_0_quadrupolar_eta:           0.17362890 (init = 0.15)
    sys_2_abundance:                        24.8249011 (init = 25)
    sys_3_site_0_isotropic_chemical_shift:  51.2660231 (init = 51)
    sys_3_site_0_quadrupolar_Cq:            5500002.73 (init = 5500000)
    sys_3_site_0_quadrupolar_eta:           0.20827675 (init = 0.15)
    sys_3_abundance:                        23.6462498 (init = 25)
    sys_4_site_0_isotropic_chemical_shift:  55.6908554 (init = 56)
    sys_4_site_0_quadrupolar_Cq:            5099895.27 (init = 5100000)
    sys_4_site_0_quadrupolar_eta:           0.29999967 (init = 0.3)
    sys_4_abundance:                        26.5387303 == '100-sys_0_abundance-sys_1_abundance-sys_2_abundance-sys_3_abundance'
    SP_0_operation_1_Gaussian_FWHM:        -0.34003342 (init = 0.15)
    SP_0_operation_2_Gaussian_FWHM:         0.13846642 (init = 0.1)
    SP_0_operation_4_Scale_factor:          42620049.0 (init = 4e+07)

The best fit solution

best_fit = sf.bestfit(sim, processor)[0]

# Plot the spectrum
plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.contour(experiment, colors="k", **options)
ax.contour(best_fit, colors="r", linestyles="--", **options)
ax.invert_xaxis()
ax.set_ylim(30, -30)
plt.grid()
plt.tight_layout()
plt.show()

The best fit solution

residuals = sf.residuals(sim, processor)[0]

fig, ax = plt.subplots(
    1, 3, sharey=True, figsize=(10, 3.0), subplot_kw={"projection": "csdm"}
)
vmax, vmin = experiment.max(), experiment.min()
for i, dat in enumerate([experiment, best_fit, residuals]):
    ax[i].imshow(dat, aspect="auto", vmax=vmax, vmin=vmin)
    ax[i].invert_xaxis()
ax[0].set_ylim(30, -30)
plt.tight_layout()
plt.show()

1

Grandinetti, P. J., Baltisberger, J. H., Farnan, I., Stebbins, J. F., Werner, U. and Pines, A. Solid-State \(^{17}\text{O}\) Magic-Angle and Dynamic-Angle Spinning NMR Study of the \(\text{SiO}_2\) Polymorph Coesite, J. Phys. Chem. 1995, 99, 32, 12341-12348. DOI: 10.1021/j100032a045