²⁷Al MAS NMR of YAG (1st and 2nd order Quad)

The following is a quadrupolar lineshape fitting example for the 27Al MAS NMR of Yttrium aluminum garnet (YAG) crystal. The following experimental dataset is a part of DMFIT [1] examples. We thank Dr. Dominique Massiot for sharing the dataset.

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

from mrsimulator import Simulator, Site, SpinSystem
from mrsimulator.method.lib import BlochDecaySpectrum
from mrsimulator import signal_processor as sp
from mrsimulator.utils import spectral_fitting as sf
from mrsimulator.utils import get_spectral_dimensions
from mrsimulator.spin_system.tensors import SymmetricTensor

Import the dataset

host = "https://nmr.cemhti.cnrs-orleans.fr/Dmfit/Help/csdm/"
filename = "27Al Quad MAS YAG 400MHz.csdf"
experiment = cp.load(host + filename)

# 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.
plt.figure(figsize=(8, 4))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, color="black", linewidth=0.5, label="Experiment")
ax.set_xlim(1200, -1200)
plt.grid()
plt.tight_layout()
plt.show()
plot 4 27Al YAG

Estimate noise statistics from the dataset

coords = experiment.dimensions[0].coordinates
noise_region = np.where(np.logical_and(coords > -570e-6, coords < -510e-6))
noise_data = experiment[noise_region]

plt.figure(figsize=(3.75, 2.5))
ax = plt.subplot(projection="csdm")
ax.plot(noise_data, label="noise")
plt.title("Noise section")
plt.axis("off")
plt.tight_layout()
plt.show()

noise_mean, sigma = experiment[noise_region].mean(), experiment[noise_region].std()
noise_mean, sigma
Noise section
(<Quantity -1.3760117>, <Quantity 0.5487051>)

Create a fitting model

Guess model

Create a guess list of spin systems.

Al_1 = Site(
    isotope="27Al",
    isotropic_chemical_shift=76,  # in ppm
    quadrupolar=SymmetricTensor(Cq=6e6, eta=0.0),  # Cq in Hz
)

Al_2 = Site(
    isotope="27Al",
    isotropic_chemical_shift=1,  # in ppm
    quadrupolar=SymmetricTensor(Cq=5e5, eta=0.3),  # Cq in Hz
)
spin_systems = [
    SpinSystem(sites=[Al_1], name="AlO4"),
    SpinSystem(sites=[Al_2], name="AlO6"),
]

Method

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

MAS = BlochDecaySpectrum(
    channels=["27Al"],
    magnetic_flux_density=9.395,  # in T
    rotor_frequency=15250,  # in Hz
    spectral_dimensions=spectral_dims,
    experiment=experiment,  # add the measurement to the method.
)

Guess Spectrum

# Simulation
# ----------
sim = Simulator(spin_systems=spin_systems, methods=[MAS])
sim.config.decompose_spectrum = "spin_system"
sim.run()

# Post Simulation Processing
# --------------------------
processor = sp.SignalProcessor(
    operations=[
        sp.IFFT(),
        sp.apodization.Gaussian(FWHM="300 Hz"),
        sp.FFT(),
        sp.Scale(factor=50),
    ]
)
processed_dataset = processor.apply_operations(dataset=sim.methods[0].simulation).real

# Plot of the guess Spectrum
# --------------------------
plt.figure(figsize=(8, 4))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, color="black", linewidth=0.5, label="Experiment")
ax.plot(processed_dataset, linewidth=2, alpha=0.6)
ax.set_xlim(1200, -1200)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()
plot 4 27Al YAG

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, include={"rotor_frequency"})
print(params.pretty_print(columns=["value", "min", "max", "vary", "expr"]))
Name                                      Value      Min      Max     Vary     Expr
SP_0_operation_1_Gaussian_FWHM              300     -inf      inf     True     None
SP_0_operation_3_Scale_factor                50     -inf      inf     True     None
mth_0_rotor_frequency                  1.525e+04 1.515e+04 1.535e+04     True     None
sys_0_abundance                              50        0      100     True     None
sys_0_site_0_isotropic_chemical_shift        76     -inf      inf     True     None
sys_0_site_0_quadrupolar_Cq               6e+06     -inf      inf     True     None
sys_0_site_0_quadrupolar_eta                  0        0        1     True     None
sys_1_abundance                              50        0      100    False 100-sys_0_abundance
sys_1_site_0_isotropic_chemical_shift         1     -inf      inf     True     None
sys_1_site_0_quadrupolar_Cq               5e+05     -inf      inf     True     None
sys_1_site_0_quadrupolar_eta                0.3        0        1     True     None
None

Solve the minimizer using LMFIT

opt = sim.optimize()  # Pre-compute transition pathways
minner = Minimizer(
    sf.LMFIT_min_function,
    params,
    fcn_args=(sim, processor, sigma),
    fcn_kws={"opt": opt},
)
result = minner.minimize()
result

Fit Result



The best fit solution

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

# Plot the spectrum
plt.figure(figsize=(8, 4))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, color="black", linewidth=0.5, label="Experiment")
ax.plot(residuals, color="gray", linewidth=0.5, label="Residual")
ax.plot(best_fit, linewidth=2, alpha=0.6)
ax.set_xlim(1200, -1200)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()
plot 4 27Al YAG

Total running time of the script: (0 minutes 9.878 seconds)

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