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 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)

# standard deviation of noise from the dataset
sigma = 0.4895381

# 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

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.
)

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

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

Fit Statistics

fitting methodleastsq
# function evals156
# data points4096
# variables10
chi-square 53118.1624
reduced chi-square 13.0000397
Akaike info crit. 10516.0329
Bayesian info crit. 10579.2105

Variables

name value standard error relative error initial value min max vary expression
sys_0_site_0_isotropic_chemical_shift 76.8375039 0.09443927 (0.12%) 76.0 -inf inf True
sys_0_site_0_quadrupolar_Cq 6023339.75 12084.9838 (0.20%) 6000000.0 -inf inf True
sys_0_site_0_quadrupolar_eta 0.04521840 0.00904216 (20.00%) 0.0 0.00000000 1.00000000 True
sys_0_abundance 47.8709270 0.45216941 (0.94%) 50.0 0.00000000 100.000000 True
sys_1_site_0_isotropic_chemical_shift 0.84494266 0.00355253 (0.42%) 1.0 -inf inf True
sys_1_site_0_quadrupolar_Cq 585070.414 4223.27692 (0.72%) 500000.0 -inf inf True
sys_1_site_0_quadrupolar_eta 0.77215757 0.01559869 (2.02%) 0.3 0.00000000 1.00000000 True
sys_1_abundance 52.1290730 0.45216941 (0.87%) 50.0 0.00000000 100.000000 False 100-sys_0_abundance
mth_0_rotor_frequency 15251.5068 0.18514030 (0.00%) 15250.0 15150.0000 15350.0000 True
SP_0_operation_1_Gaussian_FWHM 320.167956 2.04450576 (0.64%) 300.0 -inf inf True
SP_0_operation_3_Scale_factor 141.941723 1.22995032 (0.87%) 50.0 -inf inf True

Correlations (unreported correlations are < 0.100)

sys_0_abundanceSP_0_operation_3_Scale_factor0.8095
sys_1_site_0_quadrupolar_Cqsys_1_site_0_quadrupolar_eta-0.6825
sys_1_site_0_isotropic_chemical_shiftsys_1_site_0_quadrupolar_Cq0.6187
sys_0_site_0_isotropic_chemical_shiftsys_0_site_0_quadrupolar_Cq0.6029
sys_1_site_0_isotropic_chemical_shiftmth_0_rotor_frequency-0.5450
sys_1_site_0_quadrupolar_Cqmth_0_rotor_frequency-0.4216
sys_1_site_0_isotropic_chemical_shiftsys_1_site_0_quadrupolar_eta-0.2814
sys_1_site_0_quadrupolar_etamth_0_rotor_frequency0.2247
sys_1_site_0_quadrupolar_CqSP_0_operation_1_Gaussian_FWHM-0.1702
sys_0_site_0_quadrupolar_etasys_0_abundance0.1701
sys_0_abundancesys_1_site_0_quadrupolar_Cq-0.1687
SP_0_operation_1_Gaussian_FWHMSP_0_operation_3_Scale_factor0.1455
sys_0_abundancesys_1_site_0_quadrupolar_eta0.1437
sys_0_site_0_quadrupolar_etaSP_0_operation_3_Scale_factor0.1436
sys_1_site_0_quadrupolar_CqSP_0_operation_3_Scale_factor0.1408
sys_1_site_0_quadrupolar_etaSP_0_operation_3_Scale_factor-0.1277
sys_0_site_0_quadrupolar_CqSP_0_operation_3_Scale_factor0.1083
sys_0_abundanceSP_0_operation_1_Gaussian_FWHM-0.1058
sys_0_site_0_quadrupolar_Cqsys_0_abundance0.1057


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 11.677 seconds)

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