¹¹B MAS NMR of Lithium orthoborate crystal

The following is a quadrupolar lineshape fitting example for the 11B MAS NMR of lithium orthoborate crystal. The dataset was shared by Dr. Nathan Barrow.

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

from mrsimulator import Simulator, Site, SpinSystem
from mrsimulator.methods import BlochDecayCTSpectrum
from mrsimulator import signal_processing as sp
from mrsimulator.utils import spectral_fitting as sf
from mrsimulator.utils import get_spectral_dimensions

Import the dataset

filename = "https://sandbox.zenodo.org/record/814455/files/11B_lithum_orthoborate.csdf"
experiment = cp.load(filename)

# standard deviation of noise from the dataset
sigma = 0.08078374

# 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=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, "k", alpha=0.5)
ax.set_xlim(100, -100)
plt.grid()
plt.tight_layout()
plt.show()

Create a fitting model

Spin System

B11 = Site(
    isotope="11B",
    isotropic_chemical_shift=20.0,  # in ppm
    quadrupolar={"Cq": 2.3e6, "eta": 0.03},  # Cq in Hz
)
spin_systems = [SpinSystem(sites=[B11])]

Method

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

MAS_CT = BlochDecayCTSpectrum(
    channels=["11B"],
    magnetic_flux_density=14.1,  # in T
    rotor_frequency=12500,  # 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_CT.get_transition_pathways(sys)

Guess Model Spectrum

# Simulation
# ----------
sim = Simulator(spin_systems=spin_systems, methods=[MAS_CT])
sim.run()

# Post Simulation Processing
# --------------------------
processor = sp.SignalProcessor(
    operations=[
        sp.IFFT(),
        sp.apodization.Exponential(FWHM="100 Hz"),
        sp.FFT(),
        sp.Scale(factor=200),
    ]
)
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.plot(experiment, "k", linewidth=1, label="Experiment")
ax.plot(processed_data, "r", alpha=0.75, linewidth=1, label="guess spectrum")
ax.set_xlim(100, -100)
plt.grid()
plt.legend()
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)
params.pop("sys_0_abundance")
print(params.pretty_print(columns=["value", "min", "max", "vary", "expr"]))

Out:

Name                                      Value      Min      Max     Vary     Expr
SP_0_operation_1_Exponential_FWHM           100     -inf      inf     True     None
SP_0_operation_3_Scale_factor               200     -inf      inf     True     None
sys_0_site_0_isotropic_chemical_shift        20     -inf      inf     True     None
sys_0_site_0_quadrupolar_Cq             2.3e+06     -inf      inf     True     None
sys_0_site_0_quadrupolar_eta               0.03        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()
report_fit(result)

Out:

[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 74
    # data points      = 4096
    # variables        = 5
    chi-square         = 203147.651
    reduced chi-square = 49.6572113
    Akaike info crit   = 16000.4652
    Bayesian info crit = 16032.0541
[[Variables]]
    sys_0_site_0_isotropic_chemical_shift:  20.0547392 +/- 0.00152259 (0.01%) (init = 20)
    sys_0_site_0_quadrupolar_Cq:            2732039.56 +/- 431.874598 (0.02%) (init = 2300000)
    sys_0_site_0_quadrupolar_eta:           0.05154982 +/- 8.5913e-04 (1.67%) (init = 0.03)
    SP_0_operation_1_Exponential_FWHM:      68.1956038 +/- 0.70874729 (1.04%) (init = 100)
    SP_0_operation_3_Scale_factor:          199.046699 +/- 0.21335695 (0.11%) (init = 200)
[[Correlations]] (unreported correlations are < 0.100)
    C(sys_0_site_0_quadrupolar_eta, SP_0_operation_1_Exponential_FWHM)          = -0.824
    C(sys_0_site_0_isotropic_chemical_shift, sys_0_site_0_quadrupolar_Cq)       =  0.806
    C(SP_0_operation_1_Exponential_FWHM, SP_0_operation_3_Scale_factor)         =  0.561
    C(sys_0_site_0_quadrupolar_eta, SP_0_operation_3_Scale_factor)              = -0.367
    C(sys_0_site_0_isotropic_chemical_shift, SP_0_operation_1_Exponential_FWHM) = -0.127

The best fit solution

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

# Plot the spectrum
plt.figure(figsize=(4.25, 3.0))
ax = plt.subplot(projection="csdm")
ax.plot(experiment, "k", linewidth=1, label="Experiment")
ax.plot(best_fit, "r", alpha=0.75, linewidth=1, label="Best Fit")
ax.plot(residuals, alpha=0.75, linewidth=1, label="Residuals")
ax.set_xlim(100, -100)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()