¹³C MAS NMR of Glycine (CSA) [5000 Hz]

The following is a sideband least-squares fitting example of a \(^{13}\text{C}\) MAS NMR spectrum of Glycine spinning at 5000 Hz. 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, report_fit

from mrsimulator import Simulator, SpinSystem, Site
from mrsimulator.methods import BlochDecaySpectrum
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

host = "https://nmr.cemhti.cnrs-orleans.fr/Dmfit/Help/csdm/"
filename = "13C MAS 5000Hz - Glycine.csdf"
experiment = cp.load(host + filename)

# standard deviation of noise from the dataset
sigma = 3.822249

# 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(280, -10)
plt.grid()
plt.tight_layout()
plt.show()

Create a fitting model

Spin System

C1 = Site(
    isotope="13C",
    isotropic_chemical_shift=176.0,  # in ppm
    shielding_symmetric={"zeta": 70, "eta": 0.6},  # zeta in Hz
)
C2 = Site(
    isotope="13C",
    isotropic_chemical_shift=43.0,  # in ppm
)

spin_systems = [SpinSystem(sites=[C1], name="C1"), SpinSystem(sites=[C2], name="C2")]

Method

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

MAS = BlochDecaySpectrum(
    channels=["13C"],
    magnetic_flux_density=7.05,  # in T
    rotor_frequency=5000,  # in Hz
    spectral_dimensions=spectral_dims,
    experiment=experiment,  # experimental dataset
)

# 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 Model 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.Exponential(FWHM="20 Hz", dv_index=0),  # spin system 0
        sp.apodization.Exponential(FWHM="200 Hz", dv_index=1),  # spin system 1
        sp.FFT(),
        sp.Scale(factor=10),
    ]
)
processed_data = processor.apply_operations(data=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_data, linewidth=2, alpha=0.6)
ax.set_xlim(280, -10)
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, include={"rotor_frequency"})
print(params.pretty_print(columns=["value", "min", "max", "vary", "expr"]))

Out:

Name                                      Value      Min      Max     Vary     Expr
SP_0_operation_1_Exponential_FWHM            20     -inf      inf     True     None
SP_0_operation_2_Exponential_FWHM           200     -inf      inf     True     None
SP_0_operation_4_Scale_factor                10     -inf      inf     True     None
mth_0_rotor_frequency                      5000     4900     5100     True     None
sys_0_abundance                              50        0      100     True     None
sys_0_site_0_isotropic_chemical_shift       176     -inf      inf     True     None
sys_0_site_0_shielding_symmetric_eta        0.6        0        1     True     None
sys_0_site_0_shielding_symmetric_zeta        70     -inf      inf     True     None
sys_1_abundance                              50        0      100    False 100-sys_0_abundance
sys_1_site_0_isotropic_chemical_shift        43     -inf      inf     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   = 191
    # data points      = 4096
    # variables        = 9
    chi-square         = 1267.95049
    reduced chi-square = 0.31023991
    Akaike info crit   = -4785.00677
    Bayesian info crit = -4728.14688
[[Variables]]
    sys_0_site_0_isotropic_chemical_shift:  176.093349 +/- 8.2151e-04 (0.00%) (init = 176)
    sys_0_site_0_shielding_symmetric_zeta: -73.6104864 +/- 1.52263789 (2.07%) (init = 70)
    sys_0_site_0_shielding_symmetric_eta:   0.99999893 +/- 0.07549726 (7.55%) (init = 0.6)
    sys_0_abundance:                        58.2535314 +/- 0.29047642 (0.50%) (init = 50)
    sys_1_site_0_isotropic_chemical_shift:  43.3581388 +/- 0.00711644 (0.02%) (init = 43)
    sys_1_abundance:                        41.7464686 +/- 0.29047642 (0.70%) == '100-sys_0_abundance'
    mth_0_rotor_frequency:                  5033.23931 +/- 0.36010260 (0.01%) (init = 5000)
    SP_0_operation_1_Exponential_FWHM:      26.9286152 +/- 0.18909095 (0.70%) (init = 20)
    SP_0_operation_2_Exponential_FWHM:      106.899303 +/- 1.49566542 (1.40%) (init = 200)
    SP_0_operation_4_Scale_factor:          38.3645243 +/- 0.20216190 (0.53%) (init = 10)
[[Correlations]] (unreported correlations are < 0.100)
    C(sys_0_site_0_shielding_symmetric_zeta, sys_0_site_0_shielding_symmetric_eta)  =  0.899
    C(sys_0_abundance, SP_0_operation_2_Exponential_FWHM)                           = -0.600
    C(SP_0_operation_2_Exponential_FWHM, SP_0_operation_4_Scale_factor)             =  0.544
    C(SP_0_operation_1_Exponential_FWHM, SP_0_operation_4_Scale_factor)             =  0.358
    C(sys_0_abundance, SP_0_operation_4_Scale_factor)                               = -0.319
    C(sys_0_abundance, SP_0_operation_1_Exponential_FWHM)                           =  0.281
    C(sys_0_site_0_shielding_symmetric_zeta, SP_0_operation_4_Scale_factor)         = -0.266
    C(sys_0_site_0_shielding_symmetric_eta, sys_1_site_0_isotropic_chemical_shift)  =  0.182
    C(sys_0_site_0_shielding_symmetric_eta, sys_0_abundance)                        =  0.150
    C(sys_0_site_0_shielding_symmetric_zeta, sys_1_site_0_isotropic_chemical_shift) =  0.128
    C(sys_0_site_0_shielding_symmetric_eta, SP_0_operation_4_Scale_factor)          = -0.116

The best fit solution

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

# 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(280, -10)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()

1

D.Massiot, F.Fayon, M.Capron, I.King, S.Le Calvé, B.Alonso, J.O.Durand, B.Bujoli, Z.Gan, G.Hoatson, ‘Modelling one and two-dimensional solid-state NMR spectra.’, Magn. Reson. Chem. 40 70-76 (2002) DOI: 10.1002/mrc.984