¹³C MAS NMR of Glycine (CSA) [1940 Hz]

The following is a sideband least-squares fitting example of a \(^{13}\text{C}\) MAS NMR spectrum of Glycine spinning at 1940 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 1940Hz - 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
    shielding_symmetric={"zeta": 30, "eta": 0.5},  # zeta in Hz
)

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=1940,  # 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=100),
    ]
)
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"})
params["sys_1_site_0_shielding_symmetric_eta"].vary = False
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               100     -inf      inf     True     None
mth_0_rotor_frequency                      1940     1840     2040     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
sys_1_site_0_shielding_symmetric_eta        0.5        0        1    False     None
sys_1_site_0_shielding_symmetric_zeta        30     -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   = 78
    # data points      = 4096
    # variables        = 10
    chi-square         = 5247.81571
    reduced chi-square = 1.28434060
    Akaike info crit   = 1034.99309
    Bayesian info crit = 1098.17075
[[Variables]]
    sys_0_site_0_isotropic_chemical_shift:  176.133538 +/- 7.7530e-04 (0.00%) (init = 176)
    sys_0_site_0_shielding_symmetric_zeta:  72.6573659 +/- 0.18735813 (0.26%) (init = 70)
    sys_0_site_0_shielding_symmetric_eta:   0.92348112 +/- 0.00507035 (0.55%) (init = 0.6)
    sys_0_abundance:                        57.5381144 +/- 0.21265458 (0.37%) (init = 50)
    sys_1_site_0_isotropic_chemical_shift:  43.3604050 +/- 0.00500134 (0.01%) (init = 43)
    sys_1_site_0_shielding_symmetric_zeta:  23.1190164 +/- 0.50113151 (2.17%) (init = 30)
    sys_1_site_0_shielding_symmetric_eta:   0.5 (fixed)
    sys_1_abundance:                        42.4618856 +/- 0.21265458 (0.50%) == '100-sys_0_abundance'
    mth_0_rotor_frequency:                  1940.88271 +/- 0.05721410 (0.00%) (init = 1940)
    SP_0_operation_1_Exponential_FWHM:      29.6025777 +/- 0.17069359 (0.58%) (init = 20)
    SP_0_operation_2_Exponential_FWHM:      134.355218 +/- 1.06603885 (0.79%) (init = 200)
    SP_0_operation_4_Scale_factor:          170.972529 +/- 0.68520995 (0.40%) (init = 100)
[[Correlations]] (unreported correlations are < 0.100)
    C(sys_0_abundance, sys_1_site_0_shielding_symmetric_zeta)                      = -0.606
    C(sys_1_site_0_shielding_symmetric_zeta, SP_0_operation_4_Scale_factor)        =  0.555
    C(sys_0_abundance, SP_0_operation_2_Exponential_FWHM)                          = -0.446
    C(sys_0_abundance, SP_0_operation_4_Scale_factor)                              = -0.442
    C(sys_0_site_0_shielding_symmetric_zeta, sys_0_site_0_shielding_symmetric_eta) = -0.420
    C(SP_0_operation_2_Exponential_FWHM, SP_0_operation_4_Scale_factor)            =  0.409
    C(SP_0_operation_1_Exponential_FWHM, SP_0_operation_4_Scale_factor)            =  0.395
    C(sys_0_abundance, SP_0_operation_1_Exponential_FWHM)                          =  0.319
    C(sys_0_site_0_shielding_symmetric_zeta, SP_0_operation_4_Scale_factor)        =  0.144
    C(sys_0_site_0_shielding_symmetric_zeta, sys_0_abundance)                      =  0.122

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