³¹P static NMR of crystalline Na2PO4 (CSA)¶

The following is a CSA static least-squares fitting example of a \(^{31}\text{P}\) MAS NMR spectrum of \(\text{Na}_{2}\text{PO}_{4}\). 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, SpinSystem, Site
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 = "31P Phophonate Static.csdf"
experiment = cp.load(host + filename)

# standard deviation of noise from the dataset
sigma = 3.258224

# 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, color="black", linewidth=0.5, label="Experiment")
ax.set_xlim(200, -200)
plt.grid()
plt.tight_layout()
plt.show()

Create a fitting model¶

Spin System

P_31 = Site(
    isotope="31P",
    isotropic_chemical_shift=5.0,  # in ppm,
    shielding_symmetric=SymmetricTensor(zeta=-80, eta=0.5),  # zeta in Hz
)

spin_systems = [SpinSystem(sites=[P_31])]

Method

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

static1D = BlochDecaySpectrum(
    channels=["31P"],
    magnetic_flux_density=9.395,  # in T
    rotor_frequency=0,  # 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 = static1D.get_transition_pathways(sys)

Guess Model Spectrum

# Simulation
# ----------
sim = Simulator(spin_systems=spin_systems, methods=[static1D])
sim.run()

# Post Simulation Processing
# --------------------------
processor = sp.SignalProcessor(
    operations=[
        sp.IFFT(),
        sp.apodization.Gaussian(FWHM="3000 Hz"),
        sp.FFT(),
        sp.Scale(factor=4000),
    ]
)
processed_dataset = processor.apply_operations(dataset=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, color="black", linewidth=0.5, label="Experiment")
ax.plot(processed_dataset, linewidth=2, alpha=0.6, label="Guess Spectrum")
ax.set_xlim(200, -200)
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_Gaussian_FWHM             3000     -inf      inf     True     None
SP_0_operation_3_Scale_factor              4000     -inf      inf     True     None
sys_0_site_0_isotropic_chemical_shift         5     -inf      inf     True     None
sys_0_site_0_shielding_symmetric_eta        0.5        0        1     True     None
sys_0_site_0_shielding_symmetric_zeta       -80     -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()
result

Fit Statistics

fitting method	leastsq
# function evals	69
# data points	2048
# variables	5
chi-square	3405.48278
reduced chi-square	1.66690298
Akaike info crit.	1051.45512
Bayesian info crit.	1079.57821

Variables

name	value	standard error	relative error	initial value	min	max	vary
sys_0_site_0_isotropic_chemical_shift	6.32228199	0.07291435	(1.15%)	5.0	-inf	inf	True
sys_0_site_0_shielding_symmetric_zeta	-86.8324173	0.14007057	(0.16%)	-80.0	-inf	inf	True
sys_0_site_0_shielding_symmetric_eta	0.18101165	0.02313119	(12.78%)	0.5	0.00000000	1.00000000	True
SP_0_operation_1_Gaussian_FWHM	5978.91425	101.488701	(1.70%)	3000.0	-inf	inf	True
SP_0_operation_3_Scale_factor	4626.51661	6.23103856	(0.13%)	4000.0	-inf	inf	True

Correlations (unreported correlations are < 0.100)

sys_0_site_0_shielding_symmetric_eta	SP_0_operation_1_Gaussian_FWHM	-0.9747
sys_0_site_0_isotropic_chemical_shift	sys_0_site_0_shielding_symmetric_zeta	-0.5905
sys_0_site_0_shielding_symmetric_zeta	SP_0_operation_1_Gaussian_FWHM	0.3249
sys_0_site_0_shielding_symmetric_zeta	SP_0_operation_3_Scale_factor	-0.3075
sys_0_site_0_shielding_symmetric_zeta	sys_0_site_0_shielding_symmetric_eta	-0.2963
sys_0_site_0_isotropic_chemical_shift	SP_0_operation_3_Scale_factor	0.2785
sys_0_site_0_isotropic_chemical_shift	sys_0_site_0_shielding_symmetric_eta	-0.2489
sys_0_site_0_isotropic_chemical_shift	SP_0_operation_1_Gaussian_FWHM	0.2161
SP_0_operation_1_Gaussian_FWHM	SP_0_operation_3_Scale_factor	0.1824
sys_0_site_0_shielding_symmetric_eta	SP_0_operation_3_Scale_factor	-0.1179

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=(4.25, 3.0))
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, label="Best Fit")
ax.set_xlim(200, -200)
plt.grid()
plt.legend()
plt.tight_layout()
plt.show()

Total running time of the script: ( 0 minutes 1.785 seconds)

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