.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/MAF/plot_2D_2_Cs2Op4p72SiO2.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_MAF_plot_2D_2_Cs2Op4p72SiO2.py: 2D MAF data of Cs2O.4.72SiO2 glass ================================== .. GENERATED FROM PYTHON SOURCE LINES 8-17 The following example illustrates an application of the statistical learning method applied in determining the distribution of the nuclear shielding tensor parameters from a 2D magic-angle flipping (MAF) spectrum. In this example, we use the 2D MAF spectrum [#f1]_ of :math:`\text{Cs}_2\text{O}\cdot4.72\text{SiO}_2` glass. Before getting started ---------------------- Import all relevant packages. .. GENERATED FROM PYTHON SOURCE LINES 17-29 .. code-block:: default import csdmpy as cp import matplotlib.pyplot as plt import numpy as np from matplotlib import cm from csdmpy import statistics as stats from mrinversion.kernel.nmr import ShieldingPALineshape from mrinversion.kernel.utils import x_y_to_zeta_eta from mrinversion.linear_model import SmoothLasso, TSVDCompression from mrinversion.utils import plot_3d, to_Haeberlen_grid .. GENERATED FROM PYTHON SOURCE LINES 31-32 Setup for the matplotlib figures. .. GENERATED FROM PYTHON SOURCE LINES 32-45 .. code-block:: default # function for plotting 2D dataset def plot2D(csdm_object, **kwargs): plt.figure(figsize=(4.5, 3.5)) ax = plt.subplot(projection="csdm") ax.imshow(csdm_object, cmap="gist_ncar_r", aspect="auto", **kwargs) ax.invert_xaxis() ax.invert_yaxis() plt.tight_layout() plt.show() .. GENERATED FROM PYTHON SOURCE LINES 46-53 Dataset setup ------------- Import the dataset '''''''''''''''''' Load the dataset. Here, we import the dataset as the CSDM data-object. .. GENERATED FROM PYTHON SOURCE LINES 53-64 .. code-block:: default # The 2D MAF dataset in csdm format filename = "https://zenodo.org/record/3964531/files/Cs2O-4_72SiO2-MAF.csdf" data_object = cp.load(filename) # For inversion, we only interest ourselves with the real part of the complex dataset. data_object = data_object.real # We will also convert the coordinates of both dimensions from Hz to ppm. _ = [item.to("ppm", "nmr_frequency_ratio") for item in data_object.dimensions] .. GENERATED FROM PYTHON SOURCE LINES 65-69 Here, the variable ``data_object`` is a `CSDM `_ object that holds the real part of the 2D MAF dataset. The plot of the 2D MAF dataset is .. GENERATED FROM PYTHON SOURCE LINES 69-71 .. code-block:: default plot2D(data_object) .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_001.png :alt: plot 2D 2 Cs2Op4p72SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 72-92 There are two dimensions in this dataset. The dimension at index 0, the horizontal dimension in the figure, is the pure anisotropic dimension, while the dimension at index 1 is the isotropic chemical shift dimension. Prepping the data for inversion ''''''''''''''''''''''''''''''' **Step-1: Data Alignment** When using the csdm objects with the ``mrinversion`` package, the dimension at index 0 must be the dimension undergoing the linear inversion. In this example, we plan to invert the pure anisotropic shielding line-shape. In the ``data_object``, the anisotropic dimension is already at index 0 and, therefore, no further action is required. **Step-2: Optimization** Also notice, the signal from the 2D MAF dataset occupies a small fraction of the two-dimensional frequency grid. For optimum performance, truncate the dataset to the relevant region before proceeding. Use the appropriate array indexing/slicing to select the signal region. .. GENERATED FROM PYTHON SOURCE LINES 92-95 .. code-block:: default data_object_truncated = data_object[:, 290:330] plot2D(data_object_truncated) .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_002.png :alt: plot 2D 2 Cs2Op4p72SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 96-106 Linear Inversion setup ---------------------- Dimension setup ''''''''''''''' **Anisotropic-dimension:** The dimension of the dataset that holds the pure anisotropic frequency contributions. In ``mrinversion``, this must always be the dimension at index 0 of the data object. .. GENERATED FROM PYTHON SOURCE LINES 106-108 .. code-block:: default anisotropic_dimension = data_object_truncated.dimensions[0] .. GENERATED FROM PYTHON SOURCE LINES 109-111 **x-y dimensions:** The two inverse dimensions corresponding to the `x` and `y`-axis of the `x`-`y` grid. .. GENERATED FROM PYTHON SOURCE LINES 111-116 .. code-block:: default inverse_dimensions = [ cp.LinearDimension(count=25, increment="450 Hz", label="x"), # the `x`-dimension cp.LinearDimension(count=25, increment="450 Hz", label="y"), # the `y`-dimension ] .. GENERATED FROM PYTHON SOURCE LINES 117-124 Generating the kernel ''''''''''''''''''''' For MAF datasets, the line-shape kernel corresponds to the pure nuclear shielding anisotropy line-shapes. Use the :class:`~mrinversion.kernel.nmr.ShieldingPALineshape` class to generate a shielding line-shape kernel. .. GENERATED FROM PYTHON SOURCE LINES 124-134 .. code-block:: default lineshape = ShieldingPALineshape( anisotropic_dimension=anisotropic_dimension, inverse_dimension=inverse_dimensions, channel="29Si", magnetic_flux_density="9.4 T", rotor_angle="87.14°", rotor_frequency="14 kHz", number_of_sidebands=4, ) .. GENERATED FROM PYTHON SOURCE LINES 135-153 Here, ``lineshape`` is an instance of the :class:`~mrinversion.kernel.nmr.ShieldingPALineshape` class. The required arguments of this class are the `anisotropic_dimension`, `inverse_dimension`, and `channel`. We have already defined the first two arguments in the previous sub-section. The value of the `channel` argument is the nucleus observed in the MAF experiment. In this example, this value is '29Si'. The remaining arguments, such as the `magnetic_flux_density`, `rotor_angle`, and `rotor_frequency`, are set to match the conditions under which the 2D MAF spectrum was acquired. Note for this particular MAF measurement, the rotor angle was set to :math:`87.14^\circ` for the anisotropic dimension, not the usual :math:`90^\circ`. The value of the `number_of_sidebands` argument is the number of sidebands calculated for each line-shape within the kernel. Unless, you have a lot of spinning sidebands in your MAF dataset, four sidebands should be enough. Once the ShieldingPALineshape instance is created, use the :meth:`~mrinversion.kernel.nmr.ShieldingPALineshape.kernel` method of the instance to generate the MAF line-shape kernel. .. GENERATED FROM PYTHON SOURCE LINES 153-156 .. code-block:: default K = lineshape.kernel(supersampling=1) print(K.shape) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none (128, 625) .. GENERATED FROM PYTHON SOURCE LINES 157-160 The kernel ``K`` is a NumPy array of shape (128, 625), where the axes with 128 and 625 points are the anisotropic dimension and the features (x-y coordinates) corresponding to the :math:`25\times 25` `x`-`y` grid, respectively. .. GENERATED FROM PYTHON SOURCE LINES 162-167 Data Compression '''''''''''''''' Data compression is optional but recommended. It may reduce the size of the inverse problem and, thus, further computation time. .. GENERATED FROM PYTHON SOURCE LINES 167-173 .. code-block:: default new_system = TSVDCompression(K, data_object_truncated) compressed_K = new_system.compressed_K compressed_s = new_system.compressed_s print(f"truncation_index = {new_system.truncation_index}") .. rst-class:: sphx-glr-script-out Out: .. code-block:: none compression factor = 1.3195876288659794 truncation_index = 97 .. GENERATED FROM PYTHON SOURCE LINES 174-187 Solving the inverse problem --------------------------- Smooth LASSO cross-validation ''''''''''''''''''''''''''''' Solve the smooth-lasso problem. Ordinarily, one should use the statistical learning method to solve the inverse problem over a range of α and λ values and then determine the best nuclear shielding tensor parameter distribution for the given 2D MAF dataset. Considering the limited build time for the documentation, we skip this step and evaluate the distribution at pre-optimized α and λ values. The optimum values are :math:`\alpha = 5.62\times 10^{-7}` and :math:`\lambda = 3.16\times 10^{-6}`. The following commented code was used in determining the optimum α and λ values. .. GENERATED FROM PYTHON SOURCE LINES 189-218 .. code-block:: default # from mrinversion.linear_model import SmoothLassoCV # # setup the pre-defined range of alpha and lambda values # lambdas = 10 ** (-4 - 3 * (np.arange(20) / 19)) # alphas = 10 ** (-4.5 - 3 * (np.arange(20) / 19)) # # setup the smooth lasso cross-validation class # s_lasso = SmoothLassoCV( # alphas=alphas, # A numpy array of alpha values. # lambdas=lambdas, # A numpy array of lambda values. # sigma=0.002, # The standard deviation of noise from the MAF data. # folds=10, # The number of folds in n-folds cross-validation. # inverse_dimension=inverse_dimensions, # previously defined inverse dimensions. # verbose=1, # If non-zero, prints the progress as the computation proceeds. # ) # # run fit using the compressed kernel and compressed data. # s_lasso.fit(compressed_K, compressed_s) # # the optimum hyper-parameters, alpha and lambda, from the cross-validation. # print(s_lasso.hyperparameters) # # the solution # f_sol = s_lasso.f # # the cross-validation error curve # CV_metric = s_lasso.cross_validation_curve .. GENERATED FROM PYTHON SOURCE LINES 219-222 If you use the above ``SmoothLassoCV`` method, skip the following code-block. The following code-block evaluates the smooth-lasso solution at the pre-optimized hyperparameters. .. GENERATED FROM PYTHON SOURCE LINES 222-230 .. code-block:: default # Setup the smooth lasso class s_lasso = SmoothLasso( alpha=8.34e-7, lambda1=6.16e-7, inverse_dimension=inverse_dimensions ) # run the fit method on the compressed kernel and compressed data. s_lasso.fit(K=compressed_K, s=compressed_s) .. GENERATED FROM PYTHON SOURCE LINES 231-236 The optimum solution '''''''''''''''''''' The :attr:`~mrinversion.linear_model.SmoothLasso.f` attribute of the instance holds the solution, .. GENERATED FROM PYTHON SOURCE LINES 236-238 .. code-block:: default f_sol = s_lasso.f # f_sol is a CSDM object. .. GENERATED FROM PYTHON SOURCE LINES 239-246 where ``f_sol`` is the optimum solution. The fit residuals ''''''''''''''''' To calculate the residuals between the data and predicted data(fit), use the :meth:`~mrinversion.linear_model.SmoothLasso.residuals` method, as follows, .. GENERATED FROM PYTHON SOURCE LINES 246-252 .. code-block:: default residuals = s_lasso.residuals(K=K, s=data_object_truncated) # residuals is a CSDM object. # The plot of the residuals. plot2D(residuals, vmax=data_object_truncated.max(), vmin=data_object_truncated.min()) .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_003.png :alt: plot 2D 2 Cs2Op4p72SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 253-254 The standard deviation of the residuals is .. GENERATED FROM PYTHON SOURCE LINES 254-256 .. code-block:: default residuals.std() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none .. GENERATED FROM PYTHON SOURCE LINES 257-262 Saving the solution ''''''''''''''''''' To serialize the solution to a file, use the `save()` method of the CSDM object, for example, .. GENERATED FROM PYTHON SOURCE LINES 262-265 .. code-block:: default f_sol.save("Cs2O.4.72SiO2_inverse.csdf") # save the solution residuals.save("Cs2O.4.72SiO2_residue.csdf") # save the residuals .. GENERATED FROM PYTHON SOURCE LINES 266-277 Data Visualization ------------------ At this point, we have solved the inverse problem and obtained an optimum distribution of the nuclear shielding tensor parameters from the 2D MAF dataset. You may use any data visualization and interpretation tool of choice for further analysis. In the following sections, we provide minimal visualization and analysis to complete the case study. Visualizing the 3D solution ''''''''''''''''''''''''''' .. GENERATED FROM PYTHON SOURCE LINES 277-291 .. code-block:: default # Normalize the solution f_sol /= f_sol.max() # Convert the coordinates of the solution, `f_sol`, from Hz to ppm. [item.to("ppm", "nmr_frequency_ratio") for item in f_sol.dimensions] # The 3D plot of the solution plt.figure(figsize=(5, 4.4)) ax = plt.subplot(projection="3d") plot_3d(ax, f_sol, x_lim=[0, 140], y_lim=[0, 140], z_lim=[-50, -150]) plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_004.png :alt: plot 2D 2 Cs2Op4p72SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 292-295 From the 3D plot, we observe two distinct regions: one for the :math:`\text{Q}^4` sites and another for the :math:`\text{Q}^3` sites. Select the respective regions by using the appropriate array indexing, .. GENERATED FROM PYTHON SOURCE LINES 295-302 .. code-block:: default Q4_region = f_sol[0:7, 0:7, 8:34] Q4_region.description = "Q4 region" Q3_region = f_sol[0:7, 10:22, 14:35] Q3_region.description = "Q3 region" .. GENERATED FROM PYTHON SOURCE LINES 303-304 The plot of the respective regions is shown below. .. GENERATED FROM PYTHON SOURCE LINES 304-350 .. code-block:: default # Calculate the normalization factor for the 2D contours and 1D projections from the # original solution, `f_sol`. Use this normalization factor to scale the intensities # from the sub-regions. max_2d = [ f_sol.sum(axis=0).max().value, f_sol.sum(axis=1).max().value, f_sol.sum(axis=2).max().value, ] max_1d = [ f_sol.sum(axis=(1, 2)).max().value, f_sol.sum(axis=(0, 2)).max().value, f_sol.sum(axis=(0, 1)).max().value, ] plt.figure(figsize=(5, 4.4)) ax = plt.subplot(projection="3d") # plot for the Q4 region plot_3d( ax, Q4_region, x_lim=[0, 140], # the x-limit y_lim=[0, 140], # the y-limit z_lim=[-50, -150], # the z-limit max_2d=max_2d, # normalization factors for the 2D contours projections max_1d=max_1d, # normalization factors for the 1D projections cmap=cm.Reds_r, # colormap box=True, # draw a box around the region ) # plot for the Q3 region plot_3d( ax, Q3_region, x_lim=[0, 140], # the x-limit y_lim=[0, 140], # the y-limit z_lim=[-50, -150], # the z-limit max_2d=max_2d, # normalization factors for the 2D contours projections max_1d=max_1d, # normalization factors for the 1D projections cmap=cm.Blues_r, # colormap box=True, # draw a box around the region ) ax.legend() plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_005.png :alt: plot 2D 2 Cs2Op4p72SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 351-357 Visualizing the isotropic projections. '''''''''''''''''''''''''''''''''''''' Because the :math:`\text{Q}^4` and :math:`\text{Q}^3` regions are fully resolved after the inversion, evaluating the contributions from these regions is trivial. For examples, the distribution of the isotropic chemical shifts for these regions are .. GENERATED FROM PYTHON SOURCE LINES 357-391 .. code-block:: default # Isotropic chemical shift projection of the 2D MAF dataset. data_iso = data_object_truncated.sum(axis=0) data_iso /= data_iso.max() # normalize the projection # Isotropic chemical shift projection of the tensor distribution dataset. f_sol_iso = f_sol.sum(axis=(0, 1)) # Isotropic chemical shift projection of the tensor distribution for the Q4 region. Q4_region_iso = Q4_region.sum(axis=(0, 1)) # Isotropic chemical shift projection of the tensor distribution for the Q3 region. Q3_region_iso = Q3_region.sum(axis=(0, 1)) # Normalize the three projections. f_sol_iso_max = f_sol_iso.max() f_sol_iso /= f_sol_iso_max Q4_region_iso /= f_sol_iso_max Q3_region_iso /= f_sol_iso_max # The plot of the different projections. plt.figure(figsize=(5.5, 3.5)) ax = plt.subplot(projection="csdm") ax.plot(f_sol_iso, "--k", label="tensor") ax.plot(Q4_region_iso, "r", label="Q4") ax.plot(Q3_region_iso, "b", label="Q3") ax.plot(data_iso, "-k", label="MAF") ax.plot(data_iso - f_sol_iso - 0.1, "gray", label="residuals") ax.set_title("Isotropic projection") ax.invert_xaxis() plt.legend() plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_006.png :alt: Isotropic projection :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 392-402 Notice the skew in the isotropic chemical shift distribution for the :math:`\text{Q}^4` regions, which is expected. Analysis -------- For the analysis, we use the `statistics `_ module of the csdmpy package. Following is the moment analysis of the 3D volumes for both the :math:`\text{Q}^4` and :math:`\text{Q}^3` regions up to the second moment. .. GENERATED FROM PYTHON SOURCE LINES 402-421 .. code-block:: default int_Q4 = stats.integral(Q4_region) # volume of the Q4 distribution mean_Q4 = stats.mean(Q4_region) # mean of the Q4 distribution std_Q4 = stats.std(Q4_region) # standard deviation of the Q4 distribution int_Q3 = stats.integral(Q3_region) # volume of the Q3 distribution mean_Q3 = stats.mean(Q3_region) # mean of the Q3 distribution std_Q3 = stats.std(Q3_region) # standard deviation of the Q3 distribution print("Q4 statistics") print(f"\tpopulation = {100 * int_Q4 / (int_Q4 + int_Q3)}%") print("\tmean\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*mean_Q4)) print("\tstandard deviation\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*std_Q4)) print("Q3 statistics") print(f"\tpopulation = {100 * int_Q3 / (int_Q4 + int_Q3)}%") print("\tmean\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*mean_Q3)) print("\tstandard deviation\n\t\tx:\t{0}\n\t\ty:\t{1}\n\t\tiso:\t{2}".format(*std_Q3)) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Q4 statistics population = 59.18636390733823% mean x: 9.63134216393539 ppm y: 10.897496522222905 ppm iso: -104.49449285861381 ppm standard deviation x: 6.503922392419356 ppm y: 6.940197547234903 ppm iso: 5.356550091299955 ppm Q3 statistics population = 40.81363609266177% mean x: 11.188595729461198 ppm y: 87.66281729127857 ppm iso: -96.07797849647002 ppm standard deviation x: 7.226377441969985 ppm y: 10.049835899913326 ppm iso: 3.8792719778668787 ppm .. GENERATED FROM PYTHON SOURCE LINES 422-426 The statistics shown above are according to the respective dimensions, that is, the `x`, `y`, and the isotropic chemical shifts. To convert the `x` and `y` statistics to commonly used :math:`\zeta_\sigma` and :math:`\eta_\sigma` statistics, use the :func:`~mrinversion.kernel.utils.x_y_to_zeta_eta` function. .. GENERATED FROM PYTHON SOURCE LINES 426-446 .. code-block:: default mean_ζη_Q3 = x_y_to_zeta_eta(*mean_Q3[0:2]) # error propagation for calculating the standard deviation std_ζ = (std_Q3[0] * mean_Q3[0]) ** 2 + (std_Q3[1] * mean_Q3[1]) ** 2 std_ζ /= mean_Q3[0] ** 2 + mean_Q3[1] ** 2 std_ζ = np.sqrt(std_ζ) std_η = (std_Q3[1] * mean_Q3[0]) ** 2 + (std_Q3[0] * mean_Q3[1]) ** 2 std_η /= (mean_Q3[0] ** 2 + mean_Q3[1] ** 2) ** 2 std_η = (4 / np.pi) * np.sqrt(std_η) print("Q3 statistics") print(f"\tpopulation = {100 * int_Q3 / (int_Q4 + int_Q3)}%") print("\tmean\n\t\tζ:\t{0}\n\t\tη:\t{1}\n\t\tiso:\t{2}".format(*mean_ζη_Q3, mean_Q3[2])) print( "\tstandard deviation\n\t\tζ:\t{0}\n\t\tη:\t{1}\n\t\tiso:\t{2}".format( std_ζ, std_η, std_Q3[2] ) ) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none Q3 statistics population = 40.81363609266177% mean ζ: 88.37394531105538 ppm η: 0.1616324456555935 iso: -96.07797849647002 ppm standard deviation ζ: 10.010860855010964 ppm η: 0.10488989054519339 iso: 3.8792719778668787 ppm .. GENERATED FROM PYTHON SOURCE LINES 447-475 Result cross-verification ------------------------- The reported value for the Qn-species distribution from Baltisberger `et. al.` [#f1]_ is listed below and is consistent with the above result. .. list-table:: :widths: 7 15 28 25 25 :header-rows: 1 * - Species - Yield - Isotropic chemical shift, :math:`\delta_\text{iso}` - Shielding anisotropy, :math:`\zeta_\sigma`: - Shielding asymmetry, :math:`\eta_\sigma`: * - Q4 - :math:`57.7 \pm 0.4` % - :math:`-104.7 \pm 5.2` ppm - 0 ppm (fixed) - 0 (fixed) * - Q3 - :math:`42.3 \pm 0.4` % - :math:`-96.1 \pm 4.0` ppm - 89.0 ppm - 0 (fixed) .. GENERATED FROM PYTHON SOURCE LINES 477-482 Convert the 3D tensor distribution in Haeberlen parameters ---------------------------------------------------------- You may re-bin the 3D tensor parameter distribution from a :math:`\rho(\delta_\text{iso}, x, y)` distribution to :math:`\rho(\delta_\text{iso}, \zeta_\sigma, \eta_\sigma)` distribution as follows. .. GENERATED FROM PYTHON SOURCE LINES 482-490 .. code-block:: default # Create the zeta and eta dimensions,, as shown below. zeta = cp.as_dimension(np.arange(40) * 4 - 40, unit="ppm", label="zeta") eta = cp.as_dimension(np.arange(16) / 15, label="eta") # Use the `to_Haeberlen_grid` function to convert the tensor parameter distribution. fsol_Hae = to_Haeberlen_grid(f_sol, zeta, eta) .. GENERATED FROM PYTHON SOURCE LINES 491-493 The 3D plot ''''''''''' .. GENERATED FROM PYTHON SOURCE LINES 493-499 .. code-block:: default plt.figure(figsize=(5, 4.4)) ax = plt.subplot(projection="3d") plot_3d(ax, fsol_Hae, x_lim=[0, 1], y_lim=[-40, 120], z_lim=[-50, -150], alpha=0.2) plt.tight_layout() plt.show() .. image:: /auto_examples/MAF/images/sphx_glr_plot_2D_2_Cs2Op4p72SiO2_007.png :alt: plot 2D 2 Cs2Op4p72SiO2 :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 500-507 References ---------- .. [#f1] Alvarez, D. J., Sanders, K. J., Phyo, P. A., Baltisberger, J. H., Grandinetti, P. J. Cluster formation of network-modifier cations in cesium silicate glasses, J. Chem. Phys. **148**, 094502, (2018). `doi:10.1063/1.5020986 `_ .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 6.574 seconds) .. _sphx_glr_download_auto_examples_MAF_plot_2D_2_Cs2Op4p72SiO2.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/DeepanshS/mrinversion/master?urlpath=lab/tree/docs/_build/html/../../notebooks/auto_examples/MAF/plot_2D_2_Cs2Op4p72SiO2.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_2D_2_Cs2Op4p72SiO2.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_2D_2_Cs2Op4p72SiO2.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_