Modeling Pair Distribution Functions of Rare-Earth Phosphate

Article pubs.acs.org/IC

Modeling Pair Distribution Functions of Rare-Earth Phosphate Glasses Using Principal Component Analysis Jacqueline M. Cole,*,†,‡,§,∥ Xie Cheng,† and Michael C. Payne† †

Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, U.K. Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, United States § ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0QX, U.K. ∥ Department of Chemical Engineering and Biotechnology, University of Cambridge, West Cambridge Site, Philippa Fawcett Drive, Cambridge CB3 0FS, U.K. ‡

S Supporting Information *

ABSTRACT: The use of principal component analysis (PCA) to statistically infer features of local structure from experimental pair distribution function (PDF) data is assessed on a case study of rare-earth phosphate glasses (REPGs). Such glasses, codoped with two rare-earth ions (R and R′) of different sizes and optical properties, are of interest to the laser industry. The determination of structure−property relationships in these materials is an important aspect of their technological development. Yet, realizing the local structure of codoped REPGs presents significant challenges relative to their singly doped counterparts; specifically, R and R′ are difficult to distinguish in terms of establishing relative material compositions, identifying atomic pairwise correlation profiles in a PDF that are associated with each ion, and resolving peak overlap of such profiles in PDFs. This study demonstrates that PCA can be employed to help overcome these structural complications, by statistically inferring trends in PDFs that exist for a restricted set of experimental data on REPGs, and using these as training data to predict material compositions and PDF profiles in unknown codoped REPGs. The application of these PCA methods to resolve individual atomic pairwise correlations in t(r) signatures is also presented. The training methods developed for these structural predictions are prevalidated by testing their ability to reproduce known physical phenomena, such as the lanthanide contraction, on PDF signatures of the structurally simpler singly doped REPGs. The intrinsic limitations of applying PCA to analyze PDFs relative to the quality control of source data, data processing, and sample definition, are also considered. While this case study is limited to lanthanide-doped REPGs, this type of statistical inference may easily be extended to other inorganic solid-state materials and be exploited in large-scale data-mining efforts that probe many t(r) functions.

I. INTRODUCTION

PCA has been employed widely to partition extended X-ray absorption fine structure (EXAFS) spectra into distinct components that pertain to individual atomic pairwise correlations.3−9 More recently, PCA has been applied to X-ray diffraction studies on disordered materials, where data analysis uses pair distribution functions (PDFs) to extract structural information, in a similar way to XAS. To this end, Chapman et al.10 have reviewed several representative examples of the application of PCA to PDFs of inorganic nanomaterials generated by X-ray diffraction. Specific attention has been focused on the partitioning of PDFs into components of individual atomic pairwise correlations in a way that a series of such PDFs can be modeled, which differ according to sequential structural changes that are imparted by a systematic variation of an experimental control parameter, for example, the sample solution concentration, or the temperature

Statistical inference methods are becoming ever more prevalent in predicting the structure of materials at the atomic level. This holds especially true for predictions that embrace data mining on a large scale, since the structural information sought can be discriminated statistically with great accuracy. In such cases, simple, yet effective statistical measures are preferable to minimize the computational load associated with screening large data sets. For example, Ceder et al. have demonstrated an effective method to predict crystal structures via combining quantum mechanics with machine-learning tools.1 On a small scale, where spectra are the subject data, principal component analysis (PCA) has proven to be helpful in predicting information from data. For example, PCA has been used in X-ray absorption near-edge structure (XANES) studies2 to predict spectra that correspond to a certain oxidation state via a “feature vector” that is machine learnt from limited spectral data. In other X-ray absorption spectroscopy (XAS) studies, © XXXX American Chemical Society

Received: April 13, 2016

A

DOI: 10.1021/acs.inorgchem.6b00907 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry of the experiment. Meanwhile, Siddiqui et al.11 have demonstrated the application of PCA to PDFs of amorphous materials, whereby PCA has been used to determine the drug concentration within pharmaceutical formulations. In this work, the application of PCA to amorphous materials is further extended to a case study that predicts the local structure of inorganic lanthanide-doped phosphate glasses from PDF data, with the broader aim to assess the potential of this approach to corresponding large-scale data-mining predictions for PDF signatures. To this end, establishing ways that can screen hypothetical material structure information, and thus spare unnecessary synthesis and compositional characterization, represent a strong benefit of this type of prediction strategy. This work therefore focuses on the prediction of material compositions of codoped rare-earth phosphate glasses from training sets of codoped rare-earth phosphate glasses (REPGs) bearing known structural information. The study also demonstrates how one can additionally exploit such training data to predict the actual spectral profiles of PDFs in this series of compounds and partition distinct peaks in the t(r) profile, should they emanate from several atomic pairwise correlations that overlap in r. Finally, since quality control of data is also an important consideration for this type of structural prediction work, the quality of low- and high-energy synchrotron data collected on these glasses is specifically compared and contrasted. REPGs exhibit optical properties that promise potential applications in optical fiber technology12−17 and as laser-head materials.18−24 Their optical properties are directly correlated to their atomic structure, which has been characterized by a range of methods, such as X-ray and neutron diffraction,25−28 EXAFS,29,30 XANES,31 solid-state NMR,25,26 and Fourier transform IR.32 The data thus obtained have provided basic structural parameters for the glass network, such as ionic separations, coordination numbers, and Debye−Waller factors associated with various atomic pairwise correlations, especially R···O, R···P, R···R, and P···O (R = lanthanide). However, most of these studies have been focused on singly doped REPGs, while codoped REPGs, which contain two different rare-earth ions, R and R′, offer greater structural variation and therefore richer diversity with respect to the optical properties of these materials. One structural study on codoped REPGs of the general formula (R2O3)x(R′2O3)y(P2O5)1−(x+y) has been reported previously, where PDFs generated from X-ray diffraction data were used to probe a series of phosphate glasses containing laser-active Nd or Er ions (R′), diluted with other lanthanide ions, La or Ce (R).33 An accurate determination of the structural environment that immediately surrounds the rare-earth ions in these codoped REPGs (e.g., R/R′···O separations) was deemed important, since it has a marked influence on their optical properties. However, the elucidation of such structural information was limited in this study owing to the overlap of the atomic pairwise correlations in the PDFs that involve R and R′. Therefore, we decided (a) to explore if PCA could be employed to provide superior accuracy of these recently reported R/R′···O separations in codoped REPGs and (b) to assess if PCA allows accurate predictions for the R/R′···O separations in codoped REPGs of other compositions. More generally, this study focuses on applying quantitative PCA methods to understand, identify, and predict REPG properties. Before applying these PCA tools to the study of codoped REPG glasses, their development was subjected to validation checks against PDFs of singly doped REPGs, that is, reference data on structurally simpler REPGs, whereby well-established trends

such as the lanthanide contraction34 could be reproduced. With this proof-of-principle in hand, these PCA tools were employed to identify lanthanides associated with an unknown spectrum or to predict the PDF signatures of an unknown REPG. PCA was then applied to codoped REPGs, where the identification and prediction variables are the relative proportions (x:y) of the two lanthanides R and R′. These prediction strategies were used to distinguish distinct pairwise atomic correlations in REPGs, which suffer from spatial overlap in the PDF profile, to investigate structural invariance by codoping, and to compare data with different noise levels. To acquaint the reader with the nature of the data and the statistical approach, this paper begins by introducing PCA in the context of the X-ray diffraction analysis of amorphous REPGs.

II. METHODS II.A. Experimental Data. Experimental X-ray diffraction data sets for singly doped REPGs of the general composition (R2O3)x(P2O5)1−x (R = La or Er, 0.176 ≤ x ≤ 0.260)26 or of codoped REPGs with the general formula (R2O3)x(R′2O3)y(P2O5)1−(x+y) (R = La, R′ = Nd, x ≤ 0.260, y ≤ 0.099; R = Ce, R′ = Er, x ≤ 0.197, y ≤ 0.239) were obtained from a previous study.33 All X-ray diffraction data were collected using the high-energy beamline ID15 (X-ray energy = 94 keV) at the European Synchrotron Radiation Facility (ESRF, Grenoble, France), except for the codoped Ce/Er samples, which were also examined using the Synchrotron Radiation Source (SRS; Daresbury Laboratory, U.K.) using an X-ray beam of lower energy (X-ray energy = 26 keV). The analysis of the raw diffraction data has been described in detail elsewhere,26,33 but it will nevertheless be briefly outlined here for the purposes of defining and introducing the relevant data parameters in the context of this case study. The raw diffraction data were corrected for absorption effects, and the experimental interference function i(Q) was calculated via i(Q ) =

Iexp − (⟨f 2 ⟩ + Ic) ⟨f ⟩2 z ̅ 2

(1)

where Q is the magnitude of the scattering vector, Iexp is the corrected and scaled experimental intensity, Ic is the sum of the relative Compton-normalized contributions35 for each atom type, ⟨f 2⟩ is the sum of the relative contributions for each atom type of the square of the self-scattering factors,36 ⟨f⟩2 is the square of the sum of the relative contributions of the self-scattering factors for each atom type (often called the sharpening function), and z2̅ is the mean number of electrons in a scattering unit. Experimental data were scaled and fitted iteratively. An example of an i(Q) plot is illustrated in Figure 1a for the case of the reference singly doped REPGs data used in this study. A subsequent evaluation of the atomic pairwise distribution, t(r), was performed by a Fourier transform of i(Q) via t(r ) = 2π 2rρ0 +

∫Q

Q max min

MQi(Q ) sin(Qr )dQ

(2)

where r is the separation between two given pairs of atoms, ρ0 is the atomic density, and M is a Hanning window function37 that suppresses Fourier transform termination effects. Because of the statistical quality of the ESRF data at large angles, the maximum value of Q measured was 28 Å−1, which implies a real 2π space resolution in t(r) of Δr ≈ Q max = 0.22 Å. The t(r) profiles for the singly doped and codoped glasses are shown in B


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Figure 1. (a) X-ray interference functions i(Q) and (b) the Fourier transforms t(r) of eight singly doped REPGs, which are used as input “training” data for PCA, whereby subscripts denote the concentration of the lanthanide present. Figure reproduced from ref 24 with permission from the PCCP Owner Societies.

Figure 2. t(r) profiles for (a) Ce/Er codoped REPGs, collected at the ESRF and (b) at the SRS; (c) the t(r) profiles for the La/Nd codoped REPGs collected at the ESRF. Solid lines represent the experimental data, while dotted lines refer to the corresponding models. To avoid stacking, the t(r) curves are offset from zero by a constant. Stoichiometric formulas of the compounds are annotated onto the associated spectrum. Figure adapted from ref 24 with permission from the PCCP Owner Societies.

the t(r) signatures (Figure 2b).33 Their real space spectra were modeled33 as a set of Gaussian contributions, each corresponding to a pair correlation between atoms. An illustration of a fully modeled t(r) profile for a codoped REPG is shown in Figure 3.

Figure 1b and Figure 2, respectively. Atomic pair correlation functions obtained from i(Q) diffraction data acquired from the SRS beamline cover a smaller range (Q < 22.5 Å−1) and therefore exhibit higher levels of noise, which in turn broadens C


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Figure 4. Flow diagram for the use of PCA to obtain training data from experimental PDF data, which can subsequently be used for the prediction of PDF profiles, for the determination of REPG material compositions, and the resolution of overlapping structural signatures of individual atomic pairwise correlations that can manifest in PDFs.

Xia, the PCA components are the eigenvectors of a covariance matrix cij, which is defined by n

cij =

∑ a=1

Figure 3. An illustration of the fully modeled t(r) profile for the Ce/Er codoped REPG (Ce2O3)0.178(Er2O3)0.043(P2O5)0.779. Each Gaussian peak represents a contribution from an individual atomic pairwise correlation to the overall spectrum. For example, the P···O separation is modeled by two Gaussian peaks, which distinguish between terminal (P···O T) and bridging (P···O B) binding modes. This is an example of a first neighbor correlation (r ≈ 1.5−1.6 Å), which complements second (e.g., Ce−O−P) and third neighbor correlations (e.g., Ce−O−P or Ce−OP−O) that progressively overlap as a function of increasing r, thereby making the modeling approach more difficult. Figure adapted from ref 24 with permission from the Royal Society of Chemistry.

(Xia − Xia)(X ja − X ja) (n − 1)

(3)

The symmetry of the matrix allows it to be diagonalized, yielding the PCA component vectors along with the covariances in these directions. For a set of n input vectors, a maximum of (n − 1) principal components can be determined, which is especially important in the context of this paper. Here, only a limited number of feature vectors can be generated from the available spectra, while each spectrum contains a large number of dimensions. PCA has already been successfully applied to a wide range of multivariance problems, including face recognition,39 identification of faulty sensors,40 and reconstruction of structural modulations in X-ray absorption spectroscopy.2 In this study, PCA was used to predict primarily trends in R···O separations; the composition of REPGs was calculated as a validation parameter, as it can be obtained independently from electron probe microanalysis (for details, see Supporting Information). The PCA tools developed for this study were designed to fulfill the following conceptual requirements: (1) the ability to provide accurate compositional information from unknown REPG spectra on the basis of “training” spectra using REPGs with known composition (singly doped REPGs: atomic number; codoped REPGs: proportional composition x/(x + y)); (2) the ability to predict PDF profile sections of unknown REPGs on the basis of these training spectra, whereby predictions for singly- and codoped REPGs should be consistent; (3) the ability of the aforementioned processes to proceed fast with the prospect that such processes can be embedded into high levels of automation in order that they can ultimately be used to mine large databases. Such conditions typically require a minimum of human input, that is, a minimum of underlying models and assumptions. The Matlab software package was selected as the appropriate computational framework to develop such PCA tools, as it contains comprehensive statistical functions including PCA.41 When either the i(Q) or t(r) function of the raw diffraction data (Figure 1) was treated as N discrete values, its t(r) profile could be modeled as a vector in an N-dimensional space. Eight input t(r) profiles were available for the singly doped REPGs, resulting in a maximum of seven principal components after

II.B. Computational Methods. Principal component analysis (PCA)38 is a statistical inference method that uses a set of experimental data, comprising correlated variables (m + 1 vectors in an n-dimensional space), to produce a set of m (m < n) linearly uncorrelated variables (principal components) via an orthogonal matrix transformation. These principal components form the feature vector that harnesses the pattern of common information. As the feature vector is deduced from raw data, it is afforded a predictive quality, which allows analyzing similar data with unknown data signatures. In the context of this structural study, such principal components are to be generated by training them with the subject experimental X-ray diffraction data on REPGs that takes the form of PDF information. The corresponding feature vectors will manifest latent underlying structural trends that exist across the individual REPG data sets when considered together. Exposing such intrinsically latent structural trends via this form of statistical inference affords the predictive element of PCA to aid structural science. For example, once trained on relevant experimental data, PCA has the capacity to predict structural parameters that cannot be obtained by experiment alone or that may only be obtained experimentally on materials that are chemically related to the one of interest due to various practical reasons. This idea is illustrated in the form of a flow diagram in Figure 4. The mathematical process behind PCA relies on this notion of relative variation. Given an i-dimensional set of n vectors D


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Four experimental spectra with varying compositions for La/Nd or Ce/Er codoped samples were available. Owing to this limited amount of accessible training data, PCA was applied to each of the four data sets in a cyclical fashion, whereby three spectra were used as the training data at any one time to generate the feature vector that predicts the R/R′···O peak position at 50/50 composition from the fourth spectrum, which acted as the “unknown” sampling data. This procedure is illustrated in Figure 5 (left), and the results are presented in Tables 1 and 2. A back-calculation of the 50/50 rare-earth

applying PCA. Four different rare-earth codoped (La/Nd or Ce/Er) REPG compositions provided t(r) functions for analysis, with the singly doped Ce and Er REPG profiles adding two further data sets to the PCA analysis where required. All t(r) data were provided for PCA analysis in binned increments of 0.001 Å.

III. RESULTS AND DISCUSSION Preliminary tests on singly doped REPG reference data (for details, see: Supporting Information) revealed that the best results were obtained when PCA was applied to t(r) profiles. Prior to their application to codoped REPG studies, the PCA methods were validated against singly doped REPG data, which were used as training data. After the PCA tools were trained, these were able to accurately reproduce the so-called lanthanide contraction,34 to assign values for each pairwise correlation that was consistent with previously employed models, and to successfully identify lanthanides on the basis of their characteristic PDF profiles. Given that the PCA methods were able to deliver satisfactory results in all of these preliminary tests, the basic validity of this form of statistical inference for this case study was confirmed. III.A. Predicting Material Trends in Unknown Spectra. With the above basic checks in hand for the singly doped REPGs, the PCA methods were applied to their codoped analogues, where twice the number of atomic pairwise correlations involving the rare-earth ion generated more complicated spectra. PCA was assessed with respect to its ability to predict compositional information, the shape of the PDF profiles for the glasses, as well as to resolve spatial overlap of atomic pairwise correlations. In the following, we focus on predicting structural information for equal rare-earth compositions (R/R′ = 50/50), and for R/R′···O peak correlations. This reflects two materials-centered objectives: first, there is interest in assessing the extent that the greatest overall rare-earth ion size mismatch exerts upon the glass structure. This interest arises from the desire to embed the highest possible amount of optically active rare-earth ions (e.g., Nd3+ or Er3+) in optical fibers of these materials for the laser industry. Second, the optical properties of the rare-earth ion are strongly affected by its structural environment, especially within its immediate vicinity, which is best described by the R/R′···O nearest-neighbor correlation. The experimental spectra of codoped Ce/Er and La/Nd REPGs provided the training data for the generation of the principal components in the PCA. III.A.1. Prediction of the Relative Composition of Codoped Rare-Earth Phosphate Glasses. The developed PCA methods were employed to predict the R/R′···O peak position of Ce/Er or La/Nd codoped REPGs with a 50/50 rare-earth metal composition. The PCA workflow followed a three-step process: (1) the experimentally determined codoped R/R′···O peak positions for a given composition were used as the training data to produce the PCA feature vector; (2) a scale factor was introduced, which aligned the composition to the elemental nature of the rare-earth metal present by correlating the interdependent R/R′···O peak position and rare-earth metal concentration variables in the singly doped REPG spectra. These data were subsequently used to generate the scale factor, as they span most of the lanthanide series; (3) the PCA feature vector was appropriately scaled to predict the R/R′···O peak position of La/Nd or Ce/Er codoped REPGs with a 50/50 composition.

Figure 5. Schematic illustration of PCA, using a covariance matrix with (left) four input vectors to create the predicted values for 50/50 Ce/Er or La/Nd REPGs in Tables 2 and 3, respectively; or (right) six input vectors for 50/50 Ce/Er REPGs predicted for Table 4 (including 0% Ce and Er data). Black circles: data used to generate the feature vectors (red circles); arrows: eigenvectors.

Table 1. Predictions for the Relative Composition of Codoped Ce/Er REPGs with Four Input Spectra Ce (x) (exp value) Er (y) (exp value) Er [%] (exp observed) peak position [Å] (exp observed) predicted value at 50% [Å] (PCA) Er % prediction [%] (PCA) |error| [%]

0.178 0.043 19 2.41

0.111 0.108 49 2.31

0.091 0.132 59 2.28

0.047 0.190 80 2.27

2.30

2.33

2.33

2.31

−5

60

74

70

24

11

15

10

|mean error| [%] 15

Table 2. Predictions for the Relative Composition of Codoped Nd/La REPGs with Four Input Spectra Nd (x) (exp value) La (y) (exp value) La [%] (exp observed) peak position [Å] (exp observed) predicted value at 50% [Å] (PCA) La % prediction [%] (PCA) |error| [%]

0.010 0.260 96 2.46

0.019 0.226 92 2.45

0.043 0.190 82 2.45

0.099 0.147 60 2.43

2.42

2.42

2.42

2.42

87

76

74

68

9

16

8

8

|mean error| [%] 10

metal concentrations to those of the original experimental compositions y, where R represents the percentage of Er or La for an individual spectrum, yielded an estimated prediction error according to |R% (experiment) − R% (PCA prediction)|. The mean error in these predictions was 15% and 10% for codoped Er/Ce and Nd/La REPGs, respectively. To illustrate the possible improvement that can be attained from an increased number of input spectra, the Er/Ce data set E


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Inorganic Chemistry Table 3. Predictions for the Relative Composition of Ce/Er co-doped REPGs with Six Input Spectra Ce (x) (exp value) Er (y) (exp value) Er [%] (exp observed) peak position [Å] (exp observed) predicted value at 50% [Å] (PCA) Er % prediction [%] (PCA) |error| [%]

0.197 0 0 2.44 2.32 11 11

0.178 0.043 19 2.41 2.30 −5 24

0.111 0.108 49 2.31 2.33 60 11

0.091 0.132 59 2.28 2.33 74 15

0.047 0.190 80 2.27 2.31 70 10

0 0.239 100 2.25 2.32 87 13

|mean error| [%] 11

Figure 6. Predicting the profile of a peak within an unknown PDF from PCA training data of known peak profiles from experimentally determined peak profiles of other REPGs; black rectangles: training data of spectral peaks from 2, 3, 4, or 7 sets of REPG data that form the feature vector; red rectangles: unknown target spectrum; arrows: eigenvectors.

was expanded to include the singly doped Er and Ce REPGs (100/0 mixtures) as additional spectra. These six input spectra were subjected to a similar alternating cycling procedure to generate PCA feature vectors (Figure 5, right). The results, shown in Table 3, suggest a decrease of the mean prediction error by a factor of 1.36, which is consistent with a reduction according to √N = √5/3 = 1.3. III.A.2. Predicting Pair Distribution Function Profiles for Rare-Earth Phosphate Glasses. The PCA methods were subsequently used to predict profiles of an unknown t(r) function from a set of known PDF profiles. Both singly doped and Ce/Er codoped REPGs were considered sequentially. 2.1. Predicting Pair Distribution Function Profiles for Singly Doped Rare-Earth Phosphate Glasses. Applying data in a similar way to that of Section III.A.1, each of the eight singly doped REPG spectra was used in turn as the unknown spectrum, with the other seven being available as training data. The feature vector was determined using a minimum of two, and a maximum of all neighboring spectra (Figure 6). A peak profile within a specified range r was then predicted using a profile translation. The dominant principal component of the feature vector was thereby used to determine an eigenvalue from the spectra of its two closest neighboring spectra, scaled by their relative ionic radii. The relative performance of the PCA algorithm was assessed by evaluating the cosine distance of the predicted profile from that of the spectrum, which contains the lanthanide with the nearest ionic radius. The cosine distance was chosen as the

measure of spectral similarity, considering that it accounts for the vertical scaling introduced by using REPGs with different doping concentrations. A representative example of this process is shown in Figure 7, where the fraction of the t(r) function that contains the Ce···O atomic pairwise correlation is predicted

Figure 7. Predictions of the fraction of t(r) that contains the Ce···O pairwise correlation using experimental PDF profiles of the closest rare-earth metal neighbors (La and Pr) as training data. Cosine distances less than or equal to 0.001 are considered a match. Note that a clear vertical shift between the predicted and experimental spectrum of Ce is observable. F


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Table 4. Cosine Distances between Predicted and Experimental t(r) Spectra for Singly-Doped REPGs for r = 1.92−2.97 Åa

a

1.92−2.97 Å

La

Ce

Pr

closest neighbor PCA (2 spectra) PCA (3 spectra) PCA (4 spectra) PCA (7 spectra)

0.009 0.007 0.004 0.007 0.02

0.008 0.001 0.001 0.001 0.001

0.008 0.001 0.002 0.001 0.001

0.169

Sm

0.07 0.02 0.02 0.02 0.02

0.205

Sm

0.02 0.005 0.004 0.004 0.004

Gd

Tb

Er

mean

0.004 0.004 0.004 0.008 0.01

0.004 0.002 0.005 0.006 0.006

0.06 0.02 0.04 0.03 0.06

0.02 0.007 0.010 0.010 0.014

That is, the t(r) region where the R−O atomic pairwise correlation predominates.

Table 5. Cosine Distances between Predicted and Experimental t(r) Spectra for Singly-Doped REPGs for r = 1−4 Åa

a

1−4 Å

La

Ce

Pr


0.006 0.015 0.007 0.005 0.009

0.008 0.003 0.003 0.003 0.004

0.008 0.003 0.003 0.003 0.002

0.169

Sm

0.04 0.014 0.012 0.013 0.016

0.205

Sm

0.02 0.005 0.004 0.004 0.004

Gd

Tb

Er

mean

0.006 0.005 0.004 0.006 0.008

0.006 0.005 0.007 0.007 0.007

0.03 0.06 0.03 0.016 0.04

0.015 0.014 0.008 0.007 0.011

That is, the entire range of experimentally discernible atomic pairwise correlations in REPGs.

Table 6. Cosine Distances between Predicted and Experimental t(r) Spectra for Codoped Ce/Er REPGs for r = 1.92−2.97 Åa

a

1.92−2.97 Å

Er 100%

Er 80%

Er 59%

Er 49%

Er 19%

Er 0%

mean


0.03 0.005 0.005 0.007 0.01

0.03 0.001 0.001 0.001 0.001

0.007 0.002 0.002 0.003 0.004

0.007 0.002 0.002 0.002 0.002

0.02 0.004 0.004 0.004 0.003

0.02 0.01 0.008 0.009 0.01

0.02 0.004 0.004 0.004 0.005

That is, the t(r) region that features Ce−O and Er−O separations.

Table 7. Cosine Distances between Predicted and Experimental t(r) Spectra for Codoped Ce/Er REPGs for a Wider Range of r = 1−4 Åa

a

1−4 Å

Er 100%

Er 80%

Er 59%

Er 49%

Er 19%

Er 0%

mean


0.02 0.004 0.006 0.006 0.011

0.017 0.001 0.001 0.001 0.001

0.003 0.001 0.002 0.002 0.003

0.003 0.002 0.002 0.003 0.003

0.013 0.002 0.002 0.002 0.003

0.013 0.007 0.006 0.009 0.011

0.01 0.003 0.003 0.004 0.005

That is, the t(r) region that features all experimentally discernible atomic pairwise correlations in this series of compounds.

reside, seem to benefit from the use of a greater number of training spectra (mean error for two training spectra = 0.014; mean error for four training spectra = 0.007). 2.2. Predicting Pair Distribution Function Profiles for Codoped Rare-Earth Phosphate Glasses. A similar algorithm was then used to predict the PDF profiles of Ce/Er codoped REPGs across their compositional ranges. The PCA performance was observed to be similarly accurate across the two ranges of r (Tables 6 and 7), which is perhaps even slightly better than that of the singly doped REPG predictions. This is somewhat remarkable, as one might expect that the codoped prediction problem should be more challenging than that of the singly doped case, considering the high probability for significant overlap of large peaks, especially in the R···O (R = lanthanide) region. This might indicate that the ionic size differences and variation associated with the data sets could play a subtle, but nevertheless distinct, role in the predictive power of this PCA methodology. PDF profiles of the La/Nd codoped REPGs were not predicted, as singly doped Nd spectra were unavailable as subject data sets. III.A.3. Resolving Pair Distribution Function Peak Overlap. PCA can also be used to resolve atomic pairwise correlations

accurately from the translational information contained within the t(r) fraction of the singly doped La and Pr REPG diffraction data. PDF profiles were initially predicted for the eight singly doped REPGs. In each case, a PDF profile was predicted across a narrow PDF range of r (1.92−2.97 Å; Table 4), where the structural information about the R···O separation predominates, as well as for a wider range (1−4 Å, Table 5), which contains all experimentally discernible atomic pairwise correlations. For the narrow range, the PCA results based on two training spectra already provide a highly accurate performance (mean error = 0.007). In fact, adding further spectra to the singly doped REPG problem appears to yield a worse fit. This result may be ascribed to the fact that the PCA algorithm is implicitly dependent upon training data, whose rare-earth metals exhibit the closest ionic radii, while the set of singly doped REPGs span most of the rare-earth metals. In this context, it is worth noting that for the prediction of the PDF profile of (Sm2O3)x(P2O5)1−x (x = 0.169 or 0.205), Pr and Gd were used as the closest rare-earth neighbors. Accordingly, their cosine distances lean toward larger values. Meanwhile, PDF profile predictions for the wider range of r in singly doped REPGs, where more complex spectral features G


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as it demonstrates its prospective use in automated data mining of PDFs. III.B. Limitations of the Principal Component Analysis for the Elucidation of Structural Information from RareEarth Phosphate Glasses. III.B.1. Inferring the Local Structural Environment of Rare-Earth Metals in Codoped Rare-Earth Phosphate Glasses. R/R′···O separations were expected to be more readily resolved for the Ce/Er, relative to the La/Nd codoped glasses, on account of the more pronounced difference in ionic radii between the respective rare-earth ion pairs and the corresponding effect of the lanthanide contraction. Nevertheless, a direct comparison in PCA-predicted R/R′···O peak positions showed almost identical trends for Ce/Er and La/Nd codoped REPGs (Figure 10).

Figure 8. A section of the t(r) profile for codoped (Ce2O3)0.178(Er2O3)0.043(P2O5)0.779 (r ≈ 2−3 Å).

that manifest as significantly overlapping peaks within a PDF. An example can seen in Figure 8, which shows the t(r) profile for the codoped REPG (Ce2O3)0.178(Er2O3)0.043(P2O5)0.779. There, the overlap between the modeled peaks at r ≈ 2.3−2.6 Å, which represent the Ce···O and Er···O separations, is particularly pronounced, and the Er···O peak is dwarfed by the dominating Ce···O peak. An extraction of accurate structural information is thus very difficult. When PCA was applied to the PDF in the range that primarily corresponds to the R/R′···O separations (r = 2.1−2.8 Å), the highest variance component (Figure 9) evidences the presence of two peaks, whose relative heights vary with the relative proportion of Ce and Er. At high values of r and in subsets with higher proportions of Er, perturbations are observed that arise from the Er···O peak being obscured by the one corresponding to the O···(Er)···O separation. When sorted by increasing proportion of Er (%) within each subset, the contribution of the dominant PCA feature vector generally decreases in x and in z: Er = 19%: (11.2, 0.1, −11.1); Er = 49%: (9.0, −2.0, −7.0); Er = 59%: (5.2, 0.2, −5.5); Er = 80%: (7, 0.8, −7.9). Even though the eigenvalues of the y-component of the PCA vector fluctuate, this fluctuation is small, and can thus be neglected. The maxima and minima of the overall PCA components represent the average R/R′···O separations, so those for the highest Er or Ce content, with peaks centered at 2.20 Å (Figure 9, left) and ∼2.45 Å (Figure 9, right), respectively, provide close approximations for the Er···O and Ce···O separations in singly doped REPGs. This deduction could of course be reached by chemical intuition, as has been done previously via data analysis of the experimental data.19 However, the finding that PCA offers an automated way to reach this result is a key step in this study,

Figure 10. PCA-predicted trends for the average R/R′···O peak position for Ce/Er and La/Nd codoped REPGs.

With greater peak overlap, the application of PCA becomes increasingly more difficult, as all variations are correlated via the lanthanide contraction. For example, the dominant PCA component of codoped Ce/Er REPGs in the range of r = 2.9−4.1 Å (Figure 11) only manifested a monotonic t(r) change when sorted by increasing Er content. This trend is most likely due to a contribution of the tail of the O···(P)···O atomic pairwise correlation that progressively impinges upon that of the R/R′···O correlation (toward higher values of r) with increasing concentration of Er. Considering atomic pairwise correlations with progressively higher r values, the form of each of the four PCA-predicted profiles from the Ce/Er codoped REPG samples strongly suggested the presence of an Er-based second-neighbor correlation at r = 3.55 Å (Er···(O)···P), and the presence of a Ce-based one at r = 3.80 Å (Ce···(O)···P). The high atomic numbers of R/R′ and P lead to good t(r) signal for this R/R′···(O)···P correlation; as such, it can be distinguished from other, underlying, correlations in this region of the t(r) profile. However, it was not yet possible to resolve the associated O···(Er)···O and O···(Ce)···O peak profiles centered at r = 3.5−3.8 Å, or the peaks centered at 4.1 Å, which pertain to the

Figure 9. PCA components of codoped Ce/Er REPGs (r = 2.1−2.8 Å) from three PDF subsets of REPGs with increasingly higher proportions of Er (Er = 19%; covariance, cv = 99%; Er = 49%, cv = 92%; Er = 59%, cv = 85%; Er = 80%, cv = 95%). H


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Figure 11. PCA components of codoped Ce/Er REPG (r = 2.9−4.1 Å), produced from subsets of three spectra with increasing proportions of Er (Er = 19%; covariance, cv = 98%; Er = 49%, cv = 96%; Er = 59%, cv = 85%; Er = 80%, cv = 98%).

Figure 12. PCA components (r = 2.0−3.0 Å) of Ce/Er codoped REPGs, obtained from subsets of three profiles from low-energy SRS data with increasing proportions of Er (Er = 19%; covariance, cv = 90%; Er = 49%, cv = 99%; Er = 59%, cv = 96%; Er = 80%, cv = 98%).

Figure 13. PCA components (r = 3.0−4.5 Å) of Ce/Er codoped REPGs, obtained from subsets of three profiles from low-energy SRS data with increasing proportions of Er (Er = 19%; covariance, cv = 75%; Er = 49%, cv = 85%; Er = 59%, cv = 65%; Er = 80%, cv = 75%).

unknown Ce/Er spectrum and its experimental t(r) spectrum using two neighbors in the range of r = 1−4 Å was 0.011 for SRS data (cf. 0.003 for ESRF data). PCA was also applied to the SRS data to evaluate their potential for peak separation within the ranges of r = 2.0−3.0 and 3.0−4.5 Å in a similar fashion as previously discussed for the ESRF data. The respective PCA component variations are shown in Figures 12 and 13, which can be compared to the analogous PCA results for the ESRF data (Figures 9 and 11). The essential features applying to the peak separation are the same between high- and low-energy synchrotron data, even though SRS data exhibit markedly more disorder and are much less refined, especially in the region of r = 3.0−4.5 Å and beyond, where high levels of peak overlap occur. These results suggest that the t(r) profiles obtained from low-energy sources are significantly compromised with respect to their information content via the Fourier series termination errors. These are associated with their heavily restricted Q-range, used to generate the i(Q) function, together with significant levels of intrinsic data noise. All of these comparison types consistently show that PCA analysis is sufficiently sensitive to the intrinsic energy resolution of synchrotron data; that is, high-energy synchrotron source selection is a justifiable criterion for data requiring accurate findings from PCA, and the reliability of results from different synchrotron sources should be normalized accordingly. III.B.3. Limiting Factors for Data Processing and Sample Definition. The identification of lanthanides and the prediction

third-neighbor Er···(OP)···O and Ce···(OP)···O radial distributions. Their indistinguishable nature is presumably a result of two salient factors: first, relative to the first-neighbor (direct) R/R′···O correlation, the characteristics of these second- and third-neighbor atomic pairwise correlations are intrinsically less well-defined, on account of the larger standard deviation that results from higher degrees of angular freedom in the ···(R/R′)··· or ···(OP)··· chains. Second, with increasing r, progressively more atomic pairwise correlations stack upon each other, as simply more atomic correlations are possible at a separation r as a function of increasing radial distribution. The Ce···(OP)···O correlation is, in cases where low Ce proportions exist, additionally complicated by a perturbation at ∼4 Å, which corresponds to the tail of the Ce···(O)···P pairwise correlation. III.B.2. Quality Control of the Source Data. To assess the level of statistical quality required for the t(r) data to generate reliable PCA studies, a comparative PCA investigation was undertaken. This employed cognate pairs of data sets for La/Nd and Ce/Er codoped REPGs, whereby one was produced from the high-energy (high-resolution) ESRF synchrotron, and one emanated from the low-energy (low-resolution) SRS synchrotron source. The same sample batches were used to acquire both sets of data. In comparison to the aforementioned PCA analysis, which was conducted on the ERSF data, the mean error in the predicted relative Ce/Er composition for SRS data was 18% (cf. 15% for ESRF data), while the mean cosine distance between the predicted form of an I


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PCA is able to successfully distinguish overlapping atomic pairwise correlations that involve different rare-earth metal ions in codoped REPGs. For completion, we also assessed the practical limitations of this PCA analysis in its current state, especially with respect to the structural richness of a t(r) profile, the intrinsic experimental resolution of the source data, the effects of downstream data reduction processes that produce t(r) profiles from the raw experimental spectra, and material uncertainties within a sample from which experimental data were drawn. Overall, this case study on REPGs showed that the application of PCA to PDFs provides good predictive power, which should lay the foundation for applying PCA to a wider range of amorphous materials, and for developing PCA routines in a more automated fashion. Thus, PCA may become useful in large-scale data-mining efforts that could enable computationally generated predictions of structural information about a wide variety of inorganic glasses from sources of existing experimental data. Such predictions should be of particular value for cases where materials do not yet exist, or where the extraction of certain structural attributes lies beyond the capabilities of current experimental data analysis. While there is naturally a range of machinelearning algorithms that could be used to probe structural information in t(r) functions, note that PCA can be readily applied to large data volumes, as it represents an intrinsically simple statistical inference method.

of PDF profile signatures of REPGs with unknown composition are most likely limited by the quality of data processing and the intrinsic nature of the samples themselves. To this end, the scattering contributions are modeled through iterative refinement,33 which is prone to systematic errors; and the glasses may be subject to natural variations of the composition. However, it has been reported that minor contaminations of REPGs with, for example, 1−2 wt % aluminum ions do not compromise the structural integrity of the glasses significantly.25 Finally, the REPG compositions vary throughout the series of lanthanide samples; for example, singly doped REPGs exhibit a proportional range of 0.169−0.263 (56% variation), whereas codoped REPGs show only a smaller range (0.219− 0.237 for Ce/Er, 8% variation; 0.233−0.270 for La/Nd, 16% variation). While this variation is, in principle, factored into the mathematical derivation of the PDF that provides t(r), secondary structural effects such as clustering, which are associated with higher levels of lanthanide doping, may also occur;31 these are not naturally accounted for in the function derivation. The combined errors are reflected, for example, in the large compositional variation between (Sm2O3)x(P2O5)1−x (x = 0.169 or 0.205).33 The cosine distance for these two samples is 0.010, and their raw peak position, at r ≈ 2.30 Å, differs by 0.04 Å. These error values should accordingly be considered benchmarks for the evaluation of cosine distances and peak positions.

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IV. CONCLUSIONS Several PCA tools were developed to identify, evaluate, and predict structural trends in diffraction data for phosphate glasses containing rare-earth metal ions (REPGs). Initially, PCA routines were used to identify good indicators of structural information, whereby the PDF t(r) was identified as the appropriate vehicle for such a data analysis. The proof-ofprinciple that PCA may indeed be applied successfully in such data analysis methods was obtained in the form of initial tests that were able to reproduce the well-known lanthanide contraction accurately. With this validation in hand, we proceeded to develop the predictive power of these PCA methods, to explore what level of new structural information could be extracted from the statistical inference on amorphous materials, represented in this case study by REPGs. When experimentally determined t(r) functions of codoped REPGs were sourced as training and subject test data, PCA was able to successfully predict the relative proportions of rare-earth metal ions, R/R′, within several series of codoped REPGs. It was furthermore possible to predict certain structural features in a PDF profile (specified by a range of investigation, r) of these amorphous material data sets. One of these ranges (r = 1.92−2.97 Å) enveloped a PDF peak profile that corresponds to the R/R′···O atomic pairwise correlation: a Gaussian peak centered at r = 2.3−2.5 Å, depending on the nature of the rareearth metal ion. Another selected range (r = 1−4 Å) covers the full sweep of structural information that could be discerned from well-established routines of experimental data analysis. On the basis of the cosine distance, which was used as the profile similarity figure-of-merit, our results show that PCA affords good predictive power. Moreover, PCA could be successfully employed to resolve peak overlap in structural models of the t(r) function, which feature a number of closely spaced PDF peak profiles that represent distinct structural signatures of increasingly similar atomic pairwise correlations. We demonstrated that

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b00907. Prerequisites for the data analysis, basic checks for the statistical validation of the results from routine data analysis, and MATLAB code examples for the PCA analysis (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.M.C. and M.C.P. acknowledge the EPSRC Collaborative Computational Project, CCP9, (Grant No. EP/J010057/1) for funding. J.M.C. is also indebted to the Fulbright Commission for a U.K.−U.S. Fulbright Scholar Award, and to Argonne National Laboratory, where work done was supported by the Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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REFERENCES

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Modeling Pair Distribution Functions of Rare-Earth Phosphate

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