"object": "<script>\nvar beaker = bkHelper.getBeakerObject().beakerObj;\n</script>\n<style type=\"text/css\">\n/*!\n * Nomad Beaker Notebook Template\n *\n * @copyright Copyright 2017 Fritz Haber Institute of the Max Planck Society,\n * Benjamin Regler - Apache 2.0 License\n * @license http://www.apache.org/licenses/LICENSE-2.0\n * @author Benjamin Regler\n * @version 1.0.0\n *\n * Licensed under the Apache License, Version 2.0 (the \"License\");\n * you may not use this file except in compliance with the License.\n * You may obtain a copy of the License at\n * \n * http://www.apache.org/licenses/LICENSE-2.0\n *\n * Unless required by applicable law or agreed to in writing, software\n * distributed under the License is distributed on an \"AS IS\" BASIS,\n * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n * See the License for the specific language governing permissions and\n * limitations under the License.\n */\np{margin-bottom:1.3em}h1,h2,h3,h4{margin:1.414em 0 .5em;font-weight:inherit;line-height:1.2}h1{margin-top:0;font-size:3.998em}h2{font-size:2.827em}h3{font-size:1.999em}h4{font-size:1.414em}.font_small,small{font-size:.707em}.notebook-container{font-size:16px}.notebook-container .bkr{font-size:100%;font-weight:400;line-height:1.45;color:#333}.nomad--header h2{color:#20335d;font-weight:700;margin:0 0 .2em}.nomad--header h3{color:#20335d;font-weight:700;margin-top:0;text-indent:-1em;padding-left:1em}.nomad--header h3:before{content:\"\\2014\";padding-right:.25em}.nomad--header .nomad--description{margin:-1em 0 0 2em}.atomic-data--block,.nomad--last-updated{display:inline-block;margin-top:1em}.nomad--last-updated{color:grey;float:right;position:relative;z-index:1}.nomad--last-updated::before{bottom:-75%;content:attr(data-version);font-size:4em;font-weight:700;opacity:.2;position:absolute;right:0}.atomic-data label{display:block;font-size:medium;font-weight:700}.atomic-data--select,.chosen-container{width:100%!important}.atomic-data--select:disabled{color:#d3d3d3}.atomic-data--reset-buton{display:inline-block;margin-top:1.6em;width:100%}.modal-dialog{max-width:1000px;width:80%}.modal-header h1{font-size:2em;line-height:1.2}.modal-dialog h2{font-size:1.414em}.modal-dialog h2:first-child{margin-top:0}.modal-dialog h3{font-size:1.2em}.modal-dialog dt{font-size:larger;margin-top:1.414em}.modal-dialog img{width:100%}.modal-dialog .authors{text-transform:uppercase}\n</style>\n\n<div id=\"teaser\" style=\"background-color: rgba(149,170,79, 1.0); background-position: right center; background-size: 200px; background-repeat: no-repeat; \n padding-top: 20px;\n padding-right: 10px;\n padding-bottom: 50px;\n padding-left: 80px;\"> \n\n <div class=\"nomad--header\">\n <div style=\"text-align:center\">\n <h2> <img id=\"nomad\" src=\"https://nomad-coe.eu/uploads/nomad/images/NOMAD_Logo2.png\" height=\"100\" alt=\"NOMAD Logo\"> NOMAD Analytics Toolkit \n <img id=\"nomad\" src=\"https://www.nomad-coe.eu/uploads/nomad/backgrounds/head_big-data_analytics_2.png\" height=\"80\" alt=\"NOMAD Logo\"> </h2>\n </div>\n <h3>Predicting the stability of perovskite oxides and halides using a new tolerance factor</h3>\n <p class=\"nomad--description\">\n created by:\n Christopher Bartel<sup>1</sup> (<a href=\"mailto:christopher.bartel@colorado.edu\">email</a>),\n Christopher Sutton<sup>2</sup> (<a href=\"mailto:sutton@fhi-berlin.mpg.de\">email</a>)\n <br><br>\n \n <sup>1</sup> University of Colorado Boulder, 3415 Colorado Ave., Boulder, CO, USA <br>\n <sup>2</sup> Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany <br>\n \n <span class=\"nomad--last-updated\" data-version=\"v1.0.0\">[Last updated: October 16, 2017]</span>\n </p>\n</div>\n\n</div> \n\n<div style=\"text-align: right;\">\n<a href=\"https://analytics-toolkit.nomad-coe.eu/home/\" class=\"btn btn-primary\" style=\"font-size:larger;\">Back to Analytics Home</a> \n<a href=\"https://www.nomad-coe.eu/\" class=\"btn btn-primary\" style=\"font-size:larger;\">Back to nomad-coe</a> \n</div>\n"
"object": "<script>\nvar beaker = bkHelper.getBeakerObject().beakerObj;\n</script>\n<style type=\"text/css\">\n/*!\n * Nomad Beaker Notebook Template\n *\n * @copyright Copyright 2017 Fritz Haber Institute of the Max Planck Society,\n * Benjamin Regler - Apache 2.0 License\n * @license http://www.apache.org/licenses/LICENSE-2.0\n * @author Benjamin Regler\n * @version 1.0.0\n *\n * Licensed under the Apache License, Version 2.0 (the \"License\");\n * you may not use this file except in compliance with the License.\n * You may obtain a copy of the License at\n * \n * http://www.apache.org/licenses/LICENSE-2.0\n *\n * Unless required by applicable law or agreed to in writing, software\n * distributed under the License is distributed on an \"AS IS\" BASIS,\n * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\n * See the License for the specific language governing permissions and\n * limitations under the License.\n */\np{margin-bottom:1.3em}h1,h2,h3,h4{margin:1.414em 0 .5em;font-weight:inherit;line-height:1.2}h1{margin-top:0;font-size:3.998em}h2{font-size:2.827em}h3{font-size:1.999em}h4{font-size:1.414em}.font_small,small{font-size:.707em}.notebook-container{font-size:16px}.notebook-container .bkr{font-size:100%;font-weight:400;line-height:1.45;color:#333}.nomad--header h2{color:#20335d;font-weight:700;margin:0 0 .2em}.nomad--header h3{color:#20335d;font-weight:700;margin-top:0;text-indent:-1em;padding-left:1em}.nomad--header h3:before{content:\"\\2014\";padding-right:.25em}.nomad--header .nomad--description{margin:-1em 0 0 2em}.atomic-data--block,.nomad--last-updated{display:inline-block;margin-top:1em}.nomad--last-updated{color:grey;float:right;position:relative;z-index:1}.nomad--last-updated::before{bottom:-75%;content:attr(data-version);font-size:4em;font-weight:700;opacity:.2;position:absolute;right:0}.atomic-data label{display:block;font-size:medium;font-weight:700}.atomic-data--select,.chosen-container{width:100%!important}.atomic-data--select:disabled{color:#d3d3d3}.atomic-data--reset-buton{display:inline-block;margin-top:1.6em;width:100%}.modal-dialog{max-width:1000px;width:80%}.modal-header h1{font-size:2em;line-height:1.2}.modal-dialog h2{font-size:1.414em}.modal-dialog h2:first-child{margin-top:0}.modal-dialog h3{font-size:1.2em}.modal-dialog dt{font-size:larger;margin-top:1.414em}.modal-dialog img{width:100%}.modal-dialog .authors{text-transform:uppercase}\n</style>\n\n<div id=\"teaser\" style=\"background-color: rgba(149,170,79, 1.0); background-position: right center; background-size: 200px; background-repeat: no-repeat; \n padding-top: 20px;\n padding-right: 10px;\n padding-bottom: 50px;\n padding-left: 80px;\"> \n\n <div class=\"nomad--header\">\n <div style=\"text-align:center\">\n <h2> <img id=\"nomad\" src=\"https://nomad-coe.eu/uploads/nomad/images/NOMAD_Logo2.png\" alt=\"NOMAD Logo\" height=\"100\"> NOMAD Analytics Toolkit \n <img id=\"nomad\" src=\"https://www.nomad-coe.eu/uploads/nomad/backgrounds/head_big-data_analytics_2.png\" alt=\"NOMAD Logo\" height=\"80\"> </h2>\n </div>\n <h3>Predicting the stability of perovskite oxides and halides using a new tolerance factor</h3>\n <p class=\"nomad--description\">\n created by:\n Christopher Bartel<sup>1</sup> (<a href=\"mailto:christopher.bartel@colorado.edu\">email</a>),\n Christopher Sutton<sup>2</sup> (<a href=\"mailto:sutton@fhi-berlin.mpg.de\">email</a>)\n <br><br>\n \n <sup>1</sup> University of Colorado Boulder, 3415 Colorado Ave., Boulder, CO, USA <br>\n <sup>2</sup> Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany <br>\n \n <span class=\"nomad--last-updated\" data-version=\"v1.0.0\">[Last updated: October 16, 2017]</span>\n </p>\n</div>\n\n</div> \n\n<div style=\"text-align: right;\">\n<a href=\"https://analytics-toolkit.nomad-coe.eu/home/\" class=\"btn btn-primary\" style=\"font-size:larger;\">Back to Analytics Home</a> \n<a href=\"https://www.nomad-coe.eu/\" class=\"btn btn-primary\" style=\"font-size:larger;\">Back to nomad-coe</a> \n</div>\n"
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"object": "<script>\nvar beaker = bkHelper.getBeakerObject().beakerObj;\n</script>\n<legend>Descriptor for Perovskite Stability</legend>\n\n<img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> is a descriptor that takes as input the chemical composition <img src=\"http://latex.codecogs.com/svg.latex?(A,B,X)\" border=\"0\"> and outputs a prediction of perovskite stability according to the formula:\n\n<p> <img src=\"http://latex.codecogs.com/svg.latex?\\tau =\\frac{r_X}{r_B} - n_A(n_A - \\frac{\\frac{r_A}{r_B}}{ln(\\frac{r_A}{r_B})}),\" border=\"0\"><br>\n \n where <img src=\"http://latex.codecogs.com/svg.latex?r_i\" border=\"0\"> is the ionic radius of ion, <img src=\"http://latex.codecogs.com/svg.latex?i\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?n_i\" border=\"0\"> is the oxidation state of ion, <img src=\"http://latex.codecogs.com/svg.latex?i\" border=\"0\">, and <img src=\"http://latex.codecogs.com/svg.latex?\\tau<4.18\" border=\"0\"><!--4--> indicates stability in the perovskite structure.<br>\n \n</p><p> This descriptor was identified by applying the SISSO algorithm developed by R. Ouyang, S. Curtarolo, E. Ahmetick, M. Scheffler, L. Ghiringhelli: Phys. Rev. Materials 2, 083802 (2018) [<a href=\"https://journals.aps.org/prmaterials/pdf/10.1103/PhysRevMaterials.2.083802\">PDF</a>] {<a href=\"https://github.com/rouyang2017/SISSO\">Code</a>} which efficiently identifies <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> from a space of ~3,000,000,000 potential descriptors. <br><br>\n \n <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> requires the same information as Goldschmidt's famous tolerance factor (noting that <img src=\"http://latex.codecogs.com/svg.latex?r\" border=\"0\"> is an implict function of <img src=\"http://latex.codecogs.com/svg.latex?n\" border=\"0\">): <br><img src=\"http://latex.codecogs.com/svg.latex?t =\\frac{r_A+r_X}{\\sqrt{2}(r_B+r_X)}.\" border=\"0\"><br><br> \n \n While the functional forms are comparable, the accuracies are not. On a set of 576 <img src=\"http://latex.codecogs.com/svg.latex?ABX_3\" border=\"0\"> compounds characterized experimentally at ambient conditions, <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> achieves 92% accuracy in predicting whether the compound will or won't be stable as perovskite compared with 74% using <img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\">. <br><br>\n \n <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> is probabilistic, providing not only whether a given composition will crystallize as perovskite but also a probability on this prediction. \n \n Below you can input two cations - <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\"> - and one anion - <img src=\"http://latex.codecogs.com/svg.latex?X\" border=\"0\"> - and the utility will automatically assign oxidation states and radii to each ion (more on this below) and provide the probability that the <img src=\"http://latex.codecogs.com/svg.latex?ABX_3\" border=\"0\"> will form the perovskite structure. <img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\"> is also provided for context.<br><br>\n \n This result is visualized with respect to the cationic radii to show where in the space of stable and unstable perovskites the given composition sits. <br><br>\n \n The descriptor also generalizes to double perovskites - compounds with substitutions at the <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\">, <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\">, or <img src=\"http://latex.codecogs.com/svg.latex?X\" border=\"0\"> sites. Below you can explore the stability of compounds with 50/50 mixtures of ions on each or all of the sites - i.e., <img src=\"http://latex.codecogs.com/svg.latex?(AA')(BB')(XX')_6\" border=\"0\"> formulas. <br><br>\n \n More details on the identification and application of <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> are available within the <a href=\"https://arxiv.org/abs/1801.07700\">manuscript</a> and associated <a href=\"https://github.com/CJBartel/perovskite-stability\">github repository</a>.\n\n</p>"
"object": "<script>\nvar beaker = bkHelper.getBeakerObject().beakerObj;\n</script>\n<legend>Descriptor for Perovskite Stability</legend>\n\n<img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> is a descriptor that takes as input the chemical composition <img src=\"http://latex.codecogs.com/svg.latex?(A,B,X)\" border=\"0\"> and outputs a prediction of perovskite stability according to the formula:\n\n<p> <img src=\"http://latex.codecogs.com/svg.latex?\\tau =\\frac{r_X}{r_B} - n_A(n_A - \\frac{\\frac{r_A}{r_B}}{ln(\\frac{r_A}{r_B})}),\" border=\"0\"><br>\n \n where <img src=\"http://latex.codecogs.com/svg.latex?r_i\" border=\"0\"> is the ionic radius of ion, <img src=\"http://latex.codecogs.com/svg.latex?i\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?n_i\" border=\"0\"> is the oxidation state of ion, <img src=\"http://latex.codecogs.com/svg.latex?i\" border=\"0\">, and <img src=\"http://latex.codecogs.com/svg.latex?\\tau<4.18\" border=\"0\"><!--4--> indicates stability in the perovskite structure.<br>\n \n</p><p> This descriptor was identified by applying the SISSO algorithm developed by R. Ouyang, S. Curtarolo, E. Ahmetick, M. Scheffler, L. Ghiringhelli: Phys. Rev. Materials 2, 083802 (2018) [<a href=\"https://journals.aps.org/prmaterials/pdf/10.1103/PhysRevMaterials.2.083802\">PDF</a>] {<a href=\"https://github.com/rouyang2017/SISSO\">Code</a>} which efficiently identifies <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> from a space of ~3,000,000,000 potential descriptors. <br><br>\n \n <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> requires the same information as Goldschmidt's famous tolerance factor (noting that <img src=\"http://latex.codecogs.com/svg.latex?r\" border=\"0\"> is an implict function of <img src=\"http://latex.codecogs.com/svg.latex?n\" border=\"0\">): <br><img src=\"http://latex.codecogs.com/svg.latex?t =\\frac{r_A+r_X}{\\sqrt{2}(r_B+r_X)}.\" border=\"0\"><br><br> \n \n While the functional forms are comparable, the accuracies are not. On a set of 576 <img src=\"http://latex.codecogs.com/svg.latex?ABX_3\" border=\"0\"> compounds characterized experimentally at ambient conditions, <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> achieves 92% accuracy in predicting whether the compound will or won't be stable as perovskite compared with 74% using <img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\">. <br><br>\n \n <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> is probabilistic, providing not only whether a given composition will crystallize as perovskite but also a probability on this prediction. \n \n Below you can input two cations - <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\"> - and one anion - <img src=\"http://latex.codecogs.com/svg.latex?X\" border=\"0\"> - and the utility will automatically assign oxidation states and radii to each ion (more on this below) and provide the probability that the <img src=\"http://latex.codecogs.com/svg.latex?ABX_3\" border=\"0\"> will form the perovskite structure. <img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\"> is also provided for context.<br><br>\n \n This result is visualized with respect to the cationic radii to show where in the space of stable and unstable perovskites the given composition sits. <br><br>\n \n The descriptor also generalizes to double perovskites - compounds with substitutions at the <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\">, <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\">, or <img src=\"http://latex.codecogs.com/svg.latex?X\" border=\"0\"> sites. Below you can explore the stability of compounds with 50/50 mixtures of ions on each or all of the sites - i.e., <img src=\"http://latex.codecogs.com/svg.latex?(AA')(BB')(XX')_6\" border=\"0\"> formulas. <br><br>\n \n More details on the identification and application of <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> are available within the <a href=\"https://arxiv.org/abs/1801.07700\">manuscript</a> and associated <a href=\"https://github.com/CJBartel/perovskite-stability\">github repository</a>.\n\n</p>"
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"object": "<script>\nvar beaker = bkHelper.getBeakerObject().beakerObj;\n</script>\n<legend>On assigning oxidation states and radii</legend>\n\n<img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> are both functions of the site-specific radii, <img src=\"http://latex.codecogs.com/svg.latex?r_i\" border=\"0\">, yet it is not known <i>a priori</i> for a new formula which cation will sit on the 12-fold coordinated <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\">-site and 6-fold coordinted <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\">-site. To address this, we developed a systematic approach for making this determination based on the condition that <img src=\"http://latex.codecogs.com/svg.latex?r_A > r_B\" border=\"0\"> because of the larger coordination number in the perovskite structure.\n\nIonic radii have been developed by a number of researchers in the last 100 years, but R.D. Shannon's set [<a href=\"http://scripts.iucr.org/cgi-bin/paper?S0567739476001551\">link</a>] are regarded as the most comprehensive, provided as a function of both oxidation state, <img src=\"http://latex.codecogs.com/svg.latex?n_i\" ,=\"\" border=\"0\"> and coordination number, <img src=\"http://latex.codecogs.com/svg.latex?CN_i\" border=\"0\">: <img src=\"http://latex.codecogs.com/svg.latex?r_i = f(n_i, CN_i)\" border=\"0\">.\n<br><br>\nGiven a new formula, <img src=\"http://latex.codecogs.com/svg.latex?CC'X_3\" border=\"0\"> where <img src=\"http://latex.codecogs.com/svg.latex?C\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?C'\" border=\"0\"> are cations and we don't yet know which is <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\"> and which is <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\">, we apply the following scheme:\n\n<br>\n1) A list of allowed <img src=\"http://latex.codecogs.com/svg.latex?n\" border=\"0\"> is defined for <img src=\"http://latex.codecogs.com/svg.latex?C\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?C'\" border=\"0\"> based on the set of <img src=\"http://latex.codecogs.com/svg.latex?n_i\" border=\"0\"> such that <img src=\"http://latex.codecogs.com/svg.latex?r_i(n_i)\" border=\"0\"> exists within Shannon's data.\n\n<br>2) All pairs of oxidation states <img src=\"http://latex.codecogs.com/svg.latex?(n_C, n_{C'})\" border=\"0\"> that charge-balance <img src=\"http://latex.codecogs.com/svg.latex?X_3\" border=\"0\"> are considered, where <img src=\"http://latex.codecogs.com/svg.latex?n_X\" border=\"0\"> is typically known <i>a priori</i> (<i>e.g.</i>, <img src=\"http://latex.codecogs.com/svg.latex?n_O = 2^-\" border=\"0\">). \n\n<br>3) In the infrequent case where more than one charge-balanced pair exists, a single solution is chosen based on the electronegativity ratio of the two cations, <img src=\"http://latex.codecogs.com/svg.latex?\\chi_C/\\chi_{C'}\" border=\"0\">. If <img src=\"http://latex.codecogs.com/svg.latex?0.9 < \\chi_C/\\chi_{C'} < 1.1\" border=\"0\">, the pair that minimizes <img src=\"http://latex.codecogs.com/svg.latex?|n_C-n_{C'}|\" border=\"0\"> is chosen, otherwise, the pair that maximizes <img src=\"http://latex.codecogs.com/svg.latex?|n_C-n_{C'}|\" border=\"0\"> is chosen. \n\n<br>4) With <img src=\"http://latex.codecogs.com/svg.latex?(n_C, n_{C'})\" border=\"0\"> determined, the radii of each cation if they were to sit on <img src=\"http://latex.codecogs.com/svg.latex?A\\/(CN = 12)\" border=\"0\"> or <img src=\"http://latex.codecogs.com/svg.latex?B\\/(CN = 6)\" border=\"0\">is generated using Shannon's table.\n\n<br>5) The determination of which cation is larger, and therefore the <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\">-site, is made by systematically comparing these radii.\n\nThis strategy reproduces the assignment of the <img src=\"http://latex.codecogs.com/svg.latex?A\"> and <img src=\"http://latex.codecogs.com/svg.latex?B\"> cations for 100% of the 313 experimentally labeled <img src=\"http://latex.codecogs.com/svg.latex?ABX_3\" border=\"0\"> perovskites in the set of 576 used to determine <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\">. Conveniently, this approach naturally yields <img src=\"http://latex.codecogs.com/svg.latex?{n_A,n_B,n_X,r_A,r_B,r_X}\" border=\"0\"> which are the inputs to <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\">."
"object": "<script>\nvar beaker = bkHelper.getBeakerObject().beakerObj;\n</script>\n<legend>On assigning oxidation states and radii</legend>\n\n<img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> are both functions of the site-specific radii, <img src=\"http://latex.codecogs.com/svg.latex?r_i\" border=\"0\">, yet it is not known <i>a priori</i> for a new formula which cation will sit on the 12-fold coordinated <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\">-site and 6-fold coordinted <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\">-site. To address this, we developed a systematic approach for making this determination based on the condition that <img src=\"http://latex.codecogs.com/svg.latex?r_A > r_B\" border=\"0\"> because of the larger coordination number in the perovskite structure.\n\nIonic radii have been developed by a number of researchers in the last 100 years, but R.D. Shannon's set [<a href=\"http://scripts.iucr.org/cgi-bin/paper?S0567739476001551\">link</a>] are regarded as the most comprehensive, provided as a function of both oxidation state, <img src=\"http://latex.codecogs.com/svg.latex?n_i\" ,=\"\" border=\"0\"> and coordination number, <img src=\"http://latex.codecogs.com/svg.latex?CN_i\" border=\"0\">: <img src=\"http://latex.codecogs.com/svg.latex?r_i = f(n_i, CN_i)\" border=\"0\">.\n<br><br>\nGiven a new formula, <img src=\"http://latex.codecogs.com/svg.latex?CC'X_3\" border=\"0\"> where <img src=\"http://latex.codecogs.com/svg.latex?C\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?C'\" border=\"0\"> are cations and we don't yet know which is <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\"> and which is <img src=\"http://latex.codecogs.com/svg.latex?B\" border=\"0\">, we apply the following scheme:\n\n<br>\n1) A list of allowed <img src=\"http://latex.codecogs.com/svg.latex?n\" border=\"0\"> is defined for <img src=\"http://latex.codecogs.com/svg.latex?C\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?C'\" border=\"0\"> based on the set of <img src=\"http://latex.codecogs.com/svg.latex?n_i\" border=\"0\"> such that <img src=\"http://latex.codecogs.com/svg.latex?r_i(n_i)\" border=\"0\"> exists within Shannon's data.\n\n<br>2) All pairs of oxidation states <img src=\"http://latex.codecogs.com/svg.latex?(n_C, n_{C'})\" border=\"0\"> that charge-balance <img src=\"http://latex.codecogs.com/svg.latex?X_3\" border=\"0\"> are considered, where <img src=\"http://latex.codecogs.com/svg.latex?n_X\" border=\"0\"> is typically known <i>a priori</i> (<i>e.g.</i>, <img src=\"http://latex.codecogs.com/svg.latex?n_O = 2^-\" border=\"0\">). \n\n<br>3) In the infrequent case where more than one charge-balanced pair exists, a single solution is chosen based on the electronegativity ratio of the two cations, <img src=\"http://latex.codecogs.com/svg.latex?\\chi_C/\\chi_{C'}\" border=\"0\">. If <img src=\"http://latex.codecogs.com/svg.latex?0.9 < \\chi_C/\\chi_{C'} < 1.1\" border=\"0\">, the pair that minimizes <img src=\"http://latex.codecogs.com/svg.latex?|n_C-n_{C'}|\" border=\"0\"> is chosen, otherwise, the pair that maximizes <img src=\"http://latex.codecogs.com/svg.latex?|n_C-n_{C'}|\" border=\"0\"> is chosen. \n\n<br>4) With <img src=\"http://latex.codecogs.com/svg.latex?(n_C, n_{C'})\" border=\"0\"> determined, the radii of each cation if they were to sit on <img src=\"http://latex.codecogs.com/svg.latex?A\\/(CN = 12)\" border=\"0\"> or <img src=\"http://latex.codecogs.com/svg.latex?B\\/(CN = 6)\" border=\"0\">is generated using Shannon's table.\n\n<br>5) The determination of which cation is larger, and therefore the <img src=\"http://latex.codecogs.com/svg.latex?A\" border=\"0\">-site, is made by systematically comparing these radii.\n\nThis strategy reproduces the assignment of the <img src=\"http://latex.codecogs.com/svg.latex?A\"> and <img src=\"http://latex.codecogs.com/svg.latex?B\"> cations for 100% of the 313 experimentally labeled <img src=\"http://latex.codecogs.com/svg.latex?ABX_3\" border=\"0\"> perovskites in the set of 576 used to determine <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\">. Conveniently, this approach naturally yields <img src=\"http://latex.codecogs.com/svg.latex?{n_A,n_B,n_X,r_A,r_B,r_X}\" border=\"0\"> which are the inputs to <img src=\"http://latex.codecogs.com/svg.latex?\\tau\" border=\"0\"> and <img src=\"http://latex.codecogs.com/svg.latex?t\" border=\"0\">."
