Article pubs.acs.org/IC
Some Statistics on Intermetallic Compounds Julia Dshemuchadse and Walter Steurer* Laboratory of Crystallography, Department of Materials, ETH Zurich, Zurich, Switzerland S Supporting Information *
ABSTRACT: It is still largely unknown why intermetallic phases show such a large variety of crystal structures, with unit cell sizes varying between 1 and more than 20 000 atoms. The goal of our study was, therefore, to get a general overview of the symmetries, unit cell sizes, stoichiometries, most frequent structure types, and their stability fields based on the Mendeleev numbers as ordering parameters. A total of 20829 structures crystallizing in 2166 structure types have been studied for this purpose. Thereby, the focus was on a subset of 6441 binary intermetallic compounds, which crystallize in 943 structure types.
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intermetallic compounds.33 Structure maps have also been compiled in more recent years,34−37 and a new way of evaluating AETs and clusters was established by Blatov, Proserpio, and co-workers by developing the software TOPOS38,39 and demonstrating its capabilities in the field of structure screening.40−43 Another path toward “high-throughput materials discovery”combining ab initio calculations and data miningis being paved by Hart et al.44−46 The goal of the present study was to contribute to the understanding of intermetallic phases on a metalevel by identifying regularities in the distribution of data contained in the database Pearson’s Crystal Data47 (PCD) from a different angle. In the following, we will focus on the distribution of the structures of intermetallic compounds as a function of the symmetry, unit cell size, most frequent stoichiometries, and structure types, as well as on their stability fields based on Mendeleev numbers as ordering parameters. For that purpose, we had to massively filter the database, as described in the next paragraph, in order to avoid severe artifacts in our statistics by multiple entries for the same phase and other ambiguities.
INTRODUCTION When does a mixture of metallic elements with a specific chemical composition form an intermetallic compound, and which crystal structure will result? What are the main factors governing the stability and formation of a particular crystalline phase at a given temperature and pressure? How are the chemical composition, structure, and physical properties related? Once we can answer all of these questions, we can provide the materials scientist with a toolbox that allows the creation of the best metallic material based on its desired physical properties. For quite a few materials, we already have this understanding, and we know all we want to know about their crystal structures, chemical bonding, and physical properties based on firstprinciples calculations.1−4 Unfortunately, with today’s limitations in computing power, these calculations can only be done for structures of moderate size or employing additional approximations.5,6 Additionally, conceptual problems arise with structures that feature disorder or aperiodicity. It is also possible within some more limits to predict the stability of the crystal structures of intermetallic compounds for given chemical compositions and not too large unit cells.7 However, given the large number of multinary intermetallic systemsmost of them not fully explored and some of them even not at allwe still need to build on statistical data analysis in order to structure the huge amount of data already available. This kind of large-scale analysis was initiated by Pettifor, who plotted and investigated the stability regions of binary compounds of different compositions8−10 and derived a chemical scale based on these findings, which is reflected in the Mendeleev numbers.11−13 Villars et al. picked up this approach14−19also focusing on specific compound compositions20−23and continued to build a path toward a useful materials database.24,25 Villars et al. also investigated ways to extract information from the atomic environment types (AETs) of the respective structures,26−32 as well as the statistics of © XXXX American Chemical Society
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DATA MINING A statistical data analysis can only be as good as the data it is based on. Therefore, we carefully checked the database PCD47 chosen as a basis for our study on intermetallic structures. While the Inorganic Crystal Structure Database (ICSD) contains more than 169800 entries,48 the version of the PCD used in this study (2012/2013) contains a total of 227145 entries for inorganic compounds. Of these, 47192 entries describe intermetallic phases, 46071 of which contain complete structural information. We term those compounds as Special Issue: To Honor the Memory of Prof. John D. Corbett Received: October 7, 2014
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“intermetallics”, which are built solely from elements of groups 1 (except H) and 2−12, as well as from groups 13−18 underneath a zigzag line in the periodic table of elements with Al, Ge, Sb, and Po below this line and B, Si, As, Te, At, etc., above it. There are 81 elements in total. We are fully aware that this is a somewhat arbitrary border; however, there is no official definition by the IUPAC or another official body of metallic elements or intermetallic phases, and we had to decide where to draw the boundary line. The more than 46071 entries in the PCD with complete structural information are not all related to different intermetallic compounds. In many cases, they just refer to structures obtained from different structure analyses carried out on intermetallic compounds with either the same or a slightly different stoichiometry exhibiting the same structure type. Taking this into account, we reduced the data set to 20829 entries, which now largely refer to different intermetallic compounds. One has to keep in mind, however, that this data set still includes a significant number of binary and ternary intermetallic phases that are assigned to unary and binary structure types, respectively, indicating disordered structures and solid solutions. Furthermore, we have to be aware that not all structures were determined by state-of-the-art single-crystal structure analysis; many of them were identified by analogy, i.e., just by comparing their powder X-ray diffraction patterns with those of “known” structures. Also, not only are ground-state structures taken into account, but all kinds of phases reported in the database, including those that are stable at elevated temperature or pressure only or even metastable. Each one of the 2166 structure types in the reduced database is represented by a prototype compound. This compound does not necessarily have to be intermetallic (e.g., cP2-CsCl or hP3AlB2); it usually corresponds to the first compound discovered among the ones that feature the same structure type. Among these 2166 structure types, there are 80 unary, 902 binary, and 1095 ternary compounds, as well as 87 with four and 2 with five components. (The distribution over unique structure types and those with multiple representative compounds is quite even. Of the 1087 unique intermetallic structure types, 46, 436, 547, 56, and 2 have 1, 2, 3, 4, and 5 constituents, respectively. The nonunique structure types, i.e., those with representatives in at least two different intermetallic systems, are 1079 in total, divided into 34, 465, 549, 31, and 0 with 1−5 constitutents, respectively.) For our statistical analysis, we use the Mendeleev numbers M as the ordering parameter.11−13 Mendeleev numbers are assigned to the chemical elements in the following sequence: noble gases (1−6), alkali metals (7−12), alkaline-earth metals (13−16), rare-earth elements (17−33), actinoids (34−48), transition metals (49−72), and metals with full s and d orbitals (73−77, corresponding to B, Mg, Zn, Cd, and Hg), main-group metals and metalloids (78−92), and nonmetals (93−103). In our analysis, all compounds constituted only by the metallic elements (according to our working definition) with numbers M = 7−84, 87, 88, and 91 are included. The Mendeleev numbers correspond to a kind of chemical scale, taking into account the chemical similarity of groups of elements, electronegativity, atomic radii, etc., in an empirical way. Using them as order parameters allows for identification of the stability fields of intermetallics, as has been nicely demonstrated for particular binary structure types.9,10
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RESULTS The overall 20829 compounds in the reduced database crystallize in 2166 different structure types, resulting in an average number of 9.6 representatives per structure type. These numbers amount to 277/86 = 3.2 for the metallic elements, 6441/943 = 6.8 for the binary intermetallics, 13026/1391 = 9.4 for the ternary ones, 973/212 = 4.6 for the quaternary intermetallic compounds, and 112/24 = 4.7 for higher multinary ones. This ratio is with 9.6 for all intermetallics larger than any such number for the individual n-constituent subsystems because one and the same structure type can refer to compounds with different numbers of constituents (solid solutions and disordered structures). For instance, the structure type cF4-Cu can be found not only for unary but also for binary and ternary phases. How can we explain the number of representatives per structure type, which increases with the number of constituents n? If in the case of the structures of unary phases, i.e., the elements, just one kind of atom can be substituted by, on average, 3.2 other ones without changing the structure type, then this number must be lower, consequently, for the mostly more complex structure types of binary and ternary compounds. Indeed, this is the case with the numbers 6.8 = (2.6)2 and 9.4 = (2.1)3, respectively. The latter number means, for instance, that each of the three constituting elements of a ternary structure type can be substituted by, on average, 2.1 atoms of another kind, giving a total of, on average, 9.4 combinations. For multinary compounds with more than three different constituents, the database does not contain enough entries for drawing reliable conclusions. While the number of different phases of unary systems (277) is more than 3 times larger than the number of metallic elements (81) and in the case of binary intermetallics (6441) 2 times larger than the number of binary systems (81 × 80/2 = 3240), it is much smaller in the case of the ternary intermetallic compounds (13 026) than the number of possible ternary systems (81 × 80 × 79/6 = 85 320). In other words, there are, on average, three or four phases per metallic element (277/81 ≈ 3.4), two per binary system (6441/3240 ≈ 2.0), and just one out of six to seven ternary systems features a ternary compound (13026/85320 ≈ 0.15). However, one has to keep in mind that ternary and higher systems have been studied so far to a very small amount only, probably not reflecting the true numbers at all. Furthermore, many multinary systems show more than one multinary compound, quite a few even more than 10. However, this can also be discussed from a different point of view. Only 1401 out of the theoretically possible 3420 binary intermetallic systems feature at least one single binary compound in our data set. In the case of ternary systems, only for 5109 out of the 85320 theoretically possible ones, at least one ternary compound is known so far. This means that, for each binary and ternary intermetallic system forming at least one compound, there are, on average, 4.6 and 2.5 representatives, respectively, which does not differ that much from the factor 3 for the element structures. Symmetries. The distribution of all of the 20829 intermetallic compounds and 2166 structure types over the 14 Bravais lattice types is shown in Table 1. The most common lattice types are primitive hexagonal (hP) and cubic facecentered (cF). Only 5008 structures (24.0%) have low symmetries (a, m, and o). This matches our intuitive assumption of intermetallics being generally “highly-symmetric” B
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Pnma, Cmcm, I4/mmm, R3̅m, P6̅2m, P6/mmm, P63/mcm, P63/ mmc, F4̅3m, Pm3̅m, Fm3̅m, and Fd3̅m, respectively. Among them, the two space groups P6̅2m and F4̅3m are noncentrosymmetric. Unit Cell Sizes. Figure 2 depicts the distribution of unit cell sizes among intermetallic compounds. (The five largest unit-cell structures are omitted from the graph for visibility: tP906Li 28.9 Cu 6.1 Zn 6.1 Al 58.9 , hP1164-Cr 10.7 Fe 8.6 Al 80.8 , hP1192Cr 10.7 Fe 8.7 Al 80.6 , cF5908-Ta 39.5 Cu 3.9 Al 56.6 , and cF23134Ta39.1Cu5.4Al55.4.) Here it becomes apparent that a considerable number of structures are rather complex, as indicated by the large number of atoms in their unit cells. In a previous study,49 we introduced a definition in which we denominate those intermetallics that exhibit more than 100 atoms per primitive unit cell as “complex intermetallics (CIMs)”. Therein, we also specified the fraction of CIMs as approximately 2% of all intermetallic compounds. As one would expect, the frequency of the compounds decreases with increasing unit cell size of their structures. However, taking into account the logarithmic scale, the number of compounds per bin shows quite large fluctuations, breaking the general trend locally. Interestingly, the histogram for the frequency of structure types as a function of the unit cell size (not shown here) shows a distribution similar to that for the number of compounds. The Supporting Information contains separate plots of the distribution of the unit cell sizes among intermetallic compounds for each of the 14 different Bravais lattices. At first glance, the histograms look quite similar. On a closer look, one sees that the distribution of CIMs varies significantly with the symmetries. For the Bravais groups aP, mS, oF, and cI, just a few CIMs or none at all are known, in contrast to the cases of oP, oS, hP, cP, and cF. The clustering of histogram bars around 100 atoms per primitive unit cell into maxima in the cases of cF, cI, cP, tI, hR, and hP is also quite remarkable, possibly indicating cluster-based structures. Returning to the distribution of the structure types of intermetallics, it should be said that especially complex structure types tend to have very few representatives. Structure Types. Most larger structure types have just one or two representatives. The complete distribution of the number of compounds per structure type is shown in Figure 3: the structure types are arranged along the horizontal axis and ordered by decreasing numbers of representatives; the number of compounds representing each structure type is given along the vertical axis. It is amazing that half of all structure types found in intermetallics1087 to be exacthave only one representative compound so far. Also, these structure types are not necessarily the very complex ones. The percentage of CIMs in each bin (equal number of representatives/structure types) is approximately constant, except in the case of the unique structure types. There, the fraction of CIMs is significantly higher, with approximately 16%. The inset shows an equivalent plot for binary intermetallic compounds onlyin total 6441 which exhibit 943 different structure types. The trend is the same as that for the set of all intermetallics. On the left side of the histogram are the most common structure types that are all more or less well-known. They have hundreds of representative compounds and are mostly rather simple (see Table 2). The most complex structure among these is hR57-Th2Zn17 with 19 atoms per primitive rhombohedral unit cell and 57 atoms in the 3 times larger hexagonal unit cell, while the simplest structures among them, cF4-Cu and cI2-W, contain just one atom per primitive unit cell. Here it has to be noted that these twoand otherunary structure types have
Table 1. Distribution of All Intermetallic Compounds and Structure Types Included in the Reduced Database PCD47 over the 14 Bravais Lattice Typesa Bravais lattice aP mP mS oP oS oI oF tP tI hP hR cP cI cF all
no. of compounds 62 231 448 1849 1440 899 79 1433 2166 5154 940 1713 972 3443 20829
no. of structure types 31 108 171 295 247 99 33 173 184 401 141 91 81 111 2166
ave no. of compounds per structure type 2.0 2.1 2.6 6.3 5.8 9.1 2.4 8.3 11.8 12.9 6.7 18.8 12.0 31.0 9.6
a
Lattice types: a, triclinic; m, monoclinic; h, hexagonal; o, orthorhombic; t, tetragonal; c, cubic; P, primitive; S, base-centered; I, body-centered; R, rhombohedrally centered; F, face-centered. In addition to the number of compounds and structure types, also their ratio is given.
because of the isotropy of (idealized) metallic bonding. However, not even all of these high-symmetry structures, let alone the low-symmetric ones, can be regarded as being “simple”, as will be detailed below. The distribution of all of the structures and structure types over the 14 Bravais lattice types, 32 point groups, and 230 space groups is illustrated in Figure 1. It is remarkable that structure types of lower symmetries (a, m, and o) are with 984 (45.4%) entries as frequent as highly symmetric structure types (t, h, and c) with 1182 (54.6%) entries, in contrast to the compounds themselves, where the ratio is approximately 1:3. This means that the less-symmetric structure types have fewer representatives than those of higher symmetry and that they are less flexible against substitution of particular atoms. The fraction of unique structure types is significantly higher for the monoclinic and triclinic Bravais lattice types as well as for the orthorhombic face-centered one. Consequently, this applies also to the distribution over high point-group symmetries, i.e., m3̅m, 6/mmm, and 4/mmm, which have a large number of representatives but stand out much less in the case of structure types. All of the most common point groups (more than 500 representatives each) are centrosymmetric, m3̅m, 6/mmm, 3̅m, 4/mmm, mmm, and 2/m, except for one, 4̅3m. With regard to the distribution over the space groups, the structure-type histogram appears to be much “noisier” than the compounds’ histogram. On the one hand, this is related to their smaller total number; on the other hand, it also means that the ratio of the number of compounds over the number of structure types is larger for the more common space groups. Therefore, the less common space groups occur more frequently relative to the more common space groups of structure types compared with the situation for the distribution of compounds. The most common space groups (with more than 500 compounds or 100 structure types as representatives each) are found around numbers 12, 62, 63, 139, 166, 189, 191, 193, 194, 216, 221, 225, and 227, i.e., space groups C2/m, C
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Figure 1. Distribution of the compounds (left) and structure types (right) of intermetallic compounds over all Bravais lattices, point groups, and space groups. In the plots for all 2166 structure types, the 1087 unique ones are marked in green. The bins correspond to one group in all histograms (groups = Bravais lattice, point group, and space group, respectively).
ones, while in total, the numbers of binary and ternary structure types are quite similar. There are two quite common structure types, hP9-ZrNiAl and cF12-AgMgAs, with noncentrosymmetric symmetry, P6̅2m and F4̅3m, respectively, which can be of interest regarding the existence of certain physical properties. Here it should be mentioned that some of the most frequent structure types have representatives with interesting physical properties. In the case of the Laves phases, for instance, the technologically most important magnetostrictive material is Terfenol D, cF24-Dy1−xTbxFe2 (x ≈ 0.3). Because of its large (positive) magnetostriction constant, it is mainly used as an actuator, a magnetomechanical sensor, and an acoustic and ultrasonic transducer (e.g., for sonar systems). Its structure is that of the cubic Laves phase cF24-Cu2Mg, and the Curie temperature T C = 653 K. The compound cF24-SmFe 2 crystallizes in the same structure type andin contrast to Terfenol Dhas a giant negative magnetostriction constant. The magnetostrictive effect also underlies the Invar effect, i.e., an anomalously low thermal expansion, which has been discovered in a fully disordered cF4-Fe65Ni35 alloy and was
Figure 2. Distribution of the unit cell sizes of intermetallic compounds. The number of atoms per primitive unit cell is plotted for all unique structures. The red curve should serve as a guide for the eyes. The red arrow indicates the threshold value for “CIMs” at 100 atoms per primitive unit cell. The bin size is 1. Note the logarithmic vertical scale.
representatives in multinary systems as well. Such structures are, therefore, intrinsically disordered and are often solid solutions. Among the top 20 structure types, there are only six ternary ones compared to 12 binary structure types and 2 unary D
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shape-memory material because it undergoes a thermomechanical transformation to the martensitic state at Tm = 202 K. hP6-CaCu5 and hR57-Zn17Th2 are the structure types of the hard magnetic materials hP6-SmCo5 and hR57-Sm2Co17, respectively. The latter is a strong permanent magnet that can be used at elevated temperatures as well because of its relatively high Curie temperature TC = 1189 K. The most common structure types among binary intermetallics are given in Table 3. All of them are centrosymmetric. Table 3. 20 Most Common Structure Types of the 943 Identified among Binary Intermetallicsa rank
structure type
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
cF4-Cu cP2-CsCl cP4-Cu3Au hP2-Mg cF24-Cu2Mg cI2-W hP12-MgZn2 hP16-Mn5Si3 hP6-CaCu5 oS8-TlI oP16-Fe3C oP12-Co2Si hP38-Th2Ni17 cP8-Cr3Si oI12-KHg2 hP8-Mg3Cd cF8-NaCl oP36-Sm5Ge4 tP2-AuCu hP6-Co1.75Ge total
Figure 3. Distribution of the number of compounds per structure type over all structure types, ordered by decreasing frequency. The inset shows binary intermetallics only. CIMs are highlighted in red. The bin size is 1 in both histograms.
Table 2. 20 Most Common Structure Types of the 2166 Identified among All Intermetallicsa rank
structure type
space group
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
cF24-Cu2Mg cF4-Cu cP4-Cu3Au cP2-CsCl hP9-ZrNiAl hP12-MgZn2 cF16-MnCu2Al oP12-TiNiSi cI2-W hP6-CaCu5 tI10-CeAl2Ga2 hP2-Mg tI26-ThMn12 oI12-KHg2 hP6-CaIn2 hP16-Mn5Si3 hR57-Th2Zn17 cF12-AgMgAs tP10-Mo2FeB2 hP3-AlB2 total
Fd3̅m Fm3̅m Pm3̅m Pm3̅m P6̅2m P63/mmc Fm3̅m Pnma Im3̅m P6/mmm I4/mmm P63/mmc I4/mmm Imma P63/mmc P63/mcm R3̅m F4̅3m P4/mbm P6/mmm
no. of compounds 806 581 544 512 490 456 414 403 375 366 338 338 294 252 215 209 209 206 203 200 7411
% of compounds 3.9 2.8 2.6 2.5 2.4 2.2 2.0 1.9 1.8 1.8 1.6 1.6 1.4 1.2 1.0 1.0 1.0 1.0 1.0 1.0 35.6
space group Fm3m ̅ Pm3̅m Pm3̅m P63/mmc Fd3̅m Im3̅m P63/mmc P63/mcm P6/mmm Cmcm Pnma Pnma P63/mmc Pm3̅n Imma P63/mmc Fm3̅m Pnma P4/mmm P63/mmc
no. of compounds 385 290 263 248 223 210 154 153 109 99 90 67 67 66 64 64 61 58 53 52 2776
% of compounds 6.0 4.5 4.1 3.9 3.5 3.3 2.4 2.4 1.7 1.5 1.4 1.0 1.0 1.0 1.0 1.0 0.9 0.9 0.8 0.8 43.1
flag s
s s
a
These are all types representing more than 50 compounds each. Note that the sphere-packing structures cF4-Cu, hP2-Mg, and cI2-W are included in this list, although they represent mainly solid solutions or, more generally, inherently disordered and pseudobinary structures. They are marked by an “s” entry (solid solution) in the “flag” column.
Here, the compounds belonging to the structure types cF4-Cu, hP2-Mg, and cI2-W (ranks 1, 4, and 6, respectively) are intrinsically disordered and are therefore marked with an “s”flag in the table. They can generally be regarded as solid solutions of elements in the structure of the other, although this does not have to be the case across the board. There are also cases known where the unary structure type formed by the mixture of two elements does not correspond to the structure type of one of the constituting elements. Table 4 shows a few examples of such phases in binary Cu-containing systems.50 The structure type cP8-Cr3Si has many representatives of superconducting intermetallic compounds, with the highest transition temperature, Tc = 23.2 K, for cP8-Nb3Ge. Laves phases have also been found with relatively high transition temperatures, for instance, cF24-Zr0.5Hf0.5V2 with Tc = 10.1 K. Stoichiometries. Figure 4 shows how the compositions of binary intermetallics are distributed over the range A−AB−B, with the two elements A and B defined by the order of their Mendeleev numbers, M(A) < M(B). Compounds that are assigned to the sphere-packing structure types cF4-Cu, cI2-W, and hP2-Mg were excluded from these statistics. This is due to the fact that binary intermetallics adapting these structure types
a
These are all types representing at least 200 compounds, i.e., 1.0%, summing up to 7411 compounds, i.e., 35.6% of the overall 20829 intermetallic compounds.
later found to exist in ordered compounds as well, e.g., cP4Fe3Pt. The Heusler and half-Heusler phases with the structure types cF16-MnCu2Al and cF12-AgMgAs, respectively, also exhibit interesting physical properties. The half-Heusler compounds cF12-MnPtSb and cF16-MnCo2Ge exhibit large linear and quadratic magnetooptic Kerr effects, for instance. Half-Heusler phases such as cF12-MNiSn (M = Ti, Zr, Hf), which are doped with Sb on the Sn site, can be used as high-temperature thermoelectric materials (up to 1500 K). The ferromagnetic Heusler phase cF16-MnNi2Ga (TC = 376 K) can be used as a E
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structure types at all (nos. 15, 16, 18, 22, and 41) and two compositions that feature ≈90% unique structure types (nos. 42 and 43). Because 28 compositions out of the 43 most frequently occurring binary ones (≈65%) show ratios >0.5, they contain a disproportionately large number of unique structure types compared to the overall ratio of ≈0.5 for all 2166 structure types of the intermetallic phases in the database. Structure Maps. Figure 5 illustrates which combinations of two different metallic elements form compounds at all; additionally those for the most common compositions are shown: A2B, AB, A3B, and A5B3. In the latter four, the most common structure types are highlighted in blue, while CIMs are marked in the general plot in red. The regions of lanthanoidand actinoid-containing compounds are shaded light and dark gray, respectively. One has to keep in mind that these plots are not complete because not all binary phase diagrams, e.g., the actinoid-containing ones, have been fully and reliably explored so far. However, they certainly reflect the general trend properly. In general, MM plots allow one to identify the compositional stability ranges of particular structure types. They allow one to locate so far unexplored intermetallic systems where the probability would be high to find a compound crystallizing in a particular structure type. One can see that there are just a few binary compounds with Mendeleev numbers M(A) < 58 and M(B) < 42, and vice versa. The small gaps at M(A) = M(B) = 13, 29, 48, 85, and 86 as well as the extended gaps for 34 ≤ M ≤ 48 reflect the fact that there are almost no binary compounds of the radioactive elements in the database, which does not necessarily mean that they do not exist. The white regions at M(A) = M(B) = 85, 86, 89, and 90 simply correspond to compounds with the nonmetallic elements Si, B, and As, which have been excluded from our analysis. Generally, the CIMs and most frequent intermetallics show some more or less connected stability fields, which are to a large extent inside the gray-shaded regions; i.e., they contain lanthanoids (Ln) or actinoids (An). The chemical similarity of these elements explains why almost all of them within each of these two groups can replace each other easily. The stability fields of the CIMs are not symmetric along the diagonal of the plot, which means that, for these compounds AmBn, A cannot be replaced by B and vice versa. The binary Ln−Ln phases, all of them solid solutions, have unary structure types that are either those of the allotropes of the constituting elements or frequently the hR9-Sm structure type. The case of An−An compounds is similar, also in some cases with another structure type that has not been identified yet.
Table 4. Sphere-Packing Structures (belonging to Structure Types cF4-Cu, hP2-Mg, or cI2-W) belonging to Binary Intermetallic Systems, with Copper Being One of the Componentsa system
formula
structure type
comment
Cu−Ga Cu−Ge Cu−In Cu−Sb Cu−Sn Cu−Zn Cu−Zn
Cu0.78Ga0.22 Cu0.83Ge0.17 Cu0.8In0.2 Cu0.8Sb0.2 Cu0.86Sn0.14 Cu0.53Zn0.47 Cu0.2Zn0.8
hP2-Mg hP2-Mg cI2-W hP2-Mg cI2-W cI2-W hP2-Mg
≠ oS8-Ga ≠ cF8-C (Ge) ≠ tI2-In ≠ hR6-As (Sb) ≠ tI4-Sn ≠ hP2-Mg (Zn) = hP2-Mg (Zn); disconnected phase region
a
Copper itself crystallizes in the cF4-Cu structure type, while the structure of the pure second element is given in the “comment” column.
have to be intrinsically disordered, are most probably solid solutions, and are not relevant to compound formation at specific compositions. They simply provide a “background noise” to the compound structures. The frequency distribution in Figure 4 is not symmetric around the composition AB. For instance, the stoichiometries A2B and AB2 and also A3B and AB3 show similar frequencies in contrast to A4B and AB4, A6B and AB6, and A9B and AB9 (note the logarithmic scale). This means that elements A and B, with M(A) < M(B), cannot simply exchange their sites in a given structure type because of their significantly different atomic volumes or electronegativities or their different contributions to the valence electron concentration. Typical examples are the Laves phases with more than 200 representatives of the type AB2 and less than 10 of the type A2B. However, not only the frequencies of compounds for intermetallics with stoichiometries AmBn and AnBm but also the densities of different stoichiometries in the histogram differ. For instance, the density of the distribution at the left-hand side of the figure is smaller than that at the right-hand side. Compare, for instance, the densities in the ranges A6B−A4B and AB4−AB6. This means that there are more structure types possible where A atoms have a higher coordination by B atoms than vice versa. The most common 43 binary stoichiometries are listed in Table 5. Here, the pairs of compositions AmBn|AnBm [M(A) < M(B)] are compared with regard to their frequencies. One clearly sees that the number of both structure types and compounds greatly differs for that of the compositions AmBn and AnBm. It is also remarkable how strongly the number of unique structure types varies for these compositions. There are five compositions in Table 5 that do not include unique
Figure 4. Distribution of the compositions of binary intermetallics. Of the 6441 binary intermetallic compounds, all except those exhibiting the three structure types cF4-Cu, cI2-W, and hP2-Mg are depicted5598 compounds in total. The bin size is 0.001. Note the logarithmic scale. F
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Table 5. Top 43 Compositions of Binary Intermetallics, Representing 10 or More Compounds for Each Pair AmBn|AnBma rank
AmBn
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
A2B AB A3B A5B3 A3B2 A5B A17B2 A4B A4B3 A7B3 A5B4 A5B2 A7B2 A6B A13B A23B6 A12B A51B14 A11B10 A58B13 A9B4 A11B A8B3 A7B A45B11 A11B9 A11B3 A8B5 A10B7 A6B5 A24B5 A7B6 A7B4 A9B A16B11 A12B7 A23B4 A17B3 A11B2 A17B4 A19B5 A10B3 A13B7
no. of compounds 268 872 332 294 153 8 0 19 49 74 98 50 6 7 4 1 0 0 34 0 19 1 13 9 0 5 1 4 4 17 2 8 6 6 11 8 1 2 3 8 2 5 4
no. of STs 38 120 46 17 43 6 0 13 10 16 13 10 5 1 1 1 0 0 4 0 9 1 7 6 0 5 1 4 4 10 2 4 2 6 2 4 1 2 3 5 1 4 3
AnBm
no. of compounds
AB2 AB AB3 A3B5 A2B3 AB5 A2B17 AB4 A3B4 A3B7 A4B5 A2B5 A2B7 AB6 AB13 A6B23 AB12 A14B51 A10B11 A13B58 A4B9 AB11 A3B8 AB7 A11B45 A9B11 A3B11 A5B8 A7B10 A5B6 A5B24 A6B7 A4B7 AB9 A11B16 A7B12 A4B23 A3B17 A2B11 A4B17 A5B19 A3B10 A7B13
794 872 493 96 87 174 140 91 68 39 6 34 67 61 45 43 43 35 0 28 7 23 11 14 21 16 19 16 14 1 13 7 8 6 1 3 9 8 7 2 8 5 6
no. of STs 81 120 54 26 39 28 15 36 13 19 5 9 9 13 2 1 3 1 0 5 7 2 8 7 4 10 4 7 5 1 2 3 5 5 1 3 4 4 4 2 2 5 6
no. of compounds 1062 872 825 390 240 182 140 110 117 113 104 84 73 68 49 44 43 35 34 28 26 24 24 23 21 21 20 20 18 18 15 15 14 12 12 11 10 10 10 10 10 10 10
no. of STs 106 120 80 40 73 33 15 47 22 31 17 19 13 14 2 1 3 1 4 5 12 2 15 11 4 14 5 11 9 11 3 6 7 9 3 7 4 6 6 7 3 9 9
no. of uSTs 49 69 34 18 42 23 6 33 14 21 9 9 7 6 0 0 1 0 3 3 5 0 10 7 3 10 4 8 7 7 2 4 5 6 2 6 2 4 3 5 0 8 8
a
The number of compounds and of different structure types (STs) is given for each composition m,n (M(A) < M(B). In addition, the number of unique structure types (uSTs) is given for all compositions.
tions A5B3/A3B5 exhibit 41 different structure types, dominated by the hP16-Mn5Si3 type with 148 compounds.
There are 1062 compounds with compositions A2B/AB2, which crystallize in 106 different structure types. The two most frequent structures are the cubic and hexagonal Laves phases, respectively. Five other structures can be considered as derivatives of the hP3-AlB2 structure type, which itself can be found on rank 10. The 888 compounds with compositions AB exhibit 127 different structure types. A total of 9 of the top 10 structure types can be considered as rather simple close-packed structures or distorted derivative structures. Number 10, the tI64-NaPb structure type, is a Zintl phase, and all of its representatives are Zintl phases as well. The 830 compounds with compositions A3B/AB3 exhibit 82 different structure types, with cP4-Cu 3 Au by far the most frequent one (249 representatives). Finally, the 391 compounds with composi-
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CONCLUSIONS The present statistical analysis focuses on binary intermetallics. One of the most remarkable results of our present paper is the unexpected large number of unique structure types, i.e., more than 50% of all structure types. This is true even for the simple one-to-one composition AB, which features the highest number of structure types120, with 69 of them unique ones. What is also remarkable is the asymmetry of the distributions of structures of the types AmBn and AnBm with M(A) < M(B). Another important outcome of our data mining approach is that only approximately 50% of all entries refer to more or less G
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Figure 5. Distribution of intermetallic compounds AmBn as a function of their Mendeleev numbers M(A) and M(B). The upper left half of each plot contains structures with M(A) < M(B), while the lower right half contains those with M(A) > M(B). The first plot (top left) contains all 6441 binary intermetallics, with the complex compounds marked in red. The second one (bottom left) contains the 523 binary intermetallics that occur in unique structure types (the 53 compounds with compositions m = n = 0.5 are shown for both combinations of elements: AmBn and BnAm). The remaining plots show the four most common compositions: A2B, AB, A3B, and A5B3, each with the most common structure type within the respective group marked in blue. These are cF24-Cu2Mg, cP2-CsCl, cP4-Cu3Au, and hP16-Mn5Si3, respectively. In the case of compounds of the type AB, the condition M(A) < M(B) applies. The areas shaded in light and dark gray correspond to the compounds containing lanthanoids and actinoids, respectively. (5) Berger, R. E.; Lee, S.; Hoffmann, R. Chem.Eur. J. 2007, 13, 7852−7863. (6) Fredrickson, D. C.; Lee, S.; Hoffmann, R. Angew. Chem., Int. Ed 2007, 46, 1958−1976. (7) Oganov, A. R., Ed. Modern Methods of Crystal Structure Prediction; Wiley-VCH: Weinheim, Germany, 2011. (8) Pettifor, D. G.; Podloucky, R. Phys. Rev. Lett. 1984, 53, 1080− 1083. (9) Pettifor, D. G. J. Phys. C: Solid State Phys. 1986, 19, 285−313. (10) Pettifor, D. G.; Podloucky, R. J. Phys. C: Solid State Phys. 1986, 19, 315−330. (11) Pettifor, D. G. Solid State Commun. 1984, 51, 31−34. (12) Pettifor, D. G. Mater. Sci. Technol. 1988, 4, 675−691. (13) Pettifor, D. Bonding and Structure of Molecules and Solids; Oxford University Press: New York, 1995. (14) Villars, P. J. Less-Common Met. 1985, 110, 11−25. (15) Villars, P.; Hulliger, F. J. Less-Common Met. 1987, 132, 289−315. (16) Villars, P.; Brandenburg, K.; Berndt, M.; LeClair, S.; Jackson, A.; Pao, Y.-H.; Igelnik, B.; Oxley, M.; Bakshi, B.; Chen, P.; Iwata, S. Eng. Appl. Art. Int. 2000, 13, 497−505. (17) Villars, P.; Brandenburg, K.; Berndt, M.; LeClair, S.; Jackson, A.; Pao, Y.-H.; Igelnik, B.; Oxley, M.; Bakshi, B.; Chen, P.; Iwata, S. J. Alloys Compd. 2001, 317−318, 26−38. (18) Villars, P.; Cenzual, K.; Daams, J.; Chen, Y.; Iwata, S. J. Alloys Compd. 2004, 367, 167−175. (19) (a) Nianyi, C.; Wencong, L.; Ruiliang, C.; Pei, Q.; Villars, P. J. Alloys Compd. 1999, 289, 120−125. (b) Nianyi, C.; Wencong, L.; Pei, Q.; Ruiliang, C.; Villars, P. J. Alloys Compd. 1999, 289, 126−130.
reliable unique structures. The present study will be complemented by a follow-up dedicated to a statistical analysis of ternary intermetallics.
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ASSOCIATED CONTENT
S Supporting Information *
A total of 14 separate plots for the Bravais lattices, analogous to Figure 2. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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