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The AFLOW Library of Crystallographic Prototypes: Part 2

David Hicks, Michael J. Mehl, Eric Gossett, Cormac Toher, Ohad Levy, Robert M. Hanson, Gus Hart, Stefano Curtarolo

arXiv:1806.07864v1cond-mat.mtrl-sci

TL;DR

High-throughput materials discovery needs accessible structural prototypes that can be automatically decorated and generated computationally. Part 2 extends the AFLOW library with 302 prototypes, completing coverage of all 230 space groups across 590 unique prototypes and providing online AFLOW-compatible structural data.

  • Problem

    Existing crystal-structure collections were valuable prototype sources, but much of their data was not readily accessible for automatic generation in computational materials-science frameworks.

  • Method

    The authors gathered crystallographic prototypes from the literature and integrated them into the AFLOW framework, providing structural information and AFLOW-generated geometry files through an online library.

  • Results

    302 additional prototypes complete the AFLOW library at 590 unique prototypes covering all 230 space groups, with Part 2 representing the remaining 138 space groups.

  • Takeaways & Limitations

    The combined online library supports rapid prototype decoration, AFLOW-based geometry generation, and high-throughput computation of material properties.

  • Takeaways & Limitations

    Symmetry classification can depend on tolerance settings, with near-equal lattice ratios producing reasonable alternative space-group assignments related by subgroup relations.

Abstract

from arXiv · show

Materials discovery via high-throughput methods relies on the availability of structural prototypes, which are generally decorated with varying combinations of elements to produce potential new materials. To facilitate the automatic generation of these materials, we developed $\textit{The AFLOW Library of Crystallographic Prototypes}$ $\unicode{x2014}$ a collection of crystal prototypes that can be rapidly decorated using the AFLOW software. Part 2 of this work introduces an additional 302 crystal structure prototypes, including at least one from each of the 138 space groups not included in Part 1. Combined with Part 1, the entire library consists of 590 unique crystallographic prototypes covering all 230 space groups. We also present discussions of enantiomorphic space groups, Wigner-Seitz cells, the two-dimensional plane groups, and the various different space group notations used throughout crystallography. All structures $\unicode{x2014}$ from both Part 1 and Part 2 $\unicode{x2014}$ are listed in the web version of the library available at aflow.org/CrystalDatabase.

4.1.1. Plane Group #1: p1

The paper presents the AFLOW Library of Crystallographic Prototypes as an accessible resource for automated materials generation, extending the collection with 302 prototypes and graphical treatments of crystallographic symmetry and positions.

  • Computational access: The online database provides interactive Jmol views of conventional, primitive, supercell, and Wigner-Seitz representations, together with a prototype generator interfacing with AFLOW.These functions make the structural information accessible in multiple cell representations.
  • Library extension: 302 prototypes extend the library with representatives from the 138 space groups not included in Part 1.The online library combines these entries with all prototypes from Part 1.
  • Library extension: 590 unique prototypes across all 230 space groups are available through the combined AFLOW library.Part 1 contributed 288 prototypes from 92 space groups, while Part 2 adds 302 prototypes from the remaining 138.
  • Enantiomorphic space groups: Enantiomorphic space groups can represent reflected structures, with space group #78 obtained from #76 by reflection through the z = 0 plane.The Cs3P7 example illustrates the transformation between these space groups.
  • Plane groups: The paper graphically presents Wyckoff positions for the seventeen plane groups using International Tables notation.The plane-group discussion treats two-dimensional space groups and their allowed atomic positions.
  • Computational access: Each prototype entry provides structural information that can be used to generate geometry files through AFLOW for high-throughput materials-property calculations.The entries include symmetry descriptions, lattice and atomic basis vector equations, and example elements or compounds.

H3Cl (50 GPa) Structure: AB3_mC16_15_e_cf

The supplied entries document crystallographic prototypes across monoclinic, orthorhombic, and related settings, pairing space-group metadata with AFLOW labels, commands, and atomic-coordinate parameterizations. Several entries also record computationally predicted structures and their pressure ranges.

  • Coordinate construction: The coordinate tables express atomic positions through lattice vectors and Cartesian components, with Wyckoff positions and atom types specified for each site.The entries include monoclinic, simple-, base-, body-, and face-centered orthorhombic primitive-vector descriptions.
  • Reported structural examples: 300 GPa H-III data were obtained from density functional simulations, with the structure reported in good agreement with experiment and as the lowest-energy structure near 100–250 GPa including zero-point motion.The same entry notes that its primitive vectors can be approximately equal in length with angles near 60°, making the lattice close to face-centered cubic.
  • AFLOW representation: AFLOW commands encode each prototype together with lattice ratios, cell parameters, and atomic positional parameters.Examples include the command for A_mC24_15_2e2f and parameter lists for orthorhombic prototypes.
  • Reported structural examples: The library records predicted structures over pressure-dependent regimes, including H3S at 3.5–17 GPa and H2S at 40–80 GPa.The H3S entry gives data at 5 GPa, while the H2S entry gives data at 70 GPa.

κ-alumina (Al2O3) Structure: A2B3_oP40_33_4a_6a

The κ-alumina section presents AFLOW prototype entries through labels, symmetry metadata, commands, and coordinate tables. Its examples emphasize how lattice vectors and Wyckoff-position coordinates encode decorated crystal structures across several orthorhombic space groups.

  • Prototype identification: The κ-alumina prototype is labeled A2B3_oP40_33_4a_6a and assigned to space group #33, Pna21.The corresponding AFLOW command exposes lattice ratios and positional parameters for automatic generation.
  • Lattice representation: The entries use simple, base-centered, and face-centered orthorhombic primitive-vector descriptions to organize coordinate representations.Each representation is followed by lattice-coordinate, Cartesian-coordinate, Wyckoff-position, and atom-type fields.
  • AFLOW implementation: AFLOW prototype commands provide reproducible generation instructions with named lattice and positional parameters.Examples include commands for A2B8C_oP22_34_c_4c_a, A2B3_oC40_39_2d_2c2d, and A9BC_oC44_39_3c3d_a_c.

Rh5Ge3 Structure: A3B5_oP16_55_ch_agh

The section presents AFLOW prototype entries across orthorhombic structures, specifying labels, space-group metadata, lattice settings, Wyckoff positions, atomic coordinates, and related structural information.

  • Prototype metadata: The A3B5_oP16_55_ch_agh entry uses space group 55, Pbam, with simple orthorhombic primitive vectors and tabulated lattice, Cartesian, Wyckoff, and atom-type data.The entry is labeled A3B5_oP16_55_ch_agh and includes Rh and Ge sites.
  • Atomic basis: The Rh5Ge3 coordinates are specified through basis vectors at Wyckoff positions 2a, 2c, and 4g for Rh and Ge atoms.The listed basis vectors include fixed fractional coordinates and parameters x3 and y3.
  • Structural context: The entries also document chemical and structural context, including a predicted superhard carbon allotrope and experimental molecular crystals with approximated hydrogen positions.The carbon phase is discussed alongside H-carbon and O-carbon; β-toluene data were constructed from experiments at 105 K.
  • AFLOW representation: Prototype descriptions provide AFLOW labels, parameter lists, geometry-file references, and repeated symmetry-generated basis coordinates for larger structures.The A7B8_oP120_60_7d_8d entry exposes parameters a, b/a, c/a, and positional variables through an AFLOW command.

Barite (BaSO4, H02) Structure: AB4C_oP24_62_c_2cd_c

The section catalogs orthorhombic and base-centered orthorhombic prototypes, including barite’s H02 structure and extensive symmetry-generated coordinates for related entries.

  • Barite prototype: The AB4C_oP24_62_c_2cd_c prototype is assigned Strukturbericht designation H02, space group 62, and symbol Pnma.Its AFLOW label identifies the stoichiometry, orthorhombic primitive setting, atom count, and Wyckoff-position sequence.
  • Related compounds: The section records other compounds sharing related prototypes, including SrSO4, PbSO4, and KGaH4.These compounds are listed under the structure’s related-compound information.
  • Barite prototype: The barite structure is represented with simple orthorhombic primitive vectors and Wyckoff-position coordinates for the constituent atomic sites.The entry includes coordinate expressions for oxygen sites and related atom positions generated by symmetry.
  • Additional prototypes: Additional entries cover Pnma and Cmcm structures with AFLOW labels, Strukturbericht designations, space-group metadata, primitive vectors, and coordinate tables.Examples include AB_oP8_62_c_c and A2BC3_oC24_63_e_c_cg.
  • Parameterized structures: Large base-centered orthorhombic entries encode many atoms through parameterized coordinates and AFLOW commands.The A43B5C17_oC260_63_c8fg6h_cfg_ce3f2h entry uses numerous positional parameters for La, Mg, Ni, and related sites.
  • High-pressure structures: The section includes high-pressure sulfur-hydrogen structures predicted by first-principles calculations for distinct pressure ranges.H2S structures are reported as stable above 140 GPa or between 80 and 140 GPa, while an H3S structure is reported stable between 40 and 90 GPa.

LaRhC2 Structure: A2BC_tP16_76_2a_a_a

The section presents AFLOW prototype entries for tetragonal structures, specifying labels, space-group metadata, adjustable parameters, and atom positions in lattice and Cartesian coordinates.

  • Prototype metadata: The AFLOW entries identify prototypes such as A2BC_tP16_76_2a_a_a and A3B7_tP40_76_3a_7a.The entries associate these labels with space group 76 and the P41 symbol.
  • Coordinate representation: The structures use simple tetragonal primitive vectors with lattice and Cartesian coordinates, Wyckoff positions, and atom types listed for each site.Representative sites include C, I, P, and other atom-type labels assigned to Wyckoff positions.
  • AFLOW implementation: AFLOW commands encode prototype labels together with lattice-ratio and fractional-coordinate parameters for automatic structure generation.The parameter lists include a, c/a, and site coordinates such as x_i, y_i, and z_i.
  • Supporting records: The entries include geometry-file references and literature citations for representative structures such as Cs3P7 and H2S III.The listed files include CIF and POSCAR outputs, while the citations identify source literature for structural examples.

Na4Ti2Si8O22[H2O]4 Structure: A4B2C13D_tP40_90_g_d_cef2g_c

This section documents AFLOW prototype entries for tetragonal structures, including labels, space-group metadata, parameterized commands, and symmetry-generated atomic coordinates.

  • Prototype metadata: The section identifies prototypes including A4B2C13D_tP40_90_g_d_cef2g_c and AB4C17D4E_tP54_90_a_g_c4g_g_c.Both examples are associated with space group 90 and the P4212 symbol.
  • AFLOW implementation: AFLOW commands represent these prototypes using lattice parameters and fractional coordinates for the constituent sites.The commands include a, c/a, positional variables, and complete prototype identifiers.
  • Coordinate representation: The coordinate tables organize lattice and Cartesian coordinates with Wyckoff positions and atom types across symmetry-related sites.Examples include Na, Ti, O, Ba, V, Cu, and P sites expressed with tetragonal primitive vectors.
  • Space-group coverage: The section also includes prototypes for space groups 91 and 93, with symbols P4122 and P4222, respectively.The space-group entries connect the prototype records to their crystallographic numbering and notation.
  • Scope notes: Structures with space group 93 are described as rare, and the prototype omits hydrogen atoms.The passage states that the Inorganic Crystal Structure Database has no entries for space group 93 according to Hoffmann (2014).

NbTe4 Structure: AB4_tP10_103_a_d

The AFLOW entries document crystallographic prototypes through labels, space-group assignments, lattice vectors, Wyckoff positions, and executable commands. The supplied section spans tetragonal structures across multiple space groups, with one noted occupancy and a space-group classification ambiguity.

  • NbTe4 Structure: AB4_tP10_103_a_d: AB4_tP10_103_a_d is assigned to space group 103, P4cc, with simple tetragonal primitive vectors and Wyckoff-position data for Te.Its AFLOW prototype label identifies the composition and structural setting.
  • Tetragonal prototype entries: A5B5C4_tP28_104_ac_ac_c is represented by space group 104, P4nc, and an AFLOW command parameterized by lattice and positional variables.The listed parameters include a, c/a, and multiple x, y, and z coordinates.
  • Coordinate and Wyckoff descriptions: The entries use lattice-to-Cartesian coordinate relations to specify Wyckoff sites and atom types across the listed tetragonal prototypes.Examples include Ba, Ge, H, Na, O, Zn, and related species assigned to Wyckoff positions.
  • Data qualification: The AB6C4_tP22_104_a_2ac_c entry reports an occupancy of 0.92 for its second I site.This is the section’s explicit occupancy qualification.
  • Space-group classification: For one structure, FINDSYM assigns space group 111 while AFLOW-SYM and Platon assign 215 because c/a is approximately 1; both classifications are described as reasonable under subgroup relations.Lowering AFLOW-SYM’s tolerance resolves the expected space group 111.

RbGa3 Structure: A3B_tI24_119_b2i_af

The RbGa3 entry presents an AFLOW prototype in space group 119 with body-centered tetragonal vectors and explicit atomic coordinates. The surrounding entries continue this cataloging format through additional tetragonal prototypes, related compounds, and structure-specific metadata.

  • RbGa3 Structure: A3B_tI24_119_b2i_af: A3B_tI24_119_b2i_af is identified as an AFLOW prototype in space group 119, I¯4m2, with body-centered tetragonal primitive vectors.The entry label, space-group metadata, and vector definition establish its crystallographic setting.
  • Atomic coordinates: The RbGa3 coordinates place Rb and Ga atoms on specified Wyckoff sites, including Rb at 2a and Ga at 8i positions.The supplied coordinate relations include the origin site and parameterized x4 and z4 positions.
  • Related compounds: The CaKFe4As4-related compounds form a family of stoichiometric superconductors with transition temperatures Tc ranging from 26–37 K.The listed compounds include CaKFe4As4, CaCsFe4As4, SrRbFe4As4, SrCsFe4As4, and BaCsFe4As4.
  • Later prototype entries: Later entries document prototypes in space groups 127 and 128 using additional Wyckoff-coordinate descriptions and AFLOW prototype commands.The examples include A4B_tP20_127_ehj_g, A6B2C_tP18_128_eh_d_b, and A7B2C_tP40_128_egi_h_e.

C (T12 Group IV) Structure: A_tP12_138_bi

The A_tP12_138_bi prototype represents a tetragonal diamond allotrope and is identified with space group P42/ncm. Its AFLOW entry records lattice, basis, Wyckoff, and atom-type information for automatic structural generation.

  • A_tP12_138_bi is the AFLOW prototype label for the tetragonal T12 Group IV structure.
  • Space group number 138 and symbol P42/ncm identify the structure’s crystallographic symmetry.
  • The structure is a tetragonal allotrope of the diamond structure found computationally in C, Si, and Ge.
  • The authors state that the T12 polymorph accounts for experimental d spacings and Raman spectra of metastable Ge and Si-XIII phases with previously unknown structures.
  • The entry specifies simple tetragonal primitive vectors together with lattice and Cartesian coordinates, Wyckoff positions, and atom types.

K2Ta4O9F4 Structure: A2B13C4_hP57_168_d_c6d_2d

The A2B13C4_hP57_168_d_c6d_2d entry documents the K2Ta4O9F4 structure in space group 168, using extensive Wyckoff-coordinate data and an explicit AFLOW generation command.

  • A2B13C4_hP57_168_d_c6d_2d is the AFLOW prototype label for the K2Ta4O9F4 structure.
  • Space group number 168 and symbol P6 identify the crystallographic symmetry used for this prototype.
  • The AFLOW prototype command exposes the lattice, axial-ratio, and internal-coordinate parameters needed to generate the structure.
  • The oxygen sites are partially occupied with concentration 0.692O + 0.308F.
  • The structure entry lists lattice and Cartesian coordinates, Wyckoff positions, and atom types across its many basis sites.

Cu3P Structure: A3B_hP24_185_ab2c_c

The Cu3P entry records the A3B_hP24_185_ab2c_c prototype in space group 185, P63cm, while noting competing structural assignments and related compounds.

  • A3B_hP24_185_ab2c_c identifies the Cu3P prototype in space group 185, symbol P63cm.
  • Olofsson identifies this P63cm structure as the correct Cu3P structure rather than the previously assigned Strukturbericht D021 structure in P-3c1.
  • The same structure has also been proposed for AsNa3, AuMg3, IrMg3, and Mg3Pt instead of their previously reported D018 or D021 assignments.
  • The entry provides lattice and Cartesian coordinates alongside Wyckoff positions and atom types for the prototype.
  • Cu3P and Na3As share similar AFLOW prototype labels because they have the same symmetry and Wyckoff positions but different stoichiometry labels and parameters.

Mg2Zn11 Structure: A2B11_cP39_200_f_aghij

The Mg2Zn11 entry catalogs the A2B11_cP39_200_f_aghij prototype in cubic space group 200, Pm-3, with detailed Wyckoff-coordinate data and AFLOW generation files.

  • A2B11_cP39_200_f_aghij identifies the Mg2Zn11 prototype in space group 200, symbol Pm-3.
  • The structure description enumerates Mg, Zn, and other atomic sites through Wyckoff positions and Cartesian coordinates.
  • The entry supplies extensive coordinate sets for the 8i and 12j Wyckoff positions occupied by Zn species.
  • The structure is attributed to crystallographic database and literature sources including Samson and Pearson’s Crystal Data.
  • AFLOW provides both geometry files and the prototype command for generating A2B11_cP39_200_f_aghij.

AlLi3N2 (E9d) Structure: AB3C2_cI96_206_c_e_ad

The AlLi3N2 prototype is identified as AB3C2_cI96_206_c_e_ad in space group 206, Ia¯3, with an AFLOW command for generating the structure. The section also documents related prototypes and their crystallographic coordinates across several space groups.

  • Prototype identification: AB3C2_cI96_206_c_e_ad is the AFLOW prototype label for the AlLi3N2 structure.The entry assigns space group number 206 and symbol Ia¯3.
  • Prototype identification: The structure can be generated with the command aflow --proto=AB3C2_cI96_206_c_e_ad.The command is listed with the prototype entry.
  • Related compounds: Other compounds listed for this structure include GaLi3N2, ScLi3N2, TiLi3N2, ZnLi3N2, SiLi3N2, and GeLi3N2.These compounds are presented as examples sharing the structure.
  • Coordinate representations: The entries specify lattice coordinates, Cartesian coordinates, Wyckoff positions, and atom types for the prototype structures.Examples include body-centered, simple-cubic, and related coordinate listings across the documented structures.

Petzite (Ag3AuTe2) Structure: A3BC2_cI48_214_f_a_e

The Petzite structure is represented by A3BC2_cI48_214_f_a_e in space group 214, I4132, with coordinates assigned to Au, Te, Ag, and related atom types. The entry records a site-selection decision motivated by interatomic-distance consistency and links the structure to γ-brass and quaternary-Heusler descriptions.

  • Prototype identification: A3BC2_cI48_214_f_a_e is the AFLOW prototype label for the Petzite structure in space group 214, I4132.The corresponding AFLOW command is aflow --proto=A3BC2_cI48_214_f_a_e.
  • Atomic positions: The listed coordinates assign Au to the (8a) Wyckoff positions, Te to (16e), and Ag to (24f) sites.The entry provides lattice and Cartesian coordinate expressions for these atom positions.
  • Site assignment: Cu V atoms are placed on (6g) rather than (6f) sites because the (6f) assignment gives 1.8Å while (6g) agrees with the reported 2.48 Å minimum distance.The selected (6g) assignment is also reported to agree with Pearson’s description.
  • Structural relation: The text identifies the resulting structure as a variety of γ-brass comparable to the D82 structure.It further states that a specified substitution and parameter relation makes the structure identical to D82 γ-brass.
  • Related structural description: A quaternary-Heusler structure is described through occupancy of the (4a), (4b), (4c), and (4d) Wyckoff positions.The ordering is stated to be equivalent when Sn and Mg occupy paired sites and Au and Li occupy the opposite pair.

Garnet (Co3Al2Si3O12, S 14) Structure: A2B3C12D3_cI160_230_a_c_h_d

The garnet prototype is labeled A2B3C12D3_cI160_230_a_c_h_d in space group 230, Ia¯3d, and is represented with extensive lattice, Cartesian, and Wyckoff-coordinate data. The entry associates this structure with garnet compounds and records the S 14 designation.

  • Prototype identification: A2B3C12D3_cI160_230_a_c_h_d is the AFLOW prototype label for the garnet structure in space group 230, Ia¯3d.The corresponding AFLOW prototype command is listed explicitly.
  • Prototype identification: The structure carries the Strukturbericht designation S 14.The designation appears with the garnet prototype entry.
  • Related compounds: Compounds listed with this structure include Al2(Mg,Ni)3Si3O12, Al2Ca3Si3O12, Al2Co3Si3O12, and related garnet compositions.The list also includes Al2Mg3Si3O12, Al2Mn3Si3O12, and other compounds.
  • Data representation: The entry provides AFLOW and crystallographic data for garnet structures, including lattice vectors, Wyckoff coordinates, and structure-file representations.The supplied records include prototype parameters, CIF/POSCAR-related data, and symmetry operations.
  • Atomic positions: The coordinate tables assign Al atoms to (16a), Co atoms to (24c), and Si atoms to (24d) Wyckoff positions.The entry lists lattice and Cartesian coordinates for these sites.

Prototype Index

The prototype index lists crystal structures by named material and AFLOW prototype label, spanning varied compositions, symmetries, and crystallographic settings. Notes clarify when different materials share labels because they use the same symmetry operations with different parameters.

  • Named prototypes: The index pairs named materials with AFLOW prototype labels encoding composition, lattice setting, space group, and Wyckoff-position information.Examples include α-P3N5, α-PbO, Akermanite, and Al2CuIr.
  • Index coverage: The index includes multiple entries for related structure families, including distinct α-, β-, γ-, and δ-labeled compounds and repeated compounds in different prototype settings.Examples include α-, β-, and γ-PdCl2, β-RuCl3, and multiple Al[PO4] entries.
  • Shared labels: α-FeSe and α-PbO share one AFLOW prototype label despite being different materials with different parameter sets.Both are generated by the same symmetry operations.
  • Shared labels: H2S and β-SeO2 likewise share an AFLOW prototype label while using different parameter sets.The shared label reflects identical symmetry operations rather than identical chemical composition.
  • Shared labels: Al2CuIr and HoCuP2 have similar labels because alphabetic ordering produces different stoichiometry labels for the same symmetry and Wyckoff-position set.Their structures are generated by the same symmetry operations with different parameters.

Pearson Symbol Index

The Pearson symbol index organizes prototypes by lattice-centering and crystal-system symbols, then lists representative compounds with their AFLOW labels. Entries range from small-cell structures to large and chemically complex prototypes.

  • Cubic face-centered entries: The cF section includes chemically varied structures such as quaternary Heusler, spinel, pyrochlore, and fullerene prototypes.Listed examples range from cF16 and cF36 to cF416.
  • Pearson categories: Pearson-symbol categories such as cI, cP, cF, hP, and hR organize the listed prototypes by crystal-system and centering notation.The index explicitly transitions among cI, cP, cF, hP, and hR sections.
  • Cubic body-centered entries: The cI section includes prototypes from β-Hg4Pt and Th3P4 through garnet, spanning cI10 to cI160 entries.Examples include cI10, cI28, cI40, cI48, cI52, cI56, cI72, cI76, cI96, and cI160.
  • Cubic primitive entries: The cP section lists structures from SrSi2 and PH3 to Ca3Al2O6 and simple-cubic C60 Buckminsterfullerene across cP12 through cP264.The entries include cP12, cP16, cP20, cP33, cP34, cP39, cP52, cP60, cP64, cP96, cP240, and cP264.
  • Hexagonal and rhombohedral entries: The hP and hR sections contain numerous hexagonal and rhombohedral prototypes, including AuCN, β-CuI, β-Si3N4, γ-Ag3SI, and Al4C3.The hP listings extend through hP72, while the hR entries include hR3 through hR10 and beyond.
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