Nanoparticles with cubic symmetry: classification of polyhedral shapes

Structural studies of polyhedral bodies can help to analyze geometric details of observed crystalline nanoparticles (NP) where we consider compact polyhedra of cubic point symmetry as simple models. Their surfaces are described by facets with normal vectors along selected Cartesian directions (a, b, c) together with their symmetry equivalents forming a direction family {abc}. Here we focus on polyhedra with facets of families {100}, {110}, and {111}, suggested for metal and oxide NPs with cubic lattices. Resulting generic polyhedra, cubic, rhombohedral, octahedral, and tetrahexahedral, have been observed as NP shapes by electron microscopy. They can serve for a complete description of non-generic polyhedra as intersections of corresponding generic species, not studied by experiment so far. Their structural properties are shown to be fully determined by only three parameters, facet distances R 100, R 110, and R 111 of the three facet types. This provides a novel phase diagram to systematically classify all corresponding polyhedra. Their structural properties, such as shape, size, and facet geometry, are discussed in analytical and numerical detail with visualization of typical examples. The results may be used for respective NP simulations but also as a repository stimulating the structural interpretation of new NP shapes to be observed by experiment.


Introduction
Recently, the detailed characterization of polyhedral bodies, while a subject of mathematical resarch since ancient times [1], has attracted new interest in connection with crystalline nanoparticles (NPs) [2][3][4][5]. These particles come in many sizes and polyhedral shapes. Their properties have been explored both experimentally and in theoretical studies due to their exciting physical and chemical behavior, which deviates often from that of corresponding bulk material [2][3][4]. Examples are applications in medicine [6] or in catalytic chemistry where metal nanoparticles have become ubiquitous [7,8].
Many metal NPs have been observed in experiments to exhibit polyhedral shape with flat local surface areas (facets) of high atom density, reminiscent of low Miller index planes in corresponding cubic bulk crystals [9,10]. This can be associated with the geometry of the corresponding crystal lattice suggesting crystalline bulk structure inside the NP. At an atomic scale, the facets at the NP surface join to form edges and corners whose detailed structure can, however, be rather complex. This is due to the discrete distribution of atom positions giving rise to corner capping with microfacets and edge flattening leading to microstrips as discussed earlier [5]. The perturbative effect is even enhanced by local relaxation at the particle surface and as a result of chemical surface reactions.
The overall shape of experimental metal NPs often reminds of compact sections confined by simple polyhedra of Oh symmetry, such as cubes, octahedra, rhombohedra, and others, which may be attributed to the cubic lattice structure of the corresponding bulk metal lattice [5,9,10].
Here the analysis of ideal polyhedra with cubic Oh symmetry as NP envelops can be helpful to obtain further insight into possible NP shapes and their classification. Corresponding analytical results of the polyhedral structure allow estimates of NP sizes depending on the number of atoms included together with atom densites in bulk metal. They also give insight into the geometry of possible facets at NP surfaces.
In the present work, extending previous theoretical analyses [5,11], we focus on geometrical details of polyhedra of cubic Oh symmetry with their symmetry center at the origin of a Cartesian  [10] which seem to be energetically preferred. This suggests polyhedra with facet normal vectors (a, b, c) = (1, 0, 0), (1,1,0), and (1,1,1) in Cartesian coordinates together with their Oh symmetry equivalents. The analysis reveals different types of generic polyhedra which can serve for the definition of general polyhedra described as intersections of corresponding generic species. Their structural properties, such as shape, size, and surface facets, are shown to be fully determined by only three structure parameters, the facet distances R100, R110, R111. In fact, all polyhedral shapes, independent of size, can already be characterized by only two relative facet distances, such as x110 = R110/R100 and x111 = R111/R100 which provides a complete phase diagram of all polyhedral shapes.
We also consider generic polyhedra of Oh symmetry which are confined by facets of one general direction family {abc}, yielding up to 48 different facet directions. These polyhedra can be used to model metal NPs with higher Miller index facets reflecting sections of stepped and kinked facet surfaces [10]. Clearly, their structural properties are fully described by a facet distance Rabc and all components, a, b, c, determining the corresponding facet normal vector family.
All structural results of the present polyhedra are discussed in analytical and numerical detail with visualization [12] of characteristic examples. The different sections are structured identically and presented in parts with very similar phrasing to enable easy comparison. These results allow a full classification of all corresponding polyhedra which may be used as a repository to assist the interpretation of structures of real compact NPs observed by experiment. They can also be useful for corresponding nanoparticle simulations. Also mesoscopic crystallites with internal cubic lattice assuming polyhedral shape may be classified by the present scheme.
Sec. 2 introduces notations and definitions used to characterize polyhedral shape while Sec. 3 discusses many examples of polyhedra, generic and non-generic, in detail. Finally, Sec. 4 summarizes conclusions from the present work. The supplement provides further details to complement results discussed in Sec. 3.

Notation and Formal Definitions
We consider compact polyhedra of central Oh symmetry confined by finite sections of planes (facets) which can be described by facet normal vectors eabc and facet distances Rabc from the polyhedral center, resulting in facet vectors The above notations and definitions will be used in the following discussion. Note that some of the expressions of corner coordinates in Secs. 3.2 use auxiliary parameters g, h which are defined separately for each Section.

Generic Polyhedra
As discussed above, generic polyhedra are confined by facets of only one direction family {abc} and are denoted P(R{abc}). Here the simplest examples are those for {abc} = {100}, {110}, and {111} which will be discussed before the general case, which includes also the simple examples, is treated in detail.

Rhombohedral Polyhedra P(R{110})
According to (2c), (3), these polyhedra are confined by all 12 {110} facets with facet distances R110 which describes a rhombohedral polyhedron, see The 14 polyhedral corners fall in two groups of 6 and 8 each, described by vectors C{100} and C{111} relative to the center, where in Cartesian coordinates With P(R{110}) yielding 12 facets and 14 corners the number of its polyhedral edges amounts to 24 according to (5), see Fig. 2a.
The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R110), is given by Further, the area of each facet is given by F0 with Thus, the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron are given by Fig. 2b shows a nanoparticle of atom balls representing a body-centered cubic (bcc) crystal section where a polyhedron P(R{110}) serves as envelope with its corners C{100} and C{111} coinciding with atom sites.
An analysis shows that all 8 facets are of the same equilateral triangular shape where each {111} facet extends between adjacent corners C{100}, such as C(100), C(010), C(001). The resulting three edges connect corners, such as C(010) with C(100), at distances dc1 given by The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R111), is given by Further, the area of each facet is given by F0 with Thus, the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron are given by 3b shows a nanoparticle of atom balls representing a face-centerd cubic (fcc) crystal section where a polyhedron P(R{111}) serves as envelope with its corners C{100} coinciding with atom sites.

Polyhedra P(R{abc})
According to (  C{111}, and C{110} with C{111}, at different distances dd1, dd2, and dd3 given by Further, the area of each facet is given by F0 with Thus, the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron are given by

Non-generic Polyhedra
Non-generic polyhedra of Oh symmetry P(R{abc}; R{a'b'c'}; …) show facets with orientations of more than one family of facet vectors R{abc}. This can be considered as combining confinements of corresponding different generic polyhedra P(R{abc}), discussed in Sec. 3.1, which share their symmetry center. Thus, non-generic polyhedra represent mutual intersections of more than one generic polyhedron, where one cuts corners and edges of the other(s) to form additional facets.
In this section we restrict ourselves to non-generic polyhedra with up to three selected generic polyhedra, cubic P(R{100}), rhombohedral P(R{110}), and octahedral P(R{111}) which offer {100}, {110}, as well as {111} facets with facet distances R100, R110, and R111. This choice is motivated by the structure of ideal cubic metal nanoparticles whose bulk atoms form sections of cubic crystals (simple, face-, and body-centered) and where corresponding facets of {100}, {110}, and {111} families reflect crystal monolayers of highest atom density [10].
The corresponding facet distances R100, R110, and R111 can be considered as structure parameters, defining the present non-generic polyhedra, and their relations with each other determine the polyhedral shape. In the following, we discuss the three types of polyhedra, which combine two generic polyhedra each, i.e. P(R{100}; R{110}), P(R{100}; R{111}), and P(R{110}; R{111}) in Secs.
This requires and according to (7a), (13a) Thus, the two generic polyhedra intersect and yield a true polyhedron P(R{100}; R{110}) with both {100} and {110} facets only for facet distances R100, R110 with while P(R{100}; R{110}) is generic cubic for R110 ≥ 2 R100 and generic rhombohedral for R110 ≤ 1/2 R100. As a consequence, generic polyhedra P(R{100}) and P(R{110}) can be described alternatively by non-generic P(R{100}; R{110}) where The surfaces of general cubo-rhombic polyhedra P(R{100}; R{110}) exhibit 6 {100} facets and 12 {110} facets as shown in Fig. 5. four connect corners, such as C(1hh) with C(111), at distances de2 given by The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R100, R110), is given by Further, the area of each square {100} facet is given by F0 where with (37) and of each hexagonal {110} facet by F1 where with (14) This yields the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron according to
The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R100, R111), is given by Further, the area of each square {100} facet is given by F0 where with (52) and of each hexagonal {111} facet by F1 where with (52) This yields the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron according to The surfaces of truncated cubic polyhedra P(R{100}; R{111}) with 2 < 3R111/ R100 < 3 exhibit The 6 {100} facets are of the same octagonal shape where each facet extends between eight (11-g). Of the resulting eight alternating edges four connect corners, such as C(11g) with C (11-g), at distances df3 while four connect corners, such as C(11g) with C(1g1), at distances df4 given by The 8 {111} facets are of the same equilateral triangular shape where each facet extends between three adjacent C{11g} corners, such as C(11g), C(g11), C(1g1). The resulting three edges connect corners, such as C(11g) with C(g11), at distances df4 according to (61).
The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R100, R111), is given by Further, the area of each octagonal {100} facet is given by F0 where with (51), (60), (61) and of each triangular {111} facet by F1 where with (61) This yields the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron according to There are polyhedra which can be assigned to both truncated cubic and truncated octahedral type, the cuboctahedral polyhedra P(R{100}; R{111}) with 3R111/R100 = 2. They are described by vectors C{110} relative to the center, where in Cartesian coordinates which can also be derived from (51) with h = 1 or from (59) with g = 0. The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R100, R111), is given by Further, the area of each square {100} facet is given by F0 where with (68) and of each hexagonal {111} facet by F1 where with (68) Thus, the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron are given by
The largest distance from the polyhedral center to its surface along (abc) directions, sabc(R110, R111), is given by Further, the area of each hexagonal {110} facet is given by F0 where with (80) and of each triangular {111} facet by F1 where with (81) This yields the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron according to

Polyhedra P(R{100}; R{110}; R{111}) by Intersection
Intersecting cubo-rhombic P(R{100}; R{110}) with octahedral P(R{111}) polyhedra results in different polyhedral shapes depending on the relative sizes of the corresponding facet distances Rabc. Fixing R100 and R110 with (97) the shape of P(R{100}; R{110}; R{111}) is fully determined by the size of its facet distance R111. According to (97), (98), (90) R111 must always be within the range where there are two regions leading to different polyhedral shape, outer region: inner region: as illustrated in Fig. 9 The 6 {100} facets are of the same square shape where each facet extends between four adjacent corners C{1gg}, such as C(1gg), C(1-gg), C(1-g-g), C(1g-g). The resulting four edges connect corners, such as C(1gg) with C(1-gg), at distances dh1 given by The 12 {110} facets are of the same octagonal shape where each facet extends between eight adjacent corners C{1gg} and C{1hh}, such as C(1gg), C(1hh), C(hh1), C(gg1), C(g-g1), C(h-h1), C(1-hh), C (1-gg). Of the resulting eight edges two connect corners, such as C(1gg) with C(1-gg), at distances dh1 given by (104) while another two connect corners, such as C(1hh) to C(hh1), at distances dh2 and four connect corners, such as C(1gg) with C(1hh), at distances dh3 given by The 12 {110} facets are of the same rectangular shape where each facet extends between four adjacent corners C{1gg}, such as C(1gg), C(gg1), C(g-g1), C(1-gg). Of the resulting four edges two connect corners, such as C(1gg) with C(1-gg), at distances dh4 according to (115) while two connect corners, such as C(1gg) with C(gg1), at distances dh5 given by and of each triangular {111} facet by F2 where with (116) This yields the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron according to The inner region is determined by R111 values with 111 ≤ R111 ≤ 111 according to (102),  while four connect corners, such as C(1gh) with C(1hg), at distances dh7 given by The 12 {110} facets are of the same rectangular shape where each facet extends between four adjacent corners C{1gh}, such as C(1gh), C(1g-h), C(g1-h), C(g1h). Of the resulting four alternating edges two connect corners, such as C(1gh) with C(1g-h), at distances dh6 according to (124) while two connect corners, such as C(1gh) with C(g1h), at distances dh8 given by and of each rectangular {110} facet by F1 where with (115), (116) and of each hexagonal {111} facet by F2 where with (125), (126) This yields the total facet surface, Fsurf (sum over all facet areas) and the volume Vtot of the polyhedron according to There are two alternative intersection procedures to achieve a true P(R{100}; R{110}; R{111}) polyhedron which, however, lead to the same shapes and identical formulas for corner coordinates and all other structural parameters which have been discussed above. Therefore, they will be outlined only briefly in the following.

Classification of P(R{100}; R{110}; R{111})
The discussion in Secs. 3 which leads to a two-dimensional phase diagram with x110, x111 as order parameters and shown in Fig. 16. in the phase diagram, see vertical arrow in Fig. 16.
True cubo-octahedral polyhedra P(R{100}; R{111) are defined by (99)  and corresponds to the infinite horizontal strip labeled "co" (co phase) in Fig. 16. Here the dashed line, defined according to (50c) and converted to separates polyhedra of the truncated octahedral type (x111 ≤ 2, labeled "oct") from those of the truncated cubic type (x111 ≥ 2, labeled "cub"). Polyhedra of this phase along (horizontal) lines of fixed x111 differ only by the size of the rhombohedral polyhedron P(R{110}) outside P(R{100}; R{111). Therefore, they are identical with their counterparts at the co/cro phase boundary obtained by horizontal shifting according to in the phase diagram, see horizontal arrows in Fig. 16.
True rhombo-octahedral polyhedra P(R{110}; R{111}) are defined by (78) and (95) which converts to and corresponds to the triangular area labeled "ro" (ro phase) in Fig. 16. Polyhedra of this phase along (radial) lines of fixed x111/x110 differ only by the size of the cubic polyhedron P(R{100}) outside P(R{110}; R{111}). Therefore, they are identical with their counterparts at the ro/cro phase boundary obtained by radial shifting from the coordinate origin according to in the phase diagram, see diagonal arrow in Fig. 16. shared between the cro and c phases since they differ only by the size of the generic rhombohedral and octahedral polyhedra outside P(R{100}).
Generic rhombohedral polyhedra P(R{110}) are defined by (32) and (75) which converts to and corresponds to the infinite vertical strip labeled "r" (r phase) in Fig. 16. Polyhedra of this phase are identical with their counterpart at the point shared between the cro and r phases since they differ only by the size of the generic cubic and octahedral polyhedra outside P(R{110}).
Generic octahedral polyhedra P(R{111}) are defined by (47) and (77) (145) including its edges and corners which can be described as If the polyhedra are to limit nanoparticles of metal atoms, which represent sections with internal cubic bulk structure, then facet distances R100, R110, and R111 assume discrete values due to the lattice periodicity. As a result, parameters x110 and x111 become fractional forming a homogeneous network of possible values inside the phase diagram where the network mesh is square for simple cubic (sc), rectangular 1x2 for face centered cubic (fcc), and rectangular 2x1 for body centered cubic (bcc) lattices. Further, the network mesh size decreases with increasing NP size. This is illustrated in Fig. 17 showing the phase diagram of Fig. 16  There are two alternative classification schemes of polyhedral shapes which are mentioned only briefly and discussed in detail in Sec. S.2 of the Supplement. First, fixing R110 at any value allows to discriminate between all shapes of polyhedra P(R{100}; R{110}; R{111}) by considering two parameters derived from relative facet distances y100 and y111 where This leads to a two-dimensional phase diagram with y100, y111 as order parameters and shown in Fig. 18a. Second, fixing R111 at any value allows to discriminate between all shapes by considering two parameters derived from relative facet distances z100 and z110 where This leads to a two-dimensional phase diagram with z100, z110 as order parameters and shown in

Conclusions
The present theoretical analysis gives a full account of the shape and structure of compact polyhedra with cubic Oh symmetry. The polyhedral surfaces can be described by

S.1. Special Cases of Generic Polyhedra P(R{abc})
In this Section we disuss results for generic polyhedra which derive from the generic polyhe- With P(R{aac}) yielding 24 facets and 14 corners the number of its polyhedral edges amounts to 36 according to (5). Fig. S1b illustrates the general polyhedron shape.
With P(R{ab0}) yielding 24 facets and 14 corners the number of its polyhedral edges amounts to 36 according to (5). In fact, the general shape of P(R{ab0}), see Fig. S4b, can be characterized qualitatively as a cube whose six surface sides are complemented by identical square pyramids.
First, fixing R110 at any value allows to discriminate between all polyhedral shapes by considering two parameters derived from relative facet distances y100 and y111 where This leads to a two-dimensional phase diagram with y100, y111 as order parameters and shown in Fig. S7. Note that shape identical polyhedra due to generic polyhedra which do not contribute to the polyhedral shape can be discussed analogous to Sec. 3.2.4.2. Figure S7. Phase diagram of all shapes of cubo-rhombo-octahedral polyhedra P(R{100}; R{110}; R{111}) with y100 and y111 as order parameters. The different phases are shown by different colors and labeled accordingly, see text.
Altogether, the phase diagram shown in Fig. S7 covers all possible definitions of cuborhombo-octahedral polyhedra P(R{100}; R{110}; R{111}) where, however, polyhedra of truly different shape are already fully accounted for by y100, y111 values inside the rectangular area defined by (S49), (S50), and (S51) including its edges and corners which can be described as Second, fixing R111 at any value allows to discriminate between all polyhedral shapes by considering two parameters derived from relative facet distances z100 and z110 where This leads to a two-dimensional phase diagram with z100, z110 as order parameters and shown in Fig. S8. Note that shape identical polyhedra due to generic polyhedra which do not contribute to the polyhedral shape can be discussed analogous to Sec. 3.2.4.2. Figure S8. Phase diagram of all shapes of cubo-rhombo-octahedral polyhedra P(R{100}; R{110}; R{111}) with z100 and z110 as order parameters. The different phases are shown by different colors and labeled accordingly, see text.

S.3. Evaluation of Facet Edges and Corners
Vectors R pointing from the polyhedron center to a facet defined by facet vector Rabc are given by scalar products Rabc R = Rabc 2 (S87) or in Cartesian coordinates with R = (x, y, z), Rabc = (xabc, yabc, zabc) by xabc x + yabc y + zabc z = xabc 2 + yabc 2 + zabc 2 (S88) which reflects a constraint on coordinates x, y, z to yield coordinates of a two-dimensional plane.