Another structure unit of YB₆₆, shown in figure 4b, is B₈₀ cluster of 80 boron sites formed by the B10 to B13 sites. All those 80 sites are partially occupied and in total contain only about 42 boron atoms. The B₈₀ cluster is located at the body center of the octant of the unit cell, i.e., at the 8a position (1/4, 1/4, 1/4); thus, there are eight such clusters (336 boron atoms) per unit cell. Two independent structure analyses came to the same conclusion that the total number of boron atoms in the unit cell is 1584. The boron framework structure of YB₆₆ is shown in figure 5a. To indicate relative orientations of the supericosahedra, a schematic drawing is shown in figure 5b, where the supericosahedra and the B₈₀ clusters are depicted by light green and dark green spheres, respectively; at the top surface of the unit cell, the relative orientations of the supericosahedra are indicated by arrows. There are 48 yttrium sites ((0.0563, 1/4, 1/4) for YB₆₂) in the unit cell. Richards and Kasper fixed the Y site occupancy to 0.5 that resulted in 24 Y atoms in the unit cell and the chemical composition of YB₆₆. As shown in figure 6, Y sites form a pair separated by only 0.264 nm in YB₆₂. This pair is aligned normal to the plane formed by four supericosahedra. The Y site occupancy 0.5 implies that the pair has always one Y atom with one empty site.
Slack et al. reported that the total number of boron atoms in the unit cell, calculated from the measured values of density, chemical composition and lattice constant, is 1628 ± 4, which is larger than the value 1584 obtained from the structural analysis. The number of B atoms in the unit cell remains nearly constant when the chemical composition changes from YB₅₆ to YB₆₆. On the other hand, the total number of yttrium atoms per unit cell varies, and it is, for example, ~26.3 for YB₆₂ (see right table). If the total number of Y atoms stays bellow or equal to 24 then it is possible that one Y atom accommodates in each Y pair (partial occupancy). However, the experimental value of 26.3 significantly exceeds 24, and thus both pair sites might be occupied. In this case, because of the small separation between the two Y atoms, they must be repelled by the Coulomb force. To clarify this point, split Y sites were introduced in the structure analysis resulting in a better agreement with the experiment. The Y site distances and occupancies are presented in the left table.

There are twenty Y pair sites with one Y atom and three pairs with two Y atoms; there is also one empty Y pair (partial occupancy = 0). The separation 0.340 nm for the Y2 pair site (two Y atoms in the pair site) is much larger than the separation 0.254 nm for the Y1 pair site (one Y atom in the pair site), as expected. The total number of Y atoms in the unit cell is 26.3, exactly as measured. Both cases are compared in figure 7. The larger separation for the Y2 pair site is clear as compared with that for the Y1 pair site. In case of the Y2 pair, some neighboring boron sites that belong to the B₈₀ cluster must be unoccupied because they are too close to the Y2 site.

Splitting the Y site yields right number of Y atoms in the unit cell, but not B atoms. Not only the occupation of the B sites in the B₈₀ cluster must be strongly dependent on whether or not the Y site is the Y1 state or the Y2 state, but also the position of the occupied B sites must be affected by the state of the Y site. Atomic coordinates and site occupancies are summarized in table II.

REB₄₁Si_1.2

Similar to yttrium, rare-earth metals from Gd to Lu can form REB₄₁Si_1.2-type boride. The first such compound was synthesized by solid-state reaction and its structure was deduced as YB₅₀. X-ray powder diffraction

X-ray crystallography

X-ray crystallography is a method of determining the arrangement of atoms within a crystal, in which a beam of X-rays strikes a crystal and causes the beam of light to spread into many specific directions. From the angles and intensities of these diffracted beams, a crystallographer can produce a...

 (XRD) and electron diffraction

Electron diffraction

Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern...

 indicated that YB₅₀ has an orthorhombic structure with lattice constants a = 1.66251(9), b = 1.76198 and c = 0.94797(3) nm. The space group was assigned as P2₁2₁2. Because of the close similarity in lattice constants and space group, one might expect that YB₅₀ has the γ-AlB₁₂-type orthorhombic structure whose lattice constants and space group are a = 1.6573(4), b = 1.7510(3) and c = 1.0144(1) nm and P2₁2₁2. YB₅₀ decomposes at ~1750 °C without melting that hinders growth of single crystals from the melt. Small addition of silicon

Silicon

Silicon is a chemical element with the symbol Si and atomic number 14. A tetravalent metalloid, it is less reactive than its chemical analog carbon, the nonmetal directly above it in the periodic table, but more reactive than germanium, the metalloid directly below it in the table...

 made YB₅₀ to melt without decomposition, and so enabled single-crystal growth from the melt and single-crystal structure analysis.

The structure analysis indicated that YB₄₁Si_1.2 has not the γ-AlB₁₂-type lattice but a rare orthorhombic crystal structure (space group: Pbam, No. 55) with lattice constants of a = 1.674(1) nm, b = 1.7667(1) nm and c = 0.9511(7) nm. There are 58 independent atomic sites in the unit cell. Three of them are occupied by either B or Si atoms (mixed-occupancy sites), one is a Si bridge site and one is Y site. From the remaining 53 boron sites, 48 form icosahedra and 5 are bridging sites. Atomic coordinates and site occupancies are summarized in table III.

The boron framework of YB₄₁Si_1.2 consists of five B₁₂ icosahedra (I1–I5) and a B₁₂Si₃ polyhedron shown in figure 8a. An unusual linkage is depicted in figure 8b, where two B₁₂-I5 icosahedra connect via two B atoms of each icosahedron forming an imperfect square. The boron framework of YB₄₁Si_1.2 can be described as a layered structure where two boron networks (figures 9a,b) stack along the z-axis. One boron network consists of 3 icosahedra I1, I2 and I3 and is located in the z = 0 plane; another network consists of the icosahedron I5 and the B₁₂Si₃ polyhedron and lies at z = 0.5. The icosahedron I4 bridges these networks, and thus its height along the z-axis is 0.25.

The I4 icosahedra link two networks along the c-axis and therefore form an infinite chain of icosahedra along this axis as shown in figure 10. The unusually short distances (0.4733 and 0.4788 nm) between the neighboring icosahedra in this direction result in the relatively small c-axis lattice constant of 0.95110(7) nm in this compound – other borides with a similar icosahedral chain have this value larger than 1.0 nm. However, the bonding distances between the apex B atoms (0.1619 and 0.1674 nm) of neighboring I4 icosahedra are usual for the considered metal borides.

Another unusual feature of YB₄₁Si_1.2 is the 100% occupancy of the Y site. In most icosahedron-based metal borides, metal sites have rather low site occupancy, for example, about 50% for YB₆₆ and 60–70% for REAlB₁₄. When the Y site is replaced by rare-earth elements, REB₄₁Si_1.2 can have an antiferromagnetic

Antiferromagnetism

In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...

-like ordering because of this high site occupancy.

Homologous icosahedron-based rare-earth borides

Rare-earth borides REB_15.5CN, REB₂₂C₂N and REB_28.5C₄ are homologous, i.e. have a similar crystal structure, to B₄C

Boron carbide

Boron carbide is an extremely hard boron–carbon ceramic material used in tank armor, bulletproof vests, and numerous industrial applications...

. The latter has a structure typical of icosahedron-based borides, as shown in figure 11a. There, B₁₂ icosahedra form a rhombohedral lattice unit (space group: Rm (No. 166), lattice constants: a = 0.56 nm and c = 1.212 nm) surrounding a C-B-C chain that resides at the center of the lattice unit, and both C atoms bridge the neighboring three icosahedra. This structure is layered: as shown in figure 11b, B₁₂ icosahedra and bridging carbons form a network plane that spreads parallel to the c-plane and stacks along the c-axis.
These homologous compounds have two basic structure units – the B₁₂ icosahedron and the B₆ octahedron. The network plane of B₄C structure can be periodically replaced by a B₆ octahedron layer so that replacement of
every third, fourth and fifth layer would correspond to REB_15.5CN, REB₂₂C₂N and REB_28.5C₄, respectively. The B₆ octahedron is smaller than the B₁₂ icosahedron; therefore, rare-earth elements can reside in the space created by the replacement. The stacking sequences of B₄C, REB_15.5CN, REB₂₂C₂N and REB_28.5C₄ are shown in figures 12a, b, c and d, respectively. High-resolution transmission electron microscopy

Transmission electron microscopy

Transmission electron microscopy is a microscopy technique whereby a beam of electrons is transmitted through an ultra thin specimen, interacting with the specimen as it passes through...

 (HRTEM) lattice images of the latter three compounds, added to Fig. 12, do confirm the stacking sequence of each compound. The symbols 3T, 12R and 15R in brackets indicate the number of layers necessary to complete the stacking sequence, and T and R refer to trigonal and rhombohedral. Thus, REB₂₂C₂N and REB_28.5C₄ have rather large c-lattice constants.

Because of the small size of the B₆ octahedra, they cannot interconnect. Instead, they bond to the B₁₂ icosahedra in the neighboring layer, and this decreases bonding strength in the c-plane. Nitrogen atoms strengthen the bonding in the c-plane by
bridging three icosahedra, like C atoms in the C-B-C chain. Figure 13 depicts the c-plane network revealing the alternate bridging of the boron icosahedra by N and C atoms. Decreasing the number of the B₆ octahedra diminishes the role of nitrogen because the C-B-C
chains start bridging the icosahedra. On the other hand, in MgB₉N the B₆ octahedron layer and the B₁₂ icosahedron layer stack alternatively and there is no C-B-C chains; thus only N atoms bridge the B₁₂ icosahedra. However, REB₉N compounds have not been identified yet.

Sc, Y, Ho, Er, Tm and Lu are confirmed to form REB_15.5CN-type compounds. Single-crystal structure analysis yielded trigonal symmetry for ScB_15.5CN (space group Pm1 (No.164) with a = 0.5568(2) and c = 1.0756(2) nm), and the deduced atomic coordinates are summarized in table IVa.

REB₂₂C₂N was synthesized for Y, Ho, Er, Tm and Lu. The crystal structure, solved for a representative compound YB₂₂C₂N, belongs to the trigonal with space group Rm (No.166); it has six formula units in the unit cell and lattice constants a = b = 0.5623(0) nm and c = 4.4785(3) nm. Atomic coordinates of YB₂₂C₂N are summarized in table IVb.

Y, Ho, Er, Tm and Lu also form REB_28.5C₄ which has a trigonal crystal structure with space group Rm
(No. 166). Lattice constants of the representative compound YB_28.5C₄ are a = b = 0.56457(9) nm and c = 5.68873(13) nm and there are six formula units in the unit cell. Structure data of YB_28.5C₄ are
summarized in table IVc.

RE_xB₁₂C_0.33Si_3.0

Initially these were described as ternary RE-B-Si compounds, but later carbon was included to improve the structure description that resulted in a quaternary RE-B-C-Si composition. RE_xB₁₂C_0.33Si_3.0 (RE=Y and Gd–Lu) have a unique crystal structure with two units – a cluster of B₁₂ icosahedra and a Si₈ ethane

Ethane

Ethane is a chemical compound with chemical formula C2H6. It is the only two-carbon alkane that is an aliphatic hydrocarbon. At standard temperature and pressure, ethane is a colorless, odorless gas....

-like complex – and one bonding configuration (B₁₂)₃≡Si-C≡(B₁₂)₃. A representative compound of this group is Y_xB₁₂C_0.33Si_3.0 (x=0.68). It has a trigonal crystal structure with space group Rm (No. 166) and lattice constants a = b = 1.00841(4) nm, c = 1.64714(5) nm, α = β = 90° and γ = 120°.
The crystal has layered structure. Figure 15 shows a network of boron icosahedra that spreads parallel to the (001) plane, connecting with four neighbors through B1–B1 bonds. The C3 and Si3 site atoms strengthen the network by bridging the boron icosahedra. Contrary to other boron-rich icosahedral compounds, the boron icosahedra from different layers are not directly bonded. The icosahedra within one layer are linked through Si₈ ethane

Ethane

Ethane is a chemical compound with chemical formula C2H6. It is the only two-carbon alkane that is an aliphatic hydrocarbon. At standard temperature and pressure, ethane is a colorless, odorless gas....

-like clusters with (B₁₂)₃≡Si-C≡(B₁₂)₃ bonds, as shown in figures 16a and b.

There are eight atomic sites in the unit cell: one yttrium Y, four boron B1–B4, one carbon C3 and three silicon sites Si1–Si3. Atomic coordinates, site occupancy and isotropic displacement factors are listed in table Va; 68% of the Y sites are randomly occupied and remaining Y sites are vacant. All boron sites and Si1 and Si2 sites are fully occupied. The C3 and Si3 sites can be occupied by either carbon or silicon atoms (mixed occupancy) with a probability of about 50%. Their separation is only 0.413 Å, and thus either the C3 or Si3 sites, but not both, are occupied. These sites form Si-C pairs, but not Si-Si or C-C pairs. The distances between the C3 and Si3 sites and the surrounding sites for Y_xB₁₂C_0.33Si_3.0 are summarized in table Vb and the overall crystal structure is shown in figure 14.
Salvador et al. reported an isotypic terbium compound
Tb_3–xC₂Si₈(B₁₂)₃. Most parts of the crystal structure are the same as those described above; however, its bonding configuration is deduced as (B₁₂)₃≡C-C≡(B₁₂)₃ instead of (B₁₂)₃≡Si-C≡(B₁₂)₃. The authors intentionally added carbon to grow single crystals whereas the previous crystals were accidentally contaminated by carbon during their growth. Thus, higher carbon concentration was achieved. Existence of both bonding schemes of (B₁₂)₃≡Si-C≡(B₁₂)₃ and (B₁₂)₃≡C-C≡(B₁₂)₃ suggests the occpancy of the carbon sites of 50–100%. On the other hand, (B₁₂)₃≡Si-Si≡(B₁₂)₃ bonding scheme is unlikely because of too short Si-Si distance, suggesting that the minimum carbon occupancy at the site is 50%. Some B atoms may replace C atoms at the C3 site, as previously assigned to the B site. However, the carbon occupation is more likely because the site is tetrahedrally coordinated whereas the B occupation of the site needs an extra electron to complete tetrahedral bonding. Thus, carbon is indispensable for this group of compounds.

Scandium compounds

Scandium has the smallest atomic

Atomic radius

The atomic radius of a chemical element is a measure of the size of its atoms, usually the mean or typical distance from the nucleus to the boundary of the surrounding cloud of electrons...

 and ionic

Ionic radius

Ionic radius, rion, is the radius of an atom's ion. Although neither atoms nor ions have sharp boundaries, it is important to treat them as if they are hard spheres with radii such that the sum of ionic radii of the cation and anion gives the distance between the ions in a crystal lattice...

 (3+) radii (1.62 and 0.885 Å, respectively) among the rare-earth elements. It forms several icosahedron-based borides which are not found for other rare-earth elements; however, most of them are ternary Sc-B-C compounds. There are many boron-rich phases in the boron-rich corner of Sc-B-C phase diagram, as shown in figure 17. A slight variation of the composition can produce ScB₁₉, ScB₁₇C_0.25, ScB₁₅C_0.8 and ScB₁₅C_1.6; their crystal structures are unusual for borides and are very different from each other.

ScB_19+xSi_y

ScB_19+xSi_y has a tetragonal crystal structure with space group P4₁2₁2 (No. 92) or P4₃2₁2 and lattice constants of a, b = 1.03081(2) and c = 1.42589(3) nm; it is
isotypic to the α-AlB₁₂ structure type. There are 28 atomic sites in the unit cell, which are assigned to 3 scandium atoms, 24 boron atoms and one silicon atom. Atomic coordinates, site occupancies and isotropic displacement factors are listed in table VI.
The boron framework of ScB_19+xSi_y is based on one B₁₂ icosahedron and one B₂₂ unit. This unit can be observed in β-tetragonal boron and is a modification of the B₂₀ unit of α-AlB₁₂ (or B₁₉ unit in early reports). The B₂₀ unit is a twinned icosahedron made from B13 to B22 sites with two vacant sites and one B atom (B23) bridging both sides of the unit. The twinned icosahedron is shown in figure 18a. B23 was treated as an isolated atom in the early reports; it is bonded to each twinned icosahedra through B18 and to another icosahedron through B5 site. If the twinned icosahedra were independent without twinning then B23 would be a bridge site linking three icosahedra. However, because of twinning, B23 shifts closer to the twinned icosahedra than another icosahedron; thus B23 is currently treated as a member of the twinned icosahedra. In ScB_19+xSi_y, the two B24 sites which correspond to the vacant sites in the B₂₀ unit are partially occupied; thus, the unit should be referred to as a B₂₂ cluster which is occupied by about 20.6 boron atoms. Scandium atoms occupy 3 of 5 Al sites of α-AlB₁₂, that is Sc1, Sc2 and Sc3 correspond to Al4, Al1 and Al2 sites of α-AlB₁₂, respectively. The Al3 and Al5 sites are empty for ScB_19+xSi_y, and the Si site links two B₂₂ units. This phase also exists without silicon.
Figure 19a shows the network of boron icosahedra in the boron framework of ScB_19+xSi_y. In this network, 4 icosahedra form a supertetrahedron

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

 (figure 18b); its one edge is parallel to the a-axis, and the icosahedra on this edge make up a chain along the a-axis. The opposite edge of the supertetrahedron is parallel to the b-axis and the icosahedra on this edge form a chain along the b-axis. As shown in figure 19, there are wide tunnels surrounded by the icosahedron arrangement along the a- and b-axes. The tunnels are filled by the B₂₂ units which strongly bond to the surrounding icosahedra; the connection of the B₂₂ units is helical and it runs along the c-axis as shown in figure 19b. Scandium atoms occupy the voids in the boron network as shown in figure 19c, and the Si atoms bridge the B₂₂ units.

ScB₁₇C_0.25

Very small amount of carbon is sufficient to stabilize "ScB₁₇C_0.25". This compound has a broad composition range, namely ScB_16.5+xC_0.2+y with x ≤ 2.2 and y ≤ 0.44. ScB₁₇C_0.25 has a hexagonal crystal structure

Hexagonal crystal system

In crystallography, the hexagonal crystal system is one of the 7 crystal systems, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families...

 with space group P6mmm (No. 199) and lattice constants a, b = 1.45501(15) nm and c = 0.84543(16) nm.

There are 19 atomic sites in the unit cell, which are assigned to one scandium site Sc, 14 boron sites B1–B14 having 100% occupancy, two boron-carbon mixed-occupancy sites B/C15 and B/C16, and two partial-occupancy boron sites B17 and B18. Atomic coordinates, site occupancies and isotropic displacement factors are listed in table VII. Although a very small amount of carbon (less than 2 wt%!) plays an important role in the phase stability, carbon does not have its own sites but shares with boron two interstitial sites B/C15 and B/C16.

There are two inequivalent B₁₂ icosahedra, I1 and I2, which are constructed by the B1–B5 and B8–B12 sites, respectively. A "tube" is another characteristic structure unit of ScB₁₇C_0.25. It extends along the c-axis and consists of B13, B14, B17 and B18 sites where B13 and B14 form 6-membered rings. B17 and B18 sites also form 6-membered rings; however, their mutual distances (0.985 Å for B17 and 0.955 Å for B18) are too short for a simultaneous occupation of the neighboring sites. Therefore, boron atoms occupy 2nd neighbor site forming a triangle. The occupancies of B17 and B18 sites should be 50%, but the structure analysis suggests larger values. The crystal structure viewed along the a-axis is shown in figure 20, which suggests that the ScB₁₇C_0.25 is a layered material. Two layers, respectively constructed by the icosahedra I1 and I2, alternatively stack along the c-axis. However, the ScB₁₇C_0.25 crystal is not layered. For example, during arc-melting, ScB₁₇C_0.25 needle crystals violently grow along the c-axis – this never happens in layered compounds. The crystal structure viewed along the c-axis is shown in figure 21a. The icosahedra I1 and I2 form a ring centered by the "tube" shown in figure 21b, which probably governs the properties of the ScB₁₇C_0.25 crystal. B/C15 and B/C16 mixed-occupancy sites interconnect the rings. A structural similarity can be seen between ScB₁₇C_0.25 and BeB₃.
Figures 22a and b present HRTEM lattice images and electron diffraction patterns taken along the [0001] and [110] crystalline directions, respectively. The HRTEM lattice image of figure 22a reproduces well the (a, b) plane of the crystal structure shown in figure 21a, with the clearly visible rings membered by icosahedra I1 and I2 and centered by the "tube". Figure 22b proves that ScB₁₇C_0.25 does not have layered character but its c-axis direction is built up by the ring-like structure and tubular structures.

Sc_0.83–xB_10.0–yC_0.17+ySi_0.083–z

Sc_0.83–xB_10.0–yC_0.17+ySi_0.083–z (x = 0.030, y = 0.36 and z = 0.026) has a cubic crystal structure with space group F3m (No. 216) and lattice constant a = 2.03085(5) nm. This compound was initially identified as ScB₁₅C_0.8 (phase I in the Sc-B-C phase diagram of figure 17). A small amount of Si was added into the floating zone crystal growth and thus this phase is a quaternary compound. Its rare cubic structure has 26 sites in the unit cell: three Sc sites, two Si sites, one C site and 20 B sites; 4 out of 20 B sites are boron-carbon mixed-occupancy sites. Atomic coordinates, site occupancies and isotropic displacement factors are listed in table VIII.

In the unit cell, there are three independent icosahedra, I1, I2 and I3, and a B₁₀ polyhedron which are formed by the B1–B4, B5–B8, B9–B13 and B14–B17 sites, respectively.There are more than 4 sites in total among, say, B5–B8 sites, but many of them are equivalent by symmetry and thus do not have an individual label. The B₁₀ polyhedron has not been observed previously and it is shown in figure 23. The icosahedron I2 has a boron-carbon mixed-occupancy site B,C6 whose occupancy is B/C=0.58/0.42. Remaining 3 boron-carbon mixed-occupancy sites are bridge sites; C and Si sites are also bridge sites.
More than 1000 atoms are available in the unit cell, which is built up by large structure units such as two supertetrahedra T(1) and T(2) and one superoctahedron O(1). As shown in figure 24a, T(1) consists of 4 icosahedra I(1) which have no direct bonding but are bridged by four B and C20 atoms. These atoms also form tetrahedron centered by the Si2 sites. The supertetrahedron T(2) that consists of 4 icosahedra I(2) is the same as shown in figure 18b; its mixed-occupancy sites B and C6 directly bond with each other. The superoctahedron O(1) consists of 6 icosahedra I(3) and bridge sites B, C18, C1 and Si1; here Si1 and C1 exhibit a tetrahedral arrangement at the center of O(1). Interestingly, the B₁₀ polyhedra also arrange octahedrally, without the central atom, as shown in figure 24c where the B and C19 atoms bridge the B₁₀ polyhedra to form the octahedral supercluster of the B₁₀ polyhedra.
Using these large polyhedra, the crystal structure of Sc_0.83–xB_10.0–yC_0.17+ySi_0.083–z can be described as shown in figure 25. Owing to the crystal symmetry, the tetrahedral coordination between these superstructure units is again a key factor. The supertetrahedron T(1) lies at the body center and at the edge center of the unit cell. The superoctahedra O(1) locate at the body center (0.25, 0.25, 0.25) of the quarter of the unit cell. They coordinate tetrahedrally around T(1) forming a giant tetrahedron. The supertetrahedra T(2) are located at the symmetry-related positions (0.25, 0.25, 0.75); they also form a giant tetrahedron surrounding T(1). Edges of both giant tetrahedra orthogonally cross each other at their centers; at those edge centers, each B₁₀ polyhedron bridges all the super-structure clusters T(1), T(2) and O(1). The superoctahedron built of B₁₀ polyhedra is located at each cubic face center.

Scandium atoms reside in the voids of the boron framework. Four Sc1 atoms form a tetrahedral arrangement inside the B₁₀ polyhedron-based superoctahedron. Sc2 atoms sit between the B₁₀ polyhedron-based superoctahedron and the O(1) superoctahedron. Three Sc3 atoms form a triangle and are surrounded by three B₁₀ polyhedra, a supertetrahedron T(1) and a superoctahedron O(1).

ScB_14–xC_x (x = 1.1) and ScB₁₅C_1.6

ScB_14–xC_x has an orthorhombic crystal structure with space group Imma (No. 74) and lattice constants of a = 0.56829(2), b = 0.80375(3) and c = 1.00488(4) nm. The crystal structure of ScB_14–xC_x is isotypic to that of MgAlB₁₄ where Sc occupies the Mg site, the Al site is empty and the boron bridge site is a B/C mixed-occupancy site with the occupancy of B/C = 0.45/0.55. The occupancy of the Sc site in flux-grown single crystals is 0.964(4), i.e. almost 1. Solid-state powder-reaction growth resulted in lower Sc site occupancy and in the resulting chemical composition ScB₁₅C_1.6. The B-C bonding distance 0.1796(3) nm between the B/C bridge sites is rather long as compared with that (0.15–0.16 nm) of an ordinary B-C covalent bond, that suggests weak bonding between the B/C bridge sites.

Sc_4.5–xB_57–y+zC_3.5–z

Sc_4.5–xB_57–y+zC_3.5–z (x = 0.27, y = 1.1, z = 0.2) has an orthorhombic crystal structure with space group Pbam (No. 55) and lattice constants of a = 1.73040(6), b = 1.60738(6) and c = 1.44829(6) nm. This phase is indicated as ScB_12.5C_0.8 (phase IV) in the phase diagram of figure 17. This rare orthorhombic structure has 78 atomic positions in the unit cell: seven partially occupied Sc sites, four C sites, 66 B sites including three partially occupied sites and one B/C mixed-occupancy site. Atomic coordinates, site occupancies and isotropic displacement factors are listed in table IX.

More than 500 atoms are available in the unit cell. In the crystal structure, there are six structurally independent icosahedra I1–I6, which are constructed from B1–B12, B13–B24, B25–B32, B33–B40, B41–B44 and B45–B56 sites, respectively; B57–B62 sites form a B₈ polyhedron. The Sc_4.5–xB_57–y+zC_3.5–z crystal structure is layered, as shown in figure 26. This structure has been described in terms of two kinds of boron icosahedron layers, L1 and L2. L1 consists of the icosahedra I3, I4 and I5 and the C65 "dimer", and L2 consists of the icosahedra I2 and I6. I1 is sandwiched by L1 and L2 and the B₈ polyhedron is sandwiched by L2.
An alternative description is based on the same B₁₂(B₁₂)₁₂supericosahedron as in the YB₆₆ structure. In the YB₆₆ crystal structure, the　supericosahedra form 3-dimensional boron framework as shown in figure 5. In this framework, the neighboring supericosahedra are rotated 90° with respect to each other. On the contrary, in Sc_4.5–xB_57–y+zC_3.5–z the supericosahedra form a 2-dimensional network where the 90° rotation relation is broken because of the orthorhombic symmetry. The planar projections of the supericosahedron connection in Sc_4.5–xB_57–y+zC_3.5–z and YB₆₆ are shown in figures 27a and b, respectively. In the YB₆₆ crystal structure, the neighboring 2-dimensional supericosahedron connections are out-of-phase for the rotational relation of the supericosahedron. This allows 3-dimensional
stacking of the 2-dimensional supericosahedron connection while maintaining the cubic symmetry.

The B₈₀ boron cluster occupies the large space between four supericosahedra as described in the REB₆₆ section. On the other hand, the 2-dimensional supericosahedron networks in the Sc_4.5–xB_57–y+zC_3.5–z crystal structure stack in-phase along the z-axis. Instead of the B₈₀ cluster, a pair of the I2 icosahedra fills the open space staying within the supericosahedron network, as shown in figure 28 where the icosahedron I2 is colored in yellow.

All Sc atoms except for Sc3 reside in large spaces between the supericosahedron networks, and the Sc3 atom occupies a void in the network as shown in figure 26. Because of the small size of Sc atom, the occupancies of the Sc1–Sc5 sites exceed 95%, and those of Sc6 and Sc7 sites are approximately 90% and 61%, respectively (see table IX).

Sc_3.67–xB_41.4–y–zC_0.67+zSi_0.33–w

Sc_3.67–xB_41.4–y–zC_0.67+zSi_0.33–w (x = 0.52, y = 1.42, z = 1.17 and w = 0.02) has a hexagonal crystal structure with space group Pm2 (No. 187) and lattice constants a = b = 1.43055(8) and c = 2.37477(13) nm. Single crystals of this compound were obtained as an intergrowth phase in a float-zoned single crystal of Sc_0.83–xB_10.0–yC_0.17+ySi_0.083–z. This phase is not described in the phase diagram of figure 17 because it is a quaternary compound. Its hexagonal structure is rare and has 79 atomic positions in the unit cell: eight partially occupied Sc sites, 62 B sites, two C sites, two Si sites and six B/C sites. Six B sites and one of the two Si sites have partial occupancies. The associated atomic coordinates, site occupancies and isotropic displacement factors are listed in table X.

There are seven structurally independent icosahedra I1–I7 which are formed by B1–B8, B9–B12, B13–B20, B/C21–B24, B/C25–B29, B30–B37 and B/C38–B42 sites, respectively; B43–B46 sites form the B₉ polyhedron and B47–B53 sites construct the B₁₀ polyhedron. B54–B59 sites form the irregularly shaped B₁₆ polyhedron in which only 10.7 boron atoms are available because most of sites are too close to each other to be occupied simultaneously. Ten bridging sites C60–B69 interconnect polyhedron units or other bridging sites to form a 3D boron framework structure. One description of the crystal structure uses three pillar-like units that extend along the c-axis that however results in undesired overlaps between those three pillar-like units. An alternative is to define two pillar-like　structure units. Figure 29 shows the boron framework structure of　Sc_3.67–xB_41.4–y–zC_0.67+zSi_0.33–w viewed along the c-axis, where the pillar-like units P1 and P2 are colored in dark green and light green respectively and are bridged by yellow icosahedra I4 and I7.

These pillar-like units P1 and P2 are shown in figures 30a and b, respectively.　P1 consists of icosahedra I1 and I3, an irregularly shaped B₁₆ polyhedron and　other bridge site atoms where two supericosahedra can be seen above and below the B₁₆ polyhedron. Each supericosahedron is formed by three icosahedra I1 and three icosahedra I3 and is the same as the supericosahedron　O(1) shown in figure 24a.The P2 unit consists of icosahedra I2, I5 and I6, B₁₀ polyhedron and other　bridge site atoms. Eight Sc sites with occupancies between 0.49　(Sc8) and 0.98 (Sc1) spread over the boron framework.

As described above, this hexagonal phase originates from a cubic phase, and thus one may expect a similar structural element in these phases. There is an obvious relation between the hexagonal ab-plane and the cubic (111) plane. Figures 31a and b show the hexagonal (001) and the cubic (111) planes, respectively. Both network structures are almost the same that allows intergrowth of the hexagonal phase in the cubic phase.

Applications

The diversity of the crystal structures of rare-earth borides results in interesting physical properties and potential applications in thermopower

Thermopower

The thermopower, or thermoelectric power of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material...

 generation. Thermal conductivity

Thermal conductivity

In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....

 of boron icosahedra based compounds is low because of their complex crystal structure; this property is favored for thermoelectric materials. On the other hand, these compounds exhibit very low (variable range hopping type) p-type

P-type semiconductor

A P-type semiconductor is obtained by carrying out a process of doping: that is, adding a certain type of atoms to the semiconductor in order to increase the number of free charge carriers ....

 electrical conductivity. Increasing the conductivity is a key issue for thermoelectric applications of these borides.

YB₆₆ is used as a soft-X-ray

X-ray

X-radiation is a form of electromagnetic radiation. X-rays have a wavelength in the range of 0.01 to 10 nanometers, corresponding to frequencies in the range 30 petahertz to 30 exahertz and energies in the range 120 eV to 120 keV. They are shorter in wavelength than UV rays and longer than gamma...

 monochromator

Monochromator

A monochromator is an optical device that transmits a mechanically selectable narrow band of wavelengths of light or other radiation chosen from a wider range of wavelengths available at the input...

 for dispersing 1–2 keV synchrotron

Synchrotron

A synchrotron is a particular type of cyclic particle accelerator in which the magnetic field and the electric field are carefully synchronised with the travelling particle beam. The proton synchrotron was originally conceived by Sir Marcus Oliphant...

radiation at some synchrotron radiation facilities. Contrary to thermoelectric applications, high thermal conductivity is desirable for synchrotron radiation monochromators. YB₆₆ exhibits low, amorphous-like thermal conductivity. However, transition metal doping increases the thermal conductivity twice in YNb_0.3B₆₂ as compared to undoped YB₆₆.