Diamond 2H Rod Lattice Framework

• Diamond 2H Rod Lattice is a three-dimensional framework with tetrahedral coordination and ABAB hexagonal stacking that maximizes space efficiency.
• Its geometry features identical 4.90 nm rods embedded in a diamond network, achieving a 95.83% space-filling fraction through interlocking truncated tetrahedra.
• The structure is crucial for understanding molecular aggregation and optical properties of hemoglycin in astrophysical environments and related chemical systems.

The diamond 2H rod lattice is a highly regular three-dimensional framework derived by substituting each bond of the ordinary tetrahedrally coordinated diamond network with identical rigid rods. In the context of hemoglycin, each rod has a measured length $h = 4.90\,\mathrm{nm}$, producing a unique lattice topology characterized by both tetrahedral coordination at each vertex and hexagonal (2H) stacking symmetry (space group $P6_3/mmc$). The resulting structure efficiently fills nearly all of space using a minimum of molecular material, with a maximum theoretical space-filling fraction of $23/24$ (95.83%), realized by a tessellation of regular truncated tetrahedra that conform to the diamond 2H net. This geometry is significant for understanding molecular aggregation in astrophysical and chemical environments, as well as the optical properties of hemoglycin and related phases (McGeoch et al., 13 Jul 2025).

1. Symmetry and Construction of the 2H Rod Lattice

The diamond 2H rod lattice arises by embedding identical rods along the edges of a diamond network, so that four rods meet tetrahedrally at each vertex, reproducing the characteristic bond angle $\sim 109.47^\circ$. The designation “2H” denotes the hexagonal polytype of diamond known as lonsdaleite, crystallographically defined by ABAB... stacking of puckered hexagonal layers (space group $P6_3/mmc$). This stacking introduces a distinct symmetry compared to the cubic (3C) polytype, where layers stack in ABC... fashion.

A generic rod lattice may, in principle, have $N$ rods at each vertex, but only $N=4$ yields tetrahedral coordination (diamond net), while $N=6$ yields cubic coordination (simple cubic net). Among such lattices, those with tetrahedral ($N=4$) coordination maximize packing efficiency per rod (McGeoch et al., 13 Jul 2025).

2. Lattice Parameters, Unit Cells, and Layer Stacking

The conventional unit cell for the diamond 2H rod lattice is hexagonal, with experimentally derived constants:

• $$a_{\mathrm{hex}} \approx h\sqrt{3} = 8.49\,\mathrm{nm}$$,
• $P6_3/mmc$0,
• $P6_3/mmc$1, $P6_3/mmc$2.

The four basis vertices per conventional cell are located at fractional coordinates: $P6_3/mmc$3

Each vertex is connected by rods to four tetrahedral neighbors—three in the same hexagonal layer, one above or below. The alternative primitive rhombohedral cell is specified with edge length $P6_3/mmc$4 and a characteristic inter-axis angle of $P6_3/mmc$5, with an internal two-site basis at $P6_3/mmc$6 and $P6_3/mmc$7.

Layer stacking follows an ABAB... sequence, with planar puckered hexagons observable in projections along the $P6_3/mmc$8-axis. Each hexagonal layer consists of six rods; adjacent layers are rotated by $P6_3/mmc$9 and shifted by one rod length to interconnect “high” and “low” points across layers (McGeoch et al., 13 Jul 2025).

3. Space-Filling Derivation and Efficiency

The maximal space-filling geometry is achieved through a construction based on regular tetrahedral coordination. The process proceeds as follows:

• At each vertex, the largest inscribed sphere (insphere) touches the four faces of the surrounding regular tetrahedron, with $23/24$0,
• Expanding the insphere so it meets the mid-points of the six emanating rods gives $23/24$1, hence $23/24$2,
• Overlapping is avoided by replacing the full regular tetrahedron with an Archimedean truncated tetrahedron (TTA), truncated at one third of each edge.

The relevant volumes are:

Entity	Volume (in $23/24$3 units)	Notes
Truncated tetrahedron (TTA)	$23/24$4	$23/24$5
Diamond 2H quasi-cell	$23/24$6	“Half‐cell” containing one vertex
Space-filling fraction	$23/24$7	Maximum attainable by this arrangement

The truncated tetrahedra abut precisely face-to-face at rod midpoints, resulting in a tessellation where only regular tetrahedral voids (edge $23/24$8) between layers are unfilled. The empty fraction is exactly $23/24$9.

4. Combined Structure: Rods and Truncated Tetrahedra

The space-filling structure is explicitly realized by assembling truncated tetrahedra (TTA) around each vertex, interlocking through faces perpendicular to the rods. Each puckered hexagonal layer alternates TTA orientations (“up” or “down”) matching the puckering pattern. The rods themselves form the 1D edges of the skeleton, while the faces of the TTAs coincide with planes normal to the rods at their midpoints.

Between layers, the aforementioned tetrahedral voids of edge $\sim 109.47^\circ$0 are rotated by $\sim 109.47^\circ$1 and reside only for “half their thickness” due to the ABAB stacking, resulting in exactly $\sim 109.47^\circ$2 unfilled volume. This construction achieves near-complete filling compatible with the diamond 2H symmetry, representing an efficient arrangement for self-assembling rod-like molecular species such as hemoglycin (McGeoch et al., 13 Jul 2025).

5. Geometric Details and Interlocking Mechanism

The coordinates for a single regular tetrahedron (edge $\sim 109.47^\circ$3) may be taken as: $\sim 109.47^\circ$4

Truncation at $\sim 109.47^\circ$5 of each edge produces new triangular faces which, for contiguous TTAs, coalesce into hexagons lying perpendicular to each rod direction. Within any given hexagonal layer, these faces form the “puckered” hexagon (Figure 1(a–b) in the source). The interlocking rule ensures that each TTA’s hexagon face matches precisely with its neighbor’s along a rod, while the only empty space is the small residual tetrahedral voids.

Key angular relationships include:

• Rod-rod angle at each vertex: $\sim 109.47^\circ$6 (tetrahedral)
• Puckering angle in hexagon (edge rotation from flat): $\sim 109.47^\circ$7

6. Lattice Parameters and Cell Conversions

Measured parameters and relationships are as follows:

• Rod length $\sim 109.47^\circ$8 (from X-ray measurement),
• Conventional hexagonal lattice constants: $\sim 109.47^\circ$9, $P6_3/mmc$0,
• Primitive rhombohedral conversion: cell edge $P6_3/mmc$1, angle $P6_3/mmc$2,
• Four-atom basis: $P6_3/mmc$3, $P6_3/mmc$4 reproduces the 2H net.

These relationships facilitate transformations between crystallographic settings and provide explicit correspondence between the rod geometry and conventional crystallographic descriptors.

7. Significance and Physical Implications

The diamond 2H rod lattice, by achieving a $P6_3/mmc$5 ($P6_3/mmc$6) space-filling fraction through minimal use of material, provides a plausible mechanism for efficient volume coverage in the context of rod-like hemoglycin molecules. Such efficiency is argued to be essential in molecular cloud accretion processes, potentially accounting for the widespread cosmochemical occurrence of hemoglycin and its observation in meteoritic and cometary material. The structure’s minimal material maximizes accretion and supports the persistence of large-scale molecular frameworks in astrophysical environments.

Furthermore, quantum calculations of the hemoglycin lattice’s optical properties yield extinction features in close correspondence with the $P6_3/mmc$7 “UV bump” and key visible absorption features observed in astronomical extinction data, indicating that the diamond 2H rod lattice topology determines both the physical packing and the photonic response of these rod-based phases (McGeoch et al., 13 Jul 2025).