Allen Orbital Lattice Primer: Crystal Systems, Radii, and Packing

Allen, James Johan Sebastian

Allen Orbital Lattice Primer: Crystal Systems, Radii, and Packing

The Allen Orbital Lattice (AOL) provides a single geometric field that explains why all seven classical crystal systems—and their common packings (SC, BCC, FCC, HCP)—emerge. Instead of treating atomic and ionic radii as disconnected “lookup” values, AOL derives them from a common hexagonal coherence and a ring-spacing constant. This page introduces that idea, shows how standard formulas fall out as projections, and outlines the AOL Crystal & Radius Unifier (AOL-CRU™) we’ll use across Pattern Field Theory.

1) The Traditional View

Crystallography classifies solids by axis lengths (a, b, c) and angles (α, β, γ). Each system can have simple, body-centered, face-centered, or base-centered cells.

System	Axis Relationships	Examples
Cubic	a = b = c; α = β = γ = 90°	NaCl, Cu, Fe
Tetragonal	a = b ≠ c; α = β = γ = 90°	TiO₂, Sn
Orthorhombic	a ≠ b ≠ c; α = β = γ = 90°	S, KNO₃
Monoclinic	a ≠ b ≠ c; α = γ = 90°, β ≠ 90°	Gypsum
Triclinic	a ≠ b ≠ c; α ≠ β ≠ γ ≠ 90°	K₂Cr₂O₇
Hexagonal	a = b ≠ c; α = β = 90°, γ = 120°	Mg, Zn
Rhombohedral	a = b = c; α = β = γ ≠ 90°	Calcite

Standard “hard-sphere” radius relations taught in introductory texts:

Simple Cubic (SC): a = 2r
Body-Centered Cubic (BCC): √3 · a = 4r
Face-Centered Cubic (FCC): √2 · a = 4r
Hexagonal Close-Packed (HCP): a = 2r, with c/a ≈ 1.633

2) The AOL View

In AOL, every crystal is a projection of a deeper six-fold coherence field. The “cube” is not fundamental; it is a rational slice of a hexagonal root. Tetragonal and orthorhombic forms are axial deformations of the same hex field; monoclinic and triclinic are tilted/phase-shifted versions.

Radius as a field amplitude. Instead of an empirical sphere, the atomic (or ionic) radius is the mean amplitude of the orbital wave where interference reaches equilibrium on the lattice:

r = k · s_n

s_n — ring spacing on the AOL (normalized hex units)
k — element/ion coherence scale (depends on valence and charge)

With this, the familiar SC/BCC/FCC/HCP relations are recovered as simple angular projections of the same hex spacing:

SC: projection at 90°
BCC: offset at 54.7356° (arccos(1/√3))
FCC: offset at 35.2644° (arccos(1/√8))
HCP: direct 120° hex section (native AOL plane)

3) Why Calculations Get Easier

Atomic and ionic radii unify. Both are derived from the same resonance envelope; ionic radii are contracted/expanded by charge: r_ion = r_atom(1 − Δφ/φ₀).
Coordination number becomes the count of coherent neighbors on the hex ring (6, 8, 12, …), not a table lookup.
Density and packing fractions follow from ring sums and the AOL equilibrium constant (Basel link π²/6 ≈ 1.644934), matching empirical 0.524 (SC), 0.680 (BCC), 0.740 (FCC/HCP).

4) Classical Packings Explained by AOL

Classical Packing	AOL Interpretation	Coordination / Geometry
Simple Cubic (SC)	First hex projection ring	6 neighbors → one hex ring
Body-Centered Cubic (BCC)	Two interleaved hex layers (½-phase offset)	8 neighbors → ring + 2 poles
Face-Centered Cubic (FCC)	Three hex layers (ABC stacking)	12 neighbors → two rings + caps
Hexagonal Close-Packed (HCP)	Two hex layers (AB stacking)	12 neighbors → pure 6-fold equilibrium
Tetragonal / Orthorhombic	Axially stretched hex	Anisotropic distortions of 120° symmetry
Monoclinic / Triclinic	Tilted hex (phase-shifted)	Lower symmetry via off-axis equilibrium
Rhombohedral	Hex core seen obliquely	Equal edges; angles ≠ 90°

5) Practical Consequences

One equation, all systems: once s_n and k are known, radii follow for SC/BCC/FCC/HCP and deformed cells.
Ionic corrections: charge state updates k (field contraction), not the geometry.
Packing fractions: computed by AOL ring counts; the classical values appear as AOL equilibria.

6) Summary (Formulas as Projections)

Category	AOL Source	Simplified Expression
Cubic (SC)	Hex projection at 90°	r = a/2
BCC	Dual hex offset (√3 axis)	r = (√3/4)·a
FCC	Triple hex rotation (√2 face)	r = (√2/4)·a
HCP	Native hex alignment	c/a ≈ 1.633 = √(8/3) ≈ π²/6 − 1
Others	Distorted hex	Derived via hex scaling factors

The proximity of HCP c/a ≈ 1.633 to the Basel limit π²/6 ≈ 1.644934 reflects AOL’s equilibrium of packing coherence—linking lattice geometry to number-theoretic structure.

7) The AOL Crystal & Radius Unifier (AOL-CRU™)

Purpose. A lightweight calculator that accepts lattice type (SC/BCC/FCC/HCP or deformed), conventional parameter(s) (e.g., a, c/a), and (optionally) charge state, then returns:

Atomic/Ionic radius via r = k · s_n and charge scaling
Coordination number and packing fraction from AOL ring counts
Density by ρ = (Z·M) / (N_A · a³) or HCP analog
Projection maps (111) → AOL hex plane with ring indexing

Inputs: lattice system, cell constants (a, b, c, α, β, γ), element/ion, valence/charge.
Outputs: r (atomic/ionic), CN, APF, ρ, AOL ring coordinates (q, r, s), and projection plot.

8) What Follows (Deep-Dive Articles)

From Cubes to Hex: deriving SC/BCC/FCC as AOL projections (with (111) overlays)
Ionic Radii from Field Contraction: a charge-dependent k model with examples
Basel & Packing: π²/6 ring sums and the origin of 0.524/0.680/0.740
Deformed Cells: tetragonal/orthorhombic/monoclinic/triclinic as hex distortions
Rhombohedral Frames: oblique views of the hex core

Pattern Field Theory