# Single atoms, wave crossings and pure crystal geometry Element Projection Lab 0.16.0 · 21 September 2026 Open `outputs/atom_crystal_observer.html`. It works offline. The initial scene is a single hydrogen atom in a specified coherent state. Quick selections also show diamond carbon's local tetrahedron and a chosen faceted gold specimen. The element selector retains all 118 identities, 92 available crystal phases and 30 sourced optical datasets. Missing crystal coordinates and optical data remain visibly unavailable. One hydrogen crystal phase is a DFT prediction at 300 GPa, explicitly labeled; it is not ambient-pressure hydrogen. ## What the three channels mean | Channel | Geometry supplied to the renderer | Physical interpretation | |---|---|---| | Single atom, averaged envelope | Sphere at the archived r90 radius | Encloses 90% of the rotation-averaged model electron density; spherical by construction. Screened independent-orbital proxy for multielectron atoms | | Single atom, angular envelope | r(theta)=r90 abs(P_l(cos theta)), l=0,1,2,3 | Radius encodes one real canonical s/p/d/f angular amplitude; not a probability-density isosurface or full neutral-atom shape | | Hydrogen 1s, 2p, or coherent state | Numerically triangulated density level set | Standard nonrelativistic Coulomb wavefunction, not a hard atom boundary or a visible-light photograph | | Crystal, nearest sites and faces | Convex hull of reference sites within 1.10 times the nearest distance | A geometric coordination polyhedron; guides are not chemical-bond strengths or atom skins | | Crystal, primitive cell | Reference primitive parallelepiped and basis positions | A repeat cell under periodic continuation, not a smallest stable isolated crystal | | Crystal, faceted specimen | Convex hull of sites inside a chosen finite cut | Explicit display construction, not a predicted equilibrium or growth habit | | Harmonic construction | Crossings of a chosen three-variable scalar field | A demonstration of wave-defined surfaces; not an inferred density or structure of the selected element | Filled subshells have no privileged displayed orbital orientation. Displaying one canonical basis function does not imply that all electrons occupy that shape. Multielectron proxies omit correlation and relativistic effects and cannot substitute for Hartree–Fock, DFT or relativistic many-body calculations. The existing source/configuration caveats for heavy atoms remain in force. Objects are fitted to the viewing area individually; the scale bar retains angstroms, Bohr radii, or phase radians. Apparent screen size is not a shared physical size comparison. Nuclear/site markers are enlarged visual markers. ## Hydrogen field and oscillation control Coordinates below use a0, with infinite-nuclear-mass nonrelativistic hydrogen: \[ \psi_{1s}(r)=\pi^{-1/2}e^{-r},\qquad \psi_{2p_z}(x,y,z)=\frac{z}{4\sqrt{2\pi}}e^{-r/2}. \] The interference view uses the explicitly chosen normalized state \[ \psi_\phi=0.2\psi_{1s}+\sqrt{0.96}\,e^{i\phi}\psi_{2p_z}, \qquad\rho_\phi=|\psi_\phi|^2. \] Orthogonality makes its integrated density one for every phase. It is an excited coherent preparation, not the ground-state shape of ordinary hydrogen. In free evolution phi(t)=phi(0)-(E2-E1)t/hbar, modulo 2pi; the manual phase control is a slowed inspection coordinate, not a real-time display of atomic motion. A stationary 1s state has time-independent density. The displayed thresholds are 0.02/pi for 1s, 0.001 for 2p and 0.0008 for the coherent example, in a0^-3. These thresholds are display choices, not atomic surfaces. Marching tetrahedra uses 29 samples per axis for these examples. Amplitude-sign colors label the real part in the chosen phase convention; they are not emitted/reflected visible colors, and a purely imaginary contribution is not separately encoded as a third color. ## Wave crossings become surfaces The three constructions are \[ f_1=\cos(x+\phi)+\cos y+\cos z, \] \[ f_2=\cos(x+\phi)\cos y+\cos y\cos z+\cos z\cos(x+\phi), \] \[ f_3=\sin(x+\phi)\cos y+\sin y\cos z+\sin z\cos(x+\phi). \] The renderer extracts f=c inside [-pi,pi]^3 using 23 samples per axis. The phase and crossing-level controls alter the actual vertices. A triangle vertex comes from linear interpolation along an edge that brackets the chosen level; faces join these intersections in the tetrahedral subdivision. Domain clipping can leave open boundaries. The sine–cosine example is a nodal construction, not an exact minimal-surface solver. Surface triangles are numerical approximations; not every mesh is a crystal or an electron probability cloud. This makes the user's proposed operation explicit: wave values -> crossings -> coordinates -> faces -> a three-dimensional projection. It does not add a measured fourth or fifth spatial coordinate or recover chemical identity. ## Crystal faces Reference primitive vectors and all fractional basis coordinates are preserved unchanged. Fractional translations are bounded using the inverse lattice matrix so every site within the declared radial query is included. Local faces are computed from the convex hull of the chosen neighbor shell. Coplanar hull triangles are grouped into geometric faces; internal triangulation diagonals are omitted from the geometric-edge list. | Reference | Neighbor vertices | Geometric faces | Local hull | |---|---:|---:|---| | Diamond carbon | 4 | 4 | Tetrahedron | | BCC iron | 8 | 6 | Cube | | FCC gold | 12 | 14 | Cuboctahedron | | HCP magnesium | 12 | 14 | HCP neighbor polyhedron; different coordinates from FCC | Graphite's closest three neighbors are coplanar. The local view retains that fact and does not invent a nonzero hull volume. Its larger specimen includes several layers and has nonzero volume. For the specimen cut let d be the nearest distance and R=3.1d. Retain lattice sites with max(|x|,|y|,|z|)<=R and |x|+|y|+|z|<=1.65R, centered at basis site zero, then take their convex hull. Those Cartesian cuts are chosen display boundaries. Surface reconstructions, defects, temperature-dependent expansion and growth kinetics are absent. Predicting actual equilibrium facets requires orientation-dependent surface energies; equilibrium and growth shapes need not coincide. See Herring's primary analysis [1]. ## Fixed light and human color observer The camera uses orthographic projection along +z. Dragging or the angle controls rotate the object while the light and camera remain fixed. The light direction is the normalized vector (-0.55,0.8,1). This is an enlarged geometrical display, not an attempt to resolve an atom with visible light. The spectral layer retains the existing 5772 K blackbody solar approximation, unpolarized Fresnel response from sourced n(lambda),k(lambda), CIE 1931 two-degree color-matching functions [2], Bradford adaptation and sRGB conversion. There are 29 measured-input materials and one compiled-input material. Gold, silver and copper retain the Johnson–Christy optical inputs [3]. Facet-color lookups contain linear RGB at 0,15,30,45,60,75,85 degrees. No unsupported element receives an invented optical color; neutral shading displays its geometry. The shaded three-dimensional preview uses that bulk color with an explicitly illustrative facet-lighting factor 0.22+0.62 max(n.L,0)+0.75 max(n.H,0)^42, where H is the normalized light-plus-view direction. It is not an energy- normalized BRDF, a Maxwell scattering calculation, or a measured prediction of the nanocluster's appearance. It has no multiple reflections, diffraction, shadows, transmission, oxide layer, roughness model or size-dependent electronic response. Borrowed optical data need not match the selected phase or orientation. The color observer is a standard colorimetric reference, not a complete one-to-one simulation of a particular human eye, retina or perception. Atomic and synthetic wave surfaces use semantic sign/geometry colors. The bulk-material optics layer is never used to claim visible single-atom color. ## Executed checks and preserved limitations Seven new unit groups verify all 92 phase-coordinate records, known neighbor polyhedra, oriented hull volumes and Euler topology, compatibility with the previous nearest-site environments, hydrogen normalization/orthogonality, normal-incidence color preservation and graphite's planar-shell boundary. The actual interface script and native Canvas2D renderer are exercised at widths 320 and 736, including all 118 atom entries, all 92 phases in three geometry modes, missing-data handling, rotation, faces, phase and wave selections. The rendering test also verifies the hydrogen phase-reflection identity and an independent sphere level set. Its maximum radius error is 0.005118 for a unit sphere on the declared 25-point-per-axis grid; this is a bounded accuracy check, not a claim of that error bound for every atom mesh. Preview images were inspected. A browser executable is unavailable in this runtime, so these are actual-script/Canvas tests, not full browser CSS/layout verification. Counts and execution logs are in `outputs/atom_crystal_view_checks.json` and `outputs/tests_0.16.txt`. Earlier measurements, proofs, negative results and source data are unchanged. README/ledger backups retain the prior release. ## Reproduce and next physical gate ``` python3 build_atom_crystal_view.py python3 -m unittest discover -v node check_atom_crystal_view.cjs ``` The Python dependencies are already in requirements.txt. The optional renderer test additionally uses the Node package @napi-rs/canvas; it is unnecessary to open the standalone viewer, which needs no network or installation. Next physical gate: choose one specified element/phase and size, acquire sourced surface energies to predict an equilibrium habit, then use a suitable phase- and size-matched electromagnetic response and compare a frozen prediction with independent microscopy and optical data. Bulk tint on a chosen hull does not satisfy that gate. ## Sources 1. C. Herring, *Some Theorems on the Free Energies of Crystal Surfaces*, Physical Review 82, 87, published 1 April 1951. [Primary paper](https://journals.aps.org/pr/abstract/10.1103/PhysRev.82.87). 2. CIE, *CIE 1931 colour-matching functions, 2 degree observer*, 2019 dataset, 1 nm steps. [Official data](https://cie.co.at/datatable/cie-1931-colour-matching-functions-2-degree-observer). 3. P. B. Johnson and R. W. Christy, *Optical Constants of the Noble Metals*, Physical Review B 6, 4370, 15 December 1972. [Primary paper](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.6.4370). Pages checked 21 September 2026. Full prior raw inputs, citations and licensing remain in the archive's data directories and earlier method documents.