SCIENCE · INVESTIGATE AND EXPLAIN

Count shared lattice sites before calculating density

Connect conventional-cell bookkeeping, geometric radius and bulk mass per volume.

Learning outcome: Compare SC,BCC,FCC at the same edge and molar mass. Retain site shares, assigned count, radius relation and density; then increase edge with structure fixed.

Starting recap

Review signed energy transfers, dimensional consistency, reaction coefficients, state functions and phase/structural evidence boundaries.

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Try

An FCC conventional cube contains8×1/8+6×1/2=4 assigned atoms, not14 whole atoms.

Open the experiment

Predict

Does drawing an atom at every visible corner mean every atom belongs exclusively to that unit cell?

Your prediction and reason

Experiment

Compare SC,BCC,FCC at the same edge and molar mass. Retain site shares, assigned count, radius relation and density; then increase edge with structure fixed.

Count shared sites before inferring crystal density

Worked teaching model · synthetic values · Fictional monatomic cubic crystal with molar mass40.0 g/mol, exact NA=6.02214076×10²³ mol⁻¹, full occupancy and no defects. Conventional-cell counts1,2,4 atoms for SC/BCC/FCC. Touching-sphere radius is a geometric model, not a hard atomic surface.

Cubic structure:0=SC,1=BCC,2=FCC: 2 choice Cell edge: 0.4 nm

Reset keeps your prediction, captures and practice history.

Oblique schematic, not hard atomic surfaces. Hidden sites are retained. Fractions count atoms assigned to this conventional cell, not the number of drawn circles.

Site	x	y	z	Assigned fraction
corner	0	0	0	0.125
corner	0	0	1	0.125
corner	0	1	0	0.125
corner	0	1	1	0.125
corner	1	0	0	0.125
corner	1	0	1	0.125
corner	1	1	0	0.125
corner	1	1	1	0.125
face center	0	0.5	0.5	0.5
face center	1	0.5	0.5	0.5
face center	0.5	0	0.5	0.5
face center	0.5	1	0.5	0.5
face center	0.5	0.5	0	0.5
face center	0.5	0.5	1	0.5

Calculated evidence under the stated assumptions
Quantity	Value	Unit
Atoms assigned per conventional cell	4	atoms
Cell volume	6.400000e-23	cm³
Cell mass	2.656863e-22	g
Crystal density	4.151348	g/cm³
Touching-sphere radius	0.1414214	nm

Face-centered cubic

Each corner belongs to eight neighboring cells; a face center to two, and a body center to one. Do not count every displayed sphere as a whole atom assigned to this cell. Full occupancy, one atomic species and no defects are assumptions; density alone need not identify the lattice uniquely.

Retain the system, phase and structural assumptions with every number. Supplied or calculated evidence is not actual calorimetry, chemical handling or X-ray operation.

Face-centered cubic

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the repeated cell

A conventional cubic unit cell is a bookkeeping volume in a periodic structure. Atoms on its boundary are shared with neighboring cells. Each corner is shared among eight cubes and contributes one eighth of an atom to a particular cell; a face center contributes one half, and an interior body center contributes one whole atom.

Derive the counts

Simple cubic has eight corners, totaling one atom. Body-centered cubic adds one interior atom, totaling two. Face-centered cubic adds six half-shared face centers to the corner total, giving four. The diagram displays complete markers for visibility, but those markers do not each count as a whole atom assigned exclusively to the cell.

Convert mass and volume

For a fictional monatomic material of molar mass40.0 g/mol, cell mass is Ncell×40/NA. Convert an edge in nanometers to centimeters using1 nm=10⁻⁷ cm, then cube it. At edge.400 nm, volume is6.40×10⁻²³ cm³. FCC density is the four-atom cell mass divided by that volume, about4.151 g/cm³.

Link a geometric radius

A touching-sphere model gives r=a/2 for SC, r=√3a/4 for BCC along a body diagonal and r=√2a/4 for FCC along a face diagonal. These are geometry-based effective radii, not hard quantum-mechanical atomic surfaces. The same edge length does not imply the same radius in different structural models.

Test an inference

At fixed structure, composition and full occupancy, increasing edge by a factor1.25 increases volume by1.25³ and lowers density by its reciprocal. A real density discrepancy could instead involve vacancies, another composition, porosity or wrong structural assignment. One bulk number does not uniquely establish a perfect crystal. Retain the shared-site count and units before interpreting numerical agreement as material identity.

Sources and evidence boundary

Conceptual sources checked 30 October 2026: Crystalline lattice structures. All prose, cases, code and diagrams are original. The supplied models are not measured structures, a performed laboratory investigation or certification of independent mastery. No chemical handling, heating, pressure vessel or cryogenic experiment is requested. Retain predictions, contrasting evidence and a changed-case explanation.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which composition, vacancy or coordination assumptions could make a density-only identification nonunique?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Read a phase map as coexistence evidence rather than a heating trace Next: Challenge a lattice hypothesis with an independent spacing

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