Foundations — Cubic systems — SCC, BCC, FCC; packing fraction calculations

This page assumes you have seen none of the notation in the parent note. We build every letter, ratio, and picture from zero, in the order they depend on each other. Nothing is used before it is defined.

0. What a "cube edge" and "diagonal" even mean

Before any chemistry, we need to be comfortable with a plain cube — a box like a die.

WHY we need these two diagonals. Atoms don't always touch along the edge. Sometimes they touch along a face diagonal, sometimes along the body diagonal. To turn "they touch" into an equation, we must know how long these slanted lines are in terms of $a$. That length comes from Pythagoras Theorem.

1. The atom as a hard sphere — radius $r$

WHY a sphere and why $r$. To ask "how much of the box is filled," we need a shape and a size for an atom. The simplest fair model is a ball, and one number — its radius $r$ — fixes its size completely. When two atoms touch, the distance between their centres is exactly $2r$ (their surfaces just kiss). That single fact is the bridge between the atom size $r$ and the cube edge $a$.

2. Why atoms are only partly owned — the fraction rule

Real crystals repeat forever; the cube we draw shares its corners, edges and faces with neighbouring cubes. So an atom sitting on a boundary is split between cubes.

WHY count fractions. If we naïvely counted every atom we see as "1," we'd count the same corner atom eight times (once for each cube touching it). Fractions stop the double-counting so the total across the whole crystal comes out right.

3. The Greek letters and the packing symbol $\varphi$

4. Symbols that show up in the density link

The parent note also computes density, so we must define its symbols too.

5. How every symbol feeds the topic

Read it as: Pythagoras + edge give the diagonals; diagonals + radius give the touch relation; that relation, the atom volume, and the cube volume combine into $\varphi$; and $Z$, $M$, $N_A$, $a^3$ combine into $\rho$.

Equipment checklist

Test yourself — if any answer surprises you, reread that section before the parent note.

Connections

• Yeh note Hinglish mein
• Pythagoras Theorem — source of the diagonal lengths
• Density of a Unit Cell — uses every symbol from Section 4
• Avogadro Number and Molar Mass — defines $M$ and $N_A$
• Coordination Number in Crystals
• Voids — Tetrahedral and Octahedral
• Close Packing — HCP vs CCP