Almost every EEE, Physics and Materials Science exam (BCS, PGCB, BPDB, BREB, DPDC, university finals) asks at least one question from crystal structure: “How many atoms are in an FCC unit cell?”, “What is the packing factor of BCC?”, “Coordination number of simple cubic?”. These questions are easy marks if you understand where the atoms sit and how much of each atom belongs to the cell. This post builds that understanding step by step, with 3D animations, solved examples and practice MCQs.
- Crystal = Lattice + Basis. A unit cell is the smallest repeating block that rebuilds the whole crystal.
- SC: 1 atom, CN = 6, a = 2r, APF = 0.52 (52%)
- BCC: 2 atoms, CN = 8, √3a = 4r, APF = 0.68 (68%)
- FCC: 4 atoms, CN = 12, √2a = 4r, APF = 0.74 (74%) — the most closely packed cubic structure.
- Corner atom = 1/8, face atom = 1/2, edge atom = 1/4, body-centre atom = 1 (shared contribution).
1. Lattice, Basis and Unit Cell
In a crystalline solid, atoms are arranged in a regular pattern that repeats in all three directions. To describe this pattern we use three ideas:
- Lattice (space lattice): an infinite array of imaginary points in space, where every point has identical surroundings.
- Basis (motif): the atom or group of atoms attached to each lattice point.
- Crystal structure = Lattice + Basis. Put the basis on every lattice point and you get the real crystal.
A unit cell is the smallest volume of the lattice which, when repeated (translated) in three dimensions, reproduces the entire crystal — just like a single brick repeated to build a wall. If you know the unit cell, you know the whole crystal: its density, packing, slip planes and many electrical and mechanical properties.
2. Primitive vs Non-Primitive Unit Cell
| Feature | Primitive cell | Non-primitive (multiple) cell |
|---|---|---|
| Lattice points | Exactly 1 lattice point per cell (only at corners: 8 × 1/8 = 1) | More than 1 lattice point (extra points at body centre or face centres) |
| Volume | Smallest possible volume | Integer multiple of primitive volume |
| Example | Simple cubic (SC) | BCC (2 points), FCC (4 points) |
| Why used? | Mathematically minimal | Shows the full cubic symmetry clearly, so it is the “conventional” cell |
3. Lattice Parameters
Any unit cell is a parallelepiped described by six lattice parameters: three edge lengths a, b, c and three inter-axial angles α (between b and c), β (between c and a) and γ (between a and b). For the cubic system: a = b = c and α = β = γ = 90°. Based on these parameters there are 7 crystal systems and 14 Bravais lattices (see 14 Bravais Lattices and 7 Crystal Systems). The cubic system alone has three Bravais lattices — SC, BCC and FCC — which are the focus of this post.
4. How to Count Atoms in a Unit Cell
Atoms at the boundary of a unit cell are shared with neighbouring cells, so only a fraction of each belongs to the cell you are looking at:
| Position of atom | Shared by how many cells | Contribution to one cell |
|---|---|---|
| Corner | 8 | 1/8 |
| Edge centre | 4 | 1/4 |
| Face centre | 2 | 1/2 |
| Body centre (inside) | 1 | 1 |
Watch the animation below: the full atoms of an FCC cell are “cut” by the cube walls, leaving only the parts that truly belong to this cell. Adding up the pieces gives 4 whole atoms.
5. Simple Cubic (SC) Unit Cell
Structure: one atom at each of the 8 corners of the cube. Atoms touch each other along the cube edge.
- Effective atoms: n = 8 × 1/8 = 1
- Atomic radius: a = 2r ⇒ r = a/2
- Coordination number (CN): 6 (4 in the same plane + 1 above + 1 below)
- Nearest-neighbour distance: a
- Atomic packing factor:
So only 52.4% of the space is filled; 47.6% is empty. Because it wastes so much space, SC is rare in nature — α-Polonium is the classic example.
6. Body-Centred Cubic (BCC) Unit Cell
Structure: 8 corner atoms + 1 atom at the body centre. Atoms touch along the body diagonal (corner – centre – corner); corner atoms do not touch each other.
- Effective atoms: n = 8 × 1/8 + 1 = 2
- Atomic radius: body diagonal = √3a = 4r ⇒ r = √3a/4
- Coordination number: 8 (the centre atom touches all 8 corners)
- Nearest-neighbour distance: 2r = √3a/2
Examples: α-Fe (iron at room temperature), Na, K, Cr, Mo, W, V. BCC metals are typically strong and less ductile than FCC metals.
7. Face-Centred Cubic (FCC) Unit Cell
Structure: 8 corner atoms + 1 atom at the centre of each of the 6 faces. Atoms touch along the face diagonal.
- Effective atoms: n = 8 × 1/8 + 6 × 1/2 = 4
- Atomic radius: face diagonal = √2a = 4r ⇒ r = a/(2√2) = √2a/4
- Coordination number: 12 (4 in its own plane + 4 above + 4 below)
- Nearest-neighbour distance: 2r = a/√2
FCC is a close-packed structure (also called cubic close packing, CCP). 74% is the maximum packing possible with equal spheres. Examples: Cu, Al, Ag, Au, Ni, Pb, Pt, γ-Fe. FCC metals are very ductile — this is why copper and aluminium can be drawn into thin wires for electrical conductors.
Where do the atoms touch? (animated)
The key to every radius formula is finding the direction in which atoms touch. The animated drawing traces that line for each cell:
8. SC vs BCC vs FCC – Comparison Table
| Property | SC | BCC | FCC |
|---|---|---|---|
| Atoms per unit cell (n) | 1 | 2 | 4 |
| Coordination number | 6 | 8 | 12 |
| Atoms touch along | Edge | Body diagonal | Face diagonal |
| Relation between a and r | a = 2r | a = 4r/√3 | a = 2√2 r |
| Nearest-neighbour distance | a | √3a/2 | a/√2 |
| Atomic packing factor | π/6 = 0.52 | √3π/8 = 0.68 | π/(3√2) = 0.74 |
| Void (empty) space | 48% | 32% | 26% |
| Examples | α-Po | α-Fe, Na, K, Cr, W, Mo | Cu, Al, Ag, Au, Ni, Pb, γ-Fe |
9. Density from Unit Cell Data (Solved Example)
Mass of one unit cell = (number of atoms × atomic mass) / Avogadro’s number. Divide by the cell volume a³:
ρ = (n · A) / (NA · a³)Example: Copper is FCC with lattice constant a = 3.61 Å and atomic mass A = 63.55 g/mol. Find its density and atomic radius.
- n = 4 (FCC), NA = 6.022 × 10²³ mol⁻¹
- a = 3.61 Å = 3.61 × 10⁻⁸ cm ⇒ a³ = 4.705 × 10⁻²³ cm³
- ρ = (4 × 63.55) / (6.022 × 10²³ × 4.705 × 10⁻²³) = 254.2 / 28.33 ≈ 8.97 g/cm³ (measured value 8.96 g/cm³ ✔)
- r = √2a/4 = (1.414 × 3.61)/4 ≈ 1.28 Å
- Using a = 2r for BCC or FCC — that is only true for SC.
- Forgetting to convert Å to cm (1 Å = 10⁻⁸ cm) in density problems.
- Mixing up “number of lattice points” (primitive = 1) with “number of atoms” when the basis has more than one atom.
10. Practice MCQs (click to see answer)
1. The number of atoms per unit cell in an FCC structure is: (a) 1 (b) 2 (c) 4 (d) 6
2. Coordination number of a BCC crystal is: (a) 6 (b) 8 (c) 12 (d) 4
3. Atomic packing factor of simple cubic is about: (a) 0.52 (b) 0.68 (c) 0.74 (d) 0.34
4. In BCC, the relation between lattice constant a and atomic radius r is: (a) a = 2r (b) √2a = 4r (c) √3a = 4r (d) a = 4r
5. Which metal has an FCC structure? (a) Sodium (b) Tungsten (c) Copper (d) α-Iron
6. The most closely packed cubic structure is: (a) SC (b) BCC (c) FCC (d) All equal
7. A corner atom of a cubic unit cell is shared by: (a) 2 cells (b) 4 cells (c) 6 cells (d) 8 cells
8. Nearest-neighbour distance in FCC with lattice constant a is: (a) a (b) a/√2 (c) √3a/2 (d) a/2
9. Which unit cell is primitive? (a) SC (b) BCC (c) FCC (d) None
10. Percentage of void space in BCC is about: (a) 26% (b) 32% (c) 48% (d) 68%
11. বাংলায় দ্রুত রিভিশন
- ইউনিট সেল হলো ক্রিস্টালের সবচেয়ে ছোট পুনরাবৃত্ত অংশ, যা বারবার সাজালে পুরো ক্রিস্টাল তৈরি হয়। ক্রিস্টাল = ল্যাটিস + বেসিস।
- কোণার পরমাণু ১/৮, তলের কেন্দ্রের পরমাণু ১/২, ধারের পরমাণু ১/৪, এবং ভেতরের (বডি সেন্টার) পরমাণু পুরো ১টি গণনা হয়।
- SC: পরমাণু ১টি, কো-অর্ডিনেশন নম্বর ৬, a = 2r, প্যাকিং ফ্যাক্টর ৫২%।
- BCC: পরমাণু ২টি, কো-অর্ডিনেশন নম্বর ৮, √3a = 4r, প্যাকিং ফ্যাক্টর ৬৮%। উদাহরণ: α-লোহা, সোডিয়াম, টাংস্টেন।
- FCC: পরমাণু ৪টি, কো-অর্ডিনেশন নম্বর ১২, √2a = 4r, প্যাকিং ফ্যাক্টর ৭৪% (সবচেয়ে ঘন)। উদাহরণ: তামা, অ্যালুমিনিয়াম, সোনা, রূপা।
- মনে রাখার কৌশল: ১-২-৪, ৬-৮-১২, ৫২-৬৮-৭৪।
Related Posts
- What Is a Crystal? Crystal Structure Basics
- Types of Solids: Crystalline, Polycrystalline and Amorphous
- 14 Bravais Lattices and 7 Crystal Systems
- Miller Indices of a Crystal Plane
Reference values: Callister & Rethwisch, Materials Science and Engineering: An Introduction; Kittel, Introduction to Solid State Physics. Images and animations: original work by Totalecer.