GATE XE2 2027 PREP
NoteHow to use this page
This is a condensed companion to Lesson 4.2 — Diamond and Graphite: Lattice, Motif and Structure. It carries no new claims — every number and rule here is drawn from that lesson, where you'll also find the full derivations, six figures, eight worked examples and twenty practice questions. Use this page in the last day or two before your GATE 2027 XE2 paper, to fire the whole topic back into memory in a few minutes.
1The 30-Second Recap
Diamond is a face-centred cubic (FCC) lattice with a two-atom motif: every carbon atom is bonded to four neighbours arranged as a tetrahedron (sp³), giving a three-dimensional, open, very hard, electrically insulating network. Graphite is a simple hexagonal lattice with a four-atom motif: every carbon atom is bonded to three neighbours in a plane (sp²), with one electron per atom delocalised over the layer, giving soft, electrically conducting sheets held together only by van der Waals forces. At 1 atm and 298 K, graphite is thermodynamically stable and diamond is metastable.
2Core Numbers at a Glance
| Property | Diamond | Graphite |
|---|---|---|
| Lattice | Face-centred cubic (FCC) | Simple hexagonal |
| Motif | 2 atoms: (0,0,0), (¼,¼,¼) | 4 atoms: (0,0,¼), (0,0,¾), (⅓,⅔,¼), (⅔,⅓,¾) |
| Space group | Fd-3m (No. 227) | P63/mmc (No. 194) |
| Lattice parameter(s) | a = 3.567 Å | a = 2.46 Å, c = 6.71 Å |
| Atoms / conventional cell | 8 (= 4 lattice points × 2) | 4 |
| Atoms / primitive (or layer) cell | 2 | 2 per single layer |
| Coordination number | 4 (tetrahedral, 109.47°) | 3 in-plane (120°) |
| Bond length | d = √3a/4 = 1.545 Å | d = a/√3 = 1.42 Å (in-plane) |
| Atomic radius (hard-sphere) | r = √3a/8 = 0.772 Å | not a standard GATE quantity |
| Interlayer / 2nd-shell spacing | 2nd shell: 12 atoms at a/√2 | layer spacing c/2 = 3.35 Å |
| Packing factor (APF) | π√3/16 ≈ 0.340 | ≈ 0.17 (model-dependent, not load-bearing) |
| Density | 3.51 g cm−3 | 2.27 g cm−3 |
| Hybridisation | sp³ | sp² + delocalised π |
| Force holding the structure together | 3-D covalent network | covalent in-plane; van der Waals between layers |
| Stability at 298 K, 1 atm | metastable | thermodynamically stable |
3Every Formula You Need
| Eq. | Quantity | Formula |
|---|---|---|
| 5.1 | Atoms per cell (general rule) | N = Nint + Nface/2 + Nedge/4 + Ncorner/8 |
| 5.2 | Diamond bond length | d = √3a/4 |
| 5.3 | Diamond atomic radius | r = d/2 = √3a/8 |
| 5.4 | Diamond APF | π√3/16 ≈ 0.340 |
| 5.5 | Tetrahedral angle | cos θ = −⅓ → θ = 109.47° |
| 5.6 | Density, general | ρ = nM/(VcNA) |
| 5.7 | Density, diamond cubic | ρ = 8M/(NAa3) |
| 5.8 | Graphite bond length | d = a/√3 |
| 5.9 | Graphite cell volume | Vc = (√3/2)a2c |
| 5.10 | Graphite density | ρ = 4M/(NAVc) |
| 5.11 | Graphite layer spacing | c/2 |
| 5.12 | Areal density of a layer | ns = 4/(√3 a2) |
| 5.14 | Bragg's law | λ = 2dhkl sin θ |
| 5.15 | Cubic plane spacing | dhkl = a/√(h2+k2+l2) |
| 5.16 | Cubic sin²θ relation | sin²θ = (λ2/4a2)(h2+k2+l2) |
| 5.17 | Hexagonal plane spacing | 1/d2hkl = (4/3)(h2+hk+k2)/a2 + l2/c2 |
| 5.23 | Graphite–diamond boundary pressure | Peq ≈ ΔG°/(−ΔV) |
NoteConstants used throughout
NA = 6.022 × 1023 mol−1; M(C) = 12.01 g mol−1; 1 ų = 10−24 cm³; Cu Kα1, λ = 1.5406 Å. Full derivations of every line above are in Section 5 of the main lesson.
4X-ray Diffraction Cheat Sheet
| Lattice | Allowed N = h²+k²+l² (first lines) | Ratio of first three sin²θ |
|---|---|---|
| Simple cubic | 1, 2, 3, 4, 5, 6, 8 | 1 : 2 : 3 |
| Body-centred cubic | 2, 4, 6, 8, 10, 12, 14 | 1 : 2 : 3 (told apart from SC at the 7th line, N=7) |
| Face-centred cubic | 3, 4, 8, 11, 12, 16 | 3 : 4 : 8 |
| Diamond cubic | 3, 8, 11, 16, 19, 24 | 3 : 8 : 11 |
GATE trapDiamond's forbidden reflections
(hkl) is allowed only if all indices are odd, or all even with h+k+l a multiple of 4. Forbidden: (200), (222), (420) — even though FCC alone would allow them. This is the single most-tested diffraction fact on this topic.
GATE trapGraphite's basal reflections
For (00l), only even l is allowed: (001) is forbidden, (002) is the first basal line, at d = c/2 ≈ 3.35 Å, 2θ ≈ 26.5° for Cu Kα1.
5Top GATE Traps
| If you're about to write… | Stop — check this first |
|---|---|
| APF of diamond = 0.74 | It's FCC-based but not close-packed. APF = π√3/16 = 0.34. |
| Atoms per cell = 4 | That's the lattice-point count. With a 2-atom motif, it's 8. |
| Coordination number = 12 | 12 is the second-shell count. CN of diamond is 4. |
| Bond length = a/√2 or √3a/2 | Those are FCC / BCC formulas. Diamond's is √3a/4. |
| r = √3a/4 | That's the bond length d, not the radius. r = d/2 = √3a/8. |
| "the diamond lattice" | The lattice is FCC; "diamond structure" = FCC lattice + 2-atom motif. |
| Graphite unit cell = 2 or 12 atoms | 2 is a single layer; 12 is three cells (the hexagonal prism). The cell is 4. |
| Layer spacing = c | c spans two layers. Spacing = c/2 = 3.35 Å. |
| Graphite interlayer bonds are covalent | They're van der Waals only. Covalent bonds are in-plane (sp²). |
| Si's 2nd XRD line is (200) | (200) is forbidden in diamond cubic. The 2nd line is (220). |
| ΔG°>0 for graphite→diamond means diamond decays fast | It only means diamond is thermodynamically disfavoured. The activation barrier keeps it metastable for practical purposes. |
| Diamond has the higher entropy | Graphite has the higher S° (5.74 > 2.38 J mol−1K−1) — a common reversal. |
| Pressure favours graphite | Pressure favours the denser phase — diamond (3.51 > 2.27 g cm−3). |
660-Second Self-Check
Cover the page and answer each in one line before opening it.
Next
This page is deliberately bare-bones. For the full classroom-style explanation, six figures, eight fully worked examples, and all twenty MCQ/MSQ/NAT practice questions with detailed solutions, see Lesson 4.2 — Diamond and Graphite: Lattice, Motif and Structure.
Advanced Materials Lab · GATE 2027 XE2 Materials Science · Quick Revision Notes companion to Lesson 4.2.