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

2.4.13 · Chemistry › States of Matter (Quantitative)

WHAT is a unit cell?

WHY atoms ko fractions mein count karte hain? Kyunki atoms neighbouring boxes ke beech shared boundaries par baithte hain. Corner par ek atom 8 boxes ke beech share hota hai, isliye is box ka sirf $1/8$ hissa us atom ka hai.

The three simple cubic types

1. Simple Cubic (SCC)

Atoms sirf 8 corners par hain.

$ϕ_{S C C} = \frac{Z \cdot \frac{4}{3} π r ^{3}}{a ^{3}} = \frac{1 \cdot \frac{4}{3} π r ^{3}}{( 2 r ) ^{3}} = \frac{\frac{4}{3} π r ^{3}}{8 r ^{3}} = \frac{π}{6} \approx== 0.524 == (52.4%)$

2. Body-Centred Cubic (BCC)

8 corners + body centre par 1 atom.

Body diagonal derive karo. Edge $a$ wale cube ke liye: face diagonal $= a^{2} + a^{2} = a 2$ ; body diagonal $= a^{2} + (a 2)^{2} = 3 a^{2} = a 3$ . Toh $a 3 = 4 r \Rightarrow a = \frac{4 r}{3}$ .

$ϕ_{B C C} = \frac{2 \cdot \frac{4}{3} π r ^{3}}{( \frac{4 r}{3} ) ^{3}} = \frac{\frac{8}{3} π r ^{3}}{\frac{64 r ^{3}}{3 3}} = \frac{8 π \cdot 3 3}{3 \cdot 64} = \frac{π 3}{8} \approx== 0.680 == (68%)$

3. Face-Centred Cubic (FCC / CCP)

8 corners + 6 face centres.

$a 2 = 4 r \Rightarrow a = \frac{4 r}{2} = 22 r$ .

$ϕ_{F C C} = \frac{4 \cdot \frac{4}{3} π r ^{3}}{( 2 2 r ) ^{3}} = \frac{\frac{16}{3} π r ^{3}}{16 2 r ^{3}} = \frac{π}{3 2} = \frac{π 2}{6} \approx== 0.740 == (74%)$

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

Density link (exams mein kyun kaam aata hai)

Worked examples

Common mistakes

Flashcards

Packing fraction of SCC

π /6 \approx 52.4%

Packing fraction of BCC

π 3 /8 \approx 68%

Packing fraction of FCC

π 2 /6 \approx 74%

Z (atoms per cell) for SCC, BCC, FCC

1, 2, 4

BCC mein atoms kis direction mein touch karte hain?

Body diagonal ke saath, jisse

3 a = 4 r

milta hai

FCC mein atoms kis direction mein touch karte hain?

Face diagonal ke saath, jisse

2 a = 4 r

milta hai

SCC mein atoms kis direction mein touch karte hain?

Edge ke saath, jisse

a = 2 r

milta hai

Corner / face / edge / body-centre atom ka contribution

1/8

1/2

1/4

1

Unit cell ki density ka formula

ρ = Z M / (a^{3} N_{A})

FCC vs BCC vs SCC mein empty space

26% vs 32% vs 47.6%

Edge

a

wale cube ka body diagonal

a 3

Edge

a

wale cube ka face diagonal

a 2

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum ek box mein santrey pack kar rahe ho. Agar tum unhe seedhi rows mein stack karo (simple cubic), bahut saare gaps khali rehte hain — box ka sirf aadha hissa fruit hai. Agar tum beech waali jagah mein ek santrey ghusa do (body-centred), yeh zyada tight fit hota hai, 68%. Agar tum santryon ko neeche waali layer ki dimples mein nest karo (face-centred), yeh sabse tight hota hai, 74%. "Packing fraction" bas yeh hai: box mein kitna santrey hai, kitni hawa nahi. Trick yeh hai ki pata lagao kaun se santrey actually touch kar rahe hain taaki tumhe pata chale ki diye hue box ke liye woh kitne bade ho sakte hain.

Connections

Coordination Number in Crystals (SCC=6, BCC=8, FCC=12)
Voids — Tetrahedral and Octahedral
Density of a Unit Cell
Pythagoras Theorem (face & body diagonals ke liye use hota hai)
Close Packing — HCP vs CCP
Avogadro Number and Molar Mass