Friday, November 17, 2023

SC, BCC, FCC, and HCP Crystal Structures

 

All of this information can be found elsewhere on the site, but here is a quick reference sheet if you want to study the basic crystals quickly before an exam. Here are the links for the thorough comparisons of: FCC vs BCC, and FCC vs HCP.

Simple Cubic (SC)
Body-Centered Cubic (BCC)
Face-Centered Cubic (FCC)
Hexagonal Close-Packed (HCP)

Geometric Ratios of the Basic Crystal Structures

Here are the crystal geometric ratios for simple cubic, body-centered cubic, face-centered cubic, and hexagonal close-packed. This table shows the edge length (lattice parameter), face diagonal length ([110] length), body diagonal length ([111] length), c/a lattice parameter ratio, atomic packing factor (APF) and coordination number (CN).

Crystal StructureSCBCCFCCHCP
Edge Lengtha = 2Ra = 4R/√3a = 2R√2a = 2R,
c = 1.633a
Face Diagonal Length2R√2(4R√2)/√34R6R
Body Diagonal Length2R√34R2R√6(2R√11)/√3
c/a Ratio1111.633
Atomic Packing Factor (APF)52%68%74%74%
Coordination Number (CN)681212
Geometric Ratios of the Basic Crystal Structures (SC, BCC, FCC, HCP)

If you want to prove any of these numbers, check out my article about Atomic Packing Factor.

Interstitial Sites in the Basic Crystal Structures

This table summarizes the number and type of interstitial sites for simple cubic, body-centered cubic, face-centered cubic, and hexagonal close-packed crystals.

Crystal StructureSCBCCFCCHCP
(primitive)
HCP
(conventional)
Number of Tetrahedral Sites128412
Radius of Tetrahedral Sitesr = 0.291Rr = 0.225Rr = 0.225Rr = 0.225R
Number of Octahedral Sites6426
Radius of Octahedral Sitesr = 0.155Rr = 0.414Rr = 0.414Rr = 0.414R
Number of Cubic Sites1
Radius of Cubic Sitesr = 0.732R
Interstitial Sites in the Basic Crystal Structures (SC, BCC, FCC, HCP)

Check out my in-depth article about interstitial sites if you would like more diagrams, or proof of these values.

Slip Systems for the Basic Crystal Structures

Quick reminder: a slip system is a combination of a slip plane and a slip direction!

Crystal StructureSlip PlaneSlip DirectionNumber of
Slip Systems
SC\{100\}\langle 001 \rangle6
FCC\{111\}\langle 110 \rangle12
BCC\{110\}\langle 111 \rangle12
BCC\{211\}\langle 111 \rangle12
BCC\{321\}\langle 111 \rangle24
HCP\{0001\}\langle 11\bar{2}0 \rangle3
HCP\{1100\}\langle 11\bar{2}0 \rangle3
HCP\{1101\}\langle 11\bar{2}0 \rangle6
Slip Systems for the Basic Crystal Structures (SC, BCC, FCC, HCP)

Elements in Each of the Basic Crystal Structures

Here is a list of all elements that have a simple cubic, body-centered cubic, face-centered cubic, or hexagonal close-packed crystal structure at standard temperature and pressure.

Crystal StructureSimple Cubic (SC)Body-Centered Cubic (BCC)Face-Centered Cubic (FCC)Hexagonal Close-Packed (HCP)
Examples of ElementsPoLi, Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ba, Ta, W, Fr, Ra, Db, Sg, Ds, Rg, EuAl, Ca, Ni, Cu, Sr, Rh, Pd, Ag, Ir, Pt, Au, Pb, At, Mt, Fl, Yb, Ac, Th, Es, Fm, Md, NoBe, Mg, Sc, Ti, Co, Zn, Y, Zr, Tc, Ru, Cd, Hf, Re, Os, Tl, Rf, Bh, Hs, Cn, Nh, Gd, Tb, Dy, Ho, Er, Tm, Lu, Lr
Elements in Each of the Basic Crystal Structures (SC, BCC, FCC, HCP)

Summary of Properties of the Basic Crystal Structures

Crystal StructureSCBCCFCCHCP
Unit Cell TypeCubicCubicCubicHexagonal
Edge lengtha = 2Ra = 4R/√3a = 2R√2a = 2R,
c = 1.633a
Close-Packed StructureNoNoYesYes
Atomic Packing Factor (APF)52%68%74%74%
Coordination Number (CN)681212
Number of Atoms per Unit Cell1246
c/a Ratio1111.633
Face Diagonal Length2R√2(4R√2)/√34R6R
Body Diagonal Length2R√34R2R√6(2R√11)/√3
Number of Slip Systems6
(3 independent)
48
(5 independent)
12
(12 independent)
12
(3 independent)
Stacking Sequence
(Not Close-Packed)

(Not Close-Packed)
ABC-ABC…AB-AB-AB…
Number and Size of
Cubic Interstitial Sites
1 void,
r = 0.732R
Number and Size of
Octahedral Interstitial Sites
6 voids,
r = 0.155R
4 voids,
r = 0.414R
6 voids,
r = 0. 414R
Number and Size of
Tetrahedral Interstitial Sites
12 voids,
r = 0.291R
8 voids,
r = 0.225R
12 voids,
r = 0.225R
Examples of Elements (at STP)PoLi, Na, Fe, Nb, WAl, Ca, Ni, Cu, SrBe, Mg, Sc, Ti, Co
Summary of Properties of the Basic Crystal Structures (SC, BCC, FCC, HCP)

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