Comparison Of 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)

Outline

• Geometric Ratios of the Basic Crystal Structures
• Interstitial Sites in the Basic Crystal Structures
• Slip Systems for the Basic Crystal Structures
• Elements in Each of the Basic Crystal Structures
• Summary of Properties of the Basic Crystal Structures
• References and Further Reading

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 Structure	SC	BCC	FCC	HCP
Edge Length	a = 2R	a = 4R/√3	a = 2R√2	a = 2R,c = 1.633a
Face Diagonal Length	2R√2	(4R√2)/√3	4R	6R
Body Diagonal Length	2R√3	4R	2R√6	(2R√11)/√3
c/a Ratio	1	1	1	1.633
Atomic Packing Factor (APF)	52%	68%	74%	74%
Coordination Number (CN)	6	8	12	12

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 Structure	SC	BCC	FCC	HCP(primitive)	HCP(conventional)
Number of Tetrahedral Sites	—	12	8	4	12
Radius of Tetrahedral Sites	—	r = 0.291R	r = 0.225R	r = 0.225R	r = 0.225R
Number of Octahedral Sites	—	6	4	2	6
Radius of Octahedral Sites	—	r = 0.155R	r = 0.414R	r = 0.414R	r = 0.414R
Number of Cubic Sites	1	—	—	—	—
Radius of Cubic Sites	r = 0.732R	—	—	—	—

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 Structure	Slip Plane	Slip Direction	Number ofSlip Systems
SC			6
FCC			12
BCC			12
BCC			12
BCC			24
HCP			3
HCP			3
HCP			6

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 Structure	Simple Cubic (SC)	Body-Centered Cubic (BCC)	Face-Centered Cubic (FCC)	Hexagonal Close-Packed (HCP)
Examples of Elements	Po	Li, Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ba, Ta, W, Fr, Ra, Db, Sg, Ds, Rg, Eu	Al, Ca, Ni, Cu, Sr, Rh, Pd, Ag, Ir, Pt, Au, Pb, At, Mt, Fl, Yb, Ac, Th, Es, Fm, Md, No	Be, 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

Summary of Properties of the Basic Crystal Structures

Crystal Structure	SC	BCC	FCC	HCP
Unit Cell Type	Cubic	Cubic	Cubic	Hexagonal
Edge length	a = 2R	a = 4R/√3	a = 2R√2	a = 2R,c = 1.633a
Close-Packed Structure	No	No	Yes	Yes
Atomic Packing Factor (APF)	52%	68%	74%	74%
Coordination Number (CN)	6	8	12	12
Number of Atoms per Unit Cell	1	2	4	6
c/a Ratio	1	1	1	1.633
Face Diagonal Length	2R√2	(4R√2)/√3	4R	6R
Body Diagonal Length	2R√3	4R	2R√6	(2R√11)/√3
Number of Slip Systems	6(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 ofCubic Interstitial Sites	1 void,r = 0.732R	—	—	—
Number and Size ofOctahedral Interstitial Sites	—	6 voids,r = 0.155R	4 voids,r = 0.414R	6 voids,r = 0. 414R
Number and Size ofTetrahedral Interstitial Sites	—	12 voids,r = 0.291R	8 voids,r = 0.225R	12 voids,r = 0.225R
Examples of Elements (at STP)	Po	Li, Na, Fe, Nb, W	Al, Ca, Ni, Cu, Sr	Be, Mg, Sc, Ti, Co

References and Further Reading

This page is a “cheat-sheet” for quickly looking information about the common crystal structures.

If you want more information, you may be interested in my other crystallography articles. Here is that list, in recommended reading order:

Introduction to Bravais LatticesWhat is the Difference Between “Crystal Structure” and “Bravais Lattice”Atomic Packing FactorHow to Read Miller IndicesHow to Read Hexagonal Miller-Bravais IndicesClose-Packed Crystals and Stacking OrderInterstitial SitesPrimitive CellsHow to Read Crystallography Notation What are Point GroupsList of Point Groups

If you are interested in more details about any specific crystal structure, I have written individual articles about simple crystal structures which correspond to each of the 14 Bravais lattices:

1. Simple Cubic2. Face-Centered Cubic2a. Diamond Cubic3. Body-Centered Cubic4. Simple Hexagonal4a. Hexagonal Close-Packed4b. Double Hexagonal Close-Packed (La-type)5. Rhombohedral5a. Rhombohedral Close-Packed (Sm-type)6. Simple Tetragonal7. Body-Centered Tetragonal7a. Diamond Tetragonal (White Tin)8. Simple Orthorhombic9. Base-Centered Orthorhombic10. Face-Centered Orthorhombic11. Body-Centered Orthorhombic 12. Simple Monoclinic 13. Base-Centered Monoclinic 14. Triclinic