3. Crystal Structure of Metals

Table of Contents

1. Crystal Structure of Metals
2. Structural Parameters
3. Solid Solutions
4. Single v.s. Poly-Crystals

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Crystal Structure of Metals

Atomic Radii and Crystal Structures for 16 Metals

Metal	Crystal Structure${}^a$	Atomic Radius${}^b$ $(nm)$	Metal	Crystal Structure	Atomic Radius $(nm)$
Aluminum	FCC	0.1431	Molybdenum	BCC	0.1363
Cadmium	HCP	0.1490	Nickel	FCC	0.1246
Chromium	BCC	0.1249	Platinum	FCC	0.1387
Cobalt	HCP	0.1253	Silver	FCC	0.1445
Copper	FCC	0.1278	Tantalum	BCC	0.1430
Gold	FCC	0.1442	Titanium ($\alpha$)	HCP	0.1445
Iron ($\alpha$)	BCC	0.1241	Tungsten	BCC	0.1371
Lead	FCC	0.1750	Zinc	HCP	0.1332

${}^a$ FCC = face-centered cubic; HCP = hexagonal close-packed; BCC = body-centered cubic.

${}^b$ A nanometer (nm) equals $10^{-9}$ m; to convert from nanometers to angstrom units ($\mathring{A}$), multiply the nanometer value by 10.

• Number of atoms in a unit cell.
• Coordination number (CN): # of the nearest-neighbor atoms
• Atomic packing factor (APF): vol.% of atoms in a unit cell

Structural Parameters

Total Number of Atoms in a Unit Cell (N):

$N=N_i+\frac{N_f}{2}+\frac{N_c}{8}$

• $N_i$: atoms inside a unit cell.
• $N_f$: atoms on the surfaces of a unit cell.
• $N_c$: atoms at the corners of a unit cell. Atomic Packing Factor (APF):

$APF=\frac{Volumeofatomsinaunitcell}{Totalunitcellvolume}$

$APF=\frac{V_a}{V_c}=\frac{N\frac{4}{3}\pi R^3}{V_c}$

• Lattice Parameter ⇒ $a$
• Atomic Radius ⇒ $R$
• Atom Number ⇒ $N$ Theoretical Density of a Crystal

$\rho =\frac{n{A}_{element}}{V_c{N}_A}$

• $n$ ⇒ number of atoms associated with each unit cell
• $A$ ⇒ atomic weight
• $V_c$ ⇒ volume of the unit cell
• $N_A$ ⇒ Avogadro’s number ($6.023\times 10^{23}$ atoms/mol)

Solid Solutions

Substitutional Solid Solution:

• the alloying element atoms replace host atoms at the lattice sites
  • Using solvent atoms and solute atoms. Interstitial Solid Solution:
• the alloying element atoms occupy the interstitial sites in the host element lattice

Single v.s. Poly-Crystals

Single Crystals: The periodic arrangement of atoms extends throughout the entirety of the specimen without interruption. (Usually characterized by facets)

Polycrystal: The crystal is comprised of a collection of many single crystals (grains, to be elaborated in the next chapter on crystal defects).