ENSC1004 - Lecture 9

|← Back to Home

ENSC1004 - Lecture 8
Strength & Ductility 3 - Slides

Packing of Atoms in Solids

• A solid can be crystalline or amorphous
• A crystalline solid can be a single crystal or contain many crystals
  • These crystals are known as grains
  • Solids containing many grains are termed polycrystalline

---

• A single crystal is a crystalline solid in which perfect periodical arrangement of atoms extends throughout its structure
  • Difficult to grow
• Polycrystalline solids are composed of many crystals or grains
  • Gains are small and can only be seen with microscope
• Amorphous solids have randomly arranged atoms
  • Also known as non-crystalline

Crystal Structure of Metals

• Metals are crystalline
  • Their atoms pack in specific, repeating patterns

• Three most common structures:
  • FCC $-$ Face-Centred Cubic
    • One atom in every corner and centre of each face
      e.g. aluminium, copper, gold, nickel
    • Very ductile
    • 12 slip systems
  • BCC $-$ Body-Centred Cubic
    • One atom in every corner and one in the cube’s centre
    • Stronger/harder, but less ductile
    • 12 slip systems
  • HCP $-$ Hexagonal Close-Packed
    • Atoms form hexagonal prism
    • Can be ductile (but less than FCC)
    • Anisotropic $-$ different properties in different directions
      e.g. like wood being stronger along the grain
    • 3 slip systems

More slip systems → more ductility

Property	SC (Simple Cubic)	BCC (Body-Centred Cubic)	FCC (Face-Centred Cubic)
Atoms per unit cell (n)	1	2	4
Coordination number	6	8	12
Relationship between (a) and (R)	($a=2R$)	($a=\frac{4R}{\sqrt{3}}$)	($a=2\sqrt{2}R$)
Touching direction (nearest-neighbour direction)	⟨100⟩ (along cube edges)	⟨111⟩ (body diagonal)	⟨110⟩ (face diagonal)
Closest-packed plane	none (low packing)	{110}	{111}
Volume of unit cell	($a^3$)	($a^3$)	($a^3$)
Volume of one atom	($\frac{4}{3}\pi R^3$)	($\frac{4}{3}\pi R^3$)	($\frac{4}{3}\pi R^3$)
Atomic Packing Factor (APF)	($\frac{\pi }{6}=0.524$)	($\frac{\sqrt{3}\pi }{8}=0.680$)	($\frac{\pi }{3\sqrt{2}}=0.740$)
Lattice type	Primitive cubic	Primitive cubic (not FCC!)	Face-centred cubic
Number of atoms in one edge	1	2 (across body diagonal)	4 (across face diagonal)
Example metals	Polonium (rare)	Fe (α), Cr, W, Mo, Na, Li	Al, Cu, Ni, Ag, Au, Pb

Atomic Packing Factor

• APF is the fraction of volume of a unit cell that is occupied by ‘hard sphere atoms: $APF=\frac{\text{total volume of atoms in a unit cell}}{\text{volume of the unit cell}}=\frac{n(\frac{4}{3}\pi R^3)}{a^3}$

Theoretical Density

• We can calculate theoretical density ($\rho$) of a crystal if we know atomic mass ($A$), number of atoms in unit cell ($n$) and unit cell volume ($V_{UC}$)
  $\rho =\frac{nA}{V_{UC}{N}_A}$
  $N_A=6.022\times 10^{23}$ (Avogadro’s number)

---

ENSC1004 - Lecture 10