Crystal Structures of Metals

Why do metals pack together?

the more neighbors a metallic atom has, the lower it’s energy is. It tries to bond to as many other atoms as possible.

crystalline

atoms sit in a repeating, long range ordered pattern that tiles through the whole solid

ex. metals, many ceramics

amorphous

short range order only, with no repeating long range pattern (often cooled too fast to crystallize)

ex. window glass, many polymers

counting atoms - unit cell

corner atoms shared by 8 cells (1/8)

face atoms shared by 2 cells (1/2)

body atoms are fully inside (1)

atomic radius (R) and cubic edge length (a) relationship

FCC: a = 2√2 R

BCC: a = 4R√3

HCP: a = 2R

atomic packing factor (APF)

volume of atoms/ volume of cell

vol atoms: atoms per cell * (4/3)πR³

vol of cell: (2R)³

FCC

face-centered cubic

atoms per cell: 4

edge vs radius: a= 2√2 R

atomic packing factor (APF): 4(4/3)πR³ / (4R/√3)³ = 0.74

coordination #: 12

BCC

body centered cubic

atoms per cell: 2

edge vs radius: 4R√3

APF: 2(4/3)πR³ / (4R/√3)³ = 0.68

coordination #: 8

HCP

hexagonal close packed

atoms per cell: 6

edge vs radius: 2R

APF: 6(4/3)πR³ / [(3√3/2)(2R)²c] = 0.74

coordination #: 12

theoretical density of an element

p = nA/(V_C * N_A)

n = atoms per cell

V = molecular weight of element

V_C: volume of cell a³

N_A = avogadro’s # (6.022 × 10²³)

why do denser cubic structures experience larger gaps?

denser structures have less open space, but it is more concentrated, leading to larger gaps

FCC and BCC use the same-size atoms. Which

packs more densely, and why might that make it

more ductile?

FCC packs more densely. This makes it more ductile as the FCC stacking pattern is more closer together, leading to more ductility. Each plane is able to slide past each other more smoothly.

On heating, iron transforms BCC→FCC at 912

°C. What happens to its density, and is that

what you'd expect on heating?

The density increases as it goes from BCC to FCC since atomic packing factor increases. This is not what you would expect, as density typically goes down with heating.