Course A - Atomic Structure of Materials

Question 1

define a crystalline and non-crystalline solid

Answer 1

“A crystalline solid is one in which the atoms are arranged in a periodic fashion - they have long range order”

“A non-crystalline solid is one in which no ‘long range order’ occurs but short range order can occur”

Question 2

give 2 different 2D packing arrangments

Answer 2

Square, packing efficiency = 79%

Hexagonal, packing efficiency = 90.6%

Question 3

define what a simple hexagonal structure is, do any examples exist?

Answer 3

• Layers of simple 2D hexagonal stacked
• no examples exist because layers just drop into gaps, not efficient or stable

Question 4

define hexagonal close-packed (hcp) packing, give the stacking sequence, give the ideal layer separation

Answer 4

• ABAB arrangement of the 2D simple hex.
• ideally d = sqrt(2/3) a

Question 5

define cubic close-packed (ccp), give the stacking sequence

Answer 5

ABCABC stacking sequence of 2D simple hex.
cubic symmetry on rotation

Question 6

give the packing efficiencies of hcp and ccp

Answer 6

both 74%

Question 7

explain the structure of bcc, give its packing efficiency

Answer 7

one square layer, single atom in middle of square, another square layer

packing efficiency = 68%

Question 8

define goldschmidt’s packing principle

Answer 8

“The number of anions surrounding a cation tends to be as large as possible subject to the condition that all anions touch the cation”

i.e. interstitial not touching (too small) = unstable
just touching = ideal
too big = stable but not ideal

Question 9

what are the interstitial sites in ccp

Answer 9

• 8 tetrahedral sites between 1 corner atom, 3 mid-face atoms at 1/4(1,1,1) and 3/4(1,1,1) etc.
• 4 octahedral interstices - found in centre, and 12(1/4) interstices at edges

Question 10

what are the interstitial sites in hcp

Answer 10

the same as ccp

Question 11

define a lattice

Answer 11

“A lattice is an infinite array of points repeated periodically throughout space, the view from each lattice point is the same as from any other”

Question 12

what is a good way to think about constructing a structure

Answer 12

structure = lattice + motif

Question 13

give the 3 main 3D lattice types and define them

Answer 13

cubic p = 1 lattice point per cell (simple cubic)
cubic f = 4 lattice points per cell (ccp)
cubic I = 2 lattice points per cell (bcc)

Question 14

what are the lattice types called in 3D, how are they classified

Answer 14

Bravais Lattices

they are defined based on their symmetry
e.g. a cubic lattice has 3 fold symmetry

Question 15

what are the 4 possible rotational symmetries, why are we limited to only these

Answer 15

diad = 2-fold
triad = 3-fold
tetrad = 4-fold
hexad = 6-fold

we are limited to only these due to the crystallographic restriction theorem (CRT)

this is where in a lattice, any linear combination of defined lattice vectors MUST take you to another lattice point

Question 16

what is mirror symmetry

Answer 16

reflection in a mirror plane

Question 17

what is the action of a glide line

Answer 17

combines translational and mirror symmetry

reflection about a line and a translation by 1/2 a lattice spacing parallel to line

e.g. footsteps in sand

Question 18

what is a screw axis, define what each number does

Answer 18

given classification Rn
R = order of rotation/ number of rotations for 360 degrees
n = number of translations for 1 complete turn of helix/ number of repeat distances risen through in 1 full rotation

Question 19

what is a centre of symmetry in a crystal

Answer 19

a centre of symmetry in a crystal is where any line from a point to the centre (of sym.) can be repeated the other side of the centre (of sym.) and you will reach the exact same of environment

contains centre of sym = centrosymmetric
does not = non-centrosymmetric

Question 20

how is a direction vector written

Answer 20

[UVW]

for negative number, put a bar on top

Question 21

how can we find the angle between two lattice directions

Answer 21

take the dot product as normal

Question 22

what are miller indices, how do we write them

Answer 22

they define a plane

each number gives the reciprocal of the (fractional) intercepts for each unit cell

NOTE: the origin can always be moved but if the intercept is on -ve x,y,z, a bar should be placed on top as usual

Question 23

what is the notation for a set of directions or a set of planes

Answer 23

a set of directions = < > brackets (triangular)
a set of planes = {} brackets (curly)

Question 24

give the formula for interplanar spacing (IN FORMULA BOOK)

Answer 24

1/d(hkl)^2 = (h/a)^2 + (k/b)^2 + (l/c)^2

for some plane (hkl)

Note: this does not work for hexagonal crystals

Question 25

give the definition of the Weiss Zone law

Answer 25

“A zone may be defined as a set of faces or planes in a crystal whose intersections are all parallel, the common direction of the intersections is the zone axis”

“If a lattice vector [UVW] is contained within a plane of set (hkl) then
hU + kV + lW = 0 “

Question 26

what is the lattice vector to move from an atom in A position in hcp or ccp to B position in hcp or ccp

(this also applies for B to C in ccp)

Answer 26

a/6 <112>

Question 27

Define a unit cell

Answer 27

“A Parallepiped in a lattice having lattice points at all vertices”

Question 28

explain why the smallest unit cell isn’t always used

Answer 28

• A subset of the structure (not necessarily the smallest) which can be used to build the entire crystal is known as a unit cell
• It is conventional to choose a unit cell which contains the defining (rotational) symmetry of the
  crystal, even if it has more than one lattice point per cell

Question 29

although lattices cannot contain 5-fold symmetry, can motifs contain 5-fold symmetry

Answer 29

yes, they are not periodic structures