1. Next time: Diffraction
Finishing up Crystal Structures Focus: Advanced Structures Ask any crystal structure questions today!
Next time: Diffraction

2. Close packed crystals Where will the next plane of atoms go? A plane
B plane
It’s hard to visualize the marbles at the FCC or HCP so don’t worry if you don’t see it.
0.74 for both fcc and hcp
C plane
A plane
…ABCABCABC… packing
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]

3. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...
(if skipped it before, can also compare to the simple cubic structure)
Let’s start in a plane. There it’s really simple
fcc face centred cubic
hcp hexagonal close-packed
I have asked the demo person for more marbles and bonds for more groups in future.
In groups, build these two differing crystal structures.

4. In the (a) ABA and (b) ABC close-packed arrangements, the coordination number of each atom is
12.

5. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
• ABAB... Stacking Sequence
• 3D Projection
• 2D Projection
A little harder to get the APF for this one. Trick is to identify the closed packing direction (line above). Can anyone tell? Same APF as fcc.
APF = 0.74 (same as fcc)
What is the packing direction?
For ideal packing, c/a ratio of 1.633
However, in most hcp metals, ratio deviates from this value

6. Hexagonal Has Different Plane Notation (h k l m)
[010]
Hexagonal Has Different Plane Notation (h k l m)
[100]
[-1-10]
Hexagonal structure:
To better understand, consider that a-b plane (2D hexagon) can be defined by 3 vectors in plane
3D structure can be defined by 4 miller indices (h k l m)
Third miller direction and index is not independent:
h + k = -l m e
I don’t understand why the normal notation isn’t used either, but this is common to see.
Note that “l” is not the number you see if the vector notation in 2D.
Consider the (11-22) grey plane shown above. Notice the intercepts are at 1, 1, and ½ . Take reciprocals for h k and m. Then l is just found by the formula involving h and k above. k h l
Have more on HCP planes in the Additional Materials tab of website

7. Other non close-packed structures
Packing isn’t the only consideration when building a lattice.
Other non close-packed structures
In covalently bonded materials, bond direction is more important than packing
graphite
diamond
What is close packed direction?
The APF for diamond (34%) is even less than for simple cubic (52%)!
How would you do packing factors in a system with more than one element?

8. Simple Crystal Structures Diamond
Crystal class Td (tetrahedral) - Each atom has 4 nearest-neighbors (nn). What elements might benefit from 4 neighbors?
What type of lattice?
Can be interpreted as two combined fcc structures
One atom at origin
Other atom displaced along diagonal (¼, ¼, ¼)
Includes C, Si, Ge, a-Sn
What is the close packed direction?

9. Defining the Crystal Structure
Lattice  face centered cubic (fcc).
The fcc primitive lattice is generated by r = n1a1+n2a2+n3a3
with lattice vectors:
a1 = a(0,1,1)/2, a2 = a(1,0,1)/2, a3 = a(1,1,0)/2
(NOTE: The ai’s are not mutually orthogonal!)
Basis set: 2 atoms (specify element) at (000) and (¼ , ¼, ¼).
Diamond:
2 identical atoms in basis (e.g. 2 C)
fcc lattice
Zincblende:
2 different atoms in basis and fcc lattice
For FCC 2 atom ABCABC stacking, it is called zinc blende

10. Many semiconductors have the
For ABAB… stacking it is called wurzite structure (fcc zincblende was ABCABC…)
Some compounds can have either structure (i.e., GaN, SiC)
Many semiconductors have the
Wurtzite Structure
Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn). Basis set: 2 atoms. Lattice  hexagonal close packed (hcp).
A Unit Cell looks like
hcp primitive lattice vectors :
a1 = c(0,0,1)
a2 = (½)a[(1,0,0) + (3)½(0,1,0)]
a3 = (½)a[(-1,0,0) + (3)½(0,1,0)]

11. Ionic materials (Transferred Electron)
In ionic materials, different considerations can be important (electrostatics, different size of ions)
Figure shows the crystal structure of Cs+Cl-. The lattice constant a is 4.12 Å and all the bonds shown have the same length. The grey atoms are Cs and the green ones are Cl.
Group: Define crystal structure: meaning what are the primitive Bravais lattice and the associated basis for this crystal (including the locations of these atoms in terms of lattice parameter a)? a
What is the angle between the chemical bonds?
(1V3) simple cubic with two atom basis: (0 0 0) and (½ ½ ½). Need to identify which atom is at which. Though, it doesn’t matter which you pick, as you could do either, but a full definition of the crystal structure defines the lattice vectors, the atoms positions and atom types.

12. Cesium Chloride Structure Cs+Cl-
Simple cubic lattice with a basis consisting of a cesium ion at the origin 0 and a chlorine ion at the cube center
CsBr and CsI crystallize in this structure. The lattice constants are in the order of 4 angstroms.

13. Ionic Bonding
Elemental Na (atomic # 11) or Cs (#55) would be metallic, but you can change that if brought in contact with an electronegative element (e.g. Cl, 17)
What are the electronic configurations of Na & Cl?
Anion = Negative ion (atom with extra electrons)
Cation = Positive ion (atom missing electrons)
Which type of ions are Na and Cl?
Figure on page 9 of Holgate

14. NaCl (Salt) Structure
Cs+Cl-
In NaCl the small Na are in interstitial positions between the Cl ions
Group: Define the crystal structure
For comparison
The negative ions are in general much bigger than the positive ions because of the smaller nuclear charge.
So the positive ions can be fit into a close-packed lattice of negative ions as an interstitial alloy.
Can you see the fcc crystal / two fcc into each other?

15. Simple Crystal Structures NaCl
NaCl: interpenetrating fcc structures
One atom at (0,0,0)
Second atom displaced by (1/2,0,0)
NaCl

16. How many nearest neighbors (NN) does each have?
Why do these two different ionic structures have very different crystal structures?
NaCl=6, CsCl=8
How many nearest neighbors (NN) does each have?

17. Simple Crystal Structures NaCl
Majority of ionic crystals prefer NaCl structure despite lower coordination (fewer NN)
Radius of cations much smaller than anions typically (removed e-s)
For very small cations, anions can not get too close in the other typical structure (CsCl)
This favors NaCl structure where anion contact does not limit structure as much
CsCl

18. Can predict number of nearest neighbors in ionic systems
Coordination number is just the # of nearest neighbors of an ion
It varies based on the type of bonding and the size of the ions.
Let’s determine the expected CN for NaCl
Kittel doesn’t really talk about this here, but Holgate does. If you have two different types of atoms, you can predict the structure based on their size difference.
Expected

19. How many different types of atoms are in the basis?
What can you tell me about this structure? (It is at least one unit cell, maybe more.)
How many different types of atoms are in the basis?
Block in class
8 golds, others harder to count because some on outer edge of cell, but if careful get 16 grey and 32 read
While there are also some atoms shown outside, can you count the number of different types of atoms within the unit cell?

20. Spinel AB2O4 Structure (~CCP)
Oct
Alternate layers parallel (111)
Octahedral &
Octahedral - Tetrahedral
Oct & Tet
Oct
CCP with
1/8 Tetrahedral = A
¼ Octahedral = B
Perpendicular to (111)
Normal Spinel: B all Oct
Inverse Spinel: B ½ Tet

21. Group: Create Wigner-Seitz cell of this lattice
In 3-d, think about a polyhedron
Did not get to

22. Group Exercise
How many atoms are in the primitive unit cell of graphite? Identify a unit cell.
M1.4: Graphite = a staggered arrangement of stacked hexagonal layers
Answer: 4
Hard to see. Not one from corner on bottom and one inside, then halfway up, one from corners and one inside.

23. Coordination number Primitive cubic Body centered cubic8 12
Coordination number 6
Primitive cubic
Body centered cubic
Face centered cubic

24. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...

25. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...

26. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...

27. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...

28. Close-packed structures: fcc and hcp
ABABAB...
fcc
ABCABCABC...
first hcp, second fcc
very, very similar! Same packing density of 74%
funny mixtures between these would also be possible but are to my knowledge never observed for elemental solids
why is fcc a cubic structure?
The face-centred cubic (fcc) and hexagonal close-packed (hcp) structure have the same packing fraction