hexagonal close packing (hcp (ababab) recall close packing (hcp) (ababab) recall: chem 2060 lecture...

CHEM 2060 Lecture 16: Packing L16-1

Hexagonal Close Packing (hcp) (ABABAB) Recall:


Cubic Close Packing (ccp) (ABCABC) Recall:

NOTE: ccp is also known as FACE CENTERED CUBIC (fcc)


Holes in the fcc (i.e., ccp) Structure Coordination number (CN) of both close packed systems (hcp and ccp) is 12. Ø Each large ion (anions, usually) is in contact with 12 other the large ions.

There are also holes: O Holes (CN = 6) T Holes (CN = 4) Ø These are where the small ions sit (cations, usually) and the coordination

number is in reference to the number of large ions (anions) they contact Anions (large) are close packed and the cations (small) fit in the “holes”. (a) Octahedral holes are in clefts between six spheres (b) Tetrahedral holes are in the clefts between four spheres

O holes are bigger than T holes. There are twice as many T holes as O holes.


Zinc Blende (ZnS) ccp S2- with Zn2+ in half the T holes (To be precious, Zn2+ ions are in all the T sites in one of two T layers)

Coordination: 4:4 (tetrahedral) Note the similarity to the diamond structure! Each S2- anion (big) is coordinated to 4 Zn2+ cations. Each Zn2+ cation (small) is coordinated to 4 S2- anions. The Zinc Blende mineral is also called sphalerite or blackjack. It is actually (Zn,Fe)S.


Wurtzite (ZnS) hcp S2- with Zn2+ in half the T holes (To be precious, Zn2+ ions are in all the T sites in one of two T layers) Coordination: 4:4 (tetrahedral) The Wurtzite mineral is actually (Zn,Fe)S.

Other minerals that have a Wurtzite crystal structure include: CdS (Greenockite, pictured right), CdSe, and MnS.


Zinc Blende vs. Wurtzite


Fluorite (CaF2) ccp Ca2+ with F- in all T holes Coordination: Ca2+ 8 (cubic) green

F- 4 (tetrahedral) red Anti-Fluorite is a related structure in which cation and anion positions are reversed e.g., Li2O is ccp O2- with Li+ in all T holes


Rock Salt: ccp Cl- with Na+ in all the O holes

Notice the layers of Na+ (orange) and Cl- (green) ions on the diagonal (above) and illustrated has horizontal planes (left).


Comparing Structures: Perovskite (CaTiO3) Ca2+ (red) & O2- (blue) together form the ccp arrangement. The Ti4+ (green) sits in ¼ of the O holes. Note that the 4+ oxidation state of Ti is not accurate. There is significant covalent contribution to bonding.


Crystal Planes Looking closely at the ionic structures we have seen that anions and cations tend to lie in “sheets” or planes. Close packed layers Planes of atoms (or groups) are labeled using Miller Indices. [Def.] The Miller Indices are sets of 3 numbers, labeled h, k, and l, that denote a plane (hkl) in a crystal lattice. The numbers are the smallest integers which are reciprocals of the intercepts of the plane on the a, b, and c axes of the crystal. NOTE: Syntax is important! (hkl) indicates a plane {hkl} indicates the set of all planes equivalent to (hkl) by symmetry [hkl] indicates a direction (a vector, not a plane)


Crystallographic Directions [hkl] It is probably easiest to understand Miller indices by looking at how they are used to define a direction in the crystal lattice. This is done based on the origin of the unit cell. [Def] A unit cell is the smallest part of the crystal lattice that, when repeated (i.e., stacked together in all 3 directions) defines the crystal structure. We can pick an arbitrary point (usually a corner of the unit cell) as our point of origin (0) and define unit cell directions a, b, and c.

The [100] direction is along the a-axis. The [010] direction is along the b-axis. The [001] direction is along the c-axis. Question: What is [111]?


Crystallographic Planes {hkl}; Cubic system The crystal planes defined by the (hkl) or {hkl} Miller indices are those that are orthogonal to the [hkl] direction. EXAMPLES: (100) Plane in cubic system ⇒ orthogonal to the [100] direction (i.e., to the a-axis).

CUTS a axis at 1 ∴ 1 Doesn’t cut b axis ∴ 0 Doesn’t cut c axis ∴ 0 The {100} set of planes are then all those symmetry related to (100).

ab

c


(111) Plane in cubic system ⇒ orthogonal to [111] direction (i.e., to the body diagonal of the unit cell).

Cuts each axis at 1 ∴ (111) (011) Plane in cubic system

Does NOT cut a axis ∴ 0 Cuts b axis at 1 ∴ 1 Cuts c axis at 1 ∴ 1

NOTE: Another way of thinking of the number assignments: (011) = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∞ 11 ,1

1 ,1

ab

c

ab

c


Values other than 0 and 1 The negative direction isn’t called -1, it is called

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1 (read “bar one”) The numbers can be assigned as reciprocals of the unit cell length at which the plane “cuts”. So a plane that cuts at ½ the unit cell length in the a direction, but doesn’t cut the b and c axes is

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112, 1∞, 1∞

#

$ %

&

' ( = (200)


A Last Word on Crystal Planes …Miller Indices are INTEGERS

(233) plane

because

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13, 12, 12

" # $

% & ' aren’t integers

and the smallest set of integers with the same RATIO is (2,3,3).

HOMEWORK: Sketch the following planes in a cubic lattice:

(011), (012), (101), (

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1 01)


A Brief Mention of Diffraction Methods …another look at X-ray diffraction These crystal structures were determined using X-ray diffraction techniques. X-rays are (partially) reflected by planes of atoms. This is described by Bragg’s Law: When

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λ = 2d sinθ , the reflected waves have a phase difference of 2π and interfere constructively. From this, we get an interference pattern (diffraction pattern) and can mathematically solve for the atom positions.

hexagonal close packing (hcp (ababab) recall close packing (hcp) (ababab) recall: chem 2060 lecture...

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