Soild 2011 Autumn

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Crystal Structure



Can pack with irregular shapes



-

Most efficient way of packing equal sized spheres.

In 2D, have close packed layers

Coordination number

(CN) = 6. This is the2D packing.

Can stack close packed (c.p.) to give 3D structures.



If we start with one c la er two ossible wa s of addin a

second layer (can have one or other, but not a mixture) :



If we start with one c la er two ossible wa s of addin a

second layer (can have one or other, but not a mixture) :



Two possibilities:

.regular sequence …ABABABA…..Hexagonal close packing (hcp)

(2) Can have layer in C position, followed by the samerepeat, to g ve … …Cubic close packing (ccp)



Hexagonal close packed Cubic close packed



o matter w at type o pac ng, t e coor nat onnumber of each equal size sphere is always 12

possible for non-equal size spheres







FCC structure has a-b-c-a-b-c





Build up ccp layers Add construction lines(ABC… packing) - can see fcc unit cell

c.p layers are oriented perpendicular to the bodydiagonal of the cube



(hcp)











Lattices

By definition, crystals are periodic in three dimensions. A lattice is aregular infinite arrangement of points in which every point has the sameenvironment as any other point. A lattice in 2 dimensions is called a net

and a regular stacking of nets gives us a 3-dimensional lattice.

2-D net Stacks of 2-D nets

produce 3-D lattices.



Symmetry The unit cell in three dimensions.The unit cell is defined by three

Crystals are regular periodic arrays,i.e. they have long range translationalsymmetry. Crystals are often

edge vectors a, b, and c, with , , , corresponding to the anglesbetween b-c, a –c, and a-b,

cons ere to ave essent a y n n te

dimensions.a

respectively.

bOne Unit Cell

a

c

Unit cell = The smallest volume from whichthe entire crystal can be constructed bytranslation onl . All cr stals have

Unit cells are defined in terms of thelengths of the three vectors and the

translational symmetry, with the translationalvectors equal to edges of the unit cell.

ree ang es.For example, a=94.2Å, b=72.6Å,

c=30.1Å, =90°, =102.1°, =90°.



Translations (Lattices)A property at the atomic level, not of crystal

shapes

Symmetric translations involve repeatdistancesThe origin is arbitrary

1-D translations = a rowa

a is the repeat vectora is the repeat vector



Translations (Lattices)- =

a

b

, ,, ,



Translations (Lattices)- =

b

a

Every point that is exactly n repeats from that point is an equipoint to the originalEvery point that is exactly n repeats from that point is an equipoint to the original

T l ti l S tT l ti l S t



Translational SymmetryTranslational Symmetry

Every translation has a distance and a direction.

Translations are not rotated or reflected…the shape remains thesame size.



Examples of TranslationalExamples of Translational

SymmetrySymmetry

The objects simply move from one position to another retaining size and shape.



Crystal: Periodic Arrays of Atoms

ranslation Vectorsa3

Atoma1

a2a1, a2 ,a3

Primitive Cell:

• Smallest building block forthe crystal structure.

• Repetition of the primitive cell



-(sodium chloride, NaCl)

e e ne a ce po n s ; ese are po n swith identical environments



-atoms - but unit cell size should always be the same.



-it doesn’t matter if you start from Na or Cl



- or if you don’t start from an atom



same - empty space is not allowed!



. . sc er wor s c or on r - aarn- e e er an s.All rights reserved.



Seven unit cell shapes

• Cubic a=b=c == =90°• Tetragonal a=bc == =90°• Orthorhombic abc == =90°• Monoclinic abc = =90°, 90°• r c n c a c

• Hexagonal a=bc ==90°, =120°• = = = = °

Think about the sha es that these define - look at themodels provided.



Centering in Unit Cells




Note that to best indicate the symmetry of the crystal lattice, it is

necessary to choose a unit cell that contains more than one lattice point.Unit cells that contain only one lattice point are called “Primitive” and areindicated with a “P ”. There are only a limited number of unique ways toc oose centere ce s an t e num er an compos t on o poss e

centered cells depend on the crystal system.

Remember that only 1/8of an atom at the cornerof the cell is actually in

.is primitive.





For monoclinic cells, no other form of centering provides a different

solution, however for other crystal systems different types of centering arepossible. These include centering of specific faces (“A”, “B” or “C”),centering of all faces “F” or body centering “I”, in which there is a latticepo nt at t e geometr c center o t e un t ce .

All of the differentpossible lattices,including unique

centering schemeswere eterm ne y .Bravais and thus the14 unique lattices are

“

Lattices”

Primitive, P Body-centered, I Face-centered, F

Bravais Lattices



Bravais Lattices

These 14 lattices arethe unique scaffolds inw c atoms or

molecules may bearranged to form.

However there is a

symmetry in crystalswhich must beconsidered to describe

the actual arrangementof atoms in a crystal.These are the variouscrystallogaphic Point

Groups .



Name axes angles

Triclinic a b c 90o

= o

o

-

Orthorhombic a b c = 90o

Tetragonal a1 = a2 c = 90o

Hexagonal

++cc

Hexagonal (4 axes) a1 = a2 = a3 c = 90o 120o

Rhombohedral a1 = a2 = a3 90o

Isometric a1 = a2 = a3 = 90o

++aa

++bb

Axial convention:Axial convention:

“right“right--hand rule”hand rule”

c cc



b b

a P a I = Ca

b

Ponoc n c

a

b

c

a b c

c

b

POrthorhombic

a b c

C F I

c c



c c

a2

a1

P Tetragonal I

a2

a1

P or C RHexagonal Rhombohedral

a1 = a2 c

a1a2c

a1 = a2 = a3

a3

a

a2

Isometric a1 = a2 = a3

F I



Three Cubic Lattices



Three Cubic Lattices

1. Simple Cubic (SC)

a3a1 a2 a3

=

a1

2

Add one atom at the Add one atom

at the center of each face

2. Body-Centered Cubic (BCC) 3. Face-Centered Cubic (FCC)

Conventional Cell Primitive Cell



Primitive Cell of FCC

• ng e e ween a1, a2, a3:



a

aa

Body-Centered

Unit Cell Primitive Cell



a

a

a

Face-Centered

Cubic (F)

Primitive Cell

a

Unit Cell

Rotated 90º





















When silver crystallizes, it forms face-centered cubic



. .the density of silver.

m -=V

= = = .

4 atoms/unit cell in a face-centered cubic cell

m = 4 Ag atoms107.9 g

mole A

x1 mole Ag

6.022 x 1023 atoms

x = 7.17 x 10-22 g

m 7.17 x 10-22 g 3 V

6.83 x 10-23

cm3

.



















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Soild 2011 Autumn

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