1.0 crystal structure

7/29/2019 1.0 Crystal Structure

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Crystal Structures


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atoms pack in periodic, 3D arrays

typical of:

Crystalline materials...

-metals

-many ceramics

-some polymers

atoms have no periodic packing

occurs for:

Noncrystalline materials...

-complex structures-rapid cooling

crystalline SiO2

noncrystalline SiO2"Amorphous" = Noncrystalline

Materials


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Si (crystalline)

SiO2

(amorphous)


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Scanning Tunneling Microscope

Image of Iron in the (110) plane


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Crystal structures

Crystalline material: A material in which theatoms are situated in a repeating or periodic arrayover large atomic distances (i.e., long order

exists). Upon solidification, atoms will positionthemselves in a repetitive 3D pattern, in whicheach atom is bonded to its nearest-neighbor atoms.

Atomic hard sphere model is used to describeatoms.

Lattice: A 3D array of points coinciding with atompositions (or sphere centers).


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Space Lattice Atomic arrangements in crystalline solids can be described with

respect to a network of lines in three dimensions.

The intersections of the lines are called lattice sites (or latticepoints). Each lattice site has the same environment in the same

direction. A particular arrangement of atoms in a crystal structure can be

described by specifying the atom positions in a repeating unit

cell. A unit cell is shown below.


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Lattice Point- Point that contains an atom ormolecule

Unit Cell- Region defined by a,b,c which when

translated by integral multiple of these vectorsreproduces a similar region of the crystal

Basis Vector-A set of linearly independent vectors

(a,b,c) which can be used to define a unit cell


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UNIT CELL

A unit cell is the smallest component of the crystal that reproduces

the whole crystal when stacked together with purely translational

repetition

Primitive (P) unit cells contain only a single lattice point.

Internal (I) unit cell contains an atom in the body center.Face (F) unit cell contains atoms in the all faces of the planescomposing the unit cell.

Centered (C) unit cell contains atoms centered on the sidesof

the unit cell


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Primitive Unit Cell- The smallest unit cell,in volume, that can be defined for a given

lattice

Primitive Basis Vectors- A set of linearlyindependently vectors in that can be used to

define a primitive unit cell


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CrystalsWhat defines a crystal?

crystalprimary building block:

the unit celllattice:

set of points with

identical environment


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CrystalsWhich is the unit cell?

primitive cell:

smallest possible

volume

1 lattice point


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A two-dimensional lattice showing translation of a unit cell by r = 3a + 2b.

Unit Cells

F

E


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Unit Cells

Basis Vectors= 2a,2b

Unit Cell = ODEFPrimitive Unit Cell = ODEF

Primitive Basis Vectors = a,b


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Lattice Vector

R=ha+kb+lc

h,k,l are integers

A displacement of any lattice point by R will give a

new position in the lattice that has the samepositional appearance as the original position


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Single species

A B C

D

F

a

b

G

A - G : Primitive unit cells

All have same areaAll smallest unit cell

All have 1 atom/cell

a : Not a unit cell

b :Unit cell not Primitive


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Counting Lattice Points/Atoms in 2D Lattices

Unit cell is Primitive (1 lattice point) but

contains TWO atoms in the Motif

Atoms at the corner of the 2D unit cell

contribute only 1/4 to unit cell count

Atoms at the edge of the 2D unit cellcontribute only 1/2 to unit cell count

Atoms within the 2D unit cell contribute 1

(i.e. uniquely) to that unit cell


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Multi Species

Non Primitive

2 Red, 2green

Primitive

1 Red, 1 Green


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3 D, Bravis Lattices

Each unit cell is such that the entire lattice can beformed by displacing the unit cell by R with nogaps in the structure (close packed)

ie sctructures with 3, 4 and 6 fold symmetry can beclose packed. 5-fold (ie pentangles) cannot

In 3 dimensions there are only 14 ways ofarranging points symmetrically in space that cangive no gaps

These arrangements are the BRAVIS LATTICES

These can be further subdivided into 7 crystal

structures


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14 Bravais

Lattices


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14 Bravais Lattices


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Cubic Lattices

BCC and FCC are not primitive. bcc has 4 atoms/cell, fcc has 8 atoms/cell

fcc has closest packing, then bcc then sc (for cubic) (fcc and bcc more common

than sc).

Primitive have 1 atom/cell, Both are Rhombohedral (Trigonal) (McKelvey p10)


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Crystal Structures

Types of crystal structuresFace centered cubic (FCC)

Body centered cubic (BCC)

Hexagonal close packed (HCP)

Close Packed StructuresDifferent Packing of HCP and FCC

Crystallographic Directions and Planescubic systems


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Counting Atoms in 3D Cells

Atoms in different positions are shared by differing numbers

of unit cells

Vertexatom shared by 8 cells => 1/8atom per cell

Edgeatom shared by 4 cells => 1/4atom per cell

Faceatom shared by 2 cells => 1/2atom per cell

Bodyunique to 1 cell => 1atom per cell


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Body Centered Cubic

Atoms are arranged at the corners of thecube with another atom at the cube center.


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Face Centered Cubic (FCC)

Atoms are arranged at the corners andcenter of each cube face of the cell.

Atoms are assumed to touch along face

diagonals


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Triclinic (Primitive)

a b c, 90o

Monoclinic (P)

a b c, = = 90o,

90o

Monoclinic (BaseC)

a b c, = = 90o, 90o

Triclinic Monoclinic


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Primitive unit cells

Ch t i ti f S l t d El t t 20C


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Element

Aluminum

ArgonBarium

Beryllium

Boron

Bromine

Cadmium

CalciumCarbon

Cesium

Chlorine

Chromium

Cobalt

CopperFlourine

Gallium

Germanium

Gold

Helium

Hydrogen

Symbol

Al

ArBa

Be

B

Br

Cd

CaC

Cs

Cl

Cr

Co

Cu F

Ga

Ge

Au

He

H

Atomic radius

(nm)

0.143

------0.217

0.114

------

------

0.149

0.1970.071

0.265

------

0.125

0.125

0.128 ------

0.122

0.122

0.144

------

------

Density

(g/cm 3)

2.71

------3.5

1.85

2.34

------

8.65

1.552.25

1.87

------

7.19

8.9

8.94 ------

5.90

5.32

19.32

------

------

Characteristics of Selected Elements at 20C


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Some Materials have Different

Crystal Structures at DifferentTemperatures

Many elements and compounds exist in more than onecrystalline form under different conditions of temperature

and pressure.

This phenomenon is called plymorphism or allotropy. Someexamples are given below.


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Manganese

-MnTN=95K

bcc (58 atoms)

1000K

b-Mnsc (20 atoms)

1370K

-MnTN=511K

fcc

1410K

Lecture 1


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Graphite:

Graphite

3 out of the 4 electrons provide the bonding in in the hexagonal arrayThe 4th electron is free to conduct

3.4

1.4


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Diamond:

diamond

graphite


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NaCl (B1)
http://cst-www.nrl.navy.mil/lattice/struk/b1.htmlhttp://cst-www.nrl.navy.mil/lattice/struk/b1.htmlhttp://cst-www.nrl.navy.mil/lattice/struk/b1.html


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http://cst-www.nrl.navy.mil/lattice/struk.picts/b2.s.png


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1.0 crystal structure

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