lecture+2+mak crystal+structure1

7/30/2019 Lecture+2+MAK Crystal+Structure1

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PHY 3201 FIZIK KEADAAN PEPEJAL

Three-Dimensional

Lattice Types

.

There are seven crystal classes or crystal system, each

of which is a parallelepiped (pa-ra-lel-li-pi-ped). Figure

below shows the scheme whereby these seven types

are defined

3a

1a

2

a


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Three-Dimensional

Lattice Types

System Number of

lattices in

system

Lattice symbols Restrictions on

conventional cell axes and

angles

Triclinic 1 P Monoclinic 2 P, C = = 90

Orthorhombi

c

4 P, C, I, F = = = 90

Tetragonal 2 P, I = = = 90

Cubic 3 Por sc

Ior bccFor fcc

= = = 90

Trigonal 1 R = = < 120, 90Hexagonal 1 P = = 90

= 120

Table 1 The fourteen lattice types in

three dimensions


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The cubic space

lattices


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Characteristics of

cubic lattices

sc bcc fcc

Volume, conventional cell a3 a3 a3

Lattice point per cell 1 2 4

Volume, primitive cell a3 a3 a3

Lattice points per unit

volume1/a3 2/a3 4/a3

Number of nearest

neighbors6 8 12

Nearest-neighbors

distancea 31/2a/2 = 0.866a a/21/2 = 0.707a

Packing fraction 1/6 1/8 3 1/6 2


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Simple cubicThe simple cubic has 1 lattice point per unit cell, with

total area a3

Number of nearest neighbours:6

Conventional= Primitive cell

Note: a= lattice constant


7/13PHY 3201 FIZIK KEADAAN PEPEJAL

Wikipedia: Atomic packing fraction is the fraction

of volume in a crystal structure that is occupied by

atoms. It is dimensionless and always less than

unity. For practical purposes, the APF of a crystalstructure is determined by assuming that atoms are

rigidspheres. The radius of the spheres is taken to

be the maximal value such that the atoms do not

overlap.
http://en.wikipedia.org/wiki/Crystal_structurehttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Crystal_structurehttp://en.wikipedia.org/wiki/Crystal_structurehttp://en.wikipedia.org/wiki/Crystal_structure


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Body-centered Cubic

Figure 9 Body-centered cubic

lattice, showing a primitive cell.

The primitive cell shown is a

rhombohedron of edge 1/23a,and the angle between adjacent

edges is 109o28 . Here, theconventional cubic cell is not

primitive. There are two lattice

points per cubic cell due to the

extra lattice point at the body

centre of the cell.

Figure 10 Primitive translation vectors of

the body-centered cubic lattice; these

vectors connect the lattice point at the

origin to lattice points at the body centers.The primitive cell is obtained on completing

the rhombohedron. In terms of the cube

edge a the primitive translation vectors are

)(2

);(2

);(2

321 zyxa

azyxa

azyxa

a


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Body-centered

Cubic

The body-centered cubic lattice has 2 lattice points per

unit cell

Number of nearest neighbour: 8

Conventional Primitive cell


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bcc


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Three-Dimensional

Lattice Types

Figure 11 The rhombohedral

primitive cell of the face-

centered cubic crystal. The

primitive translation vectors

connect the lattice point at theorigin with lattice points at the

face centers. As drawn, the

primitive vectors are:

).(2

);(2

);(2

321 xza

azya

ayxa

a

The face-centred (fcc) in which there is a

lattice point at the centre of each cube

face. Again, the fcc conventional cubic

cell is not primitive. There are four

lattice points per cubic cell. The

primitive translation vectors for the fccare shown in fig. 7.

From these figures, you can see that the

primitive vectors/cells of both the bcc

and fcc are much too complicated. It is

much easier to classify them as a form of

cubic structure from which symmetry

operations can be visualised more

easily.


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The face-centered cubic lattice has 4 lattice points per

unit cell

Number of nearest neighbour: 12

Conventional Primitive cell


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Three-Dimensional

Lattice Types

Figure 12 Relation of the

primitive cell in the

hexagonal system (heavy

lines) to a prism of

hexagonal symmetry. Here

a = bc. Three suchrhombic prisms can be put

together to form a rightprism of hexagon, hence

the term hexagonal system

lecture+2+mak crystal+structure1

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