lecture_7, 8_chapter03_p3+p4

8/16/2019 Lecture_7, 8_chapter03_p3+p4

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Chapter 3 -

Crystal Systems

• The space lattice points in a crystal are occupied by atoms.

• The position of any atom in the 3-D lattice can be described by a vector:

r uvw = u a + v b + w c, where u, v and w are integers.

• The three unit vectors, a, b, c can define a cell as shown by the shaded region in Fig.(a).

This cell is known as unit cell (Fig. b) which when repeated in the three dimensions

generates the crystal structure.

(b) A unit cell with x, y, and z

coordinate axes, showing axial

lengths (a, b, and c) and

interaxial angles ( , ,andγ

).

Fig.a Fig.b

• The 6 parameters indicated in Fig.b, and are sometimes termed the lattice parameters of

a crystal structure. On this basis there are 7 different possible combinations of a, b, and c,

and , , and γ, each of which represents a distinct crystal system. These 7 crystal

systems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral (trigonal),

monoclinic, and triclinic. 1


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Chapter 3 -

Crystal Systems

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10 11

6 7 8 9

3

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Chapter 3 -

Crystal Systems

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12 13 14

5 6 7


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Chapter 3 -

Crystal Systems

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Some examples:


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Chapter 3 -

Crystallographic Points, Directions, And Planes

► Miller indices - A shorthand notation to describe certain crystallographic

directions and planes in a material. Denoted by [ ], < >, ( ) brackets. A negative

number is represented by a bar over the number.

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Chapter 3 -

Crystallographic Points, Directions, And Planes

• When dealing with crystalline materials, it often becomes necessary to specify a

particular point within a unit cell, a crystallographic direction, or some crystallographicplane of atoms.

• Labeling conventions have been established in which three numbers (indices) are used

to designate point locations, directions, and planes.

•

The basis for determining index values is the unit cell, with a right-handed coordinatesystem consisting of three ( x, y, and z) axes situated at one of the corners and coinciding

with the unit cell edges, as shown in the Fig. below.

Fig. A unit cell with x, y, and z coordinate

axes, showing axial lengths (a, b, and c)

and interaxial angles ( , , andγ).

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Chapter 3 -

Point Coordinates

- Point coordinates for unit cell center are

a /2, b /2, c /2 ( ½ ½ ½ )- Point coordinates for unit cell corner are 111

Translation: integer multiple of lattice

constants → identical position in another

unit cell.

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Chapter 3 -

Figure 3.5 The manner in which the q, r ,

and s coordinates at point P within the

unit cell are determined. The qcoordinate (which is a fraction)

corresponds to the distance qa along the

x axis, where a is the unit cell edge

length. The respective r and s

coordinates for the y and z axes are

determined similarly.

The position of any point located within a unit cell may be specified in terms of its

coordinates as fractional multiples of the unit cell edge lengths (i.e., in terms of a, b, and c).

The position of P in terms of the generalized coordinates q, r, and s (q is some fractional

length of a along the x-axis, r is some fractional length of b along the y-axis, and similarly

for s.

Thus, the position of P is designated using coordinates q r s with values that are ≤ to 1.

Point Coordinates

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Chapter 3 -

Question: Specify point coordinates for

all atom positions for a BCC unit cell.

Solution:

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Chapter 3 -

Crystallographic Directions

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A crystallographic direction is defined as a line between two points, or a vector.

The following steps are used to determine the three directional indices:

1. A vector of convenient length is positioned such that it

passes through the origin of the coordinate system. Any

vector may be translated throughout the crystal lattice

without alteration, if parallelism is maintained.

2. The length of the vector projection on each of the three

axes is determined; these are measured in terms of the

unit cell dimensions a, b, and c.

3. These three numbers are multiplied or divided by a

common factor to reduce them to the smallest integer

values.

4. The three indices, not separated by commas, are enclosed

in square brackets, thus: [uvw]. The u, v, and w integers

correspond to the reduced projections along the x, y,

and z axes, respectively.

Fig.3.6 The [100], [110], and[111] directions within a unit

cell.


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Chapter 3 -

• For each of the three axes, there will exist both positive and negative coordinates.

• The negative indices are represented by a bar over the appropriate index.

- For example, the direction would have a component in the (- y) direction.

- Changing the signs of all indices produces an antiparallel direction; that is,

is directly opposite to .


Question: Determine the indices for the direction

shown in the accompanying figure.

Solution:

Determination of Directional Indices

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Chapter 3 -

Construction of Specified Crystallographic Direction

Question: Draw a direction within a cubic unit cell.

Solution:

For this direction, the projections along the x, y, and z axes are a, ̶ a, and 0 a,

respectively. This direction is defined by a vector passing from the origin to point P, which

is located by first moving along the x axis a units, and from this position, parallel to the y

axis a units, as indicated in the figure. There is no z component to the vector, because the z

projection is zero.

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Chapter 3 -

Determine the Miller indices of directions A, B, and C .

Miller Indices, Directions

SOLUTION

Direction A

1. Two points are 1, 0, 0, and 0, 0, 0

2. 1, 0, 0, -0, 0, 0 = 1, 0, 0

3. No fractions to clear or integers to reduce

4. [100]Direction B

1. Two points are 1, 1, 1 and 0, 0, 0

2. 1, 1, 1, -0, 0, 0 = 1, 1, 1

3. No fractions to clear or integers to reduce

4. [111]

Direction C

1. Two points are 0, 0, 1 and 1/2, 1, 0

2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1

3. 2(-1/2, -1, 1) = -1, -2, 2

2]21[.4

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Chapter 3 -


-4, 1, 2

families of directions

z

x

where the overbar represents a

negative index

[412]=>

y

Example 2:

pt. 1 x 1 = a, y 1 = b /2 , z 1 = 0

pt. 2 x 2 = -a, y 2 = b, z 2 = c

=> -2, 1/2, 1

pt. 2head

pt. 1:

tail

Multiplying by 2 to eliminate the fraction

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Chapter 3 -


1. Vector repositioned (if necessary) to pass through

origin.

2. Read off projections in terms of unit cell dimensions

a, b, and c

3. Adjust to smallest integer values

4. Enclose in square brackets, no commas [uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

Families of directions

z

x

where overbar represents a negative index.[ 111 ]=>

y

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•For some crystal structures, several nonparallel directions with different indices are

crystallographically equivalent; this means that atom spacing along each direction is the

same.


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Chapter 3 -

Crystallographic Planes

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• The orientations of planes for a crystal structure are represented in a similar manner as the

directional indices of a crystal. The unit cell is the basis, with the three-axis coordinatesystem as represented in Figure 3.4 below.

• In all but the hexagonal crystal system, crystallographic planes are specified by three

Miller indices as ( hkl ).

•Any two planes parallel to each other are equivalent and have identical indices.

Fig.3.4 A unit cell with x, y, and z coordinate

axes, showing axial lengths (a, b, and c) and

interaxial angles ( , , andγ).


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Chapter 3 -

The procedure used to determine the h, k, and l index numbers is as follows:

1. If the plane passes through the selected origin, either another parallel plane must be

constructed within the unit cell by an appropriate translation, or a new origin must be

established at the corner of another unit cell.

2. At this point the crystallographic plane either intersects or parallels each of the three axes;

the length of the planar intercept for each axis is determined in terms of the lattice parametersa, b, and c.

3. The reciprocals of these numbers are taken. A plane that parallels an axis may be

considered to have an infinite intercept, and, therefore, a zero index.

4. If necessary, these three numbers are changed to the set of smallest integers by

multiplication or division by a common factor.

5. Finally, the integer indices, not separated by commas, are enclosed within parentheses,

thus: ( hkl ).


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Chapter 3 -

An intercept on the negative side of the origin is indicated by a bar or minus sign positioned

over the appropriate index. Furthermore, reversing the directions of all indices specifies

another plane parallel to, on the opposite side of, and equidistant from the origin.


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Summary of ( hkl ) index numbers determination procedure

• Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.

• If the plane passes through the origin, either:

– Construct another plane, or

– Create a new origin

– Then, for each axis, decide whether plane intersects or parallels the axis.

•Algorithm for Miller indices1. Read off intercepts of plane with axes in terms of a, b, c

2. Take reciprocals of intercepts

3. Reduce to smallest integer values

4. Enclose in parentheses, no commas, ( hkl )


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Chapter 3 -

One interesting and unique

characteristic of cubic crystals is

that planes and directions having

the same indices are perpendicular

to one another;

however, for other crystal systems

there are no simple geometrical

relationships between planes and

directions having the same indices.

Crystallographic planes arespecified by 3 Miller Indices ( h kl ). All parallel planes have same

Miller indices.


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Chapter 3 -


z

x

y

a b

c

4. Miller Indices (110)

example a b c z

x

y a b

c


1. Intercepts 1 1

2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2

2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

example a b c

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Chapter 3 -


z

x

y a b

c


example1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/½ 1/1 1/¾

2 1 4/33. Reduction 6 3 4

(001)(010),

Family of Planes {hkl }

(100), (010),(001),Ex: {100} = (100),

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lecture_7, 8_chapter03_p3+p4

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