Lecture 4 - Crystal Systems

7/27/2019 Lecture 4 - Crystal Systems

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Chapter 3

CRYSTAL SYSTEMS


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Unit Cell (Lattice) Parameters

a, b, c are the

edge lengths

, , are the

inter-axial angles

Seven different

combinations of

edge lengths andinter axial angles

Distinct Crystal

System

x

y

z

a

b

c


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Seven Crystal Systems

1

2

3

4

5

6

73


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1. Cubic Crystals

x

y

z

a

b

c

Unit Cell Parameters

a = b = c

= = = 90o

Halite (a form of NaCl)
http://en.wikipedia.org/wiki/Image:Cubic_crystal_shape.png


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2. Hexagonal Crystals

x

y

z

a

b

c


Calcite

a = b c

= = 90o, = 120o
http://mineral.galleries.com/minerals/carbonat/calcite/calcite.htmhttp://en.wikipedia.org/wiki/Image:Hexagonal.png


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3. Tetragonal Crystals

x

y

z

a

b

c


a = b c

= = = 90o

Ruby Hematite
http://en.wikipedia.org/wiki/Image:Tetragonal.png


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4. Rhombohedral (Trigonal)

Crystals

x

y

z

a

b

c


a = b c

Quartz

= = 90o
http://en.wikipedia.org/wiki/Image:Rhombohedral.png


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5. Orthorhombic Crystals

x

y

z

a

b

c


BariteTopaz

= = = 90oa b c
http://en.wikipedia.org/wiki/Image:Orthorhombic.png


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6. Monoclinic Crystals

x

y

z

a

b

c


Gypsum Muscovite

a b c

= = 90o
http://mineral.galleries.com/minerals/silicate/muscovit/mus-17.jpghttp://mineral.galleries.com/minerals/sulfates/gypsum/gypsum.htm


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7. Triclinic Crystals

x

y

z

a

b

c


Oligoclase

(Na-Ca-Al Silicate)

Kyanite

(Al Silicate)

a b c
http://en.wikipedia.org/wiki/Image:Triclinic.png


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Crystallographic Points,

Directions, Planes

Needs either a point within the unit cell, acrystallographic direction or a

crystallographic plane of atoms to describe

the crystal

Becomes more important when the co-

ordinate axes are not perpendicular to each

other

HexagonalRhombohedral

Monoclinic

Triclinic


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Crystallographic Points

(point coordinates)

Point P (q,r,s) (q,r,s) 1

q corrsponds to

distance qa

rcorresponds to

distance rb

s corresponds to

distance sc

x

y

z

a

b

cP (q,r,s)

qa

rb

sc


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Crystallographic Points

Find Point P (, 1, ) Substitute q = , r = 1,

and s =

x

y

z

a

b

cP (, 1, )

a

1b

c


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Crystallographic Directions

A line between two points, or a vector.1. Draw a vector that passes through the origin of the

co-ordinate system

2. Determine the length of the vector projection on

three axesmeasured in terms of unit celldimensions a, b, c

3. Reduce them to smallest integer values

4. Represent as [uvw] no commas.

u, v, w correspond to reduced projections along x, y, z

axes


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Example for direction [uvw]

Step 1: A vector is positioned such that it passesthrough the origin of the coordinate system

X

Y

Z

What are the indices for theVector shown?

0.4 nm

0.5 nm

0.3 nm


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Step 2: Lengths of the vector projection on each of thethree axes are determined; these are indicated in

terms of unit cell dimensions a,b,c

X

Y

Z

0.4 nm

0.5 nm

0.3 nm

Projections

X Y Z

0a 0.5b 0.5c


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Step 3: These projections are multiplied to reducethem to the smallest integer value

X

Y

Z

0.4 nm

0.5 nm

0.3 nm

Projections

X Y Z

0a 0.5b 0.5c

0 0.5 0.5Reduction

0 1 1Reduction


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Step 4: The reduced integers are enclosed in squarebrackets without comma

X

Y

Z

0.4 nm

0.5 nm

0.3 nm

Projections

X Y Z

0a 0.5b 0.5c

0 0.5 0.5Reduction

0 1 1Reduction

[011]


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Negative directions get a hat

X

Y

Z

0.4 nm

0.5 nm

0.3 nm

Projections

X Y Z

0a 0.5b 0.5c

0 0.5 0.5Reduction

0 1 1Reduction

[011]


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Negative directions get a hat

X

Y

Z

0.4 nm

0.5 nm

0.3 nm

Projections

X Y Z

0a 0.5b 0.5c

0 0.5 0.5Reduction

0 1 1Reduction

[011]


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Indices for the Direction

1. Take the projection on the X-axis: a/2

2. Take the projection on the Y-axis: b

3. Take the projection on the Z-axis: 0c

b

c

a

x

y

z

Reduce them into smallest integers

[ 1 0] or [1 2 0] ?


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Crystallographic Planes

In all crystal structuresother than hexagonal,

crystallographic planes

are specified by three

Miller Indices(hkl)

Any two planes parallel

to each other are

equivalent and have

identical indices

The unit cell system is

the basis

x

y

z

a

b

c


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23

Crystallographic Planes

Adapted from Fig. 3.26,

Callister & Rethwisch 4e.


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Determining h,k,l

1. If the plane passes through selected origin,select another origin

2. Read off intercepts of plane with axes interms ofa, b, c

3. Take the reciprocals of these numbers

A plane that parallels an axis has infinite intercept

hence zero index

4. Reduce to set ofsmallest integers5. Represent as (hkl)

An intercept on the negative side denoted with a

bar on the top of the index

D t i Mill I di f th


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Determine Miller Indices for the

Plane

b

c

a

x

y

z

c/2

x

Step 1: Redefine Origin

b

c

ay

z

c/2y

z

x


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Plane

Step 2: Determine the plane intercepts

x b

c

a y

z

c/2y

z

x

x y z

Intercepts inf a -b c/2

Intercepts

(Lattice par.) inf -1 1/2



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Plane

Step 3: Determine the reciprocals

x b

c

a y

z

c/2y

z

x

x y z

Reciprocals 0 -1 2

Step 4: Reduction to integersnot needed

Step 5: (012)


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Miller Indices for the Plane

x

y

z

O

O

z

x

-b

c/2

1. Plane parallel to X-axis is a, intercept

is , index is 0

2. Intercept along Y-axis isb,

intercept is -1, index is -1

3. Intercept along Z-axis is c/2,

intercept is , index is 2

Miller Indices for this plane : [012]


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Miller Indices for the Plane

z

y

x

1. Plane parallel to X-axis is a, interceptis , index is 0

2. Intercept along Y-axis is b/2,intercept is 1/2, index is 2

3. Plane parallel to Z-axis is c, interceptis , index is 0

Miller Indices :

(020)


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Constructing a Plane (011)

0 indicates that the

plane is parallel to X-

axis

-1 indicates that the

intercept along Y axisisb


intercept along Z axis

is cx

y

z

a

b

c

-b


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Constructing a (110) Plane

1 indicates that theintercept along X-axisis a


intercept along Y axisis b

0 indicates that theplane is parallel to

the Z-axis

x

y

z

Lecture 4 - Crystal Systems

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