lecture+3+mak crystal+structure
TRANSCRIPT
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PHY 3201 FIZIK KEADAAN PEPEJAL
Directions in crystal
1
Fig. Shows
[111] direction
We choose one lattice point on the lineas an origin, say the point O. Choice oforigin is completely arbitrary, since everylattice point is identical.
Then we choose the lattice vector joiningO to any point on the line, say point T.This vector can be written as;
R = n1 a + n2 b + n3c
To distinguish a lattice direction from alattice point, the triple is enclosed insquare brackets [ ...] is used.[n1n2n3]
[n1n2n3] is the smallest integer of thesame relative ratios.
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Directions in crystal
DirectionA
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 B1. 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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Directions in crystal
210
X = , Y = , Z = 1
[ 1] [1 1 2]X = 1 , Y = , Z = 0
[1 0] [2 1 0]
More examples
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Directions in crystal
X = -1 , Y = -1 , Z = 0 [110]
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Directions in crystal
X =-1 , Y = 1 , Z = -1/6
[-1 1 -1/6] [6 6 1]
We can move vector to the origin.
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Crystal directions may be grouped in families. To avoid confusion
there exists a convention in the choice of brackets surrounding
the three numbers to differentiate a crystal direction from a family
of direction. For a direction, square brackets [hkl] are used to
indicate an individual direction.Angle brackets indicate afamily of directions. A family of directions includes any directions
that are equivalent in length and types of atoms encountered. For
example, in a cubic lattice, the [100], [010], and [001] directions
all belong to the family of planes because they are
equivalent. If the cubic lattice were rotated 90 , the a, b, and c
directions would remain indistinguishable, and there would be no
way of telling on which crystallographic positions the atoms are
situated, so the family of directions is the same.
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Equivalency of crystallographic directions of a form in cubic
systems.
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The position of basis or atoms in the
conventional cell is often expressed in
terms of the axes defining the cell. For
instance, the position of the body-centredatoms is 1/2, 1/2, 1/2 and the face-centred
atom is 1/2, 1/2, 0.
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INDEX SYSTEM FOR
CRYSTAL PLANES
Figure 13. This plane
intercepts the a1, a2 , a3. axes
at 3a, 2b, 2c. The reciprocals
of these numbers are . Thesmallest three integers having
the same ratio are 2, 3, 3, and
thus the Miller indices of the
plane are (233).
MILLER INDICES
In Solid State Physics, it is important to
be able to specify a plane or a set of
planes in the crystal. This is normally
done by using the Miller indices. The
use and definition of these Miller indicesare as follows
Find the intercepts on the axes in terms
of the lattice constants a1, a2 , a3. The
axes may be those of a primitive or
nonprimitive cell.
Take the reciprocals of these numbersand then reduce to three integers having
the same ratio, usually the smallest
integers. The results, enclosed in
parentheses (hkl), is called the index of
the plane.
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MILLER INDICES
If the plane cuts an axis at infinity, thecorresponding index will be zero. By
convention, if the intercept has a negative
value, the corresponding index is alsonegative. A minus sign is normally
placed above that index in the bracket.
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MILLER INDICES
Axis X Y Z
Interceptpoints 1
Reciprocals 1/1 1/ 1/ Smallest
Ratio 1 0 0
Miller ndices (100)(1,0,0)
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MILLER INDICES
Axis X Y Z
Interceptpoints 1 1
Reciprocals 1/1 1/ 1 1/ Smallest
Ratio 1 1 0
Miller ndices (110)(1,0,0)
(0,1,0)
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MILLER INDICES
Axis X Y Z
Interceptpoints 1/2 1
Reciprocals 1/() 1/ 1 1/ Smallest
Ratio 2 1 0
Miller ndices (210)(1/2, 0, 0)
(0,1,0)
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MILLER INDICES
Miller indices are also used to denote a setof planes which are parallel. For instance,the plane (200) is parallel to (100). The
former cuts the x-axis at /2. Also bysymmetry, many sets of planes, e.g. all thefaces of a cube, may be represented by asingle set of Miller indices (100). In this
case the curly bracket is used, hence {100}.In other words the {100} automaticallyincludes the planes (100), (010) and (001).
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PlaneA
1. x= 1, y= 1, z= 1
2.1/x= 1, 1/y = 1,1 /z= 1
3. No fractions to clear
4. (111)
Plane B
1. The plane never intercepts the z axis, so x= 1, y= 2, andz=
2.1/x= 1, 1/y =1/2, 1/z= 0
3. Clear fractions:1/x= 2, 1/y = 1, 1/z = 0
4. (210)
Plane C
1. We must move the origin, since the plane passes through
0, 0, 0. Lets move the origin one lattice parameter in the y-direction. Then, x= , y= -1, and z =
2.1/x= 0, 1/y = 1, 1/z = 0
3. No fractions to clear.
)010(.4