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  • 7/30/2019 Lecture+3+MAK Crystal+Structure

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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