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    Tutorial - 5

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    keshav bulia

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    Tutorial - 5

    Uploaded by

    keshav bulia
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    9 views16 pages

    Tutorial - 5

    Uploaded by

    keshav bulia

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    Download as pptx, pdf, or txt
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    TUTORIAL - 6

    MM 318
    Question 1

    For the direct lattice shown in the figure, draw a representative


    reciprocal lattice and find the reciprocal lattice vector representing
    the set of planes drawn in direct lattice.
    Hint

    a) draw the reciprocal lattice vector

    b) Find a direction normal to the set of planes in direct lattice

    c) Draw a vector along one of the planes by taking intercepts of these planes on direct
    lattice vectors

    d) Convert this vector to reciprocal lattice vector which represents the set of planes
    Solution
    Question 2

    Consider the 2D-direct lattice has lattice vectors in


    cartesian coordinates as, a⃗ = 3x̂ + ŷ and b⃗ = 2x̂ + 6y.
    a) Find the angle between the lattice vectors
    b) Draw the direct lattice
    c) Calculate the reciprocal lattice vectors and angle between them
    d) Draw the reciprocal lattice
    e) Draw 1st and 2nd Brillouin zones of the reciprocal lattice
    Hint

    a)From the given lattice vectors, the angle between can be obtained by taking the dot
    product of the vectors

    b) Use the cartesian coordinates to draw the direct lattice

    c)For calculating the reciprocal lattice vectors for 2D lattice, assume c⃗ = ẑ and use the
    3D lattice formulas

    d) Draw the reciprocal lattice from the reciprocal lattice vectors

    e) Draw Brillouin zones by calculating the reciprocal lattice vector and the condition

    G ⃗ ⋅ G ⃗+ 2 K ⃗ ⋅ G ⃗ = 0
    Solution

    a)From the given lattice vectors, the angle between can be obtained by taking the dot
    product of the vectors

    a⃗ ⋅ b⃗ = | a⃗| | b⃗ | cosθ

    6+ 6= 10 40 cosθ

    θ = cos−1(12/20) = 53.13∘

    b) Direct lattice
    Solution

    c) For calculating the reciprocal lattice vector for 2D lattice, assume c⃗ = ẑ .


    Then,
    b⃗ × ⃗ ⃗= c⃗ × a⃗ a⃗ × b⃗
    ⃗
    a* = b* ⃗
    c* =
    c
    a⃗ ⋅ (b⃗ × a⃗ ⋅ (b ×
    ⃗
    a⃗ ⋅ (b⃗ ×
    c⃗) c⃗)
    This gives, c⃗)

    3 1 1 3
    a* ⃗ = x̂ − y ̂ b* ⃗ = − x̂ + y 8 8
    16 16

    Then angle between them is,

    θ* = 126.87∘
    Solution

    d) Then reciprocal lattice can be drawn as below,


    Solution

    e) 1st and 2nd Brillouin zones of the reciprocal lattice are


    Question 3

    For the given reciprocal lattice and X-ray scattering from a crystal
    with mono-atomic basis as shown, what are the possible scattering
    vectors k / when k ⃗ = 2b1
    Hint

    Use the relation between the incident wave vector and the reciprocal lattice vector G
    to identify the scattering wave vectors
    Solution

    We know that,

    k / ⃗ = k⃗ +
    and G

    Then, | k/ ⃗ | = |
    k ⃗|
    k / ⃗ = 2b1⃗ +
    G
    Solution

    We can draw this as,

    Hence

    k / ⃗ = ± 2b1⃗, ± 2b2⃗, ±

    2(b1⃗ + b2⃗)
    A 2D projection of the reciprocal lattice of a cubic system (lattice
    parameter = a) and the superimposed Ewald’s sphere for X-ray
    diffraction is shown in the figure.

    a) Give a generic expression for the reciprocal lattice vector(s)


    corresponding to the plane that satisfies the Laue’s condition for
    constructive interference. (Only one is sufficient)

    b) For a cubic material with a = 3.68Å, find the wavelength of the


    incoming X-ray.

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