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Questions and Answers
What defines a polycrystalline material?
What defines a polycrystalline material?
What is a grain boundary?
What is a grain boundary?
What does anisotropy in crystalline materials refer to?
What does anisotropy in crystalline materials refer to?
How does structural symmetry affect anisotropy in materials?
How does structural symmetry affect anisotropy in materials?
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What happens to the behavior of polycrystalline materials due to the random orientations of individual grains?
What happens to the behavior of polycrystalline materials due to the random orientations of individual grains?
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Which of the following is true about isotropic materials?
Which of the following is true about isotropic materials?
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Which factor influences the extent and magnitude of anisotropic effects in crystalline materials?
Which factor influences the extent and magnitude of anisotropic effects in crystalline materials?
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Which property can exhibit anisotropic behavior in single crystals?
Which property can exhibit anisotropic behavior in single crystals?
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What is the first step in determining crystallographic directions?
What is the first step in determining crystallographic directions?
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Which statement about crystallographic planes is true?
Which statement about crystallographic planes is true?
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How are crystallographic direction indices represented?
How are crystallographic direction indices represented?
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Which of the following correctly describes the format for Miller indices of crystallographic planes?
Which of the following correctly describes the format for Miller indices of crystallographic planes?
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What must be done with the lengths of the planar intercept for each axis when determining Miller indices?
What must be done with the lengths of the planar intercept for each axis when determining Miller indices?
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What indicates that two crystallographic planes are equivalent?
What indicates that two crystallographic planes are equivalent?
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In which situation is an infinite intercept considered for a crystallographic plane?
In which situation is an infinite intercept considered for a crystallographic plane?
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What does a bar or minus sign over an index in crystallographic planes indicate?
What does a bar or minus sign over an index in crystallographic planes indicate?
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What is done to the three values of vector projections to determine the crystallographic direction indices?
What is done to the three values of vector projections to determine the crystallographic direction indices?
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Which statement about planes and directions in cubic crystals is true?
Which statement about planes and directions in cubic crystals is true?
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What characterizes a 'family' of planes in crystallography?
What characterizes a 'family' of planes in crystallography?
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What is a defining feature of a single crystal?
What is a defining feature of a single crystal?
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Which environment condition is crucial for growing single crystals?
Which environment condition is crucial for growing single crystals?
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What happens to a single crystal when it grows without external constraints?
What happens to a single crystal when it grows without external constraints?
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What is one of the major uses of single crystals in modern technology?
What is one of the major uses of single crystals in modern technology?
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What is true about the atomic packing of (110) planes in FCC and BCC crystal structures?
What is true about the atomic packing of (110) planes in FCC and BCC crystal structures?
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Study Notes
Crystallographic Directions
- A crystallographic direction is a line between two points, also known as a vector.
- Determining crystallographic directions involves these steps:
- Position a vector of a convenient length through the origin of the coordinate system. Maintaining parallelism throughout the crystal lattice is crucial.
- Measure the vector's projections onto each of the three axes (a, b, and c) in terms of the unit cell dimensions.
- Reduce the three values to their smallest integer values by multiplication or division by a common factor.
- Enclose the reduced indices within square brackets, e.g., [uvw], where u, v, and w correspond to the reduced projections along the x, y, and z axes, respectively.
Negative Indices
- Negative coordinates are possible and are represented by a bar over the appropriate index.
- Reversing the signs of all indices defines an antiparallel direction.
- For example, [111] is the direct opposite of [111].
Equivalent Directions
- Some crystal structures have multiple non-parallel directions with different indices but have equivalent atomic spacing.
- These equivalent directions are grouped together in angle brackets, e.g., (100).
- In cubic crystals, directions with the same indices (irrespective of order or sign), are equivalent. For example, [123] and [213] are equivalent.
Crystallographic Planes
- The orientations of crystal planes are also represented similarly.
- In most crystal systems (except hexagonal), crystallographic planes are specified by three Miller indices (hkl).
- Determining indices follows these steps:
- If the plane passes through the chosen origin, a parallel plane must be constructed within the unit cell. Otherwise, a new origin must be created.
- Determine the intercepts of the plane with each of the three axes (x, y, and z) in terms of the lattice parameters (a, b, and c).
- Calculate the reciprocals of these intercepts.
- Simplify the reciprocals to their smallest integral values via multiplication or division by a common factor.
- Enclose the integer indices in parentheses to represent the plane, e.g., (hkl).
- Intercepts on the negative side of the origin are indicated by a bar (e.g., (123)). Reversing all the indices gives the opposite plane.
- Planes with the same indices (e.g., (100)) in a cubic crystal are perpendicular to each other
Atomic Arrangement
- Atomic arrangements on crystallographic planes depend on the crystal structure (e.g., FCC or BCC).
- Circles in diagrams represent atoms located at the centers of full-sized hard spheres in the crystallographic planes.
- A "family" of planes includes all equivalent planes having the same atomic packing.
- In cubic crystals, equivalent planes share the same indices, irrespective of order and any sign. For example both (123) and (312) belong to the {123} family.
Crystalline and Non-crystalline Materials
- A single crystal has a perfect, repeating atomic arrangement throughout the entire specimen.
- Single crystals can occur naturally or be created artificially.
- Growth of single crystals is often difficult and requires careful control of the environment.
- Polycrystalline materials are composed of many small crystals/grains that grow during solidification.
- These crystals have random orientations, indicated by square grids.
- Adjacent grains in a polycrystalline material impinge on one another and have some atomic mismatch forming grain boundaries.
Anisotropy
- Anisotropy describes the phenomenon where physical properties of single crystals depend on the crystallographic direction in which the measurements are taken.
- Properties like elastic modulus, electrical conductivity, and refractive index can vary based on direction.
- Isotropic materials have properties that are independent of direction.
- The extent and magnitude of anisotropic effects increase with decreasing structural symmetry in crystalline materials. Triclinic structures are highly anisotropic.
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