Lec7_Role of crystals II(Mlller indices & defects)

Lec7_Role of crystals II(Mlller indices & defects) -...

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The Role of Crystal Structure II (Miller Indices)
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Content Significance of crystallographic direction and plane Miller indices Linear & Planar density
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Deformation under loading (eg. slip) occurs on certain crystalline planes and in certain crystallographic directions . Helps to predict modes of material failure . Other properties of materials (electrical conductivity, thermal conductivity, elastic modulus, magnetic property, piezoelectric property etc.) can vary in a crystal with orientation . Significance of Crystallographic directions and planes
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Example Efficient strong Ni based single crystal turbine blades withstand high temperature due to absence of grain boundaries because blades are grown on close packed direction [1 1 0] FCC Ni. Computer chips come from Si wafers sliced from single crystal Si grown along [1 0 0] direction. GaAs lasers in CD players are grown from GaAs single crystals grown along [1 0 0] direction. Image : Turbine blade CD player laser Silicon ingots and wafers in various sizes Image :
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Name Symbol Meaning Half-moon Bracket ( ) Individual Plane - (1 0 0) Curly Bracket { } Family of Planes {1 0 0} = (1 0 0), (0 1 0), (0 0 1) Square bracket [ ] Individual direction - [ 1 0 0] Carrot Bracket < > Family of Directions <1 0 0> = [1 0 0], [0 1 0], [0 0 1] Miller Indices Miller Indices are the designation of the planes and direction in the unit cell. The number of indices will match with the dimension of the lattice or the crystal. Example: In 3-D, unit cell has 3 indices namely (h k l) . (h, k, l) represents a point – but “ no comma ” is used while representing indices. Meaning of the different brackets
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Steps involved Choose origin O in unit cell such that Plane does not touch the origin. Plane is closest to the origin. Mark intercepts from chosen origin O along X- axis as ‘ a’ , from O along Y- axis as ‘ b’ and from O along Z-axis as ‘c’ . Take reciprocal of the intercepts, i.e.,1/a, 1/b, 1/c. Modify the reciprocal Reduce multiples. Eliminate fraction Put bar above the negative integer in indices (if any) Place round bracket around indices (no commas) How to find Miller Indices of a given plane? a b c Answer : (1 1 1)
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Thus, Miller Indices (h j k) are the reciprocals of the parameters (l, m, n) of each crystal face. Green Face = (1/1, 1/ , 1/ ) = (1 0 0) Brown Face = (1/ , 1/ , 1/1) = (0 0 1) Blue Face = (1/ , 1/1, 1/ ) = (0 1 0) ( 1, ∞, ∞ ) ( ∞, ∞, 1 ) ( ∞, 1, ∞ ) O
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Determine the indices for the given plane. 1.Choose new Origin (O’) 1 1 0 Unit cell intercepts a b c 2. Intercept from origin (O ) -1 −1 2 1 3. Reciprocal value -1 -2 1 4. Miller Indices ( 1 2 1) Plane cannot touch origin Choose closest origin Z’ a, b, c are intercepts along X’, Y’ & Z’
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Determine the indices for the given plane.
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