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In a cubic unit cell, the angle between normal to the planes (111) and (121) is:
- a)19.47º
- b)18.47º
- c)45º
- d)90º
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
In a cubic unit cell, the angle between normal to the planes (111) and...
cosθ=h1h2+k1k2+l1l2h21+k21+l21−−−−−−−−−√h22+k22+l22−−−−−−−−−√
For the (111) and (121) planes:
cosθ=1+2+11+1+1−−−−−−−√1+4+1−−−−−−−√=43–√6–√=432–√
From this,
θ=
19.47Digree
Most Upvoted Answer
In a cubic unit cell, the angle between normal to the planes (111) and...
Cubic Unit Cell and Angle Calculation
Cubic Unit Cell: A cubic unit cell is a type of crystal lattice structure where the unit cell is in the shape of a cube. It has atoms or molecules at each of the eight corners of the cube and one in the center of the cube.
Miller Indices: Miller indices are used to represent crystal planes in a lattice structure. They are written in the form (hkl), where h, k, and l are integers.
Angle Calculation: The angle between two planes in a crystal lattice can be calculated using the formula:
cosθ = (h1h2 + k1k2 + l1l2) / sqrt(h1^2 + k1^2 + l1^2) * sqrt(h2^2 + k2^2 + l2^2)
where h1, k1, l1 and h2, k2, l2 are the Miller indices of the two planes.
Given: The two planes are (111) and (121).
Solution:
h1=1, k1=1, l1=1 (for plane (111))
h2=1, k2=2, l2=1 (for plane (121))
cosθ = (1*1 + 1*2 + 1*1) / sqrt(1^2 + 1^2 + 1^2) * sqrt(1^2 + 2^2 + 1^2)
cosθ = 4 / sqrt(3) * sqrt(6)
cosθ = 4 / 3.464 * 2.449
cosθ = 1.1547
θ = cos^-1(1.1547)
θ = 19.47 degrees
Therefore, the angle between normal to the planes (111) and (121) is 19.47 degrees (option A).
Cubic Unit Cell: A cubic unit cell is a type of crystal lattice structure where the unit cell is in the shape of a cube. It has atoms or molecules at each of the eight corners of the cube and one in the center of the cube.
Miller Indices: Miller indices are used to represent crystal planes in a lattice structure. They are written in the form (hkl), where h, k, and l are integers.
Angle Calculation: The angle between two planes in a crystal lattice can be calculated using the formula:
cosθ = (h1h2 + k1k2 + l1l2) / sqrt(h1^2 + k1^2 + l1^2) * sqrt(h2^2 + k2^2 + l2^2)
where h1, k1, l1 and h2, k2, l2 are the Miller indices of the two planes.
Given: The two planes are (111) and (121).
Solution:
h1=1, k1=1, l1=1 (for plane (111))
h2=1, k2=2, l2=1 (for plane (121))
cosθ = (1*1 + 1*2 + 1*1) / sqrt(1^2 + 1^2 + 1^2) * sqrt(1^2 + 2^2 + 1^2)
cosθ = 4 / sqrt(3) * sqrt(6)
cosθ = 4 / 3.464 * 2.449
cosθ = 1.1547
θ = cos^-1(1.1547)
θ = 19.47 degrees
Therefore, the angle between normal to the planes (111) and (121) is 19.47 degrees (option A).
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