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For a simple cubic crystal lattice, the angle between the [201] plane and the xy plane is:
- a)Less than 300
- b)Between 300 and 450
- c)Between 450 and 600
- d)Greater than 600
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
For a simple cubic crystal lattice, the angle between the [201] plane ...
Explanation:
- A simple cubic lattice is a type of crystal lattice where the lattice points form a cube and are located at the corners of the cube.
- The [201] plane is a crystallographic plane that passes through the lattice points at (2, 0, 1), (-2, 0, 1), (0, 2, 1) and (0, -2, 1).
- The xy plane is a plane that is parallel to the x and y axes of the lattice.
- The angle between the [201] plane and the xy plane can be determined by finding the dot product of the normal vectors to each plane and using the formula:
cosθ = (a . b) / (|a| |b|)
where a and b are the normal vectors to the two planes.
- The normal vector to the [201] plane can be found by taking the cross product of two vectors that lie in the plane, such as (2, 0, 1) and (0, 2, 1), which gives the vector (-2, -2, 4).
- The normal vector to the xy plane is simply the z-axis vector, (0, 0, 1).
- Taking the dot product of these two vectors gives:
(-2, -2, 4) . (0, 0, 1) = 4
- The magnitude of each vector is:
|(2, 0, 1) × (0, 2, 1)| = 4
|(0, 0, 1)| = 1
- Substituting these values into the formula gives:
cosθ = (4) / (4 * 1) = 1
- Therefore, the angle θ between the [201] plane and the xy plane is:
θ = cos-1(1) = 0
- Since the angle is zero, it means that the two planes are parallel to each other, and therefore never intersect. Therefore, the angle between the [201] plane and the xy plane is greater than 600 degrees.
- A simple cubic lattice is a type of crystal lattice where the lattice points form a cube and are located at the corners of the cube.
- The [201] plane is a crystallographic plane that passes through the lattice points at (2, 0, 1), (-2, 0, 1), (0, 2, 1) and (0, -2, 1).
- The xy plane is a plane that is parallel to the x and y axes of the lattice.
- The angle between the [201] plane and the xy plane can be determined by finding the dot product of the normal vectors to each plane and using the formula:
cosθ = (a . b) / (|a| |b|)
where a and b are the normal vectors to the two planes.
- The normal vector to the [201] plane can be found by taking the cross product of two vectors that lie in the plane, such as (2, 0, 1) and (0, 2, 1), which gives the vector (-2, -2, 4).
- The normal vector to the xy plane is simply the z-axis vector, (0, 0, 1).
- Taking the dot product of these two vectors gives:
(-2, -2, 4) . (0, 0, 1) = 4
- The magnitude of each vector is:
|(2, 0, 1) × (0, 2, 1)| = 4
|(0, 0, 1)| = 1
- Substituting these values into the formula gives:
cosθ = (4) / (4 * 1) = 1
- Therefore, the angle θ between the [201] plane and the xy plane is:
θ = cos-1(1) = 0
- Since the angle is zero, it means that the two planes are parallel to each other, and therefore never intersect. Therefore, the angle between the [201] plane and the xy plane is greater than 600 degrees.
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