If a plane has intercepts at 2a, 3a, and 4a along the three Cartesian coordinates where a is the lattice constant of the cubic cell, (i) find the Miller indices of the plane and (ii) find the distance between the neighboring planes parallel to this plane
(i) Well, if a plane has intercepts at 2a, 3a, and 4a, it must have a really good sense of direction! Anyway, let's get to the point.
To find the Miller indices of the plane, we need to determine the reciprocals of the intercepts. In this case, the reciprocals are 1/(2a), 1/(3a), and 1/(4a).
To convert these to integers, we take the reciprocal of the smallest value. In this case, the smallest value is 1/(4a), so we multiply all the reciprocals by 4a. This gives us (4a)/(2a) = 2, (4a)/(3a) = 4/3, and (4a)/(4a) = 1.
So the Miller indices of the plane are (2, 4/3, 1).
(ii) Ah, the distance between neighboring planes parallel to this plane! Well, it's like trying to find your way out of a corn maze, but we'll try our best.
The distance between neighboring planes parallel to this plane is given by the formula:
d = a / sqrt( h² + k² + l² )
where d is the distance, a is the lattice constant, and h, k, and l are the Miller indices of the plane.
Substituting the values we found earlier, we get:
d = a / sqrt( 2² + (4/3)² + 1² )
Now, that 4/3 is a bit tricky, but I assure you, it's not as tricky as trying to juggle bananas while riding a unicycle.
Calculating the expression inside the square root, we have:
d = a / sqrt( 4 + 16/9 + 1 )
Simplifying further:
d = a / sqrt( 9/9 + 16/9 + 9/9 )
d = a / sqrt( 34/9 )
Finally, the distance between neighboring planes parallel to this plane is:
d = a / (sqrt(34)/3)
Or, in simpler terms, it's like trying to measure a paper clip with a rubber band. You can try, but it won't be very precise.
To find the Miller indices of the plane, you can follow these steps:
(i) Find the reciprocals of the intercepts along the Cartesian coordinates:
- Let's denote the reciprocals as h, k, and l. Therefore, h = 1/2a, k = 1/3a, and l = 1/4a.
(ii) To remove any fractions, multiply each reciprocal by the lowest common denominator of all reciprocals (12a in this case):
- Multiply h by 12a: h = 1/2a * 12a = 6
- Multiply k by 12a: k = 1/3a * 12a = 4
- Multiply l by 12a: l = 1/4a * 12a = 3
(iii) The final step is to normalize the Miller indices. Look for the smallest non-zero value in the set (h, k, l), and divide all values by it:
- In this case, l is the smallest non-zero value. Divide all indices by 3:
h_normalized = 6/3 = 2
k_normalized = 4/3
l_normalized = 3/3 = 1
(iv) The Miller indices are given by the normalized indices (h_normalized, k_normalized, l_normalized):
Miller indices = (2, 4/3, 1)
To find the distance between neighboring planes parallel to this plane, you can use the formula:
Distance between neighboring parallel planes = a / sqrt(h^2 + k^2 + l^2)
Given that the lattice constant a is the same as the cubic cell constant a, and the Miller indices are (2, 4/3, 1), we can substitute the values into the formula:
Distance = a / sqrt(2^2 + (4/3)^2 + 1^2)
Distance = a / sqrt(4 + 16/9 + 1)
Distance = a / sqrt(37/9 + 16/9 + 9/9)
Distance = a / sqrt(62/9)
Distance = a / (sqrt(62)/3)
Distance = 3a / sqrt(62)
Therefore, the distance between neighboring planes parallel to this plane is (3a / sqrt(62)).