How Do You Calculate Miller Indices for Complex Crystal Planes?

[Akitirija]

Homework Statement

Determine the Miller indices of the cubic crystal plane that intersects the position coordinates
C (1, 1/4, 0), A (1, 1, 1/2), B (3/4, 1, 1/4), and all coordinate axes.

The Attempt at a Solution

This is an example problem with solution from Foundations of Materials Science and Engineering by Smith and Hashemi, so I already have the solution and I know the way to solve it. My only problem is the "and all coordinate axes" part of the problem.

The authors create a point D, which of course has the y-coordinate 1 and z-coordinate 0. However, I have no idea how to find the x-coordinate, which they just write is 1/2, without any explanation. I tried thinking about this in a trigonometric way, but I cannot see that I have enough to solve it trigonometrically either.

If anyone can explain to me how they conjured up the x-coordinate I would be very grateful!

 

[shahd yasser]

Hello,

It seems like you are trying to determine the Miller indices of a cubic crystal plane using given position coordinates. This is a common problem in materials science and can be solved using a few steps.

First, we need to determine the three lattice vectors of the cubic crystal system. These are usually represented as a1, a2, and a3. In this case, we can assume that the a1 vector is parallel to the x-axis, a2 is parallel to the y-axis, and a3 is parallel to the z-axis.

Next, we need to find the intercepts of the plane with the three axes. For the x-axis, we can see that the plane intersects at the point C, which has coordinates (1, 1/4, 0). This means that the plane intercepts the x-axis at a distance of 1 unit from the origin. Similarly, for the y-axis, we can see that the plane passes through the point A, which has coordinates (1, 1, 1/2). This means that the plane intercepts the y-axis at a distance of 1 unit from the origin.

Now, for the z-axis, we can see that the plane passes through the point B, which has coordinates (3/4, 1, 1/4). This means that the plane intercepts the z-axis at a distance of 1/4 unit from the origin.

Using these intercepts, we can now determine the Miller indices of the plane. The Miller indices are given by (hkl), where h, k, and l are the reciprocals of the intercepts on the x, y, and z axes respectively. In this case, we have (1, 4, 4) as the Miller indices (since the intercepts are 1, 1, and 1/4).

Now, coming to the "and all coordinate axes" part of the problem, we need to find the intercepts of the plane with the coordinate axes. This means that we need to find the coordinates of a point on the plane that also lies on the x, y, and z axes. This can be done by finding the common factors of the intercepts we already have. In this case, the common factor is 1/4. So, we can create a point D with coordinates (1/4, 1/4, 0) which

hello, how did you get the last part of your solution? i do not get it.