For Miller Indices of planes if we encounter a case where the plane passes through the origin then I have heard that the only remedy can be arrived by shifting the origin to a different point.However I do not understand whether or not I am supposed to change the index of the other planes whose origin was not passing through the previous origin.
hkl index belongs not to a particular plane, but to a family of equally distant planes. Therefore hkl is independent regarding the origin.
Rajorshi Koyal : The fundamental method of planes with miller indices involves figuring out the intercepts a plane makes with all the three lattice vectors. There is also a definition which involves the use of reciprocal lattice space but I will not go into it. Typically, once you get the intercepts, the indices are given as 1/intercept. This is true for all the values of (h k l). This notation has an advantage of converting fractions into whole numbers. For instance if a plane intercepts the coordinate y =1/2 you can assign the index 2 to k. This is just a notation. However, in the special case where the planes pass through the origin, instructors are asked to tell students that they have to shift the origin. Note that this is just an exercise to simplify things for students who struggle with the notation. You shift the planes in your mind to get the index. In reality, you are just providing the index of an equivalent plane that belongs to the unit cell.
To understand what this means, I will consider the example of the (100) plane in a simple cubic system. When I mention this plane, your first reaction should be to think about intercepts. Realize that this plane intersects the y and z axes at infinity and x axis at 1. Hence its index is (100), as 1/infinity = 0. This plane has 4 atoms on its corners. However, since the unit cell is composed of 8 atoms, there will be an equivalent plane that will also pass through the origin of the coordinate system. Hence this place is also indexed as (100). To make sure that people understand this equivalence, they are asked to shift the axis and see the intercept it makes with any of the lattice vectors. In doing so you don't change the frame of reference for the whole crystal system. Now, you might immediately ask, why should one not assign the index (200) to the plane that passes through the origin. One can argue that, instead of shifting the origin by 1 we can also shift it by 1/2. However, note that in the system considered, (200) is not equivalent to (100). This is because (100) has 4 atoms on its corners but (200) does not. The only other plane, which is equivalent to the plane that actually intercepts the x axis at (1, 0, 0) is the one which passes through the origin as it also contains 4 atoms. Hope this makes sense to you. One should be careful while assimilating learning tools and not assume that we are mathematically moving things around. A lot of these definitions are based on symmetry.
By the way, (-100) and (100) will be equivalent planes in this case as well. So you will ask why can't I index the plane on the origin as (-100)? The answer is you can. This is because it is an equivalent plane of (100). When you are drawing a unit cell which has planes (001) (010) and (-100) You have shifted the frame of reference while mentioning the coordinates. Nothing more. It is now your repsonsibility to maintain this frame of reference and report the indices of other arbitrary planes. Otherwise you could confuse the reader.
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