Planes perpendicular to reciprocal vectors problems

[ngkamsengpeter]

Homework Statement

Given the shortest reciprocal lattice of fcc is (2PI/a)(+-x+-y+-z).
How can i show that (111) is perpendicular to the shortest reciprocal lattice?
And how to show that other planes such as (001),(010),(110) is not perpendicular to the shortest reciprocal lattice?

Homework Equations

Is it the plane will perpendicular to all the 6 reciprocal vector if it perpendicular to anyone of the reciprocal vectors,said 2Pi/a (x-y+z)?

The Attempt at a Solution

I can get the answer by imagination, but i want to know how to prove it mathematically. I think it should use dot products but don't know how.

 

[Jhero]

To show that (111) is perpendicular to the shortest reciprocal lattice, we can use the dot product between the two vectors. The dot product of two vectors is defined as the product of their magnitudes multiplied by the cosine of the angle between them. If the dot product is zero, it means the vectors are perpendicular to each other.

So, let's take the shortest reciprocal lattice vector, which is (2PI/a)(+-x+-y+-z), and the (111) plane normal vector, which is (1,1,1). The dot product between these two vectors can be calculated as:

(2PI/a)(+-x+-y+-z) * (1,1,1) = (2PI/a)(+-1)(+-1)(+-1) * (1,1,1) = (2PI/a) * (1)(1)(1) = 2PI/a

Since 2PI/a is not equal to zero, it means that the two vectors are not perpendicular to each other. Therefore, we can conclude that (111) is not perpendicular to the shortest reciprocal lattice.

To show that other planes such as (001), (010), and (110) are also not perpendicular to the shortest reciprocal lattice, we can use the same method. The dot product between the shortest reciprocal lattice vector and the normal vectors of these planes will also result in a non-zero value, indicating that they are not perpendicular to each other.

I hope this helps to answer your question and provide a mathematical proof for the perpendicularity of planes to the shortest reciprocal lattice. Keep up the curiosity and keep exploring the wonders of science!