Crystal plane perpendicularity to direction

[dikmikkel]

Homework Statement

Show that the direction [hkl] is perpendicular to the plane (hkl) in a cubic system.

Homework Equations

None, maybe cross product

The Attempt at a Solution

I've tried to define the crystal translation vectors(cubic means a=b=c):
$\vec{a}_1 = a\hat{x}\\ \vec{a}_2 = a\hat{y}\\ \vec{a}_3 = a\hat{z}$
I just can't imagine how to obtain the vector normal to the plane. I cannot use the reciprocal space as we haven't gone over that yet. Please help me, it is not to be handed in i just need to understand it.

My question is: What is the connection between Miller Indicies of a plane and the plane's normal vector.

 

[mark726]

Hello,
To show that the direction [hkl] is perpendicular to the plane (hkl) in a cubic system, we can use the definition of Miller indices.

Miller indices represent the reciprocals of the intercepts of the plane on the crystallographic axes. For a plane with intercepts a, b, and c on the x, y, and z axes respectively, the Miller indices are (1/a, 1/b, 1/c).

Now, the normal vector of a plane can be defined as the vector perpendicular to the plane's surface. In a cubic system, the plane (hkl) has a normal vector that is parallel to the direction [hkl]. This can be seen by considering the intercepts of the plane on the crystallographic axes.

For example, if we have a plane with Miller indices (1,2,3), its normal vector will be parallel to the direction [1,2,3]. This can also be verified by taking the dot product of the two vectors, which will result in a value of 0, indicating that they are perpendicular.

In summary, in a cubic system, the direction [hkl] is perpendicular to the plane (hkl) because the normal vector of the plane is parallel to the direction [hkl], and the Miller indices of a plane are reciprocals of its intercepts on the crystallographic axes.

I hope this helps to clarify the connection between Miller indices and a plane's normal vector. Let me know if you have any further questions.