Identify the [221] vector.(crystallographic directions qns)
The answer given corresponds to 0.5 in x direction,0.5 in y and 1 in z. Why is it not [1 1 0.5], the simply reduced form. The ans is the reciprocal of [221], which puzzles me since this is not a plane.
In crystallography, the notation [hkl] is used to represent crystallographic directions. The numbers h, k, and l are the Miller indices and denote the intercepts of a given direction with the crystallographic axes, usually labeled as a, b, and c. To identify the [221] vector, we need to understand how to interpret these Miller indices.
To determine the direction, Miller indices are usually presented as integers. However, when these directions are expressed in fractional form, it means they are normalized or rationalized by a common denominator.
In the case of [221], it may not be immediately apparent that the normalized form is [0.5 0.5 1]. This is because the direction vector is parallel to the a and b axes, meaning that it only intersects those axes at a distance of 0.5. On the other hand, it intersects the c-axis at a distance of 1, hence the fractional component of the direction.
Now, let's consider why the reciprocal of [221] is involved here. The reciprocal of a direction vector is used to determine a family of planes that are perpendicular to the original direction.
Reciprocals are obtained by inverting the fractional components of the direction vector. In this case, the reciprocal of [0.5 0.5 1] is [2 2 1]. The exact mathematical interpretation of reciprocals and their relationship to planes in crystallography is beyond the scope of this explanation.
However, it's worth noting that the reciprocal of a direction vector can also represent a plane. In this case, [2 2 1] corresponds to a crystallographic plane, although it may not be immediately apparent from the original [221] vector description.
In summary, the notation [221] corresponds to the direction vector [0.5 0.5 1], while its reciprocal, [2 2 1], represents a corresponding plane. The use of reciprocals in crystallography allows for a unified description of both lines and planes in terms of their corresponding Miller indices.