# Using Reciprocal to Determine Miller Indices

• Chillguy
In summary, the conversation discusses the process of finding the shortest reciprocal lattice vector and its corresponding miller index for a BCC lattice. The reciprocal lattice basis vectors are given, and it is suggested that the corresponding miller index should be {111}, but is actually ({110}). It is explained that this is because for cubic materials, the conventional cubic unit cell is used instead of the primitive basis vectors. The relationship between the two forms of HKL is discussed, and examples of HKL for different primitive indices are given.
Chillguy
If we know the reciprocal space basis of a BCC lattice $$b_1=\frac{2\pi}{a}(\vec{x}+\vec{y}),b_2=\frac{2\pi}{a}(\vec{z}+\vec{y}),b_3=\frac{2\pi}{a}(\vec{x}+\vec{z})$$ how do we go about finding the shortest reciprocal lattice vector and its corresponding miller index?

To me all the constants in from of all reciprocal vectors are the same so the corresponding miller index should be {111} but it is apparently supposed to be ({110}). I conceptually can't make sense of this and any help would be appreciated.

What you have written are the primitive basis vectors for the reciprocal lattice. For cubic materials one generally uses the conventional cubic unit cell. The corresponding reciprocal lattice unit cell is also cubit and has the basis vectors

a*=2pi/a x, b*=2pi/a x, c*=2pi/a z.

A reciprocal lattice vector is then G_HKL = H a* + K b* + L c*

Try to work out the relation between HKL_cubic and HKL_primitive, and then see what the HKL_cubic of, say, (100)_primitive, (110)_primitive and (111)_primitive are.

## 1. What is the purpose of using reciprocal to determine Miller indices?

Using reciprocal to determine Miller indices is a mathematical method that helps identify the crystal planes and directions in a crystal lattice. This information is useful in understanding the symmetry and properties of crystals in materials science and mineralogy.

## 2. How do you calculate the reciprocal of a Miller index?

The reciprocal of a Miller index is calculated by taking the inverse of the index values. For example, if the Miller index is (hkl), the reciprocal would be (1/h, 1/k, 1/l). This is done to simplify the identification and comparison of different crystal planes and directions.

## 3. Can reciprocal indices have negative values?

Yes, reciprocal indices can have negative values. In fact, the presence of negative values in the reciprocal indices is an important factor in determining the orientation of a crystal plane or direction in relation to the crystal lattice. Negative values indicate that the plane or direction is parallel to the negative axis of the crystal system.

## 4. Are Miller indices unique to a specific crystal lattice?

Yes, Miller indices are unique to a specific crystal lattice. Each type of crystal lattice has its own set of Miller indices that are used to identify the crystal planes and directions in that lattice. For example, the Miller indices for a cubic crystal lattice would be different from those of a hexagonal crystal lattice.

## 5. How can reciprocal indices be used to determine the spacing between crystal planes?

Reciprocal indices can be used to calculate the interplanar spacing between crystal planes using the equation d = a/√(h² + k² + l²), where d is the interplanar spacing, a is the lattice constant, and (hkl) are the Miller indices of the crystal plane. This information is important in understanding the structure and properties of crystals in various applications.

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