Using Reciprocal to Determine Miller Indices

[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.

 

[M Quack]

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.