Confusion regarding calculation of Miller indices for planes

Question

Let's calculate the Miller indices of the yellow plane.

The intercepts are $(2,2,1)$ . Taking the reciprocals we get $1/2,1/2,1$ . Clearing the fractions I get the indices as $(1,1,2)$. Why then does the figure show a different indices? Am I supposed to clear fractions as I've done? I'm finding conflicting answers on the net.

Could anyone please help me understand this? A similar way to calculate Miller indices as I've done is shown here

Image credits: Mr porphyrin.

Answer 1

You are right! It's hard to say what's the author's intention to do this. The planes shown in the figure are parallel to each other and therefore possess the same miller indices i.e. $(1,1,2)$.

For reference, the following is a statement from M.A.Omar's book Elementary solid-state physics :

To determine the indices for the plane, we find its intercepts with the axes along the basis vectors. Let these intercepts be $x,y$ and $z$. Usually, $x$ is a fractional multiple of $a$, $y$ a fractional multiple of $b$ ,and so forth. We form the fractional triplet $$\left(\frac{x}{a},\frac{y}{b},\frac{z}{c}\right)$$ invert it to obtain the triplet $$\left(\frac{a}{x},\frac{b}{y},\frac{c}{z}\right)$$ and then reduce this set to a similar one having the smallest integers by multiplying by a common factor. This last set is called the Miller indices of the place and is indicated by $(h,k,l)$.

Answer 2

The indices are $(224)$ and not $(112)$ is to imply that the family of planes of which the yellow plane is a part is twice as squeezed as the family of planes of which the grey plane is a part.

Note that for every family of lattice planes, there corresponds a reciprocal vector $\bf{g}_{hkl}$=$(h,k,l)$ (w.r.t. the basis of reciprocal lattice), and the distance between these planes is $$d=\frac{2\pi}{|\bf{g}_{hkl}|}$$

So that when $(h,k,l)=(224)$, the distance between the set of planes is $1/2$ of the distance between set of planes whose corresponding indices is $(112)$.