Answer
$(1,1,-2,0)or(1,1,\bar2,0)$
and $(0,0,0,1)$
Work Step by Step
To convert the [110] and [001] directions into the four-index Miller–Bravais scheme for hexagonal unit cells.
If $u=1$, $v=1$ and $w=0$
then
$u'=\frac{1}{3}(2u-v)$
$v'=\frac{1}{3}(2v-u)$
$t'=-(u'+v')$
$w'=w$
So the required term would be
$u'=\frac{1}{3}(2-1)=\frac{1}{3}$
$v'=\frac{1}{3}(2-1)=\frac{1}{3}$
$t'=-(\frac{1}{3}+\frac{1}{3})=-\frac{2}{3}$
$w'=0$
So the term of the indices in hexagonal unit cell would be $(\frac{1}{3},\frac{1}{3},-\frac{2}{3},0)$
On dividing the term with 3
The required term of the indices is
$(1,1,-2,0)$
Now the term [001] directions into the four-index Miller–Bravais scheme for hexagonal unit cells.
Let given that the term $u=0$, $v=0$ and $w=1$ then
$u'=\frac{1}{3}(2u-v)$
$v'=\frac{1}{3}(2v-u)$
$t'=-(u'+v')$
$w'=w$
So the required term is
$u'=\frac{1}{3}(0-0)=0$
$v'=\frac{1}{3}(0-0)=0$
$t'=-(0+0)=0$
$w'=1$
So the term of the indices in hexagonal unit cell is $(0,0,0,1)$
The required term of the indices is
$(0,0,0,1)$
