a$.$ List the coordinates for points A$1-$A$8,$ inclusive $($in basic translation units of a $=1)$;

b$.$ List the coordinates for points B$1-$B$8,$ inclusive $($in basic translation units of a $=1)$;

c$.$ Write an explicit expression for the following vectors in the coordinate system given

$($in basic translation units of a $=1)$:

$$

$1$

$$

$2$

$$

$1$

$$

$4$

$$

$1$

$$

$5$

$$

$1$

$$

$1$

d$.$ Show that vectors $$

$1$

$$

$2,$

$1$

$$

$4,$

$1$

$$

$1,$ can serve as a set of basic

translation vectors for the BCC structure. Convince yourself this by showing that

points A$1-$A$8($inclusive$)$ and B$1$ can all be expressed in the form of $$

$,$ where $,,$ are integers;

e$.$ Sketch the unit cell as defined by $,,.$ Is this unit cell primitive or non$-$primitive?

f$.$ The conventional BCC unit cell is defined by the following set of basic translation

vectors:

$$

$1$

$$

$2$

$$

$1$

$$

$4$

$$

$1$

$$

$5$

Show that this set of basic translation vectors cannot be primitive, e$.$g$.,$ B$1$ cannot be

expressed in the form of $,$ where $,,$ are integers;

g$.$ Show that the BCC structure can be generated if the set of basic translation vectors

shown in f is associated with a two$-$atom motif $($or the basis$).$ Give the coordinates of

these two atoms.