Chapter 3.15, Problem 29AAP

(a)

To determine

Show the cubic direction $\left[010\right]$ in a BCC unit cell. Write the position coordinates of the atoms with their centers intersected by each plane. Find the repeat distance in plane direction $\left[010\right]$ in terms of lattice constant.

Explanation of Solution

Show the cubic direction $\left[010\right]$ in the BCC unit cell as in Figure (1).

Position coordinates:

For the cubic direction $\left[010\right]$, the position coordinates of the atoms are $\left(0,0,0\right)$, and $\left(0,1,0\right)$.

Repeat distance:

For the cubic direction $\left[010\right]$, the repeat distance is the distance between the corner atoms and the center of the atoms which is $a$. Therefore, the repeat distance is $a$.

(b)

To determine

Show the cubic direction $\left[011\right]$ in the BCC unit cell. Write the position coordinates of the atoms with their centers intersected by each plane. Find the repeat distance in plane direction $\left[011\right]$ in terms of lattice constant.

Explanation of Solution

Show the cubic direction $\left[011\right]$ in the BCC unit cell as in Figure (2).

Position coordinates:

For the cubic direction $\left[011\right]$, the position coordinates of the atoms are $\left(0,0,0\right)$, and $\left(0,1,1\right)$.

Repeat distance:

For the cubic direction $\left[011\right]$, the repeat distance is the distance between the centers of the corner atoms which is $\sqrt{2}a$. Therefore, the repeat distance is $\sqrt{2}a$.

(c)

To determine

Show the cubic direction $\left[111\right]$ in the BCC unit cell. Write the position coordinates of the atoms with their centers intersected by each plane. Find the repeat distance in plane direction $\left[111\right]$ in terms of lattice constant.

Explanation of Solution

Show the cubic direction $\left[111\right]$ in the BCC unit cell as in Figure (3).

Position coordinates:

For the cubic direction $\left[111\right]$, the position coordinates of the atoms are $\left(0,0,0\right)$, and $\left(1,1,1\right)$.

Repeat distance:

For the cubic direction $\left[111\right]$, the repeat distance is the distance between the center of BCC center atom and the center of the corner atoms which is $\frac{\sqrt{3}}{2}a$. Therefore, the repeat distance is $\frac{\sqrt{3}}{2}a$.

(d)

To determine

The angle between the cubic directions $\left[011\right]$ and $\left[111\right]$.

Answer to Problem 29AAP

The angle between the cubic directions $\left[011\right]$ and $\left[111\right]$ is $\text{35}\text{.26}°$.

Explanation of Solution

Write the expression to calculate angle between the cubic directions $\left(\theta \right)$.

  $\theta ={\cos }^{-1}\left[\frac{h_1{h}_2+k_1{k}_2+l_1{l}_2}{\sqrt{h_1^2+k_1^2+l_1^2}\sqrt{h_2^2+k_2^2+l_2^2}}\right]$                                                                    (I)

Here, Miller indices of the cubic plane 1 are $h_1$, $k_1$ and $l_1$ respectively and Miller indices of the cubic plane 2 are $h_2$, $k_2$ and $l_2$ respectively.

Conclusion:

Substitute 0 for $h_1$, 1 for $k_1$, 1 for $l_1$, 1 for $h_2$, 1 for $k_2$ and 1 for $l_2$ in Equation (I).

 $\begin{array}{c}\theta ={\cos }^{-1}\left[\frac{\left(0\right)\left(1\right)+\left(1\right)\left(1\right)+\left(1\right)\left(1\right)}{\sqrt{0^2+1^2+1^2}\sqrt{1^2+1^2+1^2}}\right]\\ ={\cos }^{-1}\left[\frac{2}{\left(\sqrt{2}\right)\left(\sqrt{3}\right)}\right]\\ ={\cos }^{-1}\left(0.8165\right)\\ =\text{35}\text{.26}°\end{array}$

Thus, the angle between the cubic directions $\left[011\right]$ and $\left[111\right]$ is $\text{35}\text{.26}°$.