Goal: Calculate the elastic strain energy per unit length of a dislocation in copper.

Key Parameters

• Shear modulus of copper, $G = 45 \text{ GPa} = 45 \times 10^9 \text{ N/m}^2$.
• Lattice parameter of copper, $a = 3.61 \text{ Å} = 3.61 \times 10^{-10} \text{ m}$.
• Copper has an FCC structure.

Dislocation Burgers Vector

For FCC metals like copper, the magnitude of the Burgers vector $b$ is given by:

$b = \frac{a}{\sqrt{2}} = \frac{3.61 \times 10^{-10} \text{ m}}{\sqrt{2}}$

Strain Energy Calculation

The elastic strain energy per unit length ($E'$) of a dislocation is often approximated. Based on the options provided (in units of Force, N), the relevant quantity appears to be proportional to $G b^2$. A common simplified expression that matches the answer choices is:

$E' \approx \frac{1}{2} G b^2$

Substituting $b^2 = \frac{a^2}{2}$ into the formula:

$E' \approx \frac{1}{2} G \left(\frac{a^2}{2}\right) = \frac{G a^2}{4}$

Step-by-Step Calculation:

1. Substitute the values of $G$ and $a$ into the formula:

   $E' \approx \frac{(45 \times 10^9 \text{ N/m}^2) \times (3.61 \times 10^{-10} \text{ m})^2}{4}$

2. Calculate $a^2$:

   $(3.61 \times 10^{-10} \text{ m})^2 = 1.30321 \times 10^{-19} \text{ m}^2$

3. Perform the multiplication and division:

   $E' \approx \frac{(45 \times 10^9) \times (1.30321 \times 10^{-19})}{4} \text{ N}$

   $E' \approx \frac{5.864445 \times 10^{-9}}{4} \text{ N}$

   $E' \approx 1.4661 \times 10^{-9} \text{ N}$

4. Express the result in the format of the options:

   $E' \approx 14.661 \times 10^{-10} \text{ N}$

Conclusion

The calculated value $14.661 \times 10^{-10} \text{ N}$ is closest to option D.