Strain Energy of Ni Screw Dislocation

Calculation of the strain energy per unit length for a screw dislocation in Ni (FCC).

Dislocation Energy Formula

The strain energy per unit length ($E$) for a screw dislocation is often approximated using the formula:

$E = \frac{G b^2}{2}$

Here, $G$ represents the shear modulus and $b$ is the magnitude of the Burgers vector.

Burgers Vector Calculation (FCC)

Nickel (Ni) has a face-centered cubic (FCC) structure. For FCC crystals, the Burgers vector magnitude $b$ is determined by the lattice parameter $a$ as follows:

$b = \frac{a}{\sqrt{2}}$

Given the lattice parameter $a = 0.35$ nm, which is equal to $0.35 \times 10^{-9}$ m.

$b = \frac{0.35 \times 10^{-9}}{\sqrt{2}} \text{ m}$

Calculating Energy

First, calculate the square of the Burgers vector magnitude ($b^2$):

$b^2 = \left(\frac{0.35 \times 10^{-9}}{\sqrt{2}}\right)^2 = \frac{(0.35)^2 \times 10^{-18}}{2} = \frac{0.1225 \times 10^{-18}}{2}$

$b^2 = 0.06125 \times 10^{-18} \text{ m}^2$

The given shear modulus is $G = 76$ GPa, which is $76 \times 10^9$ Pa.

Now, substitute $G$ and $b^2$ into the energy formula:

$E = \frac{1}{2} \times G \times b^2$

$E = \frac{1}{2} \times (76 \times 10^9 \text{ Pa}) \times (0.06125 \times 10^{-18} \text{ m}^2)$

$E = \frac{1}{2} \times (76 \times 0.06125) \times 10^{(9 - 18)} \text{ J/m}$

$E = \frac{1}{2} \times 4.655 \times 10^{-9} \text{ J/m}$

$E = 2.3275 \times 10^{-9} \text{ J/m}$

Result

Rounding the calculated strain energy to two decimal places gives:

$E \approx 2.33 \times 10^{-9} \text{ J/m}$