Young's modulus for cubic Van der Waals crystal

[ZetaOfThree]

Homework Statement

Calculate the Young’s moduli for Van der Waals solids with sc, bcc, and fcc structures.

Homework Equations

Definition of Young's Modulus
$$Y=\frac{stress}{strain}$$
Total energy due to Van der Waals interaction:
$$U_{tot}=\frac{1}{2}N(4 \epsilon) \left( \sum_j{ \left( \frac{ \sigma}{p_{j}R}\right)}^{12} - \sum_j{ \left( \frac{ \sigma}{p_{j}R}\right)}^{6} \right)$$
Where ##p_{j}R## is the distance between the atom at the origin and the ##j^{th}## atom, expressed in terms of the nearest neighbor distance ##R##.

The Attempt at a Solution

I am considering using the fact that ##X_x=\frac{\partial U}{\partial e_{xx}}## (where ##X_x## is stress) and dividing by ##e_{xx}##, but I'm not sure how to do that with the expression for ##U## given above. Perhaps I can get ##R## in terms of the strain and then take the derivative? Are there any tips or hints you can give me so I can start this problem?

Thanks in advanced.

 

[mark726]

Hello,

To calculate the Young's modulus for Van der Waals solids, we need to use the definition of Young's modulus, which is the ratio of stress to strain. In this case, we can use the expression for total energy due to Van der Waals interaction to calculate the stress and strain.

First, we need to express the nearest neighbor distance ##p_{j}R## in terms of the strain. We can do this by considering the change in distance due to strain. Let's assume that the original distance between the atoms is ##d## and the strain is ##\epsilon##. The new distance can be expressed as ##d' = d(1 + \epsilon)##. Therefore, we can write ##p_{j}R = d' = d(1 + \epsilon)##.

Next, we can take the derivative of the total energy with respect to strain and divide by the strain to get the stress. This will give us the expression for stress in terms of the strain and other constants. Finally, we can plug this expression for stress into the definition of Young's modulus to get the expression for Young's modulus in terms of the strain and other constants.

To calculate the Young's modulus for Van der Waals solids with sc, bcc, and fcc structures, we need to substitute the values of the nearest neighbor distance for each structure into the expression for Young's modulus. For sc structure, the nearest neighbor distance is ##\sqrt{3}R##, for bcc structure, it is ##\sqrt{3}R/2##, and for fcc structure, it is ##\sqrt{2}R/2##.

I hope this helps you get started on the problem. Let me know if you have any further questions.