- #1

BRN

- 108

- 10

1. Homework Statement

1. Homework Statement

Use the Lennard-Jones function:

$$ V(R)=4\epsilon[(\frac{\sigma}{R})^{12}- (\frac{\sigma}{R})^6] $$

like as a adiabatic potential energy model in function of separetion R between cople of argon nuclei. With ## \epsilon=1.68*10^{-21} ##, in the first neighbors approximation, calculates the coesive energy for 1 mole of crystal argon with these structures: fcc, bcc, hcp and sc. For fcc case, estimates the effect of the interaction with second neighbors.

(Solution:fcc→6070 J/mol; bcc→4047 J/mol; hcp→6070 J/mol; sc →3035 J/mol; fcc(including 2nd neighbor interaction at fixed lattice spacing)→7398 J/mol; fcc(including 2nd neighbor interaction at optimal lattice spacing)→7627 J/mol)

## The Attempt at a Solution

The atom disposition in the lattice is due to the minimum energy condiction, where atoms are spaced by a minimum radius ## R_{0} ##. This occurs when ## \frac{\partial V}{\partial R}=0 ##

**:**

$$ \frac{\partial }{\partial R}[4\epsilon[(\frac{\sigma}{R})^{12}- (\frac{\sigma}{R})^6]]=4\epsilon \sigma^6[-12\sigma^6R^{-13}+6R^{-7}]=0\Rightarrow R_0^6=2\sigma^6 $$

then

$$ V(0)=4\epsilon[(\frac{\sigma}{R_0})^{12}- (\frac{\sigma}{R_0})^6]=4\epsilon[\frac{\sigma^{12}}{(2\sigma^6)^2}- \frac{\sigma^6}{2\sigma^6}]=-\epsilon $$

The coesive energy is ## E_c=-NN_AV(0) ##, with N number of atoms in the lattice.

So

- sc case: ## N= (\frac{1}{8}*8)=1 \Rightarrow E_c=1011.71 [J/mol] ##
- fcc case: ## N= (\frac{1}{2}*6+\frac{1}{8}*8)=4 \Rightarrow E_c=4046.85 [J/mol] ##
- bcc case: ## N= (1+\frac{1}{8}*8)=2 \Rightarrow E_c=2023.42 [J/mol] ##
- hcp case: ## N= (3+\frac{1}{2}*2+\frac{1}{6}*12)=6 \Rightarrow E_c= 6070.13 [J/mol] ##

$$ N= (\frac{1}{2}*6+\frac{1}{8}*8+\frac{1}{4}*12)=7 \Rightarrow E_c= 7081.18 [J/mol]$$

Only hcp case is OK... what am I doing wrong?