Young's modulus in microscopic terms

[Mathsboi]

Homework Statement

By considering the force-separation curve for two adjacent atoms in a solid, f(x), show that the Young’s modulus can be expressed on the microscopic scale as:
$$Y = - \frac{1}{x_0} \frac{df}{dx}\right| |_{x=x_0}$$
(the | is meant to go allt he way form the top to bottom of df/dx)
where $$x_0$$ is the equilibrium separation of the atoms

Homework Equations

$$f(x) = - \left(\frac{A}{x}\right)^7 + \left(\frac{B}{x}\right)^{13}$$
(I'm assuming A and B should be 1 angstrom so 1E-10m)

The Attempt at a Solution

$$x_0$$ is found by doing f(x).dx = 0 to find where f(x) crosses the x axis.
$$E \equiv \frac{\mbox {tensile stress}}{\mbox {tensile strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L}$$
Y is the gradient of stress/straing graph

hmmmm
:(

 

[conway]

I can't exactly read your equations but a long time ago I remember doing an analysis based on a typical diamond lattice structure whereby you could show that Posson's ration came out correctly ( approximatley 1/3) if you assumed that the bond lengths stayed the same and only the bond angles were deformed.

 

[Mapes]

Homework Statement

By considering the force-separation curve for two adjacent atoms in a solid, f(x), show that the Young’s modulus can be expressed on the microscopic scale as:
$$Y = - \frac{1}{x_0} \frac{df}{dx}\right| |_{x=x_0}$$
(the | is meant to go allt he way form the top to bottom of df/dx)
where $$x_0$$ is the equilibrium separation of the atoms

Homework Equations

$$f(x) = - \left(\frac{A}{x}\right)^7 + \left(\frac{B}{x}\right)^{13}$$
(I'm assuming A and B should be 1 angstrom so 1E-10m)

The Attempt at a Solution

$$x_0$$ is found by doing f(x).dx = 0 to find where f(x) crosses the x axis.
$$E \equiv \frac{\mbox {tensile stress}}{\mbox {tensile strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L}$$
Y is the gradient of stress/straing graph

hmmmm
:(

What's $E =F L_0/A_0 \Delta L$ in differential form? What's a reasonable estimate for $A_0$ on the atomic scale?