Young's Modulus Small

If in a metallic lattice, the stretching of the material is the stretching of the electron \(B\) orbits, then graph of \(F_a\) would give the Young's Modulus per atom.  The following plot of

\(F_a=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } } \cfrac { r-r_e }{ \left\{ (2a_{ e })^{ 2 }+(r-r_e)^{ 2 } \right\} ^{ 3/2 } } d\,r\)

focuses on the positive increase to \(r_e\), the radius of the electron \(B\) orbit,

Force per atom per unit extension,

\(Y_{cell}=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } }\cfrac{1}{(2a_e)^3}\)

approximately.  The negative sign indicates that the force is attractive.

And we have Young's Modulus at the atomic scale. 

To obtain the yield point when the material begins to give (ie. when the opposing force to stretching starts to decrease), we consider the extrema of  \(F_a\),

\(\cfrac{d\,F_a}{d\,r}=-\cfrac { n_{ e }q^{ 2 } }{ 4\varepsilon _{ o } }\cfrac{(2a_e)^2-2(r-r_e)^2}{\left[(2a_e)^2+(r-r_e)^2\right]^{5/2}}\)

\(F^{'}_a=0\)    when

\((2a_{ e })^{ 2 }-2(r-r_{ e })^{ 2 }=0\)

\( \pm \sqrt { 2 } a_{ e }=r-r_{ e }\)

\( r=r_{ e }\pm \sqrt { 2 } a_{ e }\)

Since we are interested in the increase in \(r_e\), the yield point is given by,

\( r_y=r_{ e }+\sqrt { 2 } a_{ e }\)

in the atomic scale, where \(a_e\) is the radius of the electron and \(r_e\) is the radius of the \(B\) orbit.  Previously, it was assumed that the orbits stacked closely at a distance of  \(2a_e\).  If we replace the distance between stacked orbits with, \(h_s\) then,

 \( r_y=r_{ e }+\cfrac{h_s}{\sqrt { 2 }}\)

per atom.