Impact parameter dependence of classical scattering angle

[otg]

Homework Statement

The problem is to get the classical turning point as a function of the impact parameter b for the Lennard-Jones potential.

Homework Equations

The Lennard-Jones potential is given as [itex]V(r)=4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^6][\itex].
The effective potential is [itex]V_{eff}(r)=V(r)+E\frac{b^2}{r^2}[\itex].
Also, [itex]\frac{m}{2}\dot{r}+V_{eff}(r)=E[\itex].

The Attempt at a Solution

Since the turning point occurs when [itex]\dot{r}=0[\itex], we have that [itex]4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^6]+E\frac{b^2}{r^2}=E[\itex]. Even though [itex]\sigma,\,\epsilon\,\text{and }\,E[\itex] are given, I find it quite hard if not impossible to solve the equation to get [itex]r(b)[\itex]. There has to be some other way even if this was the indicated strategy.

The exercise says that the equation [itex]V_{eff}(r)=E[\itex] can be rewritten as a polynomial and solved with a computer program, but since it becomes (at least that's what I get) a twelve degree polynomial in [itex]r[\itex] it's not possible to get a decent answer.

Grateful for any kind of help how to proceed. And also how to get the latex working in the posts...

 

[andrien]

Homework Statement

The problem is to get the classical turning point as a function of the impact parameter b for the Lennard-Jones potential.

Homework Equations

The Lennard-Jones potential is given as $V(r)=4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^6]$.
The effective potential is $V_{eff}(r)=V(r)+E\frac{b^2}{r^2}$.
Also, $\frac{m}{2}\dot{r}+V_{eff}(r)=E$.

The Attempt at a Solution

Since the turning point occurs when $\dot{r}=0$, we have that $4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^6]+E\frac{b^2}{r^2}=E$. Even though $\sigma,\,\epsilon\,\text{and }\,E$ are given, I find it quite hard if not impossible to solve the equation to get $r(b)$. There has to be some other way even if this was the indicated strategy.

The exercise says that the equation $V_{eff}(r)=E$ can be rewritten as a polynomial and solved with a computer program, but since it becomes (at least that's what I get) a twelve degree polynomial in $r$ it's not possible to get a decent answer.

Grateful for any kind of help how to proceed. And also how to get the latex working in the posts...

use / sign not \ after ending( i don't think there is a simple way)

 

[otg]

use / sign not \ after ending( i don't think there is a simple way)

ha ha ok my fingers must've gone into LaTeX mode :)

To reply on my own post then, no there is no simple way, I think I interpreted the question "solve for r as function of b" as "find an expression" but the task was as simple as plot the roots to find the functional behavior. Thanx for pointing out the /-thing though (probably would have worked next time without me knowing what went wrong) :)