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Impact parameter dependence of classical scattering angle

  1. Nov 11, 2012 #1

    otg

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    1. The problem statement, all variables and given/known data
    The problem is to get the classical turning point as a function of the impact parameter b for the Lennard-Jones potential.

    2. Relevant 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].

    3. 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...
     
    Last edited: Nov 11, 2012
  2. jcsd
  3. Nov 12, 2012 #2
    use / sign not \ after ending( i don't think there is a simple way)
     
  4. Nov 19, 2012 #3

    otg

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    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) :)
     
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