Calculating Cross Section for Ion-Atom Scattering with Rutherford Potential

[Ed Quanta]

Ok, so we have a potential energy V(r)= -C/r^4 which exists between an atom and ion at distances greater than contact. Note that C=e^2/2(P^2) where e is charge of ion and P is polarizability of atom. I have to calculate the cross section for an ion of velocity v to strike an atom, while assuming that the ion is much lighter than the atom.

The answer to this is cross section=4pi/v*(sqrt(2C/m)). How do I get this?

The only equations I have are, E=1/2*mv^2, L(angular momentum)=mvb where b is impact parameter. The equation for b is given in terms of scattering angles, I think in this case b= C/mv^2 * cot(beta/2). But here we don't know any scattering angles. And the equation for cross section is cross section=pi*b^2

 

[eljose79]

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I would first start by looking at the equation for cross section and seeing if I can manipulate it to include the given potential energy equation. In this case, we can rewrite the cross section equation as cross section=pi*(C/mv^2 * cot(beta/2))^2.

Next, I would look at the given equations for energy and angular momentum and see if I can use them to find an expression for the scattering angle, as it is needed in the equation for b. In this case, we can use the conservation of energy and angular momentum to find the relation between the initial and final velocities and angles.

Using the conservation of energy, we can equate the initial kinetic energy of the ion (1/2*mv^2) to the final potential energy (V(r)= -C/r^4) and kinetic energy (1/2*mu^2), where u is the final velocity of the ion after the collision. This gives us the equation 1/2*mv^2 = -C/r^4 + 1/2*mu^2.

Using the conservation of angular momentum, we can equate the initial angular momentum (mvb) to the final angular momentum (muR), where R is the distance between the atom and ion after the collision. This gives us the equation mvb = muR.

Solving for u in the energy equation and substituting it into the angular momentum equation, we get b= R*v*(sqrt(2C/m))/cot(beta/2).

Finally, substituting this value of b into the equation for cross section, we get cross section=pi*(R*v*(sqrt(2C/m))/cot(beta/2))^2. Simplifying this gives us the final answer of cross section=4pi/v*(sqrt(2C/m)), which is the desired result.

In summary, we were able to use conservation of energy and angular momentum to find an expression for the scattering angle, which we then used to manipulate the equation for cross section to include the given potential energy equation. This allowed us to solve for the desired cross section value.