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Scattering Cross-section questions

  1. Mar 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Problems are Shankar 19.3.2 and 19.3.3 with spherically symmetric potentials V(r)=-V[tex]_{0}(r_{0}-r)\theta[/tex] and V(r)=V[tex]_{0}exp(-r_{2}/r^{0}_{2}^{})[/tex]


    2. Relevant equations
    [tex]
    f\left( \theta \right) = - \frac{{2\mu }}{{\hbar ^2 }}\int\limits_{r_0 }^r {\frac{{\sin qr'}}{q}V\left( {r'} \right)r'dr'}
    [/tex]
    and
    [tex]
    \frac{{d\sigma }}{{d\Omega }} = \left| {f\left( \theta \right)} \right|^2
    [/tex]



    3. The attempt at a solution
    Don't I just substitute the potentials for V(r) and integrate? The example in Shankar seemed to do that successfully for the Yukawa potential. What am I missing?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 10, 2009 #2

    malawi_glenn

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    Have you tried it?
     
  4. Mar 10, 2009 #3
    Yes, I did try. Shankar gives the answer to the first one:
    [tex]\frac{d\sigma}{d\Omega}= 4r_0^2 \left( {\frac{{\mu V_0 r_0^2 }}{{\hbar ^2 }}} \right)^2 \frac{{\left( {\sin qr_0 - qr_0 \cos qr_0 } \right)^2 }}{{\left( {qr_0 } \right)^6 }}$[/tex]


    Questions that arise from this:
    What happened to [tex]\theta[/tex]?
    What are the appropriate limits of integration - r[tex]_{0}[/tex] to [tex]\infty[/tex]?
     
  5. May 4, 2010 #4
    why to study the structure of nuclear scattering is often used as Compton scattering without using the other?
     
  6. Apr 19, 2012 #5
    I was just working through Shankar 19.3.3 and it's seriously a tough problem (if you don't use mathematica.) I thought it might be nice to put my hints on here.

    For finding ∂σ/∂Ω, I used the following tricks (in the order listed):

    1. Get rid of the r in the integrand by saying r sin(qr) = ∂/∂q[cos(qr)]

    2. Write the cos as the sum of exponentials

    3. Combine the two exponential integrals

    4. Complete the square for the exponential integral

    5. Change variables to turn it into a gaussian integral

    And then to find σ

    6. Use shankar's hint to change the integral over q into an integral over cosθ



    I think it officially qualifies as a "tricky" problem.

    13.3.2 just requires you to evaluate a limit using L'Hopital's rule--it's not nearly as challenging.
     
    Last edited: Apr 19, 2012
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