GATE 2021 PH – Question 50
A particle scatters from the Yukawa potential $U(r)=-U_0e^{-\alpha r}/r$, with positive $U_0,\alpha$. In the first Born approximation, how does the scattering amplitude vary at large momentum transfer $q$? You may use $\int_0^\infty e^{-\alpha r}\sin(qr)\,dr=q/(\alpha^2+q^2)$.
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Correct answer: (A) $q^{-2}$
Explanation
For a central potential the Born amplitude is proportional to $q^{-1}\int_0^\infty rU(r)\sin(qr)\,dr$. Substituting the Yukawa form cancels the factor $r$, and the supplied integral gives $f(q)\propto q^{-1}q/(\alpha^2+q^2)=1/(\alpha^2+q^2)$. For $q\gg\alpha$, this is proportional to $q^{-2}$. The differential cross section, the square of the amplitude, would fall as $q^{-4}$.
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