Question

Elastic scattering from some central potential V may be adequately calculated using the first Born approximation. Experimental results give the following general behavior of the cross section as a function of momentum transfer q = |k - k'|.

In terms of the parameters shown in Fig:

(a) What is the approximate size (extension in space) of the potential V?

(Expand the Born approximation for the scattering amplitude for small q.)

(b) What is the behavior of the potential V at very small distances?

Answer 1

(a) The Born approximation gives

where qh is the magnitude of the momentum transfer and q = 2k sin \(\frac\theta 2\). For q → 0,

replacing V by \(\bar V\), some average value of the potential in the effective force range R. For a small momentum transfer q₀, we have

Note that |f(q₀)l² is the measured value of \(\frac{d\sigma}{s\Omega}|_{q_0}\) for some small q₀, |f(0)l² is the value of \(\frac{d\sigma}{s\Omega}\) for a set of small q₀ extrapolated to q = 0. From these values the effective range of the potential can be estimated.

(b) In view of the behavior of the scattering cross section for large q, we can say that the Born integral consists mainly of contributions from the region qr ≤ π, outside which, on account of the oscillation between the limits \(\pm1\) of the sine function, the contributions of the integrand are nearly zero. Thus we need only consider the integral from qr = 0 to π. Assuming V(r) \(\sim\) rⁿ for small T, where n is to be determined, we have

A comparison with the given data gives \(\frac N2\) = 3 + n. Hence \(\bar V\) behaves like \(r^{(\frac N2 - 3)}\).