9.9: Resonance Scattering

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There is a significant exception to the independence of the cross-section on energy. Suppose that the quantity $ \pi/2$ . As the incident energy increases, $ \pi/2$ . In this case, $ k'a = \pi/2$ , it follows from Equation \ref{1013} that $ \delta_0 \simeq \pi/2$ (because we are assuming that $ \sigma_{\rm total} = \frac{4\pi}{k^2} \sin^2\delta_0 = 4\pi \,a^2 \left(\frac{1}{k^2 \,a^2}\right).$ \ref{1020}

Note that the cross-section now depends on the energy. Furthermore, the magnitude of the cross-section is much larger than that given in Equation \ref{1017} for $ k\,a\ll 1$ ).

The origin of this rather strange behaviour is quite simple. The condition
$ V_0$ possesses a bound state at zero energy. Thus, for a potential well that satisfies the above equation, the energy of the scattering system is essentially the same as the energy of the bound state. In this situation, an incident particle would like to form a bound state in the potential well. However, the bound state is not stable, because the system has a small positive energy. Nevertheless, this sort of resonance scattering is best understood as the capture of an incident particle to form a metastable bound state, and the subsequent decay of the bound state and release of the particle. The cross-section for resonance scattering is generally far higher than that for non-resonance scattering.

We have seen that there is a resonant effect when the phase-shift of the $ \pi/2$ . There is nothing special about the $ l$ th partial wave is $ \delta_l$ attains the value $ E_0$ , so that

$ \cot \delta_l$ in the vicinity of the resonant energy: $\cot \delta_l(E) = \cot \delta_l(E_0) +\left( \frac{ d \cot\delta... ...ft(\frac{1}{\sin^2\delta_l}\frac{d\delta_l}{d E}\right)_{E=E_0} (E-E_0)+\cdots.$ \ref{1023}

Defining

\( \cot\delta_l(E) = - \frac{2}

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\,(E-E_0) + \cdots.\) \ref{1025}

Recall, from Equation \ref{984}, that the contribution of the $ \sigma_l = \frac{4\pi}{k^2} \,(2\,l+1)\,\sin^2\delta_l = \frac{4\pi}{k^2} \,(2\,l+1)\,\frac{1}{1+\cot^2\delta_l}.$

\ref{1026}

Thus,

$ \sigma_l$ with the incident energy has the form of a classical resonance curve. The quantity $ {\mit\Gamma}$ is the width of the resonance (in energy). We can interpret the Breit-Wigner formula as describing the absorption of an incident particle to form a metastable state, of energy $ E_0$ , and lifetime $\tau = \hbar/ {\mit\Gamma}$ .

Contributors

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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