For elastic scattering off a central potential, the scattering phase shift for partial wave \(l\) can be written \(\operatorname{as} \delta_{l}(E)=\delta_{\mathrm{bg}}(E)+\delta_{\mathrm{res}}(E),\) where the resonant part of the phase shift is given by $$\tan \delta_{\mathrm{res}}(E)=\frac{\Gamma}{2\left(E_{\mathrm{res}}-E\right)}$$ and the background phase shift is often a slowly varying function of energy. (a) Sketch the behaviour of \(\delta_{l}\) as a function of energy in the vicinity of \(E_{\text {res }}\) if \(\delta_{\mathrm{bg}}\) is taken to be independent of energy with a constant value of (i) \(0 ;\) (ii) \(\pi / 4\) (iii) \(\pi / 2 ;\) (iv) \(3 \pi / 4\) (b) The partial wave cross-section \(\sigma_{l}(E)\) is proportional to \(\sin ^{2} \delta_{l}(E) .\) Sketch the dependence of the latter on energy in the vicinity of \(E_{\mathrm{res}}\) for the four values of \(\delta_{\mathrm{bg}}\) given in part (a). Note that for \(\delta_{\mathrm{bg}}=0, \sin ^{2} \delta_{l}(E)\) has the Breit-Wigner form (eqn 14.67).

Step by step solution

01

Understand the phase shift formula

First, understand the phase shift function \( \delta_l(E)\) is composed of two shifts: \( \delta_{bg}(E)\), which is usually a slowly varying function of energy, and \( \delta_{res}(E)\), which has resonant behavior. The resonant part has a formula as follows: \( tan \delta_{res}(E) = \frac{\Gamma}{2(E_{res}-E)}\). Here,\( \Gamma\) and \(E_{res}\) are constants. If \(E\) is close to \(E_{res}\), the term \(2(E_{res}-E)\) will approach zero, yielding a large magnitude for resonant phase shift.

02

Considering a constant background phase shift

The quotient \( \delta_{bg}(E)\) can be regarded as a constant because it is said to be a slowly varying function of energy. Now, the behaviors of \( \delta_l(E)\) under different background phase shifts \( \delta_{bg}\) will be drawn visually. The function will be dissected under four conditions: (i) \(0 ;\) (ii) \(\frac{\pi}{4} ;\) (iii) \(\frac{\pi}{2} ;\) (iv) \(\frac{3\pi}{4}\). Each case will cause a vertical shift of the tangent graph which represents the resonance part.

03

Calculate the dependence of cross-section

Now we calculate the cross-section \(\sigma_l(E)\), proportional to \( \sin^2 \delta_l(E)\). According to this relationship, we will depict the behaviors of \( \sigma_l(E)\) under the four mentioned situations. Apply the sine function to the phase shift function \( \delta_l (E)\) and graph the \( \sin^2 \delta_l(E)\) is to see the variation as energy changes.

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