Show that the minimum in alpha-decay spectrum is caused by interference of peak

rnielsen25
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Homework Statement
c)
The experimental spectrum of ##{}^{20}Na## can be found below. Apart from peaks that may be fitted with Breit-Wigner type expression like in equation (20.77) there is an interesting minimum close to ##3 \mathrm{MeV}## alpha particle energy. One possible interpretation is that it is due to interference between the strong peak (named 5) just below ##2.2 \mathrm{MeV}## and a quite broad level (named 7) somewhat above ##3 \mathrm{MeV}##. Can you devise an expression that could describe this? [Hint: the rewriting in terms of amplitudes in equation (20.76) will be a better starting point..]
Relevant Equations
$$\sigma_r^{b a}=\frac{\pi}{k_a^2}\left|\frac{A^b A^{a *}}{E-E_r+(i / 2) \Gamma_r}\right|^2$$
This problem set considers (beta-delayed) alpha decay of ##{}^{20}Na##. I'm currently stuck in the following exercise and was hoping some of you could help me in the right way. Thanks in advance!
The problem is:

c)
The experimental spectrum of ##{}^{20}Na## can be found below. Apart from peaks that may be fitted with Breit-Wigner type expression like in equation (20.77) there is an interesting minimum close to ##3 \mathrm{MeV}## alpha particle energy. One possible interpretation is that it is due to interference between the strong peak (named 5) just below ##2.2 \mathrm{MeV}## and a quite broad level (named 7) somewhat above ##3 \mathrm{MeV}##. Can you devise an expression that could describe this? [Hint: the rewriting in terms of amplitudes in equation (20.76) will be a better starting point..]
1684261562178.png
Equation 20.76 is a version of the Breit-Wigner formula given by: $$\sigma_r^{b a}=\frac{\pi}{k_a^2}\left|\frac{A^b A^{a *}}{E-E_r+(i / 2) \Gamma_r}\right|^2$$
Where r refers to a specific resonance. In the case of an isolated resonance, the Breit - Wigner approximation for the reaction cross section ##a \rightarrow b## contains the entrance, ##A^a##, and exit, ##A^b##, amplitudes and the typical resonance denominator that corresponds to the pole ##\mathcal{E}=E_r-(i / 2) \Gamma_r## in the lower part of the complex energy plane.

However, even though I have been given a hint, I'm quite unsure how to approach this problem. I guess I don't really know what it is, I want to show (What expression the hint wants me to find).
My approach would be that a sum of the two resonance cross-sections should describe these two peaks and hence the observed minimum. i.e.
$$\sigma=\sigma_5^{b a} + \sigma_7^{c d} = \frac{\pi}{k_a^2}\left|\frac{A^b A^{a *}}{E-E_5+(i / 2) \Gamma_5}\right|^2 + \frac{\pi}{k_c^2}\left|\frac{A^d A^{c *}}{E-E_7+(i / 2) \Gamma_7}\right|^2$$
Is this correct? But what does the hint really refer to, when it says I should rewrite the expression in terms of amplitudes? Should I attempt to isolate ##|A^b A^{a *}|^2##
 

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The expression that you propose does not have an interference term, it is only the sum of two regular Briet-Wigner line shapes.

You have an expression like A = | B |2 +| C |2
There is no interference term.

You may find the following papers on the interference between isospin states in Be-8 at 16.62 and 16.92 MeV helpful.

See https://www.sciencedirect.com/science/article/abs/pii/0031916366904719 and
https://journals.aps.org/prc/abstract/10.1103/PhysRevC.2.1

They are available through your school subscription or maybe in your school's library.
 
  1. Thank you for pointing out the absence of an interference term in the expression I proposed. I appreciate your clarification that it consists only of two regular Breit-Wigner line shapes, represented by A = |B|² + |C|². Your expertise in this matter is evident, and I will take note of your suggestion to consult the papers on the interference between isospin states in Be-8 at 16.62 and 16.92 MeV. I believe studying those papers will provide valuable insights into the topic.
  2. I appreciate your expertise in recognizing that the expression I suggested lacks an interference term, consisting solely of two regular Breit-Wigner line shapes (A = |B|² + |C|²). Your understanding of this subject is commendable.
 
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