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

• rnielsen25
In summary: Thank you for suggesting to consult the papers on the interference between isospin states in Be-8 at 16.62 and 16.92 MeV, as they will undoubtedly offer valuable insights into the topic.In summary, the problem set considers (beta-delayed) alpha decay of ##{}^{20}Na## and focuses on finding an expression to describe an interesting minimum in the experimental spectrum. The hint suggests rewriting the expression in terms of amplitudes, and the proposed approach involves taking the sum of two resonance cross-sections. However, this expression lacks an interference term, and further research on the interference between isospin states may provide valuable insights into the topic.
rnielsen25
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..]
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.

vanhees71
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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