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Physics / Chapter 42: Nuclear Physics / Section 42-5

Beta Decay — Question 10

Section 42-5 · Physics pages 7090–7091

Question 10 Allowed beta selection rules distinguish spin and parity changes Use the following allowed-transition rules: Fermi transitions have ∆J = 0 and no parity change; Gamow–Teller transitions have ∆J = 0, 1 and no parity change, but exclude 0 →0. Consider 0+ →0+, 0+ →1+, and 0+ →1−. Jπf f Jπi i β Schematic; labels, not drawing lengths, specify the data. (a) Classify which allowed mechanism can contribute to each transition. (b) A supplied long-wavelength model gives an extra forbidden amplitude factor qR/ℏ= .0200. Estimate the corresponding rate suppression at equal phase space and otherwise equal matrix-element scale.
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Question 10 - Solution Allowed beta selection rules distinguish spin and parity changes Jπf f Jπi i β (a) Method and calculation 0+ →0+ : Fermi allowed; Gamow–Teller excluded . 0+ →1+ : Gamow–Teller allowed; 0+ →1−: neither allowed type . (b) Interpretation and further calculation Mforb/Mallow ∼qR/ℏ, Γ ∝|M|2. Γforb/Γallow ∼(.0200)2 = 4.00 × 10−4 . Physical check Forbidden denotes suppression of the leading allowed term, not an impossible decay. The estimate is model-dependent and does not replace a nuclear matrix-element calculation.

Question 10

Allowed beta selection rules distinguish spin and parity changes

Use the following allowed-transition rules: Fermi transitions have ΔJ=0\Delta J=0 and no parity change; Gamow–Teller transitions have ΔJ=0,1\Delta J=0,1 and no parity change, but exclude 0→00\to0. Consider 0+→0+0^+\to0^+, 0+→1+0^+\to1^+, and 0+→1−0^+\to1^-.

Diagram shown in the original SVG worksheet.

  1. Classify which allowed mechanism can contribute to each transition.

  2. A supplied long-wavelength model gives an extra forbidden amplitude factor qR/ℏ=.0200qR/\hbar=.0200. Estimate the corresponding rate suppression at equal phase space and otherwise equal matrix-element scale.

Question 10 - Solution

Allowed beta selection rules distinguish spin and parity changes

Diagram shown in the original SVG worksheet.

(a) Method and calculation

0+→0+:Fermi allowed; Gamow–Teller excluded.\boxed{0^+\to0^+:\ \text{Fermi allowed; Gamow--Teller excluded}}.0+→1+:Gamow–Teller allowed;0+→1−:neither allowed type.\boxed{0^+\to1^+:\ \text{Gamow--Teller allowed};\quad0^+\to1^-:\ \text{neither allowed type}}.

(b) Interpretation and further calculation

ℳforb/ℳallow∼qR/ℏ,Γ∝|ℳ|2.\mathcal M_{\mathrm{forb}}/\mathcal M_{\mathrm{allow}}\sim qR/\hbar,\quad\Gamma\propto|\mathcal M|^2.Γforb/Γallow∼(.0200)2=4.00×10−4.\boxed{\Gamma_{\mathrm{forb}}/\Gamma_{\mathrm{allow}}\sim(.0200)^2=4.00\times10^{-4}}.

Physical check

Forbidden denotes suppression of the leading allowed term, not an impossible decay. The estimate is model-dependent and does not replace a nuclear matrix-element calculation.

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