Key Takeaways — Chapter 17
Core Concepts
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Reaction notation $a(b,c)d$: target$($ projectile, ejectile $)$ residual. The entrance channel ($a + b$) and exit channel ($c + d$) are specified by the particle identities and their quantum numbers.
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Conservation laws constrain which reactions are possible: energy, momentum, charge ($Z$), baryon number ($A$), lepton number ($L$), angular momentum ($J$), and parity ($\pi$, for strong/EM interactions).
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The Q-value $Q = (M_{\text{initial}} - M_{\text{final}})c^2$ determines the energetics: - $Q > 0$: exothermic (kinetic energy released) - $Q < 0$: endothermic (kinetic energy absorbed, threshold exists) - Calculated from atomic mass excesses: $Q = \Delta_a + \Delta_b - \Delta_c - \Delta_d$
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Threshold energy for endothermic reactions: $$T_{\text{th}} = -Q \cdot \frac{M_a + M_b + M_c + M_d}{2 M_a}$$ Always exceeds $|Q|$ because momentum conservation requires the products to carry kinetic energy.
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Center-of-mass energy $T_{\text{CM}} = T_{\text{lab}} \cdot M_a/(M_a + M_b)$ is the kinetic energy available for the reaction. The rest goes into CM motion.
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The cross section $\sigma$ is the fundamental reaction observable: - Dimensions of area, measured in barns ($1\,\text{b} = 10^{-24}\,\text{cm}^2$) - NOT a geometric area — it is a quantum-mechanical probability - Differential cross section $d\sigma/d\Omega$ gives the angular distribution - Ranges from $10^{-19}\,\text{b}$ (neutrino reactions) to $10^3\,\text{b}$ (thermal neutron capture)
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The Rutherford scattering formula: $$\frac{d\sigma}{d\Omega} = \left(\frac{a}{2}\right)^2 \frac{1}{\sin^4(\theta/2)}, \quad a = \frac{kz_1 z_2 e^2}{2E_{\text{CM}}}$$ Derived classically from the Coulomb orbit; exact quantum result for distinguishable spinless charges. Total cross section is infinite (long-range Coulomb force).
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Partial wave analysis decomposes scattering into angular momentum components $l$: - Phase shifts $\delta_l$ encode the effect of the potential - $\sigma_{\text{el}} = (4\pi/k^2)\sum(2l+1)\sin^2\delta_l$ - Unitarity limit: each $l$ contributes at most $(2l+1) \cdot 4\pi/k^2$ - Low energy ($kR \ll 1$): only $s$-wave matters
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The optical model $V(r) = V_R(r) + iW(r)$: - Real part $\to$ elastic scattering (refraction) - Imaginary part $\to$ absorption (reactions) - Predicts diffraction patterns, size resonances, and total cross sections - Woods-Saxon form factor with $\sim 10$ parameters fits data across broad energy and mass ranges
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Ericson fluctuations arise when compound-nucleus resonances overlap ($\Gamma > D$). The energy autocorrelation of cross section fluctuations has Lorentzian form with width $= \Gamma$.
Essential Equations
| Quantity | Formula |
|---|---|
| Q-value | $Q = (M_a + M_b - M_c - M_d)c^2$ |
| Threshold | $T_{\text{th}} = -Q(M_a + M_b + M_c + M_d)/(2M_a)$ |
| CM energy | $T_{\text{CM}} = T_{\text{lab}} M_a / (M_a + M_b)$ |
| Invariant mass | $s c^4 = M_a^2 c^4 + M_b^2 c^4 + 2M_a c^2(M_b c^2 + T_{\text{lab}})$ |
| Reaction rate | $R = I \cdot \sigma \cdot n$ |
| Rutherford | $(d\sigma/d\Omega) = (a/2)^2 / \sin^4(\theta/2)$ |
| Partial wave $\sigma_{\text{el}}$ | $(4\pi/k^2)\sum(2l+1)\sin^2\delta_l$ |
| Reaction $\sigma$ | $(\pi/k^2)\sum(2l+1)(1-|\eta_l|^2)$ |
Common Pitfalls
- Confusing lab and CM frames: always check which frame a formula applies in.
- Forgetting that $T_{\text{th}} > |Q|$: the threshold is not just the Q-value magnitude.
- Treating the cross section as a geometric area: $\sigma$ can be thousands of times larger or smaller than $\pi R^2$.
- Applying the Rutherford formula where it does not apply: it fails when the projectile touches the nuclear surface.
- Neglecting the Jacobian when converting $d\sigma/d\Omega$ between lab and CM frames.
Connections
- Backward: Rutherford scattering was introduced qualitatively in Chapter 1; the full derivation here puts it on rigorous quantitative footing. The quantum mechanics tools from Chapter 5 (angular momentum, partial waves) are essential.
- Forward: Chapter 18 uses the optical model wavefunctions to describe compound nucleus formation and develops the Breit-Wigner resonance formula. Chapter 19 uses distorted waves (optical model) for direct reaction calculations. Chapter 21 applies the Coulomb barrier penetration from the Rutherford derivation to fusion cross sections.