Nuclear physics: fission and fusion

1. Binding energy and reaction Q values

Definitions & inputs. m denotes rest mass, c light speed; one atomic mass unit u has uc²≈931.494 MeV.

1. Mass missing from separated nucleons is binding energy.

   $$B=[Zm_p+Nm_n-m_{nucleus}]c^2$$

2. Rest-mass decrease becomes kinetic energy and radiation when Q is positive.

   $$Q=(\sum m_{initial}-\sum m_{final})c^2$$

3. Convert nuclear energy per event into SI units before calculating power.

   $$E[\mathrm J]=E[\mathrm{MeV}](1.602176634\times10^{-13})$$

Interpretation. Binding per nucleon explains why both splitting heavy nuclei and combining light nuclei can release energy.

• U.S. NRC — Reactor Concepts Manual ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

2. Radioactive decay and activity

Definitions & inputs. N atom count, λ decay constant, τ mean lifetime, t1/2 half-life, A activity in becquerels.

1. Integrate the exponential rate equation.

   $$dN/dt=-\lambda N\Rightarrow N=N_0e^{-\lambda t}$$

2. Half-life and mean lifetime differ by ln2.

   $$\lambda=\ln2/t_{1/2},\quad\tau=1/\lambda$$

3. Activity counts decays per second, not emitted energy or biological dose.

   $$A=\lambda N$$

Interpretation. Decay chains and exposure assessments need additional models and isotope data.

• U.S. NRC — Reactor Concepts Manual ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

3. Fission energy and conversion budgets

Definitions & inputs. Qf prescribed recoverable energy per fission, Rf fission rate, ηth electric conversion efficiency.

1. Multiply heat per reaction by reaction rate.

   $$P_{th}=R_fQ_f$$

2. A heat engine rejects part of its thermal input.

   $$P_e=\eta_{th}P_{th},\quad P_{rejected}=P_{th}-P_e$$

3. A prescribed multiplication factor summarizes an idealized neutron-generation balance; reactor prediction needs transport and kinetics.

   $$N_{g+1}=kN_g$$

Interpretation. Delayed neutrons, feedback, heat removal, and material behavior are central to actual reactor dynamics.

• U.S. NRC — Reactor Concepts Manual ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

4. Fusion reaction rate and plasma energy balance

Definitions & inputs. D,T deuterium/tritium ions, nD,nT densities, ⟨σv⟩ reactivity, W stored plasma energy, τE energy confinement time.

1. A common fusion reaction releases energy mainly to the neutron and alpha particle.

   $$D+T\to{}^4He+n+17.6\ \mathrm{MeV}$$

2. For distinct reactants multiply both densities, reactivity, volume, and energy per event.

   $$\mathcal R=n_Dn_T\langle\sigma v\rangle,\quad P_f=\mathcal R VQ$$

3. Stored energy loss and plasma gain differ from facility electrical efficiency.

   $$P_{loss}=W/\tau_E,\quad Q_{plasma}=P_f/P_{external}$$

Interpretation. Scientific plasma gain is not wall-plug or net-electric gain. Continue through sections 5–8 below to derive the cross section, its energy dependence, and the thermal reactivity used in this rate equation.

• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

5. Fusion cross section: effective area, energy and channel

Definitions & inputs. σab(E) is the fusion cross section for a specified reaction channel, in m²; 1 barn (b)=10⁻²⁸ m². J is incident particle flux (m⁻² s⁻¹), nt target number density, ℓ path length, and Γ the reaction frequency for one target particle. E is centre-of-mass relative kinetic energy, μ=mamb/(ma+mb) the reduced mass, and vrel the relative speed.

1. Operational definition: divide the number of reactions per target particle per second by the incident flux. Fusion and elastic scattering have different channel cross sections.

   $$\Gamma=J\sigma(E),\qquad [J\sigma]=\mathrm{s}^{-1}$$

2. Multiply effective area by the number of targets per unit transverse area. Only this dimensionless product approximates a per-projectile fusion probability in a thin target.

   $$P_{fus}\simeq n_t\ell\sigma(E)\quad(n_t\ell\sigma\ll1)$$

3. For a stationary target b, convert projectile laboratory energy into centre-of-mass energy before using a table or fit. A deuteron on a stationary triton has Ecm≈3Elab/5 using mass numbers.

   $$E=\tfrac12\mu v_{rel}^2,\quad E_{cm}=\frac{m_b}{m_a+m_b}E_{lab}$$

4. Worked unit check: use a hypothetical supplied cross section, convert barns to square metres, and multiply by the target column density. This does not assign a measured cross section to any real reaction or energy.

   $$\sigma=0.20\,\mathrm b=2.0\times10^{-29}\,\mathrm{m}^2,\quad n_t\ell=10^{24}\,\mathrm{m}^{-2}\Rightarrow P_{fus}\simeq2.0\times10^{-5}$$

Interpretation. Always identify the reacting species, product branch, energy frame, energy units, and whether the value is measured, evaluated, or fitted. Total D–D fusion combines the neutron and proton branches; a branch-specific cross section counts only that product channel. Experimental uncertainties and fit ranges matter.

• IAEA EXFOR — experimental nuclear reaction data ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

6. Why the cross section depends strongly on energy

Definitions & inputs. Za,Zb are nuclear charge numbers, α the fine-structure constant, c light speed, μ reduced mass, S(E) the astrophysical S factor (energy times area), EG a Coulomb energy scale, and θ=kBT the thermal energy. S(E) here is not the liquid structure factor S(k).

1. Separate the approximate Coulomb tunneling suppression from the remaining nuclear physics. At low collision energy, repulsion makes reaching short-range nuclear interactions unlikely even when the reaction releases energy.

   $$\sigma(E)=\frac{S(E)}{E}\exp[-\sqrt{E_G/E}],\quad E_G=2\mu c^2(\pi\alpha Z_aZ_b)^2$$

2. The thermal reaction integral balances a decreasing high-energy population against increasing barrier penetration. It is not dominated simply by the mean ion energy.

   $$E\sigma(E)e^{-E/\theta}=S(E)\exp[-E/\theta-\sqrt{E_G/E}]$$

3. For slowly varying S(E), differentiate the exponent to locate the most strongly weighted collision energy.

   $$\frac{d}{dE}\left(E/\theta+\sqrt{E_G/E}\right)=\frac1\theta-\frac{\sqrt{E_G}}{2E^{3/2}}=0$$

4. Solve the stationary-point condition. The contributing range around E0 is called the Gamow window; a resonant S(E) changes the weighting.

   $$E_0=(E_G\theta^2/4)^{1/3}$$

Interpretation. Temperature does not set one collision energy. Fusion can occur by tunneling below a classical Coulomb-barrier estimate, but tunneling alone does not determine the reaction probability. D–T has important nuclear resonance structure, so a constant-S estimate must not replace a validated D–T parametrization.

• Adelberger et al. — Solar Fusion Cross Sections: low-energy factors and reaction rates ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

7. Derive thermal reactivity from the cross section

Definitions & inputs. fa(v),fb(v) are velocity probability densities normalized to one; σ(E) is in m², vrel in m/s, θ=kBT in joules, μ in kg, and ⟨σv⟩ in m³/s. na,nb are number densities; δab=1 for identical reactants and zero for distinct species.

1. Average the product of reaction area and relative speed over all reacting pairs. Multiplying a cross section at a chosen mean energy by a mean speed generally gives a different answer.

   $$\langle\sigma v\rangle=\int d^3v_a\,d^3v_b\,f_a(\mathbf v_a)f_b(\mathbf v_b)\,\sigma(E)|\mathbf v_a-\mathbf v_b|$$

2. The difference of independent Maxwellian velocities has a Maxwellian distribution with reduced mass μ and the common thermal energy θ.

   $$f_{rel}(\mathbf v)=\left(\frac{\mu}{2\pi\theta}\right)^{3/2}e^{-\mu v^2/(2\theta)}$$

3. Integrate over directions, contributing 4πv²dv, and retain the extra v from collision rate.

   $$\langle\sigma v\rangle=4\pi\left(\frac{\mu}{2\pi\theta}\right)^{3/2}\int_0^\infty\sigma(E)v^3e^{-\mu v^2/(2\theta)}\,dv$$

4. Change integration variable from relative speed to collision energy and collect the prefactor. Use joules consistently in both the exponent and dimensional prefactor, or convert an explicitly unit-specific fit.

   $$E=\mu v^2/2,\quad v^3dv=2E\,dE/\mu^2\quad\Rightarrow\quad\langle\sigma v\rangle=\sqrt{\frac8{\pi\mu}}\,\theta^{-3/2}\int_0^\infty\sigma(E)E e^{-E/\theta}\,dE$$

5. Count distinct pairs once. D–T needs no factor of one half; identical D–D pairs do. Multiply reactions per cubic metre per second by joules per reaction to obtain fusion power density.

   $$\mathcal R_{ab}=\frac{n_an_b}{1+\delta_{ab}}\langle\sigma v\rangle,\quad p_f=\mathcal R_{ab}Q$$

Interpretation. The existing prescribed-rate example uses ⟨σv⟩=10⁻²² m³/s; it is a rate coefficient, not a cross section of 10⁻²² m². For a measured prediction, select the channel and valid energy interval, integrate σ(E) with the distribution, or use a validated thermal-reactivity fit such as Bosch–Hale within its stated range.

• Fitzpatrick — Nuclear Fusion Reactions and Maxwellian reactivity ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

8. Worked analytical check: a constant-cross-section toy model

Definitions & inputs. Prescribe σ0=1 barn=10⁻²⁸ m² independent of energy, μ=2.0×10⁻²⁷ kg, θ=10 keV=1.602176634×10⁻¹⁵ J, and two distinct species with na=nb=5×10¹⁹ m⁻³. This toy σ0 is not a measured D–T cross section.

1. Substitute u=E/θ; integration by parts gives the dimensionless integral equal to one.

   $$\int_0^\infty E e^{-E/\theta}\,dE=\theta^2\int_0^\infty u e^{-u}du=\theta^2$$

2. Insert the exact integral into the Maxwellian formula. Only because σ is constant can it be taken outside the average.

   $$\langle\sigma v\rangle=\sigma_0\sqrt{\frac{8\theta}{\pi\mu}}=\sigma_0\langle v_{rel}\rangle$$

3. Compute the mean relative speed using the reduced mass and thermal energy in SI units.

   $$\langle v_{rel}\rangle=\sqrt{\frac{8(1.602176634\times10^{-15})}{\pi(2.0\times10^{-27})}}\simeq1.4283\times10^6\ \mathrm{m\,s}^{-1}$$

4. Multiply by σ0 and then the distinct-species density product. This completes the chain from area to rate coefficient to reactions per volume per time.

   $$\langle\sigma v\rangle\simeq1.4283\times10^{-22}\ \mathrm{m^3\,s^{-1}},\quad\mathcal R\simeq(5\times10^{19})^2(1.4283\times10^{-22})=3.5707\times10^{17}\ \mathrm{m^{-3}\,s^{-1}}$$

Interpretation. This benchmark isolates unit conversion, the Maxwellian prefactor and pair counting. In real fusion, substitute the energy-dependent channel cross section; do not treat this constant-area benchmark as an evaluated nuclear datum.

• Fitzpatrick — Nuclear Fusion Reactions and Maxwellian reactivity ↗
• U.S. Department of Energy — deuterium-tritium fusion fuel ↗

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