Fusion Reactions and Confinement

Fusion is the joining of two light nuclei into a heavier one with release of energy. The energy comes from the same binding-energy curve that drives fission from the opposite end: $B/A$ rises steeply from hydrogen to the iron peak near $A=56$, so combining nuclei below that peak moves the system to more tightly bound configurations and liberates the difference. Per unit mass the yield exceeds fission, and the fuel — hydrogen isotopes — is abundant. The obstacle is the Coulomb repulsion that keeps two positively charged nuclei apart until they are within range of the strong force. At the temperatures where a thermal plasma can supply that energy, the reacting pair still lacks the classical energy to reach contact, and fusion proceeds by barrier tunneling, exactly as alpha decay proceeds outward.¹

The Coulomb barrier and the cross section

Two nuclei of charges $Z_{1}e$ and $Z_{2}e$ approaching to a separation $R\approx R_1+R_2$ of a few femtometres feel a Coulomb barrier of height

For two hydrogen isotopes ($Z_1=Z_2=1$) with $R\approx 5\mathrm{fm}$ the barrier is about $0.3\mathrm{MeV}$. A plasma hot enough to fuse at appreciable rates has a temperature of order $10\mathrm{keV}$ ($10^8\mathrm{K}$), so a typical pair carries a thousand times less energy than the barrier top. Classically nothing fuses. Fusion happens because the relative-motion wavefunction penetrates the barrier, and the penetrability is governed by the same Gamow exponent that controls alpha emission, now for two nuclei approaching rather than one departing.

The transmission through the Coulomb barrier at relative energy $E$ far below its top is $e^{-2\pi \eta }$, where $\eta$ is the Sommerfeld parameter

with $v=\sqrt{2E/m_r}$ the relative velocity, $m_r$ the reduced mass, and $E_G$ the Gamow energy. Because the barrier penetration sets the strong energy dependence, the cross section is written as

isolating the tunneling factor $e^{-\sqrt{E_G/E}}$ and the geometric $1/E\propto {\lambda }^2$ from the slowly varying astrophysical $S$-factor $S(E)$, which carries the nuclear-physics part of the matrix element. Away from resonances $S(E)$ changes little over the energy range that matters, so measured cross sections are extrapolated to low energy through $S(E)$ rather than through the violently varying $\sigma (E)$ itself.²

The thermonuclear reaction rate and the Gamow peak

In a plasma at temperature $T$ the reacting nuclei have a Maxwell-Boltzmann distribution of relative energies. The reaction rate per pair is the thermal average of $\sigma v$,

Substituting $\sigma =(S/E)e^{-\sqrt{E_G/E}}$ leaves the integrand proportional to

Two competing exponentials shape it. The Maxwell factor $e^{-E/kT}$ falls with energy: few particles are far out on the thermal tail. The tunneling factor $e^{-\sqrt{E_G/E}}$ rises with energy: penetration improves as the pair approaches the barrier top. Their product is sharply peaked at the Gamow energy $E_0$ where the sum of the exponents is stationary,

Fusion in a thermal plasma is dominated by a narrow band of relative energies around $E_0$, well above the mean thermal energy $kT$ but far below the barrier. For deuterium-tritium at $kT=10\mathrm{keV}$, the reduced mass gives $E_G\approx 1.18\mathrm{MeV}$ and $E_0\approx 31\mathrm{keV}$, three times the mean energy. Only the fast tail of the distribution reacts, and it reacts almost entirely within a few tens of keV of $E_0$.

Approximating the peak by a Gaussian gives the width and height directly. The exponent at $E_0$ equals $\tau =3E_0/kT$, and expanding to second order yields a $1/e$ full width

The rate rises with temperature not through the mean energy but through $\tau \propto T^{-1/3}$ in the exponent: raising $T$ shifts $E_0$ outward and fattens the Gamow peak, so $⟨\sigma v⟩$ climbs steeply, roughly as a large power of $T$ over the range of interest. The one-third power keeps the growth from being as violent as the bare tunneling factor would suggest, because the Maxwell tail must also be populated at $E_0$.

The fusion fuels

Three reactions carry practical interest, distinguished by their barriers and $Q$-values. The barrier scales as $Z_1{Z}_2$, so single-charge pairs are easiest.¹

• Deuterium-tritium. ${}^2\mathrm{H}+{}^3\mathrm{H}\to {}^4\mathrm{He}+n$, with $Q=17.6\mathrm{MeV}$. The alpha particle carries $3.5\mathrm{MeV}$ and the neutron $14.1\mathrm{MeV}$. The cross section peaks near $5\mathrm{barn}$ at a deuteron energy of about $65\mathrm{keV}$, boosted by a broad $J^{\pi }=3/2^{+}$ resonance in the ${}^5\mathrm{He}$ compound system. It has the lowest ignition temperature of any fuel and is the reaction of every near-term reactor design.
• Deuterium-deuterium. Two nearly equal branches, ${}^2\mathrm{H}+{}^2\mathrm{H}\to {}^3\mathrm{H}+p(Q=4.03\mathrm{MeV})$ and ${}^2\mathrm{H}+{}^2\mathrm{H}\to {}^3\mathrm{He}+n(Q=3.27\mathrm{MeV})$. The fuel needs no radioactive tritium, but the cross section is roughly a hundred times smaller than D-T at reactor energies and the required temperature is higher.
• Deuterium-helium-3. ${}^2\mathrm{H}+{}^3\mathrm{He}\to {}^4\mathrm{He}+p$, with $Q=18.3\mathrm{MeV}$. All products are charged, so the energy is recoverable without a neutron-activated blanket, but the doubled charge product $Z_1{Z}_2=2$ raises the barrier and demands still higher temperature.

Ignition and the Lawson criterion

A reactor must release more fusion energy than it takes to heat and confine the plasma. Let a $50/50$ D-T plasma have total ion density $n$ (so $n_D=n_T=n/2$) at temperature $T$. The fusion power density is

The plasma also radiates and conducts energy away. Collecting all losses into an energy confinement time ${\tau }_E$, the loss power density is the stored thermal energy divided by ${\tau }_E$. With electrons and ions each contributing $\frac{3}{2}kT$ per particle at equal densities, the stored energy density is $W=3nkT$ and

Ignition is the point at which the charged fusion products alone sustain the temperature against losses, requiring no external heating. In D-T only the alpha particle stays confined (the neutron escapes to the blanket), depositing $Q_{\alpha }=3.5\mathrm{MeV}$. Setting the alpha heating equal to the losses,

This is the Lawson criterion. Multiplying by $T$ gives the more temperature-robust triple product, because $⟨\sigma v⟩/T^2$ is nearly constant for D-T over $T\approx 10$–$20\mathrm{keV}$:

The triple product has a minimum near $14\mathrm{keV}$: below it $⟨\sigma v⟩$ falls too fast, above it the stored energy $\propto T$ grows faster than the rate. The value $3\times 10^{21}\mathrm{keV}\mathrm{s}{\mathrm{m}}^{-3}$ is the target every confinement scheme aims to reach.³

Confinement approaches

The triple product can be reached in opposite corners of the density-time plane, since only the product $n{\tau }_E$ is fixed.³

Magnetic confinement holds a dilute plasma, $n\sim 10^{20}{\mathrm{m}}^{-3}$, for a long time, ${\tau }_E$ of order seconds, at $T\sim 10$–$20\mathrm{keV}$. Charged particles spiral along magnetic field lines but drift across them, so a field that closes on itself is required. The tokamak wraps the plasma into a torus. A strong toroidal field from external coils runs the long way around; a poloidal field generated by a large current driven through the plasma itself runs the short way around. Their sum is a helical field whose lines wind around nested toroidal surfaces, averaging out the vertical drifts that would otherwise carry particles to the wall. The pitch of the winding is set by the safety factor $q$, the number of toroidal turns per poloidal turn, which must stay above about $2$–$3$ for stability against kink and tearing modes.

Inertial confinement takes the opposite limit: a millimetre pellet of frozen D-T is compressed by laser or ion beams to $n\sim 10^{31}{\mathrm{m}}^{-3}$, a thousand times solid density, and burns in the time it takes the fuel to fly apart, $\tau \sim 10^{-11}\mathrm{s}$. No external field confines it; the fuel's own inertia holds it together for the disassembly time. The relevant figure of merit is the areal density $\rho R$: the burn fraction rises with $\rho R$, and reaching a useful yield requires $\rho R\gtrsim 3\mathrm{g}\mathrm{c}{\mathrm{m}}^{-2}$, which is why the fuel must be compressed so far before it is heated. A central hot spot ignites first and a burn wave propagates outward through the cold, dense shell.

The two schemes meet the same $n{\tau }_E$ from opposite ends. Their comparison, and the burn physics they share with a fission chain, tie fusion energetics back to the reaction kinematics and cross-section language developed for nuclear reactions. The same barrier-tunneling rate integral, applied not to a laboratory plasma but to a stellar core, powers the reaction networks of the next lesson.