---
title: Fusion Reactions and Confinement
module: Fusion and Nucleosynthesis
moduleNumber: 10
lessonNumber: 1
order: 1001
summary: >
  Light nuclei release energy when they fuse because binding per nucleon rises steeply
  toward the iron peak, but the Coulomb barrier suppresses the rate at reactor temperatures.
  The thermonuclear rate is a convolution of the Maxwell distribution with the tunneling
  probability, sharply peaked at the Gamow energy. The deuterium-tritium reaction has the
  lowest barrier and largest cross section; sustained energy gain requires the Lawson
  triple product of density, temperature, and confinement time, reached by magnetic or
  inertial confinement.
topics: [Fusion and Nucleosynthesis]
sources:
  - book: Krane
    ref: "Ch. 14 — Nuclear Fusion; §14.1 Basic Fusion Processes, §14.2 Characteristics of Fusion, §14.3 Controlled Fusion Reactors"
  - book: Wong
    ref: "Ch. 6 — Nuclear Collective Motion and Reactions; §6-3 Nuclear Fusion"
draft: false
---

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.[^krane-basic]

## 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

$$
V_C = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0\,R}
= \frac{Z_1 Z_2 (1.44\ \mathrm{MeV\,fm})}{R}.
$$

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](/nuclear-physics/alpha-decay/alpha-decay-gamow-theory), 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

$$
2\pi\eta = \frac{2\pi Z_1 Z_2 e^2}{4\pi\epsilon_0\,\hbar v}
= 2\pi Z_1 Z_2\,\alpha\,\frac{c}{v} = \sqrt{\frac{E_G}{E}},
\qquad
E_G = 2 m_r c^2\,(\pi \alpha Z_1 Z_2)^2,
$$

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

$$
\sigma(E) = \frac{S(E)}{E}\,e^{-\sqrt{E_G/E}},
$$

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.[^krane-char]

> **Definition (Astrophysical $S$-factor).** The $S$-factor removes the Coulomb-barrier
> and de Broglie energy dependence from the fusion cross section,
> $$
> S(E) = \sigma(E)\,E\,e^{+\sqrt{E_G/E}}.
> $$
> It is nearly constant for a non-resonant reaction, which is why fusion data taken at
> laboratory energies of tens of keV can be extrapolated down to the few-keV energies of a
> stellar or reactor plasma.

## 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 $\sigma v$,

$$
\langle \sigma v\rangle
= \sqrt{\frac{8}{\pi m_r}}\,\frac{1}{(kT)^{3/2}}
\int_0^\infty \sigma(E)\,E\,e^{-E/kT}\,\d E.
$$

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

$$
S(E)\,\exp\!\left(-\frac{E}{kT} - \sqrt{\frac{E_G}{E}}\right).
$$

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,

$$
\frac{\d}{\d E}\!\left(\frac{E}{kT} + \sqrt{\frac{E_G}{E}}\right) = 0
\quad\Longrightarrow\quad
E_0 = \left(\frac{\sqrt{E_G}\;kT}{2}\right)^{2/3}
= \left(\frac{E_G}{4}\right)^{1/3}(kT)^{2/3}.
$$

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 $E_0$.

$$
% caption: The reaction rate integrand is the product of the falling Maxwell factor and the
% rising tunneling probability; their overlap forms the Gamow peak at E0, far out on the
% thermal tail but well below the Coulomb barrier top.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {relative energy $E$};
  \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {probability};
  % Maxwell factor e^{-E/kT}, decreasing (dashed)
  \draw[black, thick, dashed]
    (0.5,3.7) .. controls (1.6,2.3) and (2.8,1.15) .. (4.4,0.55)
    .. controls (5.8,0.24) and (7.2,0.12) .. (8.0,0.09);
  % tunneling factor e^{-sqrt(EG/E)}, increasing (densely dotted)
  \draw[black, thick, densely dotted]
    (0.5,0.05) .. controls (1.8,0.12) and (2.6,0.35) .. (3.6,0.75)
    .. controls (5.0,1.4) and (6.4,2.1) .. (8.0,3.0);
  % product: Gamow peak (solid acc)
  \fill[acc!12]
    (1.3,0) .. controls (2.0,0.35) and (2.6,1.6) .. (3.2,2.5)
    .. controls (3.6,3.05) and (4.0,3.05) .. (4.4,2.5)
    .. controls (5.0,1.6) and (5.6,0.4) .. (6.3,0) -- cycle;
  \draw[acc, very thick]
    (1.3,0.02) .. controls (2.0,0.35) and (2.6,1.6) .. (3.2,2.5)
    .. controls (3.6,3.05) and (4.0,3.05) .. (4.4,2.5)
    .. controls (5.0,1.6) and (5.6,0.4) .. (6.3,0.02);
  \draw[black, dashed] (3.8,0) -- (3.8,2.95);
  \node[anchor=north, black!70] at (3.8,-0.05) {$E_0$};
  \node[black, anchor=south east, font=\scriptsize] at (2.0,2.6) {Maxwell};
  \node[black, anchor=south west, font=\scriptsize] at (6.4,2.6) {tunneling};
  \node[acc, anchor=south, font=\scriptsize] at (3.8,3.05) {Gamow peak};
\end{tikzpicture}
$$

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

$$
\Delta = \frac{4}{\sqrt 3}\sqrt{E_0\,kT},
\qquad
\langle \sigma v\rangle \propto
S(E_0)\,\tau^{2}\,e^{-\tau},
\qquad
\tau = \frac{3E_0}{kT} = 3\left(\frac{E_G}{4kT}\right)^{1/3}.
$$

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
$\langle\sigma v\rangle$ 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.[^krane-basic]

- **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.

$$
% caption: Fusion cross sections rise from threshold as the Gamow factor turns on; D-T
% peaks near 5 barn at the lowest energy because of its low barrier and a compound-nucleus
% resonance, while D-D and D-He3 stay smaller and peak higher.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {deuteron energy};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {cross section};
  % D-T: rises fast, peaks then falls (solid acc)
  \draw[acc, very thick]
    (0.6,0.05) .. controls (1.4,0.3) and (1.9,2.8) .. (2.7,3.7)
    .. controls (3.5,4.2) and (4.6,3.6) .. (6.0,2.6)
    .. controls (7.0,1.95) and (7.7,1.6) .. (8.1,1.45);
  \node[acc, anchor=south, font=\scriptsize] at (2.7,3.75) {D + T};
  % D-3He: peaks higher energy, lower (dashed)
  \draw[black, thick, dashed]
    (2.0,0.04) .. controls (3.2,0.3) and (4.4,1.2) .. (5.8,1.85)
    .. controls (6.8,2.25) and (7.6,2.4) .. (8.1,2.4);
  \node[black, anchor=south west, font=\scriptsize] at (6.6,2.4) {D + He3};
  % D-D: smallest, densely dotted
  \draw[black, thick, densely dotted]
    (1.2,0.03) .. controls (2.6,0.15) and (4.2,0.5) .. (6.0,0.9)
    .. controls (7.0,1.1) and (7.7,1.2) .. (8.1,1.25);
  \node[black, anchor=north west, font=\scriptsize] at (6.3,0.85) {D + D};
\end{tikzpicture}
$$

## 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

$$
P_{\text{fus}} = n_D n_T\,\langle\sigma v\rangle\,Q
= \frac{n^2}{4}\,\langle\sigma v\rangle\,Q.
$$

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 $\tfrac{3}{2}kT$ per particle
at equal densities, the stored energy density is $W = 3\,n\,kT$ and

$$
P_{\text{loss}} = \frac{W}{\tau_E} = \frac{3\,n\,kT}{\tau_E}.
$$

**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,

$$
\frac{n^2}{4}\,\langle\sigma v\rangle\,Q_\alpha = \frac{3\,n\,kT}{\tau_E}
\quad\Longrightarrow\quad
n\,\tau_E \ge \frac{12\,kT}{\langle\sigma v\rangle\,Q_\alpha}.
$$

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

$$
n\,T\,\tau_E \ge \frac{12\,(kT)^2}{\langle\sigma v\rangle\,Q_\alpha}
\approx 3\times10^{21}\ \mathrm{keV\,s\,m^{-3}}
\quad(T \approx 14\ \mathrm{keV}).
$$

The triple product has a minimum near $14\ \mathrm{keV}$: below it $\langle\sigma v\rangle$
falls too fast, above it the stored energy $\propto T$ grows faster than the rate. The value
$3\times10^{21}\ \mathrm{keV\,s\,m^{-3}}$ is the target every confinement scheme aims to
reach.[^krane-reactor]

> **Definition (Lawson triple product).** The product of ion density, temperature, and
> energy confinement time that must be exceeded for the fusion self-heating to balance the
> plasma energy losses. For D-T ignition it is roughly $n T \tau_E \gtrsim 3\times10^{21}\
> \mathrm{keV\,s\,m^{-3}}$, minimized near $T \approx 14\ \mathrm{keV}$; the milder
> breakeven condition (fusion power equal to input heating) sits a factor of several below.

$$
% caption: The ignition condition traces a curve in the plane of confinement parameter
% against temperature with a minimum near 14 keV; magnetic confinement operates at low
% density and long time, inertial confinement at extreme density and short time.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {temperature};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {Lawson parameter};
  % ignition U-curve
  \draw[acc, very thick]
    (1.1,4.0) .. controls (2.0,2.2) and (2.7,1.35) .. (3.7,1.2)
    .. controls (5.0,1.0) and (6.4,1.35) .. (7.9,2.4);
  \node[acc, anchor=south west, font=\scriptsize] at (3.4,1.25) {ignition};
  \fill[acc] (3.7,1.2) circle (1.6pt);
  \node[black!70, anchor=north, font=\scriptsize] at (3.7,1.1) {minimum};
  % region above curve labelled
  \node[black, anchor=west, font=\scriptsize] at (4.6,3.4) {net energy gain};
  \node[black, anchor=west, font=\scriptsize] at (0.9,0.5) {sub-ignition};
  % operating points
  \fill[black!70] (5.0,2.4) circle (1.8pt);
  \node[black!70, anchor=west, font=\scriptsize] at (5.15,2.4) {tokamak};
  \fill[black!70] (6.6,3.6) circle (1.8pt);
  \node[black!70, anchor=south east, font=\scriptsize] at (6.6,3.7) {inertial};
\end{tikzpicture}
$$

## 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.[^krane-reactor]

**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.

$$
% caption: In a tokamak an external toroidal field and the poloidal field of the plasma
% current combine into a helical field winding around nested toroidal flux surfaces, which
% confines the drifting charged particles.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % outer torus outline (two ellipses)
  \draw[black, thick] (0,0) ellipse (3.4 and 1.9);
  \draw[black, thick] (0,0) ellipse (1.5 and 0.8);
  % plasma tube cross-section on the right
  \draw[very thick] (2.45,0) ellipse (0.55 and 0.95);
  \draw[very thick] (-2.45,0) ellipse (0.55 and 0.95);
  % helical field line winding on right cross-section
  \draw[acc, thick] (2.45,0.95) .. controls (2.9,0.5) and (2.9,-0.5) .. (2.45,-0.95);
  \draw[acc, thick, densely dotted] (2.45,0.95) .. controls (2.0,0.5) and (2.0,-0.5) .. (2.45,-0.95);
  % toroidal direction arrow along top
  \draw[black, ->, thick] (-1.6,1.72) .. controls (-0.5,2.05) and (0.5,2.05) .. (1.6,1.72);
  \node[black, anchor=south, font=\scriptsize] at (0,2.0) {toroidal B};
  % plasma current arrow
  \draw[black, ->, thick] (1.9,-1.7) .. controls (0.6,-2.05) and (-0.6,-2.05) .. (-1.9,-1.7);
  \node[black, anchor=north, font=\scriptsize] at (0,-2.05) {plasma current};
  \node[black, anchor=west, font=\scriptsize] at (3.1,0) {poloidal B};
\end{tikzpicture}
$$

**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
$\rho R \gtrsim 3\ \mathrm{g\,cm^{-2}}$, 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](/nuclear-physics/nuclear-reactions/reaction-kinematics-cross-sections).
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](/nuclear-physics/fusion-nucleosynthesis/stellar-nucleosynthesis).

[^krane-basic]: **Krane**, _Introductory Nuclear Physics_, §14.1 (Basic Fusion Processes):
the binding-energy-curve origin of the fusion yield, the $Q$-values of the D-T, D-D, and
D-${}^3$He reactions, and the Coulomb barrier estimate $V_C = Z_1 Z_2 e^2/4\pi\epsilon_0 R$.
Cross-section and $S$-factor data are compiled by the IAEA Nuclear Data Services,
[https://www-nds.iaea.org/](https://www-nds.iaea.org/).

[^krane-char]: **Krane**, §14.2 (Characteristics of Fusion), and **Wong**, _Introductory
Nuclear Physics_, §6-3. The Sommerfeld parameter, the $S$-factor decomposition of $\sigma$,
and the thermal average $\langle\sigma v\rangle$ with the Gamow peak at $E_0 =
(E_G/4)^{1/3}(kT)^{2/3}$ and peak exponent $\tau = 3E_0/kT$.

[^krane-reactor]: **Krane**, §14.3 (Controlled Thermonuclear Reactions): the Lawson criterion
$n\tau_E \ge 12kT/\langle\sigma v\rangle Q_\alpha$, the D-T triple product $n T \tau_E
\gtrsim 3\times10^{21}\ \mathrm{keV\,s\,m^{-3}}$ near $14\ \mathrm{keV}$, and the magnetic
(tokamak) and inertial confinement routes. Reactor plasma parameters follow the ITER-class
design values summarized by the IAEA.