Nuclear Energy and Reactions (Fission and Fusion)

Nuclear Energy

The concept of nuclear binding energy explains that the nucleons within a stable nucleus are held together by the strong nuclear force, and forming a nucleus from free nucleons releases a large amount of energy (equivalent to the mass defect). Conversely, energy must be supplied to break a nucleus apart. Nuclear energy refers to the energy released from the nucleus of an atom during nuclear reactions, primarily nuclear fission and nuclear fusion. These processes involve transformations of atomic nuclei and result in the conversion of a small amount of mass into a very large amount of energy, as described by Einstein's mass-energy equivalence, $E=mc^2$.

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Fission

Nuclear fission is a nuclear reaction in which a heavy nucleus (such as Uranium-235 or Plutonium-239) splits into two or more lighter nuclei (fission fragments), usually accompanied by the emission of neutrons, gamma rays, and a large amount of energy. Fission can occur spontaneously for some very heavy isotopes, but it is most commonly induced by bombarding the heavy nucleus with neutrons.

Nuclear fission of Uranium-235 by absorption of a slow neutron.

A typical example is the fission of Uranium-235 ($_{92}^{235}U$) when it absorbs a slow (thermal) neutron:

$ _0^1n + _{92}^{235}U \longrightarrow _{92}^{236}U^* \longrightarrow _{56}^{141}Ba + _{36}^{92}Kr + 3 _0^1n + \text{Energy (}\gamma, KE\text{)} $

In this reaction, Uranium-235 absorbs a neutron to form a highly unstable compound nucleus Uranium-236 ($U^*$), which immediately undergoes fission, splitting into Barium-141 and Krypton-92, and emitting three neutrons. Other fission products are also possible (e.g., Strontium and Xenon, or other pairs).

The key features of fission are:

• Release of a large amount of energy (typically around 200 MeV per fission event). This is much larger than the energy released in chemical reactions (a few eV per reaction). The energy comes from the mass defect; the total mass of the fission products and emitted neutrons is less than the mass of the original nucleus and the incident neutron. The binding energy per nucleon of the fission fragments is higher than that of the heavy parent nucleus.
• Emission of several fast neutrons (typically 2 to 3 neutrons per fission event). These neutrons can go on to induce further fission reactions in other fissile nuclei, potentially leading to a chain reaction.
• Formation of radioactive fission fragments. The fission fragments are typically unstable and undergo radioactive decay ($\beta^-$ and $\gamma$ emission), producing heat and radiation over time.

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Nuclear Reactor (Chain Reaction, Moderation, Control Rods)

A nuclear reactor is a device used to initiate and control a sustained nuclear chain reaction for the purpose of generating energy or producing radioisotopes. The energy released from controlled nuclear fission in a reactor is primarily used to generate heat, which is then used to produce steam to drive turbines and generate electricity.

Simplified diagram of a nuclear reactor core.

Key components and concepts in a nuclear reactor:

1. Fissile Fuel: The fuel typically used is Uranium enriched in the fissile isotope Uranium-235 ($_{92}^{235}U$), or Plutonium-239 ($_{94}^{239}Pu$). Natural uranium contains only about 0.7% $^{235}U$; most is non-fissile $^{238}U$. Enrichment increases the concentration of $^{235}U$ to typically 3-5% for power reactors.
2. Chain Reaction: The neutrons released during fission can cause further fission reactions. If, on average, at least one neutron from each fission event causes another fission, the reaction becomes self-sustaining, leading to a chain reaction. The neutron multiplication factor ($k$) is defined as the average number of neutrons from one fission that cause another fission. For a sustained, controlled chain reaction (critical state), $k=1$. If $k>1$, the reaction rate increases (supercritical), and if $k<1$, the reaction rate decreases (subcritical).
3. Moderator: The neutrons emitted during fission are fast neutrons (high kinetic energy). However, Uranium-235 is more likely to undergo fission when it absorbs slow (thermal) neutrons. A moderator material is used to slow down the fast neutrons without absorbing them significantly. Common moderators are light water (H$_2$O), heavy water (D$_2$O), and graphite. The neutrons lose energy through collisions with the moderator atoms.
4. Control Rods: Control rods are made of neutron-absorbing materials (like Cadmium or Boron). They are inserted into the reactor core to control the rate of the chain reaction. By adjusting the position of the control rods, the number of neutrons available to cause fission can be regulated, thereby maintaining the chain reaction at a critical level ($k=1$) for steady power generation, or shutting down the reactor ($k<1$).
5. Coolant: A coolant (like water, heavy water, gas, or liquid metal) circulates through the reactor core to remove the heat generated by fission. This heat is then used to produce steam to drive turbines for electricity generation. The coolant also helps in moderating neutrons in some reactor designs.
6. Reflector: A reflector material (like graphite or beryllium) surrounds the core to scatter escaping neutrons back into the core, improving neutron economy.
7. Shielding: Thick layers of concrete and lead surround the reactor to protect personnel from harmful radiation (neutrons and gamma rays).

Nuclear reactors offer a source of large-scale, low-carbon electricity, but they also produce radioactive waste that requires careful handling and disposal.

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Nuclear Fusion – Energy Generation In Stars

Nuclear fusion is a nuclear reaction in which two or more light nuclei combine to form a heavier nucleus, releasing a tremendous amount of energy. Fusion is the process that powers stars, including our Sun.

Nuclear fusion reaction between Deuterium and Tritium, releasing energy and a neutron.

A common fusion reaction involves isotopes of hydrogen, Deuterium ($_1^2H$) and Tritium ($_1^3H$):

$ _1^2H + _1^3H \longrightarrow _2^4He + _0^1n + \text{Energy} $

In this reaction, a Deuterium nucleus fuses with a Tritium nucleus to form a Helium nucleus and a neutron, releasing about 17.6 MeV of energy. While this energy per reaction is less than fission, the energy released per unit mass of fuel is much higher for fusion.

Key aspects of fusion:

• Fusion occurs between light nuclei. The binding energy per nucleon increases as light nuclei combine to form heavier ones (up to the peak of the binding energy curve).
• A large amount of energy is released, originating from the mass defect (the mass of the products is less than the mass of the reactants).
• Fusion requires extremely high temperatures (millions of degrees Celsius, hence called thermonuclear reactions) to overcome the electrostatic repulsion between the positively charged nuclei and allow them to come close enough for the strong nuclear force to initiate fusion. At these temperatures, matter exists as a plasma (ionized gas).

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Energy Generation in Stars

Stars like the Sun are giant fusion reactors. The immense gravitational pressure in the core of a star creates the necessary high temperatures and densities for nuclear fusion to occur. The primary energy-producing process in the Sun is a sequence of reactions called the proton-proton chain, which ultimately converts four protons into a helium nucleus, releasing energy in the process.

Simplified representation of the proton-proton chain in the Sun's core.

This continuous fusion process in the core of stars provides the energy that makes them shine and counteracts the gravitational collapse.

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Controlled Thermonuclear Fusion

Developing controlled nuclear fusion on Earth as a practical energy source is a major scientific and engineering challenge. The goal is to replicate the fusion processes that power stars, but in a controlled manner to generate electricity.

The main challenges are:

• Achieving Extremely High Temperatures: Raising and maintaining a plasma at temperatures of tens or hundreds of millions of degrees Celsius is necessary to overcome the Coulomb repulsion between nuclei.
• Plasma Confinement: Containing this super-hot plasma, which would vaporise any material container. Magnetic fields are used to confine the plasma in devices like tokamaks and stellarators (magnetic confinement fusion). Inertial confinement fusion involves using lasers or particle beams to rapidly heat and compress a fuel pellet.
• Achieving Sufficient Density and Confinement Time: There must be enough nuclei in a sufficiently small volume (density) and kept together long enough (confinement time) at high temperature for fusion reactions to occur frequently enough to produce a net gain in energy (ignition).

Despite the enormous technical difficulties, research into controlled fusion continues worldwide (e.g., the ITER project) because it holds the promise of a virtually inexhaustible, clean, and safe energy source (fuel is readily available, and it produces less long-lived radioactive waste compared to fission). However, a commercially viable fusion power plant is still likely several decades away.

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Answer:

Given:

Energy released per fission event, $E_{fission} = 200 \, MeV$

Desired power output, $P = 1 \, Watt = 1 \, J/s$

First, convert the energy per fission from MeV to Joules. $1 \, MeV = 1.602 \times 10^{-13} \, J$.

$ E_{fission} = 200 \, MeV \times (1.602 \times 10^{-13} \, J/MeV) = 320.4 \times 10^{-13} \, J = 3.204 \times 10^{-11} \, J $

Let $N_f$ be the number of fission events required per second to produce a power of 1 Watt.

Total energy produced per second = (Number of fissions per second) $\times$ (Energy per fission event).

$ P = N_f \times E_{fission} $

Rearrange to find $N_f$:

$ N_f = \frac{P}{E_{fission}} $

Substitute the values:

$ N_f = \frac{1 \, J/s}{3.204 \times 10^{-11} \, J} $

$ N_f = \frac{1}{3.204} \times 10^{11} \, s^{-1} $

$ N_f \approx 0.3121 \times 10^{11} \, s^{-1} = 3.121 \times 10^{10} \, s^{-1} $

Approximately $3.121 \times 10^{10}$ fission events per second are required to produce a power of 1 Watt. This shows how much energy is released in a single nuclear fission event, as billions of fissions per second are needed for just 1 Watt of power.