Nuclear Reactions And Scattering

• Type of physical science: Nuclear physics
• Field of study: Nuclei

The collision of an atomic nucleus with another nucleus or other subatomic particle can result in a nuclear reaction, in which the interacting particles change their identities, or a scattering event, in which the particles simply rebound. These processes are important in stellar energy production, production of radioisotopes for medical uses, trace element analysis, and nuclear power generation.

Overview

A variety of processes can occur when two or more atomic nuclei or other subatomic particles approach each other to within about one nuclear diameter. In elastic scattering, the particles retain their identity and essentially bounce off each other with a total kinetic energy before the collision that is equal to that afterward. In inelastic scattering, the particles rebound with less final total kinetic energy: The difference is retained as excitation energy in one or more of the recoiling nuclei. In a nuclear reaction, the particles after the collision are different from those before. There may be one, two, or many final particles, depending on the process and the amount of energy available. Other fundamental nuclear reaction types include fusion, in which two light nuclei combine to form a heavier, more stable nucleus—releasing energy in the process—and fission, where a heavy nucleus splits into two or more lighter fragments, typically accompanied by the release of neutrons and a large amount of energy. These basic interaction types—elastic scattering, inelastic scattering, fusion, and fission—represent the principal ways that nuclear matter can transform during collisions and energy exchanges.

Reactions are described in terms of reactants and reaction products, but they are also often characterized by the mechanism that describes the reaction process. For example, deuterium and tritium nuclei, two isotopes of hydrogen, can be forced into close proximity by heating or by other means. Deuterium contains one neutron and one proton in its nucleus; tritium contains two neutrons and one proton. These particles can combine to form an α particle (a helium nucleus that contains two neutrons and two protons) and a free neutron. In addition, energy is released, which is shared as the energy of motion of the reaction products. The reaction is a “fusion” reaction because two light nuclei fuse to produce a heavier one. It is also called a “thermonuclear” reaction when thermal (heat) energy has initiated it. Reactions of this type are the primary energy source in thermonuclear weapons.

All nuclei are electrically charged in proportion to the number of protons they contain.

An electric repulsion, called the Coulomb barrier, therefore exists between them. This barrier must be overcome before the nuclei can react. This can be accomplished by projecting one of the particles toward the other, known as the target, with enough kinetic energy to surmount the barrier or to bring the particles sufficiently close together that they can quantum-mechanically tunnel through the barrier. If the particles have insufficient energy to overcome the barrier, elastic scattering is usually the only possible interaction.

Projectiles come in many forms. Cosmic rays, which consist of high-energy protons and other particles and which constantly bombard the Earth, can produce nuclear reactions.

Alpha particles from naturally radioactive elements were used as projectiles for early scattering and reaction studies. Charged projectiles such as protons, helium nuclei, and heavier ions are usually obtained from atoms that are ionized and then accelerated by machines (accelerators) to energies sufficient to induce reactions. Fewer than 1 million electronvolts of kinetic energy may be sufficient to produce reactions with very light nuclei, but because barrier height increases as the product of the electrical charges of the projectile and the target nucleus, several hundred million electronvolts may be necessary with very heavy ions. Neutrons, which are uncharged, do not encounter a Coulomb barrier, but they must be ejected from nuclei by means of nuclear reactions before they are used as projectiles. Although neutrons are stable in many nuclei, they live only for about 15 minutes as free particles.

Target nuclei are usually contained in samples of ordinary material, preferably in solid or liquid form to maximize the density of nuclei. They are usually considered to be at rest, since velocities of thermal motion are small compared to projectile velocity. Projectiles are usually incident on the material from a well-defined direction. The reaction products can be observed directly if the target is thin enough to allow them to emerge without loss of energy or identity.

This may limit target thickness to several micrometers for low-energy charged reaction products.

In some cases, the distinction between projectile and target is not clear. To obtain very high energies for elementary particle studies, for example, colliding beams of particles approach each other with equal speeds near the speed of light. In thermonuclear reactions, both reactants move randomly, but sometimes they collide with relative thermal velocities sufficient to overcome the Coulomb barrier. The distinction is even more blurred in stars that have high central densities, in which more than two particles can combine to form a heavier nucleus.

Nuclear interactions take place over very short time intervals during which the interacting particles are essentially isolated from their environment. Certain quantities, such as total energy, momentum, angular momentum, and electric charge, must, therefore, have precisely the same values immediately before and after the collision. Such quantities are said to be conserved.

Total energy consists of the sum of the kinetic and the rest mass energies of the particles. (Nuclear excitation energy, if any, is added to the rest mass energy). The value of a nuclear reaction is defined as the total mass-energy of the reactants minus the mass-energy of the reaction products. If the Q value is positive, the total kinetic energy of the particles after the reaction must be greater than that of the particles before the reaction to keep the total energy constant. Such reactions, in which mass-energy is converted to kinetic energy, are called exoergic or exothermic. If the Q value is negative, kinetic energy must be supplied by the reactants before the reaction can occur. Those reactions are termed “endoergic” or “endothermic.”

For projectile kinetic energies lower than about 1 billion electronvolts, the number of nucleons before the reaction is observed to be the same as that afterward. At higher energies, which make possible the creation of nucleons, antinucleons, and other nucleonlike particles, this conservation law must be revised. The number of nucleonlike particles, or baryons, is conserved, but particle-antiparticle pairs are excluded from the count. Electric charge is conserved regardless of the number of particles created.

Reactions can also be characterized by the time scale over which they occur. Direct reactions occur in a time comparable to the time it takes for the projectile to traverse a target nucleus. A “knockout” reaction is one example in which the incident particle knocks an individual nucleon out of the target nucleus. The original particle may also promptly emerge but with reduced energy. “Stripping” reactions can occur when a composite projectile like deuterium passes close to a nucleus. One of the nucleons can be stripped from the projectile and incorporated directly into a nuclear-excited state. The remaining nucleons continue with sufficient energy to keep total energy constant.

Compound nuclear reactions occur over time scales one thousand to one million times longer than those required for direct reactions. The projectile is absorbed by the target, and its energy is distributed over the nucleons of the composite system. Eventually, the energy may be concentrated again in one, two, or more nucleons or particles, which then emerge. The process takes so long on a nuclear time scale that the mode of formation of the compound nucleus is independent of its mode of decay. Fission reactions are an example. They may be induced when a heavy nucleus such as uranium absorbs a neutron. The compound system has excessive energy and begins to undergo shape oscillations and distortions, and may eventually split into two (or more) fragments, usually with the release of additional neutrons. The fragments can gain substantial energy because of the electric repulsion between them. In an appropriate environment, the neutrons that are released can produce additional fission reactions and more neutrons. The continuation of this process is called a chain reaction.

The compound nucleus model describes a reaction in which the incoming projectile is fully absorbed by the target nucleus, creating a temporary, highly excited intermediate system. This compound nucleus has time to redistribute the excitation energy among its nucleons before decaying into final products. Because the energy levels of this intermediate state are quantized, reactions often exhibit sharp peaks in their cross sections at specific projectile energies—these are known as resonances, and they provide critical insight into the internal structure and energy levels of the nucleus.

Nuclei are quantum-mechanical systems, and one cannot predict with certainty what will happen when a target is bombarded. One can only specify the probability that a particular reaction will occur. Probabilities are expressed in terms of an effective cross-sectional area that the target nucleus presents to a projectile. In a thin slab of target material in which all nuclei are exposed to projectiles, the probability per incident particle for a specific reaction is the ratio of the total effective cross-sectional area of the nuclei to the front surface area of the target slab.

One might expect the cross-section for a nuclear process to equal the geometrical cross-section of the nucleus. This is not the case. Effective cross-sections range from zero (when the process does not occur) to thousands of times larger than the geometrical area. This reflects the wave nature of the particles and the fact that nuclei can exist only in certain limited quantum states. Reaction and scattering cross sections are not easily predictable and can be determined only through theoretical calculations, which are often inaccurate, or measurements. Hybrid quantum-classical algorithms have been developed to simulate nuclear reactions with improved efficiency and accuracy, extending traditional computational approaches.

Applications

Measurements of nuclear reactions and scattering are important sources of knowledge about nuclei. In a typical set of measurements, selected projectiles bombard specific nuclei in a thin target. Reaction products that leave the target are detected, and data, such as the numbers, types, and energies of particles, are recorded. Cross sections for each reaction are calculated.

These quantities are often determined for a number of different projectile energies and at a number of different angles with respect to the projectile direction.

The data are used to characterize the quantum states of nuclei in terms of parameters such as excitation energy, angular momentum, lifetime, decay modes, and decay probabilities.

Recurrent patterns in the behavior of these parameters from nucleus to nucleus, and in the cross-section data themselves, are often the inspiration for theoretical models of reaction mechanisms and types of excitations. Recognized excitation models range from those involving a single nucleon to collective vibrational and rotational excitations of the nucleus as a whole.

As a result of analyzing nuclear reaction data and models of stars, scientists know that nuclear reactions in stars are primarily responsible for the existence and abundance of chemical elements and isotopes. Fusion reactions of primordial hydrogen and helium head a series of fusion reactions that lead to elements as heavy as iron. Neutron-capture reactions involving these light elements, followed by radioactive decays, contribute to the abundance of heavier elements. Most of the naturally occurring radioactive isotopes on Earth most likely originated in the evolution and explosion of ancient stars.

Nuclear reactions play an important role in various methods of age-dating diverse specimens. For example, radioactive carbon is continuously produced in the atmosphere by cosmic-ray bombardment and incorporated into living organisms in small amounts. After an organism dies, it can no longer assimilate carbon, but the radiocarbon already there continues to decay. The amount of radioactive carbon, as opposed to ordinary carbon, therefore, decreases with the time elapsed since death and can be used to determine that time. This particular technique is well suited for dating archaeological specimens, since about 5,730 years is required for half the radiocarbon in a static sample to decay.

Short-lived radioactive isotopes of elements such as carbon, oxygen, and phosphorus, which are important in biological materials, can be produced by means of accelerator-induced reactions. They can be chemically separated from the target and incorporated into various organic compounds. When such labeled compounds are involved in chemical reactions or biological processes, the fate of the element can be traced by means of its radioactive emissions. These isotopes are commonly produced in cyclotrons or linear accelerators, which provide controlled, high-energy environments for inducing nuclear reactions tailored to medical isotope generation.

Labeled compounds are also used in medicine for therapeutic and diagnostic purposes. One of the most widely used diagnostic techniques is positron emission tomography (PET), which uses short-lived positron-emitting isotopes to generate detailed images of metabolic activity within tissues.

Some pharmaceutical compounds tend to concentrate in specific organs. Iodine, for example, concentrates in the thyroid gland. If a compound labeled with radioactive iodine is ingested, an image of the gland can be formed from the gamma (γ) rays emitted by the iodine concentrated there. In addition to imaging, theranostic approaches use radioisotopes like Lutetium-177 and Actinium-225, which emit radiation capable of destroying cancer cells while allowing for imaging and tracking of treatment response. In 2022, the US Food and Drug Administration (FDA) approved Pluvicto, one of the first radiopharmaceuticals for certain advanced prostate cancers, while in 2024, the FDA approved a lutetium-177 radiopharmaceutical for a specific group of pediatric patients with neuroendocrine tumors, establishing the growing relevance of radioisotopes in medicine.

There are many related imaging techniques that can be used to assess organ function, identify tumors, and trace metabolic activity.

Neutron activation analysis is a useful technique for determining the trace-element content of a wide variety of materials. The specimen, a mineral sample, for example, is exposed to an intense neutron flux from a nuclear reactor or other source. Neutrons are captured by nuclei in the specimen, inducing reactions that may result in radioactive isotopes. The isotopes can be identified by their characteristic decay modes, energies, and lifetimes. The rate of radioactive decay is proportional to the number of radioactive atoms and can be related through cross-section data to the number of target nuclei in the original specimen.

There are other techniques that use nuclear reactions and scattering directly for trace-element surface analysis. One example is Rutherford backscattering analysis. Charged particles, usually protons or α particles with energies below the barrier, bombard the target. In nuclear forensics, similar reaction-based techniques are applied to identify and trace the origin of illicit nuclear materials.

The energy and number of particles elastically scattered back toward the source are detected.

Particles backscattered from light elements at the surface will have less energy than those scattered from heavy elements because light target nuclei will have more recoil energy. The energy distribution of scattered particles, therefore, reflects elemental surface composition.

The importance of fission reactions in nuclear power generation is well known, but nuclear reactions are also important in the production of transuranic elements that can be used as fuel, and as a source of neutrons for applied and basic research in solid-state and nuclear physics.

Because the fuel supply for fission reactions is limited, controlled fusion reactions continue to be explored as an alternative source of power.

Context

The study of nuclear reactions and nuclei has its roots in the discovery of natural radioactivity by French physicist Antoine-Henri Becquerel in 1896. Within the following fifteen years, α, β, and γ decay were discovered, and the α particle was identified as a helium nucleus, the β particle as an electron, and the γ ray as electromagnetic energy. The transmutation of one chemical element into another as a result of α and β decay was also established.

New Zealand physicist Ernest Rutherford used α particles as projectiles with which to bombard a variety of materials. In particular, he found that when metallic foils were bombarded, some of the α particles were elastically scattered back toward the source. In 1911, he concluded from this observation that most of the mass and the positive charge of the atom were concentrated in a tiny nucleus at the center of the atom. Subsequent measurements of the dependence of scattering cross section on scattering angle verified this hypothesis and indicated that the nucleus is about 100,000 times smaller in diameter than the atom itself.

Rutherford also observed the first artificially induced nuclear reaction in 1919, when he bombarded nitrogen with α particles and produced protons and oxygen nuclei as reaction products. In 1932, English physicist James Chadwick proposed that electrically neutral particles, known as neutrons, were produced when beryllium nuclei were bombarded by α particles.

From 1932, it was clear that nuclei contained both protons and neutrons, and the understanding of nuclei rapidly increased. John Douglas Cockcroft and Ernest Thomas Sinton Walton observed the first nuclear reaction to be induced by machine-accelerated protons in 1932.

Irene Joliot-Curie and Frederic Joliot were the first to produce artificial radioactivity by means of a nuclear reaction. Otto Hahn and Fritz Strassmann discovered nuclear fission in 1939. In 1942, Enrico Fermi produced the first self-sustaining chain reaction.

After World War II, more powerful accelerators were developed, and knowledge of nuclei and nuclear forces extended to higher energies and shorter distances. Whereas reactions induced at random by cosmic rays were once the only way to produce exotic short-lived particles, such particles began to be produced in quantity in the laboratory. From studies of these particles, physicists have developed a detailed understanding of the nature of the forces that hold nuclei together despite the electric repulsion that tends to tear them apart. It is generally accepted that the electromagnetic interaction between charged particles is unified with the weak interaction that governs the β decay of nuclei.

Principal terms

ALPHA PARTICLE: a particle containing two protons and two neutrons (a helium nucleus) that is emitted in radioactive decay

BETA DECAY: a form of radioactive decay in which a neutron, for example, is transformed into a proton, an electron, and an antineutrino

ELECTRONVOLT: a unit of energy equivalent to the energy gained by an electron that is accelerated through an electrical potential difference of 1 volt

GAMMA RAY: electromagnetic radiation emitted when a nuclear excited state decays to a lower energy state

ION: an atom that has had one or more electrons added or removed in order to produce an electrically charged particle

ISOTOPE: nuclei that have the same number of protons, and therefore belong to the same chemical element, but have different numbers of neutrons

KINETIC ENERGY: energy of motion, which, at low speeds, is proportional to mass times speed squared

MASS ENERGY: the energy equivalent of mass is the amount of mass times the speed of light squared

NUCLEONS: a generic term for neutrons and protons, the particles that compose the atomic nucleus

RADIOACTIVITY: the spontaneous decay of a nucleus, usually by the emission of α, β, or γ rays

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