Definitions

# Nuclear reaction

In nuclear physics, a nuclear reaction is the process in which two nuclei or nuclear particles collide to produce products different from the initial particles. In principle a reaction can involve more than two particles colliding, but because the probability of three or more nuclei to meet at the same time at the same place is much less than for two nuclei, such an event is exceptionally rare. While the transformation is spontaneous in the case of radioactive decay, it is initiated by a particle in the case of a nuclear reaction. If the particles collide and separate without changing, the process is called an elastic collision rather than a reaction.

In the symbolic figure shown to the right, and deuterium react to form the highly excited intermediate nucleus which then decays immediately into two alpha particles. Protons are symbolically represented by red spheres, and neutrons by blue spheres.

## The reaction equation

A nuclear reaction can be written in terms of a formula just like a chemical reaction. Nuclear decays can be written in a similar way, but with only one nucleus on the left side.

Every particle partaking in the reaction is written with its chemical symbol, with the mass number at the upper left and the atomic number at the lower left. The neutron is written "n"; the proton can be written "1H" or "p".

The equation is correct only if the sums of the mass numbers on both sides are identical (as required by the conservation law for baryon number), and if the sums of the atomic numbers on both sides are identical (as required by the conservation law for electric charge). In the example shown above, this leads to (assuming we would know only one particle to the right):

:

→    ?
To make the sums correct, the second nucleus to the right must have atomic number 2 and mass number 4; it is therefore also Helium-4. The complete equation therefore reads: :
→
or more simply: :
→

## Notation

Instead of using the full equations as shown in the previous section, in many situations a compact notation is used to describe nuclear reactions. This is A(b,c)D, which is equivalent to A + b gives c + D. Common light particles are often abbreviated in this shorthand, typically p for proton, n for neutron, α representing an alpha particle or Helium-4, etc. The reaction above would be written as Li-6(d,α)α.

## Energy conservation

Kinetic energy may be released during the course of a reaction (exothermic reaction) or kinetic energy may have to be supplied for the reaction to take place (endothermic reaction). This can be calculated by reference to a table of very accurate particle rest masses (see http://physics.nist.gov/PhysRefData/Compositions/index.html), as follows. According to the reference tables, the nucleus has a relative atomic mass of 6.015 atomic mass units (abbreviated u), the deuteron has 2.014 u, and the helium-4 nucleus has 4.0026 u Thus:

• Total rest mass on left side = 6.015 + 2.014 = 8.029 u
• Total rest mass on right side = 2 × 4.0026 = 8.0052 u
• Missing rest mass = 8.029 - 8.0052 = 0.0238 atomic mass units.

In a nuclear reaction, the total (relativistic) energy is conserved. The "missing" rest mass must therefore reappear as kinetic energy released in the reaction; its source is the nuclear binding energy. Using Einstein's mass-energy equivalence formula E = mc², the amount of energy released can be determined. We first need the energy equivalent of one atomic mass unit:

1 u c2 = (1.66054 × 10-27 kg) × (2.99792 × 108 m/s)2
= 1.49242 × 10-10 kg (m/s)2 = 1.49242 × 10-10 J (Joule)
× (1 MeV / 1.60218 × 10-13 J)
= 931.49 MeV,

so 1 u c2 = 931.49 MeV.

Hence, the energy released is 0.0238 × 931 MeV = 22.4 MeV.

Expressed differently: the mass is reduced by 0.3 %, corresponding to 0.3 % of 90 PJ/kg is 300 TJ/kg.

This is a large amount of energy for a nuclear reaction; the amount is so high because the binding energy per nucleon of the helium-4 nucleus is unusually high, because the He-4 nucleus is doubly magic. (The He-4 nucleus is unusually stable and tightly-bound for the same reason that the helium atom is inert: each pair of protons and neutrons in He-4 occupies a filled 1s nuclear orbital in the same way that the pair of electrons in the helium atom occupy a filled 1s electron orbital). Consequently, alpha particles appear frequently on the right hand side of nuclear reactions.

The energy released in a nuclear reaction can appear mainly in one of three ways:

When the product nucleus is metastable, this is indicated by placing an asterisk ("*") next to its atomic number. This energy is eventually released through nuclear decay.

A small amount of energy may also emerge in the form of X-rays. Generally, the product nucleus has a different atomic number, and thus the configuration of its electron shells is wrong. As the electrons rearrange themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.

## Q-value and energy balance

In writing down the reaction equation, in a way analogous to a chemical equation, one may in addition give the reaction energy on the right side:

Target nucleus + projectile -> Final nucleus + ejectile + Q.

For the particular case discussed above, the reaction energy has already been calculated as Q = 22.4 MeV. Hence:

:

→    22.4 MeV

The reaction energy (the "Q-value") is positive for exothermal reactions and negative for endothermal reactions. On the one hand, it is the difference between the sums of kinetic energies on the final side and on the initial side. But on the other hand, it is also the difference between the nuclear rest masses on the initial side and on the final side (in this way, we have calculated the Q-value above).

## Reaction rates

If the reaction equation is balanced, that does not mean that the reaction really occurs. The rate at which reactions occur depends on the particle energy, the particle flux and the reaction cross section.

## Neutrons vs ions

In the initial collision which begins the reaction, the particles must approach closely enough so that the short range strong force can affect them. As most common nuclear particles are positively charged, this means they must overcome considerable electrostatic repulsion before the reaction can begin. Even if the target nucleus is part of a neutral atom, the other particle must penetrate well beyond the electron cloud and closely approach the nucleus, which is positively charged. Thus, such particles must be first accelerated to high energy, for example by:

• particle accelerators
• nuclear decay (alpha particles are the main type of interest here, since beta and gamma rays are rarely involved in nuclear reactions)
• very high temperatures, on the order of millions of degrees, producing thermonuclear reactions
• cosmic rays

Also, since the force of repulsion is proportional to the product of the two charges, reactions between heavy nuclei are rarer, and require higher initiating energy, than those between a heavy and light nucleus; while reactions between two light nuclei are the most common ones.

Neutrons, on the other hand, have no electric charge to cause repulsion, and are able to effect a nuclear reaction at very low energies. In fact at extremely low particle energies (corresponding, say, to thermal equilibrium at room temperature), the neutron's de Broglie wavelength is greatly increased, possibly greatly increasing its capture cross section, at energies close to resonances of the nuclei involved. Thus low energy neutrons may be even more reactive than high energy neutrons.

## Notable types

While the number of possible nuclear reactions is immense, there are several types which are more common, or otherwise notable. Some examples include:

• Fusion reactions - two light nuclei join to form a heavier one, with additional particles (usually protons or neutrons) thrown off to conserve momentum.
• Fission reactions - a very heavy nucleus, spontaneously or after absorbing additional light particles (usually neutrons), splits into two or sometimes three pieces. (α decay is not usually called fission.)
• Spallation - a nucleus is hit by a particle with sufficient energy and momentum to knock out several small fragments or, smash it into many fragments.
• Induced gamma emission belongs to a class in which only photons were involved in creating and destroying states of nuclear excitation.

### Direct reactions

An intermediate energy projectile transfers energy or picks up or loses nucleons to the nucleus in a single quick (10−21 second) event. Energy and momentum transfer are relatively small. These are particularly useful in experimental nuclear physics, because the reaction mechanisms are often simple enough to calculate with sufficient accuracy to probe the structure of the target nucleus.

#### Inelastic scattering

Only energy and momentum are transferred.

• (p,p') tests differenced between nuclear states
• (α,α') measures nuclear surface shapes and sized. Since α particles that hit the nucleus react more violently, elastic and shallow inelastic α scattering are sensitive to the shapes and sizes of the targets, like light scattered from a small black object.
• (e,e') is useful for probing the interior structure. Since electrons interact less strongly than do protons and neutrons, they reach to the centers of the targets and their wave functions are less distorted by passing through the nucleus.

#### Transfer reactions

Usually at moderately low energy, one or more nucleons are transferred between the projectile and target. These are useful in studying outer shell structure of nuclei.

• (α,n) and (α,p) reactions. Some of the earliest nuclear reactions studied involved an alpha particle produced by alpha decay, knocking a nucleon from a target nucleus.
• (d,n) and (d,p) reactions. A deuteron beam impinges on a target; the target nuclei absorb either the neutron or proton from the deuteron. The deuteron is so loosely bound that this is almost the same as proton or neutron capture. A compound nucleus may be formed, leading to additional neutrons being emitted more slowly. (d,n) reactions are used to generate energetic neutrons.
• The strangeness exchange reaction (K,π) has been used to study hypernuclei.

Reactions with neutrons are important in nuclear reactors and nuclear weapons. While the best known neutron reactions are neutron scattering, neutron capture, and nuclear fission, for some light nuclei, the most probable reaction with a thermal neutron is a transfer reaction:

(n,α) 6Li + n → T + α 10B + n → 7Li + α 17O + n → 14C + α 21Ne + n → 18O + α 37Ar + n → 34S + α
(n,p) 3He + n → T + p 7Be + n → 7Li + p 14N + n → 14C + p 22Na + n → 22Ne + p

Some reactions are only possible with fast neutrons:

### Compound nuclear reactions

Either a low energy projectile is absorbed or a higher energy particle transfers energy to the nucleus, leaving it with too much energy to be fully bound together. On a time scale of about 10−19 seconds, particles, usually neutrons, are "boiled" off. That is, it remains together until enough energy happens to be concentrated in one neutron to escape the mutual attraction. Charged particles rarely boil off because of the coulomb barrier. The excited quasi-bound nucleus is called a compound nucleus.

• Low energy (e, e' xn), (γ, xn) (the xn indicating one or more neutrons), where the gamma or virtual gamma energy is near the Giant dipole resonance These increase the need for radiation shielding around electron accelerators

## Calculation

Applying the methods of scattering by two potentials, the plane wave of each free charged particle is replaced by the exact solution for a charged particle moving in the presence of another point charge.

Direct nuclear reactions are most often calculated by some form of distorted wave Born approximation. Applying again scattering by two potentials, the coulomb solutions and neutron plane waves are replaced by the optical model wave functions for the incident and outgoing particles moving in and near the nucleus. These are obtained mostly from elastic scattering experiments, and from inelastic scattering to vibrational and rotational collective excitations. The reaction itself is then modeled by the Born approximation. That is, the excitation or transfer process is treated as a first order perturbation on elastic scattering. An early improvement on this was to exactly treat the coupling between a small number of excited states, known as coupled channels Born approximation.