Nuclear reaction
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In nuclear physics, a nuclear reaction is a process in which two nuclei or nuclear particles collide, to produce different products than the initial particles. In principle a reaction can involve more than two particles colliding, but such an event is exceptionally rare. If the particles collide and separate without changing, the process is called an elastic collision rather than a reaction.
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[edit] Representation
A nuclear reaction can be represented by an equation similar to a chemical equation, and balanced in an analogous manner. Nuclear decays can be represented in the same way.
Each particle taking part in the reaction is written with its chemical symbol, then atomic number subscripted, and atomic mass superscripted. The neutron and electron, not being chemical elements, are given the symbols n and e respectively. The proton may be denoted by "H" (as a hydrogen nucleus) or as "p".
To balance the equation, we must ensure that the sum of the atomic numbers on each side of the equations are equal (required by the conservation law of electric charge), and that the sum of the atomic masses on each side are also equal (required by the law of conservation of baryon number). For example:
- 63Li + 21H → 42He + ?
Balancing:
- Atomic masses on the left are 6 and 2, giving 8 total. Therefore another 4 is required on the right side.
- Atomic numbers on the left are 3 and 1, giving 4 total. Therefore another 2 is required on the right side.
- Thus the missing particle has mass 4 and number 2, which means it is also a helium nucleus.
The complete reaction is thus:
- 63Li + 21H → 42He + 42He
which could also be written:
- 63Li + 21H → 2 42He
[edit] Simplified representation
Many particles appear in reactions so often that they are usually abbreviated. Thus, a helium nucleus (also known as an alpha particle) is written with the Greek letter "α". Deuterons (heavy hydrogen, 21H) are written simply as "d". Also, because the atomic numbers are implied by the chemical symbols, they are redundant after balancing, and often omitted. Finally, many common reactions take the form of a relatively heavy nucleus being struck by one of a small group of common, reactive particles, and emitting another common particle, to produce another nucleus. For these reactions, the notation can be greatly condensed into the form:
- <target nucleus> ( incoming particle, outgoing particle(s) ) <product nucleus>
So we could rewrite the preceding example by first abbreviating symbols:
- 63Li + d → α + α
then dropping atomic numbers:
- 6Li + d → α + α
and finally using the condensed form:
- 6Li(d,α)α
[edit] Energy
Energy may be released during the course of a reaction or energy may have to be supplied for the reaction to take place. This can be calculated by reference to a table of very accurate particle masses, as follows. According to the reference tables, the 63Li nucleus has an atomic weight of 6.015 atomic mass units, the deuteron is 2.014 a.m.u. and the helium nucleus is 4.0026 a.m.u. Thus:
- Total mass on left side = 6.015 + 2.014 = 8.029
- Total mass on right side = 2 × 4.0026 = 8.0052
- Missing mass = 8.029 - 8.0052 = 0.0238 atomic mass units.
This "missing" mass comes from energy released from the reaction; its source is the nuclear binding energy. Using Einstein's famous "E=mc²" formula, we can work out how much energy has been released. In fact, one atomic mass unit is equivalent to 931 MeV, so 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 the pair of helium electrons 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:
- kinetic energy of the product particles
- emission of very high energy photons, called gamma rays
- some energy may remain in the nucleus, as a metastable energy level.
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.
[edit] Reaction rates
The rate at which reactions occur depends on the particle flux and the reaction cross section.The number of reactions per unit volume per unit time can be obtained if we make the assumptions that all the neutrons are traveling with the same speed
[edit] Neutrons versus 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 commoner still.
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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] Quasi-elastic knock out reactions
The projectile interacts closely with only one nucleon of the target.
- Intermediate energy (p,2p) reactions were used as early tests of shell structure.
- d(n,2n)p and d(p,2p)n, analyzed with Faddeev equations calculations, have been used to look for three nucleon effects.
- (e, e'p) and (γ,p) reactions are easier to interpret but harder to perform.
- p(p,p'γ)p, known as p,p bremsstrahlung, has been used to study the off the energy shell p,p interaction.
[edit] 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.
[edit] 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.
