**Pair Production and Annihilation**

**Introduction**

The concept of the wave-particle duality which means all matter has both “particle-like” and “wave-like” nature has been the inception of the quantum physics. Especially, for the “particle-like” nature, there are some examples that can be an empirical proof: first two processes as proves are the photoelectric effect and the Compton effect. Those two effects are well-known processes of photon absorption or scattering that photon loses its energy by interacting with other matter as a “particle-like” things.

The final one for proving “particle-like” nature of all matter is pair production. It was first observed by English experimental physicist, Patrick Blackett, who was the winner of the 1948 Novel Prize in physics. While both the two processes mentioned above—the photoelectric effect and the Compton effect—usually occur in low energy and middle energy condition, pair production interaction is known to occur when photon has high enough energy before its collision.

**Pair Production**

The pair production is a crucial example that photon energy can convert into kinetic energy as well as rest mass energy. Schematic diagram about the process of pair production is shown in figure 1. The high-energy photon that has energy *hν* loses its entire energy when it collides with nucleus. Then, it makes pair of electron and positron and gives kinetic energy to each particle.

Figure 1. Pair Production Process

Basically, these interactions are ruled by three kind of the law of conservation: total energy, momentum, and electric charge. After the collision, pair of electrons and positron occurs. In this collision, the positron, *+e*, as a particle, has the same properties which electron has, except its charge sign; the two particle, electron and positron has the opposite charge, and thus its magnetic momentum sign is also opposite. Having an opposite charge sign means that the sum of net charge of pairs is zero, which is actually equal to photon before the collision. Therefore, the conservation of electric charge will be conserved.

Now, what counts is conservation of total energy and momentum. First of all, momentum in this process can be ignored because atomic nucleus is thousands of times more massive than pair of electron and positron, and thus, the photon momentum can be absorbed; thus, it is possible to anticipate that absorbing momentum occur without absorbing much energy. So, it is can be represented by an equation that shows the conservation of total energy only:

*hν* = *E*_{-} + *E*_{+} = (*m*_{0}*c*^{2} + *K*_{-}) + (*m*_{0}*c*^{2} + *K*_{+}) = *K*_{-} + *K*_{+} + 2*m*_{0}*c*^{2} ------—eq.1

Here, *E*_{-} and *E*_{+} represent total energy of the electron and positron, and *K*_{-} and *K*_{+} represent the kinetic energy of the electron and positron. Also, *m*_{0}*c*^{2} = 0.511 MeV, and it is the rest mass energy of an electron, which is equal to that of the positron. That is, both the two particles have the same amount of rest mass energy, and from this, we can get the sum, 2*m*_{0}*c*^{2} = 1.02 MeV (1 MeV = 10^{6} eV). In the kinetic energy, because of the Coulomb interaction between positively-charged nucleus and both the two particles—the positron and electron, positron would be accelerated and electron would be decelerated; thus, the kinetic energy of positron is actually little larger than electron’s.

If the photon energy before colliding to nucleus were exactly same with 2*m*_{0}*c*^{2} , namely, if the photon energy were 1.02 MeV, the two particles would be created at rest with zero kinetic energy. In this case, we are able to think that the complete conversion of energy into mass occurs in this process. However, if the photon energy were large than 1.02 MeV, then kinetic energy would occur. Also, the process cannot occur when photon energy is below 1.02 MeV. Consequently, From equation 1, we are able to get minimum photon energy that need to produce pair electron and positron, and “2*m*_{0}*c*^{2} = 1.02 MeV” would be a kind of threshold energy for the pair production that are able to decide whether the pair production process can occur or not, and further, if it occurs, how much kinetic energy each particle has.

The wavelength of 2m0c2 is 0.012Å. If the wavelength of photon is shorter than 0.012Å, it has larger than threshold energy, and total photon energy convert into kinetic energy as well as rest mass energy. This wavelength, 0.012Å, is in the range of high energy of γ-ray or X-ray.

**Pair Annihilation**

Pair Annihilation means the reverse process of pair production. In the pair annihilation, the electron and positron in the stationary state combine with each other and annihilate. Surely, the particles are disappeared and radiation energy will occur instead of two particles. For the momentum conservation, the most frequent process in pair annihilation is making two photons that have exactly opposite direction and the same amount of momentum. (Sometimes it produces three photons in the pair annihilation process.)

Figure 2. Pair Annihilation Process

Figure 2 is shown the annihilation of pair electron and positron which is making two photons. In the case of Figure 2, the energy balance can be represented as:

*K*_{-} + *K*_{+} + 2*m*_{0}*c*^{2} = 2 *hν* --------—eq.2

*K*_{-} and *K*_{+} represent the kinetic energy of the electron and positron before the collision. Also, 2*m*_{0}*c*^{2} means the rest mass energy of both particles. From the equation 2, if the initial kinetic energy was zero, then,

*hν* = *m*_{0}*c*^{2} = 0.511 MeV --------—-eq.3

Therefore, in the equation 3, photons produced by pair annihilation have 0.511 MeV energy, and it correspond to 0.024Å of the wavelength in γ-ray. However, if the initial kinetic energy is not the same with zero, photon’s energy is larger than 0.511 MeV, and its wavelength might be shorter than 0.024Å.

The positron is produced by the process of pair production. Generally, this positively-charged particle loses their energy by colliding with other particles in the path within matter, and finally combines with electron. We call those, “combined things”, “positroium”. The positronium collapse within about 10-10 second and produce two photons(pair annihilation).

**Reference**

http://www.britannica.com/EBchecked/topic/438692/pair-production

http://physics.pdx.edu/~egertonr/ph311-12/pair-p&a.htm

http://en.wikipedia.org/wiki/Pair_production

http://www.upscale.utoronto.ca/PVB/Harrison/Flash/Nuclear/PairProduction/PairProduction.html

Robert Eisberg, Robert Resnick, "Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles", John Wiley & Sons, New York

Raymond A. Serway, Clement J. Moses, and Curt A. Moyer, "Modern Physics", Saunders College Publishing