pair-p&a.htm(Ó R. Egerton)

Figure references in this section are to the alternate second edition of* Elementary Modern Physics *

by R.T. Weidner and R.L. Sells (Allyn and Bacon, 1973).

Pair Production and Annihilation

The photoelectric and Compton effects represent two mechanisms of **photon absorption**, the process in which a
photon gives up some or all of its energy to a material particle (contained usually within an atom in a solid).
Photon absorption can be *quantified* in terms of an **absorption coefficient** m
,commonly expressed in units of m^-1 or cm^-1 and defined by the eq uation:

I(x) = I(0) exp(-(mu) x)

where I(0) and I(x) are the photon intensities (W/m^2) at the *entrance* surface and at a depth x below the surface (measured in the direction of the incident beam).
The intensity therefore decays exponentially within the material, as shown in Fig. 4-26. We can also take x as being the entire thickness of the material, in which case I(x) represents the photon intensity at the *exit* surface. By placing radiation detectors in front and behind a sheet of material of known thickness x, the absorption coefficient can be measured.

If the experimentally determined (mu)
is plotted as a function of increasing photon frequency f (or photon energy hf), its value falls continuously within the visible, ultraviolet and x-ray regions but eventually reaches a minimum and then starts to increase in the g
-ray region; see Weidner & Sells Fig. 4.24. The reason for the initial decrease is that the absorption processes of photoelectric and Compton scattering become less probable as the photon energy increases; in other words, higher-energy radiation is more **penetrating**.

The eventual rise in (mu)
indicates that a third process occurs at high photon energy; this is **pair production,** in which a pair of **elementary particles** (a particle and its **antiparticle** of the same mass but opposite electrostatic charge) are created from the energy (hf) of the original photon. In this case, the two particles are an electron and an antielectron (more commonly known as a **positron**, whose rest mass m0 is the same as that of an electron but whose charge is +e ).

Pair production can be represented by an equation which represents the conservation of total energy (or mass-energy):

hf = 2(m0 c^2) + K(-e) + K(+e)

Here, (m0 c^2) = 0.511 MeV is the rest energy of an electron, which is equal to that of the positron, so the factor of 2 represents the fact that two particles of identical rest mass are created. K(-e) and K(+e) represent the kinetic energy of the electron and positron, immediately after their creation.

If the photon energy were exactly 2m0 c^2 = 1.02 MeV, the two particles would be created at rest (with zero kinetic energy) and this would be an example of the *complete* conversion of energy into mass. For photon energies below 2m0c^2, the process cannot occur; in other words, 1.02 MeV is the **threshold energy** for pair production. For photon energies *above* the threshold, a photon has more than enough energy to create a particle pair and the surplus energy appears as kinetic energy of the two particles.

There is, however, a further condition which must be satisfied during the pair-production process: conservation of **momentum**. Taking this requirement into account, we can anticipate that pair production *cannot take place in empty space*; something must absorb the momentum (p=h/l
=hf/c) of the initial photon. (To see this, consider the threshold situation where the particles have to be created *at rest* and cannot themselves absorb *any* momentum). The photon momentum can be absorbed by an **atomic nucleus**, which is thousands of times more massive than an electron or positron and can therefore absorb momentum *without* absorbing much energy; therefore the energy-conservation equation above remains approximately valid. Consequently, pair production is observed when high-energy gamma rays *enter a solid*, where a high density of atomic nuclei is present.

There exists an **inverse process** to pair production called **pair annihilation**,
in which a particle and its antiparticle collide and annihilate each other, the total energy
of the two particles appearing as electromagnetic radiation. In the case of an electron and
positron, the energy balance can be represented as:

2m0c^2 + K(-e) + K(+e) = 2 hf

The first term represents the rest energy of *both* particles, the second and third terms are the kinetic energies just before the collision, and the term on the right-hand side of the equation represents the creating of two photons, each having a the same frequency f and energy hf . The equation indicates that hf must be at least equal to m0c^2=0.51 MeV, so the photons produced correspond to g
-ray radiation. Like pair production, this process usually takes place inside a solid, which provides the electrons which are needed. The positrons enter the solid as a beam of particles produced by a particle accelerator.

If the kinetic energies of the two initial particles were both small (<< m0c^2), the total momentum before the collision would be close to
zero. From *conservation of momentum*, the momentum after annihilation must also be approximately zero, and the only way that this can happen is for the two photons to be emitted in *opposite directions* such that their individual momenta cancel. More typically, the positrons entering the solid have appreciable momentum whereas the electrons (originally present in the solid) are almost at rest. Then the paths of the two g
-ray photons are at some angle to each other, and these paths intersect at the position of the pair-annihilation event (see Fig. 4-xx).

This is the principle behind **positron emission tomography** (PET), a medical
diagnostic technique which provides images showing the internal structure of body organs.
By detecting the direction of travel of g
-ray photons which are produced *simultaneously* (**coincidence** detection) and
counting the number produced in different regions of space, the PET machine builds up an
image of the *electron density* within a plane (actually a narrow slice) within the
tissue. Moving the detectors (or the patient) forward, the process can be repeated so that
three-dimensional information (a **tomograph**) is obtained. The positrons are provided by
a radioisotope, injected into the bloodstream. PET is now a
commercially developed technique, complementary to CT and MRI scanning.

Figure 4.19 (from Weidner & Sells) summarizes schematically the photon-electron interactions which we have dealt with recently. Note that in some cases, the "particle" (electron, positron or photon) which enters the solid is completely destroyed in the interaction; in other cases (Compton and bremstrahlung effects) the particle just loses some of its energy.

Lead contains a higher density of electrons (and more highly-charged atomic nuclei) than an element such as aluminum
(of lower atomic number); therefore the photon-absorption processes are stronger (higher (mu)
; see Figs. 4-24b and 4-26), making lead suitable as a radiation-shielding material. However, the decrease in (mu)
with increasing photon energy (below 1 MeV) means that relatively thick sheets are needed to absorb **hard** (short-wavelength) x-rays or gamma
-rays, compared to **soft** (long-wavelength) x-rays for example.