bei48482_FM

hv/c

p

p

θ

θ

p cos θ

p cos θ

Figure 2.26Vector diagram of the momenta involved if a photon were to materialize into an electron-

positron pair in empty space. Because such an event cannot conserve both energy and momentum, it

does not occur. Pair production always involves an atomic nucleus that carries away part of the initial

photon momentum.

The rest energy mc^2 of an electron or positron is 0.51 MeV, hence pair production

requires a photon energy of at least 1.02 MeV. Any additional photon energy becomes

kinetic energy of the electron and positron. The corresponding maximum photon wave-

length is 1.2 pm. Electromagnetic waves with such wavelengths are called gamma rays,

symbol , and are found in nature as one of the emissions from radioactive nuclei and

in cosmic rays.

The inverse of pair production occurs when a positron is near an electron and the

two come together under the influence of their opposite electric charges. Both parti-

cles vanish simultaneously, with the lost mass becoming energy in the form of two

gamma-ray photons:

eeS 

The total mass of the positron and electron is equivalent to 1.02 MeV, and each pho-

ton has an energy hof 0.51 MeV plus half the kinetic energy of the particles relative

to their center of mass. The directions of the photons are such as to conserve both en-

ergy and linear momentum, and no nucleus or other particle is needed for this pair

annihilationto take place.

Example 2.5

Show that pair production cannot occur in empty space.

Solution

From conservation of energy,

h 2

 mc^2

where his the photon energy and mc^2 is the total energy of each member of the electron-

position pair. Figure 2.26 is a vector diagram of the linear momenta of the photon, electron,

and positron. The angles are equal in order that momentum be conserved in the transverse

direction. In the direction of motion of the photon, for momentum to be conserved it must

be true that

 2 pcos

h 2 pc cos

h



c

Particle Properties of Waves 81

bei48482_ch02.qxd 1/16/02 1:53 PM Page 81