The transmitted energy of the beta particles to orbital electrons depends on the energy of the beta particle. The expressions describing the transmitted energy are different whether the velocity of the beta particle is below or above the velocity of light in a vacuum.

At E(j < mec2 (E$ is the energy of the beta particle), the energy used up for ioni­zation is:

dE 4ne4n, 1.66mevi

= —— 2 Z ln

dx ion meve 2/

Equation (5.37) is similar to Eq. (5.26), indicating that the ionization is similar for both alpha and beta particles. The numerical factors signify the differences in the size of the alpha and beta particles.

At Ep > mec2, the energy used up for ionization is:

dE _ 2ne4n / E3 + 1

dx ion mec2 2mec2/2 8

Equation (5.38) takes into consideration the relative mass increase because of the high energy of the beta particle.

The decrease of the energy of the beta particles as a result of ionization is shown in Figure 5.11.

Some of the beta particles interact with the nuclear field, producing Bremsstrahlung. As discussed previously, Bremsstrahlung is a continuous X-ray. The energy of beta particles producing Bremsstrahlung can be expressed as:

Figure 5.11 Specific energy loss of beta particles versus energy for different absorbers.

Source: Adapted from Kiss and Vtsrtes (1979), with permission from Akademiai Kiado.

The total energy loss of the beta particle is the sum of the energies transmitted to orbital electrons (ionization) and producing Bremsstrahlung (X-rays):

dE = / dE + /dE

to tot dx ion dx X-ray

The ratio of the energies producing X-rays and ionization is expressed as follows:

dE

dx X-ray ~ EZ dE ~ 800

dx ion

where E is the energy of the beta particle, and Z is the atomic number of the absorber.