The beta particles may scatter both on the orbital electrons and in the nuclear field. Since the beta particles are much lighter than the alpha particles, the degree of the scattering of the beta particles is much higher than that of the alpha particles, resulting in very important measuring and analytical consequences.

Layer of the sample (g/cm2)

Figure 5.18 Intensity as a function of thickness in the method of constant specific activities.

The backscattering of the beta radiation as a function of the thickness of the scattering medium can be described as follows. Let us take a beta emitter on the bottom of a ring of lead-shielding, and a scattering medium with d thickness and arrange them as illustrated in Figure 5.19.

Then let us irradiate the surface area (F) of the medium with a beta radiation with I0 intensity. Because of the absorption of the beta radiation, the intensity decreases when passed through a distance x, and the intensity reaching the dx unit thickness is:

dIx = I0e2^ (5.66)

Let v be the ratio of beta particles that are backscattered from the dx thickness:

v dIx dx = v I0 e~Fx dx (5.67)

The backscattered beta particles are absorbed again when returned through the x thickness. Therefore, the intensity of the backscattered beta particles reaching the surface (F) can be expressed again by the absorption law. The energy of the back — scattered beta particles may be lower than the energy of the original beta particles, so the values of the mass absorption coefficients may be different when the beta particles pass in (Fin) or out (^out). In backscattering studies, the resultant effect of the two mass absorption coefficients is observed, so we can assume that

Fin 1 Fout Fb:

dI = vI0 e2(Fin+F“t)x dx = vI0 e2Fbx dx (5.68)

The total backscattered intensity can be obtained by the integration of Eq. (5.68) for the total thickness (d):

■d V

dI = —10[1 — e-M>d] 0 Mb

As seen from Eq. (5.69), the backscattered intensity tends to a limit as a function of the thickness. This limit for the infinite thickness of the sample is:

The backscattering of beta radiation can be characterized by the backscattering coefficient (Rf):

Rf can also be expressed in percent.

The energy of the backscattered beta particles is less than the energy of the orig­inal particles (Figure 5.20).

Similar to Eq. (5.65), the half-thickness of the backscattered beta radiation can be defined. The backscattered intensity of beta radiation of aluminum, zinc, and lead is shown in Figure 5.21 as a function of the half-thickness. As seen, the back — scattered intensity depends on the atomic number of the scattering media.

d1/2 Figure 5.21 Backscattered intensity of beta radiation of aluminum, copper, and lead as a function of half-thickness.

Table 5.5 Constants of the Muller Formula for Backscattering of Beta Radiation Period Z a b R II 2-10 1.2311 -2.157 0.3-10.2 III 10-18 0.96731 0.476 10.2-17.9 IV 18-36 0.68582 5.556 17.9-30.3 V 36-54 0.34988 17.664 30.3-36.6 VI 54-86 0.26225 22.396 36.6-45 Source: Adapted from Muller (1957), with permission from the American Chemical Society.

The values of Rf and the atomic number (Z) are in strict correlation. The back — scattered intensity, or Rf versus Z function, cannot be calculated exactly; empirical correlations are usually applied. One of them is as follows:

In = ki Zk2 (5.72)

where Inj is the scattered intensity at an angle, k1 = 0.0415 INJ2n and k2 = 2/3. Another Rf versus Z function is the so-called Muller formula:

R = aZ 1 b (5.73)

where a and b are constants for the elements in a given period of the periodic table (Table 5.5). Hydrogen is a special element; it can be fitted into the system by a hypothetical atomic number, which is —7.434. This can be explained by the fact that the ratio of nucleons to electrons is usually 2, while in the case of hydrogen, this ratio is only 1.

Equations (5.73) and (5.74) are also valid for compounds and mixtures if the mean atomic number is applied. The mean atomic number can be defined as:

n

niAiZi n

Z = i51 = XiZi (5.74)

niAi i51

i=1

where Z, is the atomic number of the constituents, n, is the number of the atoms, A, is the mass of the atoms, and x, is the mass ratio of the ith atom in the compound or mixture.

Equations (5.73) and (5.74) can also be applied for solutions; the Rf versus x, function is linear. Therefore, the Rf versus x, function is suitable for the concentra­tion measurement of solutions. In addition, by extrapolating the Rf versus x, func­tion to x, = 1, the backscattering coefficient (Rf) of the pure solid substance is obtained.

In conclusion, the measurements of the backscattered intensities of beta radia­tion give information on:

1. The thickness of the scattering matter or the thickness of thin layers on a thick plate (Section 11.3.4).

2. The mean atomic number.

3. The concentration of solutions.