In a sample containing the beta emitter, the beta particles can also be absorbed by the sample itself in a process called “self-absorption.” The precise and accurate measurement of the samples containing beta emitters requires taking into consider­ation the effect of self-absorption, except when the sample is “infinitely thin.” In every other case, the radioactive intensities measured for the same radioactivity

depend on the thickness of the sample, the quantity of the carrier (an excess of inactive atoms of the same elements in the same chemical state), or an inactive matrix (every other substance). In some cases, the molecule itself containing the radioactive isotope can absorb a part of the beta radiation. In solutions, the solvent can absorb the radiation to such a high degree that it becomes impossible to mea­sure the activity. For this reason, beta emitters are usually measured in the solid phase. (It is important to note here that there is a special technique for the measure­ment of beta emitters, namely, the liquid scintillation technique, which utilizes this absorption. It will be discussed in Section 14.2.1.)

When the beta energy is low (so-called weak beta, e. g., 14C, 35S), the self­absorption is significant at thin layers. However, it cannot be ignored at high beta energies, either, especially when the sample is thick because of the presence of the inactive matrix. Depending on the quantity of inactive matrix, the measured inten­sity of the same radioactive substance can differ.

The effect of self-absorption can be corrected in two ways: the method of con­stant activities for high beta energies, and the method of constant specific activities for low beta energies. In the method of constant activities, the intensity is extrapo­lated for the infinitely thin layer, while in the method of constant specific activities, the intensity is measured at the so-called saturated thickness (>10 X d1/2). The two methods of the correction of self-absorption are discussed as follows.

In the method of constant activities, the total radioactivity of the sample is con­stant and the quantity of the matrix changes. The samples are arranged as shown in Figure 5.16.

The thickness of the sample is d (g cm-2), and the total intensity of the radiation (the intensity without matrix) is I0. From here, the intensity of the radiation in a unit thickness is I<jd. Consider a dx elementary thickness at a distance x from the upper sur­face of the sample; the intensity in this elementary thickness is I0 dx/d. Passing through the distance x, the radiation is absorbed and the intensity decreases. The intensity reach­ing the upper surface (dl) is expressed by the radiation absorption law (Eq. (5.48)):

dl = — exp(- nx)dx (5.59)

d

where д is the mass absorption coefficient.

By integrating Eq. (5.59) for the total thickness of the sample (d), we obtain the total intensity reaching the surface:

I = — [1 — exp(—pd)] (5.60)

pd

If the intensities of samples with constant radioactivity in different quantities of matrix are measured, the intensity decreases as the quantity of the matrix, i. e., the thickness of the samples increases. The I0 can be obtained by extrapolating to zero thickness. It can be done graphically or by a parameter-estimating computer pro­gram from the I versus d function. The measurements can be done up to 1 —2 half­thickness (Figure 5.17).

For thin layers (up to the 30% of the half-thickness), I0 can be determined by the series expansion of Eq. (5.60):

I = I0 1 — 2 pd (5.61)

In the method of constant specific activities, the characteristic properties of self­absorption (mass absorption coefficient and half-thickness) can be determined as follows. Samples with different thickness are produced from a substance having the same specific activity. In this case, the intensity of a unit thickness is defined as I0 (its dimension is intensity/surface density). The intensity reaching the surface decreases because of the absorption as described by the radiation absorption law (Eq. (5.48)): I0 exp(—pd). The total intensity reaching the surface is:

Since

from Eq. (5.62), we obtain the following:

I = I®[1 — exp(-pd)]

where IN is the intensity at the saturation thickness. This is the maximal intensity, which does not increase even if the thickness increases. The intensity as a function of the thickness is shown in Figure 5.18.

In the method of constant specific activities, the half-thickness can be defined (dm) as:

The half-thickness can be determined from Eq. (5.64) graphically or by a param­eter-estimating computer program.

The method of constant specific activities can be used if the thickness is at least 7—10 times greater than the half-thickness. In this case, the specific intensities of the samples can be compared since they are proportional to the radioactivity.