The overall efficiency E of a given device for the measurement of radioactivity depends on the geometric efficiency of the physical setup E_g and the intrinsic efficiency E_i of the detector. Thus, the overall efficiency can be expressed as

E = E_g × E_i

Intrinsic Efficiency. The intrinsic efficiency E_i (see Chapter 8) is defined as

It primarily depends on the linear attenuation coefficient µ (linear) of the material of which a detector is made and on the thickness of the detector.

In a scintillation counter, when one restricts oneself to the photo peak counts only, the resulting intrinsic efficiency is known as the photo peak efficiency.

Geometric Efficiency. Geometric efficiency E_g can be understood as follows. Let a small sample of radioactive compound be located at a distance from a detector with a cross-sectional radius r as shown in Figure 9.1 (top). Because there is no preferred direction for the emission of radiation from a radionuclide, the radiations (α, β, or γ) from a radioactive sample will be emitted with equal probability in all directions. Therefore, of the total number of nuclear radiations emitted in all directions, only a fraction of these will be incident on the sensitive volume of the detector (Fig. 9.1, top). Geometric efficiency E_g, then, is defined as the ratio of the number of radiations incident on the detector from a radioactive sample
to the total number of radiations emitted by the sample, or

The geometric efficiency Eg, for the device shown in Figure 9.1 (top), depends on two factors: the distance x between the source and the detector, and the radius r of the cross-sectional area of the detector. For distances that are sufficiently larger than the radius of the detector (i.e., x >> r), the geometric efficiency varies as 1/x² and r², respectively, with the distance x and the radius r; that is, if one increases the distance between the sample and the detector twofold, the geometric efficiency is reduced fourfold (Fig. 9.1, center). If the radius of the detector is doubled, the geometric efficiency will be increased fourfold (Fig. 9.1, bottom).

When the sample is very close to the detector (i.e., x << r), the above relationship E_g ∝ (1/x²) does not hold true. The exact relationship between E_g and the distance x, in this case, is of no practical concern to us here except to know that the maximum obtainable geometric efficiency is achieved when x = 0 (i.e., the source is directly against the face of the detector). In this case, E_g approaches 50%, because even in this situation, one-half of the radiations are emitted away from the detector.

To increase E_g to more than 50%, one has to either use more than one detector or somehow surround the sample from all sides by the detector. The second approach is used in well-type NaI(Tl) scintillation detectors (well counters) and liquid scintillation detectors described later and is also used in dose calibrators as is described in Chapter 8.

In terms of its operation and associated electronics, the well-type NaI(Tl) scintillation detector (well counter) is exactly the same as the NaI(Tl) scintillation detector described in Chapter 8, except that in a well counter, a small cylindrical hole is constructed in the NaI(Tl) crystal to allow a radioactive sample to be positioned very close to the center of the crystal (Fig. 9.2). In this type of detector, only a small fraction (<5%) of the radiations emitted by the sample, escapes from the sensitive volume of the crystal. Thus, E_g approaches 95%. The intrinsic efficiency of an NaI(Tl) detector is dependent on the size of the crystal—the larger the crystal, the higher the intrinsic efficiency for a given energy γ-ray. Well counters are commercially available in various sizes. The so-called standard well counter (1.75-inch diameter, 2 inches in height with a hole 0.75 inches in diameter and 1.5 inches deep) enjoys the most popularity in nuclear medicine. For high-energy γ-rays (>500 keV), however, one achieves a far better count rate with a 3 × 3-inch well counter
than with the standard model. Table 9.1 lists the intrinsic efficiencies of standard and 3 × 3-inch well counters for various γ-ray energies. It can be seen from columns 2 and 3 that at higher energies, one gains a factor of 1.5 in intrinsic efficiency by using a 3 × 3-inch rather than a standard well counter. For many procedures, one is interested only in the photo peak counts (i.e., one rejects γ-ray interactions via the Compton scattering). In this case, the advantage of a 3 × 3-inch over a standard well counter for higher energy γ-rays becomes even greater (see columns 4 and 5 in Table 9.1).