Some Issues in Choosing the Optimal Geometric Dimensions of Cylindrical Collimators for Spectrometry of Intense Extended $\gamma$-Sources

V. M. Kantselyarskii, L. N. Lazarev, I. G. Minaev, N. I. Sushko,
V. I. Shulev

Introduction

Collimation of gamma radiation is usually used to obtain a spectrum convenient for processing. In this case the source of gamma radiation has point geometry. The collimator prevents scattered $\gamma$-quanta from reaching the detector, and the spectrum has a more or less uniform Compton distribution, whereas the absence of a collimator will cause a sharp increase in the spectrum in the region of 100–200 keV, against which photopeaks cannot be detected.

If the source of gamma quanta has dimensions large enough that it cannot be regarded as a point source, then the collimator, in addition to the role indicated above, can be used to isolate some part of the source volume in order to determine the total activity of the entire volume (if the activity is distributed uniformly, one measurement will be required), as well as the distribution of activity over the volume.

Collimator Designs

If we assume that the material of the collimator is impermeable to gamma quanta, then the problem of choosing a collimator is determined only by geometric factors [1]. The principal characteristic of such a collimator is the geometric efficiency, $G$. It is defined as the ratio of the number of quanta passing through the collimator channel to the number of quanta emitted by a point source located on its axis at some distance $L$. For a single-channel cylindrical or conical collimator, the geometric efficiency can be determined, from the dimensions of the collimator and the distance $L$, by the formula [1]:

\[ G=\left[\frac{d}{4(L+b)}\right]^2, \tag{1} \]

where
$d$ is the diameter of the collimator channel;
$b$ is the thickness of the collimator;
$L$ is the distance from the source to the face of the collimator.
(All dimensions are expressed in the same units.)

However, when choosing a collimator one cannot be guided by geometric factors alone. The edge effect of the collimator plays a large role, i.e., when gamma quanta pass through the material of the colli-

of the collimator material predominantly at the edges. This effect is characterized by the transmittance of the collimator, which is defined as

\[ P=\frac{N_{\text{tot}}-N_0}{N_0}, \tag{2} \]

where \(P\) is the transmittance of the collimator;
\(N_{\text{tot}}\) is the number of gamma quanta incident on the detector;
\(N_0\) is the number of gamma quanta incident on the detector only through the collimator channel.

Fig. 1.

If the number of gamma quanta is expressed in terms of their density per unit solid angle, then the transmittance is expressed as:

\[ P=\frac{\Omega_{\text{eff}}-\Omega_0}{\Omega_0}, \tag{3} \]

where \(\Omega_{\text{eff}}\) is the effective solid angle of the collimator;
\(\Omega_0\) is the optical solid angle of the collimator.

Figure 1 shows the two cases of collimator use discussed above.

In case 1a the transmittance is expressed by the formula

\[ P= \frac{ \displaystyle \int_{\operatorname{arctg}\frac{r}{b}}^{\pi/2} \sin\Theta\, e^{-\mu\left(\frac{b}{\cos\Theta}-\frac{r}{\sin\Theta}\right)} \,d\Theta }{ \displaystyle 1-\cos\left(\operatorname{arctg}\frac{r}{b}\right) }. \tag{4} \]

(For \(r \ll L\), the detector “\(D\)” may be regarded as a point detector).
\(\mu\) is the linear attenuation coefficient of the material. The remaining parameters are shown in Fig. 1.

In case 1b the transmittance is expressed by

\[ P= \frac{ \displaystyle \int_{\operatorname{arctg}\frac{r}{b}}^{\operatorname{arctg}\frac{R}{b}} \sin\Theta\, e^{-\mu\left(\frac{b+L}{\cos\Theta}-\frac{r}{\sin\Theta}\right)} \,d\Theta }{ \displaystyle 1-\cos\left(\operatorname{arctg}\frac{r}{b+L}\right) }, \tag{5} \]

where \(R\) is the detector radius. Formula (5) is valid for the case in which \(L\) is comparable in magnitude with \(r\).

The results of calculations of the transmittance \(P\), determined by formula (4) and carried out on the “Ural-1” computer, are presented in Table 1.

Table 1

It is evident from the table that the transmission of the collimator decreases sharply with an increase in the length of the lead collimator up to \(17 \div 20\) cm for the maximum value of the gamma-quantum energy, 1.5 MeV. The radius of the collimator has a weaker effect on the transmission within the range \(0.2 \div 1\) cm, especially for the lengths indicated above. Thus, the choice of collimator is determined mainly by its optimal length, when the transmission does not exceed 0.2–0.3. At this length, any collimator radius may be specified in order to select the required value of the geometric efficiency.

Fig. 2.

Experimental studies

Considering formulas (4) and (5), it may be noted that in the first case the transmission has a larger value than in the second. (The limits of integration are wider and the values of the exponential function are larger for equal \(r\) and \(b\).) Formula (5) will have the value closest to formula (4) when \(L = 0\), and if \(R\) is taken significantly larger than \(r\), then the formulas may be considered almost identical, since the integrand decreases very rapidly. For the experiment we choose the case of Fig. 1 б under the above-described conditions (\(L = 0\), \(2r = 8\) mm, \(2R = 40\) mm), since case 1 а is very difficult to carry out under laboratory conditions. We vary the collimator thickness \(b\) from 8 to 28 cm at intervals of 4 cm.

Fig. 3.

As the source we use \( \mathrm{Co}^{60} \); the detector is a NaI(Tl) crystal \(40 \times 30\) mm. Figure 2 shows the course of the spectrometer intensity curve (curve 4) and the calculated values of transmission as a function of collimator thickness (curve 1 is calculated for the mean energy of \( \mathrm{Co}^{60} \)—1.25 MeV; curves 2 and 3, respectively, for energies of 1.17 and 1.33 MeV). The shaded region corresponds to the choice of the optimal collimator thickness. For the experimental and theoretical data, the region is approximately the same.

Figure 3 shows \( \mathrm{Co}^{60} \) spectrograms taken with different collimators (the spectrograms are normalized to the principal photopeaks). It is clearly seen that the difference between the spectrograms becomes smaller after the value \( b = 120 \) mm. Thus, the optimal collimator thickness for \( \mathrm{Co}^{60} \) lies within the range 15–17 cm.

Conclusions

1. The calculations performed make it possible to choose the optimal dimensions of the collimator for each specific gamma-quantum energy.

2. For the experimental geometry indicated in Fig. 1a, corrections for transmission can be introduced when calculating the total activity of the emitter.

REFERENCES

1. “Nucleonics” 14, No. 11, 1956. J. A. W. van der Doec de Bye, p. 128.

2. “Phys. Med. Biol.” 1965, vol. 10, No. 1, pp. 51–65. By A. D. Rotenberg, Ph.D., and H. E. Jons, Ph. L. F. D. S. C.

3. K. L. Stolyarova. Applied Spectrometry of Ionizing Radiations. Moscow, Atomizdat, 1964.