DETERMINATION OF THE OPTIMAL DIMENSIONS OF A SCINTILLATION CRYSTAL IN BETATRON FLAW DETECTION OF MATERIALS

V. I. GORBUNOV, V. B. KUZNETSOV, V. K. KULESHOV, Yu. B. YANKELEVICH

(Submitted by the scientific seminar of the Scientific Research Institute of Electrical Measurements)

When measuring small intensities of betatron bremsstrahlung behind an absorber, the selection and justification of the optimal dimensions of the scintillation crystal play an important role. The dimensions of the crystal affect the efficiency of registration of gamma quanta and, consequently, the minimum radiation intensity that can be measured by a given crystal with sufficient accuracy.

It is necessary, however, to distinguish between the efficiency of interaction of a beam of gamma quanta with a crystal and the efficiency of registration of these quanta.

The interaction efficiency is determined by the usual formula

\[ \varepsilon_b = 1 - e^{-\mu d}, \tag{1} \]

where \(\mu\) is the total linear absorption coefficient of the crystal material;

\(d\) is the thickness of the crystal along the path of the beam.

The efficiency of registration of quanta will be lower than the interaction efficiency, since not all quanta that interact with the crystal will be registered. This is explained by the fact that some of the scintillations are lost as a result of optical losses and, in addition, some quanta produce scintillations lying below the sensitivity threshold of the photomultiplier.

The latter factor is significant in the registration of high-energy quanta (on the order of \(5 \div 10\) MeV and higher). In this energy range, Compton scattering through small angles and comparatively small energy losses of the quantum in the crystal are characteristic.

When measuring the radiation intensity of radioactive isotopes using the counting-spectrometric method [1], the most important characteristic of the efficiency of a scintillation detector is the area under the “total energy” peak and the level of the continuous distribution in the instrumental spectrum. In the case of measuring the intensity of pulsed bremsstrahlung from a betatron, however, it is more convenient to introduce the concept of “energy losses of quanta outside the crystal.” The point is that the scintillation crystal does not “resolve” the quanta in a betatron radiation pulse, since the duration of this pulse is \(1 \div 5\ \mu\text{s}\), and the number of quanta per \(1\ \text{cm}^2\) is \(10^4 \div 10^6\). In this case the photomultiplier responds

on the integral light flux emitted by the crystal during a radiation pulse. Therefore it makes sense to speak of fluctuations in the number of light photons produced by a pulse of quanta. From this point of view, the concept of “energy losses outside the crystal” serves as a kind of indicator of the “efficiency” of the crystal. The smaller these losses, the higher the registration efficiency of the scintillation detector

\[ \varepsilon_{p}=\varepsilon_{b}\cdot h \cdot \vartheta, \tag{2} \]

where \(\varepsilon_{p}\) is the registration efficiency;
\(h\) is a coefficient taking into account the energy losses of quanta outside the crystal;
\(\vartheta\) is a coefficient taking into account the optical losses in the crystal.

Table 1 gives the interaction efficiencies of gamma quanta with energies of \(10\ \mathrm{MeV}\), calculated by formula (1) for crystals of different lengths.

Table 1

As will be shown below, crystals of small length (\(1 \div 5\ \mathrm{cm}\)) and small diameter, in addition to a low interaction efficiency, have an extremely high percentage of energy losses outside the crystal, which substantially lowers their registration efficiency.

The choice of a scintillation crystal of large dimensions inevitably leads to an increase in optical losses in the crystal and to an increase in the level of “noise pulses” from gamma quanta passing through the walls of the detector shielding. In addition, the considerable cost of crystals and a number of technical difficulties associated with the use of large scintillators compel one to refrain from increasing scintillation crystals in betatron flaw detection of materials. The authors set themselves the goal of developing a method for determining the optimal dimensions of a scintillation crystal for detecting the bremsstrahlung radiation of a betatron.

In view of the complexity of calculations by the Monte Carlo method, it was not the “white” spectrum of betatron bremsstrahlung that was considered, but monoenergetic quanta with an energy of \(10\ \mathrm{MeV}\). It may be assumed that this corresponds to the effective energy of the bremsstrahlung spectrum behind an absorber of \(50\ \mathrm{cm}\) of aluminum (Fig. 1). With an increase in the effective energy of the spectrum behind the absorber, the qualitative conclusions regarding the optimal dimensions of the scintillation crystal will be analogous. Conversely, when the effective energy of the spectrum behind the absorber is decreased to \(3 \div 5\ \mathrm{MeV}\) (which corresponds to a \(Pb\)-type absorber), the conclusions about the optimal dimensions of the NaI(Tl) crystal should be different, in particular with respect to the crystal diameter. This can be explained by the fact that, for gamma-quantum energies of \(3 \div 5\ \mathrm{MeV}\) in NaI(Tl), the Compton-scattering process is most characteristic. In this case the radial dimensions of the crystal will have a more substantial effect on the magnitude of the energy losses outside the crystal than at a quantum energy of \(10\ \mathrm{MeV}\), where the pair-production process predominates. An increase in the crystal diameter will accordingly increase the contribution of multiple Compton scattering and reduce the total energy losses of quanta outside the crystal.

In a number of works [2, 3, etc.] an experimental and theoretical evaluation has been made of the registration efficiency, of energy losses outside the crystal, and of the energy resolution for NaI(Tl) crystals and gamma quanta with energies above 10 MeV. However, this evaluation is valid for crystals of large dimensions—more than 12.7 cm in length and

Fig. 1.

more than 10 cm in diameter. Crystals of such dimensions are not yet used at present in flaw detection, and it is not possible to evaluate crystals of smaller dimensions from these data.

The authors calculated the mean value of the energy losses of quanta outside NaI(Tl) crystals of dimensions \(30 \times 10\) and \(50 \times 50\) mm, which made it possible to estimate the registration efficiency of scintillation crystals over a wide range of their dimensions.

1. NaI(Tl) crystal \(30 \times 10\) mm. By the method of statistical trials (Monte Carlo), the spectrum of energy losses of quanta outside a crystal 30 mm in diameter and 10 mm long was determined. A total of 2200 histories of quanta with energy \(E_0 = 10\) MeV were considered. In the calculations it was assumed that all quanta fall on the crystal strictly along its axis. In addition, because of the small thickness of the crystal and the high energy of the gamma quanta, multiple scattering of quanta was not taken into account, since its contribution in the present case is negligibly small.

In considering the histories of the quanta, energy losses outside the crystal due to the small radius and losses due to the small length of the crystal were separated. As a result of the calculations, the following values of the energy losses of quanta outside the crystal were found:

radial losses \(\Delta E_r = 0.012E_0\);

losses along the length \(\Delta E_l = 0.52E_0\);

losses along the length due to the escape of secondary particles (components of pairs and Compton electrons) \(\Delta E_э = 0.14E_0\);

total losses \(\Delta E_n = \Delta E_r + \Delta E_l = 0.53E_0\).

Thus, the registration efficiency for a NaI(Tl) crystal of dimensions \(30 \times 10\) mm, owing to the large value of the energy losses of quanta outside the crystal, is more than twice as small as the interaction efficiency indicated in Table 1.

The high level of energy losses in this case is due to the fact that the range of the secondary particles at \(E_0 = 10\) MeV is, for the most part, greater than the length of the crystal.

1. NaI(Tl) crystal \(30 \times 50\) mm. To investigate the course of the dependence \(\Delta E_n = f(r,l)\), the Monte Carlo method was used to determine the magnitudes of energy losses outside a NaI(Tl) crystal of diameter 30 mm and thickness 50 mm.

A total of 1000 histories of quanta with \(E_0 = 10\) MeV incident along the axis of the crystal were considered. In view of the increased dimensions of the crystal, multiple Compton scattering of quanta was taken into account in the calculations. The mean values of the energy losses are as follows:

radial losses \(\Delta E_r = 0.03E_0\);

losses along the length \(\Delta E_l = 0.21E_0\);

losses along the length due to the escape of secondary particles \(\Delta E_e = 0.05E_0\);

total energy losses outside the crystal \(\Delta E_n = 0.24E_0\).

It should be noted that the level of multiple Compton scattering is comparatively low. Thus, the ratio of the number of quanta scattered two or more times to the total number of Compton-scattered quanta is 0.09.

1. Extrapolation for crystals of other dimensions. A detailed examination of the process of energy exchange of quanta in NaI(Tl) crystals of dimensions \(30 \times 10\) and \(30 \times 50\) mm showed that for these crystals the radial energy losses are caused mainly by annihilation quanta with energy 0.511 MeV. As a result, the values of the radial losses for crystals \(76 \times 10\) and \(76 \times 50\) mm can be found from the usual exponential law. Knowing the magnitude of these losses, one can, using the data of [2] and [3] for large crystals, construct the dependences:

\[ 1)\quad \frac{\Delta E_n}{E_0} = f(l) \quad \text{for } R = \mathrm{const}; \]

\[ 2)\quad \frac{\Delta E_l}{E_0} = f(l) \quad \text{for } R = \mathrm{const}; \]

\[ 3)\quad \frac{\Delta E_r}{E_0} = f(l) \quad \text{for } R = \mathrm{const}; \]

\[ 4)\quad \frac{\Delta E_r}{E_0} = f(r) \quad \text{for } l = \mathrm{const}. \]

The data for constructing dependences (1) and (2) are given in Tables 2 and 3, and for dependences (3) and (4) in Tables 4 and 5.

The diameter of the NaI(Tl) crystal for dependences (1) and (3) is 76 mm.

Table 2

Table 3

Table 4

Table 5

As is seen in Figs. 2 and 3, for crystals up to 127 mm long, the principal contribution to the energy losses is made by losses along the length. With an increase in the crystal length, radial losses predominate, although their absolute value is quite small.

Fig. 2.

The rapid decrease in the total energy losses with an increase in the crystal length to 10–12 cm makes it possible to conclude that a further increase in the crystal length is irrational, since the absolute value of the energy losses is already small (5–7%). In addition, optical losses inevitably arise as the crystal length is increased. With regard to radial losses, it should be noted that at small crystal lengths (10–50 mm) an increase in the radial losses with increasing length is observed (Fig. 3). This is explained by the fact that the secondary particles (electrons and positrons) produced by quanta with \(E_0 = 10\) MeV have a comparatively large range—up to 1.5 cm. Therefore, for a crystal length of 10 mm, a large number of

positrons leave the crystal without annihilation. They carry away with them part of the kinetic energy and rest energy, which in crystals of great length is converted into the energy of annihilation quanta.

Fig. 3. Instrument spectrum of betatron radiation with \(E = 30\) MeV beyond an Al layer \(= 50\) cm. Histogram—true spectrum.

Analysis of dependences 1–4 shows that, at a quantum energy \(E_0 = 10\) MeV, the radial dimensions of the scintillation crystal are not especially critical, and for purposes of flaw detection one may limit oneself to a diameter of 40–50 mm. Of course, this is valid only in the case when the cross-sectional area of the crystal is many times greater than the cross section of the beam of gamma quanta incident on the crystal. Otherwise, the value of the radial losses, as well as of the total losses, increases significantly.

It should be noted that fulfillment of this condition in actual flaw-detector designs is achieved by strict collimation of the radiation beam at the detector entrance. Further investigations, taking into account optical losses in the crystal for the actual inspection geometry, will make it possible to give more complete recommendations for selecting the scintillation crystal in betatron flaw detection of materials and products.

REFERENCES

1. V. I. Gorbunov, V. B. Kuznetsov, V. K. Kuleshov, Yu. B. Yankelevich. The counting-spectrometric method in scintillation flaw detection (present collection).
2. R. S. Foote, H. W. Koch. Rev. Sci. Instr., 25, 746–758 (1954).
3. Nucl. Instrum. and Methods. 10, No. 1, 31–36, 1961.