Estimation of the Degree of Signal-to-Noise Ratio Degradation by Single-Crystal Screens
Abstract Generated abstract
This paper evaluates how several 0.5 cm single-crystal scintillation screens degrade signal-to-noise ratio in radiation introscopes, supplementing earlier comparisons based on energy efficiency alone. The authors define a signal-to-noise degradation measure from the mean absorbed energy and its variance, then estimate these quantities to second-collision accuracy while including annihilation quantum contributions. Calculations over 0.01 to 30 MeV indicate that cadmium tungstate and calcium tungstate screens provide the best signal-to-noise performance, whereas zinc sulfide and cadmium sulfide, despite their high energy efficiency, perform worst by this criterion. The results suggest that low optical loss introscopes should favor tungstate screens to improve contrast sensitivity, while the estimated total error is generally below 10 percent except at high energies, where electron energy escape can make it much larger.
Full Text
Vol. 248
1975
ESTIMATION OF THE DEGREE OF DEGRADATION OF THE SIGNAL-TO-NOISE RATIO BY SINGLE-CRYSTAL SCREENS
A. P. Shpagin, G. A. Kucher
(Submitted by the scientific seminar of the Research Institute of Electron Introscopy)
Results are presented for estimating the degradation of the signal-to-noise ratio by single-crystal screens 0.5 mm thick.
In [1] we calculated the energy efficiencies of several single-crystal screens in the energy range \(10\ \text{keV} \div 10\ \text{MeV}\).
Figure 1 gives the values of the energy efficiency \(\varphi_E(E)\) for various single-crystal screens. It is seen from the figure that, in terms of energy efficiency, the best screen is zinc sulfide ZnS; it is followed by cadmium sulfide CdS and cadmium tungstate CdWO\(_4\) screens.
Fig. 1. Energy efficiencies of single-crystal screens 0.5 cm thick: 1—CdS, 2—ZnS, 3—NaJ(Tl), 4—CsJ(Tl), 5—CdWO\(_4\), 6—CaWO\(_4\).
Energy efficiency is far from always decisive in evaluating the quality of screens. Sometimes screens should be compared according to the degree of degradation of the signal-to-noise ratio.
Thus, for example, in screens of practical importance, when one gamma quantum with an energy greater than a kiloelectron-volt is absorbed, a large number of photons is emitted; therefore the signal-to-noise ratio at the output of the introscope, with sufficiently efficient optics, will be determined only by the signal-to-noise ratio at the output of the screen. The same effect can be obtained by increasing the radiation energy even with ordinary optics. Therefore in such cases screens must be chosen on the basis that they should, as far as possible, change the signal-to-noise ratio as little as possible.
The change in the signal-to-noise ratio in a screen can be characterized by the formula
\[ \varphi_{\mathrm{э}}(E)=\frac{\Pi(E)}{\sqrt{\Pi_{\Delta}(E)}} , \]
where \(\Pi(E)\) is the mean energy absorbed in the converter under normal incidence on its surface of a radiation quantum with energy \(E\), and
\(\Pi_{\Delta}(E)\) is the variance of this energy.
An exact calculation of the functions \(\Pi(E)\) and \(\Pi_{\Delta}(E)\) is not at present possible, but with accuracy sufficient for practical application these functions can be computed approximately, taking into account that the screens used have a small thickness. In the present paper the functions \(\Pi(E)\) and \(\Pi_{\Delta}(E)\) were calculated to second-collision accuracy, taking into account the contribution of annihilation quanta, by the formulas [2]:
\[ \Pi(E)=\Pi_1(E)+\Pi_2(E)+\Pi_a(E) \]
\[ \Pi_{\Delta}(E)=\Pi_{1\Delta}(E)+\Pi_{2\Delta}(E)+\Pi_{a\Delta}(E), \]
where
\[ \Pi_1(E)=\frac{1}{\mu(E)}(\tau E+\sigma E_{\sigma}+\chi E_{\chi}) \{1-\exp[-\mu(E)x_0]\}= \]
\[ = T(E)\{1-\exp[-\mu(E)x_0]\} \]
is the function \(\Pi(E)\) to first-collision accuracy; here
\(T(E)\) is the mean energy transferred to an electron in one collision;
\(\tau,\chi,\sigma\) are the linear attenuation coefficients due to the photoelectric effect, pair production, and the Compton effect, respectively;
\(\mu\) is the total linear coefficient;
\(E_{\chi}=E-2mc^2\) is the energy left by quanta absorbed in the converter through pair production;
\(mc^2\) is the rest energy of the electron;
\(x_0\) is the screen thickness.
\[ E_{\sigma}=\frac{mc^2\pi r_0^2}{\sigma_e} \left[ \left(1-\frac{2E_{\gamma}+3}{E_{\gamma}^2}\right)\ln(2E_{\gamma}+1) +\frac{6}{E_{\gamma}}-\frac{5}{6} \right. \]
\[ \left. -\frac{1}{2E_{\gamma}+1} -\frac{1}{2(2E_{\gamma}+1)^2} +\frac{1}{3(2E_{\gamma}+1)^3} \right], \]
\(E_{\sigma}\) is the energy transferred to the recoil electron, on average, in one Compton collision;
\(\sigma_e\) is the cross section for Compton scattering on one electron;
\[ E_{\gamma}=\frac{E}{mc^2} \]
is the quantum energy in units of \(mc^2\).
Similarly,
\[ \Pi_{1\Delta}(E)=\frac{1}{\mu(E)}(\tau E^2+\chi E_{\chi}^2+\sigma f_e) \{1-\exp[-\mu(E)x_0]\}= \]
\[ = T_{\Delta}(E)\{1-\exp[-\mu(E)x_0]\}, \]
where
\[ f_e=\frac{(mc^2)^2\pi r_0^2}{\sigma_e} \left\{ \left[E_\gamma-\frac{2(2+E_\gamma)}{E_\gamma}\right]\ln(2E_\gamma+1) -\frac{17}{12}E_\gamma+6.6(6)+ \frac{1.16(6)}{1+2E_\gamma} +\frac{0.5}{(1+2E_\gamma)^2} -\frac{11}{24(1+2E_\gamma)^3} +\frac{1}{8(1+2E_\gamma)^4} \right\} \]
is the variance of the energy transferred to the recoil electron in Compton scattering of quanta with energy \(E\).
The energy left by quanta after the second collision was calculated from the formula
\[ \Pi_2(E)= \int_{E_r}^{E} \frac{d\sigma}{d\varepsilon}\,T(\Sigma)\{1-\exp[-\mu(E)x_0]\}\,d\varepsilon+ \]
\[ +\int_{E''}^{E} \frac{d\sigma}{d\varepsilon}\,T(\Sigma)\, \frac{1}{\dfrac{\mu(\varepsilon)}{\cos\Theta}-\mu(E)} \left\{ \exp\left[\frac{-\mu(\varepsilon)x_0}{\cos\Theta}\right] -\exp[-\mu(E)x_0] \right\}\,d\varepsilon+ \]
\[ +\int_{E_r}^{E''} \frac{d\sigma}{d\varepsilon}\,T(E)\, \frac{ 1-\exp\left\{x_0\left[\mu(E)-\dfrac{\mu(\varepsilon)}{\cos\Theta}\right]\right\} }{ \mu(\varepsilon)/\cos\Theta-\mu(E) }\,d\varepsilon, \]
where
\[ \varepsilon \]
is the energy of a quantum that has undergone one collision; \(E''\) and \(E_r\) are the energies of quanta scattered, respectively, through \(90^\circ\) and \(180^\circ\):
\[ E''=\frac{mc^2E}{mc^2+E},\qquad E_r=\frac{mc^2E}{mc^2+2E}. \]
The contribution of annihilation quanta was taken into account by the formula
\[ \Pi_a(E)= \frac{2\varkappa(E)T^+}{\mu(E)} \{1-\exp[-\mu(E)x_0]\}- \]
\[ -\varkappa(E)mc^2T^+ \left\{ \int_{E''}^{E} \frac{ \exp(-\mu^+x_0/\cos\Theta)-\exp[-\mu(E)x_0] }{ \varepsilon^2[\mu^+/\cos\Theta-\mu(E)] }\,d\varepsilon+ \right. \]
\[ \left. +\int_{E_r}^{E''} \frac{ 1-\exp\{x_0[\mu(E)-\mu^+/\cos\Theta]\} }{ \varepsilon^2[\mu^+/\cos\Theta-\mu(E)] }\,d\varepsilon \right\}, \]
where
\[ T^+=T(0.511),\qquad \mu^+=\mu(0.511). \]
The formulas that take into account the contribution of the second collision and of annihilation quanta for \(\Pi_{\Delta}(E)\) are analogous to \(\Pi_2(E)\) and \(\Pi_a(E)\), except that \(T(E)\) must be replaced by \(T_{\Delta}(E)\) and \(T_{\Delta}^{+}(0.511)\).
Figure 2 gives the functions \(\varphi_\phi(E)\) for all the screens studied. As can be seen from Fig. 2, the best screen from the standpoint of obtaining the maximum signal-to-noise ratio over the entire energy range studied (from \(0.01\) MeV to \(30\) MeV) is a cadmium tungstate screen, \(\mathrm{CdWO_4}\), and the second best is calcium tungstate, \(\mathrm{CaWO_4}\).
Screens that are best in energy efficiency, ZnS and CdS, turn out, on the contrary, to be the worst with respect to the signal-to-noise ratio. Third with respect to signal-to-noise ratio will be CsI(Tl), while NaI(Tl) in the low-energy region coincides with CdS; in the medium-energy region sodium iodide is the worst of all with respect to signal-to-noise ratio, and at energies above 10 MeV it rises slightly above ZnS.
Fig. 2. Change in the signal-to-noise ratio for single-crystal screens 0.5 cm thick: 1—CdS, 2—ZnS, 3—NaI(Tl), 4—CsI(Tl), 5—CdWO₄, 6—CaWO₄
Since the screens that are best in energy efficiency are the worst with respect to the signal-to-noise ratio, one may conclude that in introscopes with small optical losses, i.e., when the number of light quanta reaching the photocathode upon absorption of one radiation quantum is much greater than unity, for example when using a betatron as the radiation source or when using fiber optics, in order to increase contrast sensitivity it is necessary to employ precisely screens of the CaWO₄, CdWO₄ type, which will make it possible to appreciably increase the sensitivity of introscopes.
The method used has the following errors: 1) computational error; 2) error due to inaccurate knowledge of interaction cross sections; 3) error in calculating the functions \(P(E)\), \(P_{\Delta}(E)\), due to the fact that the third and all subsequent collisions were not taken into account in the calculation; 4) error in calculating \(P(E)\), \(P_{\Delta}(E)\), due to the fact that the energy carried out of the screen by fast electrons was not taken into account.
The total error for various screens does not exceed 10%, but it increases greatly at high energies because of energy carried away by electrons; thus, for example, at an energy equal to 10 MeV, the total error reaches 50%.
REFERENCES
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A. P. Shpagin, G. A. Kucher. Comparison of the efficiency of single-crystal screens. Defektoskopiya, No. 1, 112, 1971.
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A. P. Shpagin, G. A. Kucher, V. A. Malyshev. Estimation of the efficiency of a single-crystal screen. Izv. TPI, vol. 156, 1968.