DETERMINATION OF THE EFFICIENCY OF A NEUTRON DETECTOR BASED ON A PLASTIC SCINTILLATOR Ø100×200 mm

V. M. KUZNETSOV, A. P. POTYLITSYN, E. V. REPENKO, O. I. STUKOV

(Presented by the Scientific and Technical Seminar of the High-Energy Laboratory of the Nuclear Physics Research Institute)

To determine the efficiency of the neutron detector, two methods were used: a theoretical calculation and a direct estimate from measurements on recoil neutrons of the photoproduction process of \(\pi^+\)-mesons.

For calculating the efficiency of a neutron counter made on the basis of a large-size plastic scintillator, the method proposed by R. Kurz [1] was used. Since the neutron is uncharged, the detector efficiency for neutrons is determined by charged particles arising from their interaction with the nuclei of the scintillator, that is, with hydrogen and carbon nuclei. For energies below 15 MeV (Table 1), the interaction is due only to \((np)\)-scattering reactions, whose cross sections are well known, and therefore the efficiency can be calculated with good accuracy. For neutron energies above 15 MeV, calculation of the efficiency is complicated by the increasing contribution of inelastic \((nC)\)-interactions. The neutron registration efficiency due to all possible reaction channels is determined, as shown in [1], in the following way:

\[ \varepsilon(E_n,T)=\sum_{i=1}^{n} g_i(E_n,T)\frac{\sigma_i(E_n)}{\sigma(E_n)} W(E_n,\bar{l}), \tag{1} \]

where

\[ W(E_n,\bar{l})=\sigma\gamma\int_{0}^{\bar{l}} e^{-\gamma\sigma(E_n)z}\,dz \]

is the total probability of interaction of a neutron with the scintillator material;

\[ g_i(E_n,T) \]

is the probability of registering charged particles produced as a result of neutron interaction in the \(i\)-th reaction channel;

\[ \sigma_i(E_n) \]

is the cross section of the reaction in the \(i\)-th channel;

\[ \gamma \]

is the number of hydrogen (or carbon) nuclei per cm\(^3\);

\[ \bar{l} \]

is the effective length of the crystal, averaged over the directions of the neutrons incident on the detector;

\[ z \]

is the neutron free-path length in the crystal;

\[ \sigma(E_n)=\sigma_{np}(E_n)+\sigma_{nC}(E_n) \]

is the sum of the cross sections of \((np)\)- and \((nC)\)-interactions.

It is seen from formula (1) that the efficiency depends on the threshold. In practice, calculations use the detector sensitivity threshold for electrons. This is the minimum electron energy starting from which the neutron detector registers an electron with probability equal to unity. In this case, protons and α-particles (products of interactions of neutrons with scintillator nuclei) that produce in the scintillator the same light output as electrons with energy \(T\) will also have minimum energies \(E_p(T)\) and \(E_\alpha(T)\). These energies were determined from the data of [2]. Since the light output and sensitivity of the photomultiplier are random quantities, depending on the design of the counter and on a number of random causes, the probability of registration at threshold energy \(T\) depended on the energy resolution of the counter, which was \(\tau \approx 50\%\). In this connection, the mean value of the efficiency takes the form

\[ \varepsilon(E_n)=W(E_n\bar l)\sum_{i=1}^{n}\sigma_i(E_n) \frac{ \displaystyle \int_{0}^{\infty} g_i(T,E_n)\exp\left\{-\frac{1}{2}\left(\frac{T-T_0}{\tau T_0}\right)^2\right\}\,dT }{ \displaystyle \int_{0}^{\infty}\exp\left\{-\frac{1}{2}\left(\frac{T-T_0}{\tau T_0}\right)^2\right\}\,dT }, \tag{2} \]

where \(T_0\) is the mean threshold energy, determined experimentally, which in our case was equal to 1.3 MeV for electrons.

The effective length \(\bar l\) was calculated from the geometry of the crystal and the direction of the neutrons being registered, as shown in Fig. 1, according to the formula

\[ \bar l=\frac{1}{\cos\theta_2} \left\{ \int_{\theta_1}^{\theta_2} \left(\frac{R-S\tg\theta}{\cos\theta}\right)d\theta + \int_{0}^{\theta_1}\frac{L}{\cos\theta}\,d\theta \right\}, \tag{3} \]

\[ \theta_1=\operatorname{arc\,tg}\frac{R}{L+S},\qquad \theta_2=\operatorname{arc\,tg}\frac{R}{S}. \]

To determine \(g_i(E_nT)\), entering into expressions (1) and (2), the reactions making a noticeable contribution [1] were included in the calculation. These reactions are given in Table 1.

Table 1

Reactions of the type \(\mathrm{C}(n,n')\mathrm{C}\) and \(\mathrm{C}(n,n'\gamma)\mathrm{C}\) were not taken into account in the calculation, since they made a contribution to the efficiency of less than 2% [1].

For the first reaction channel \(\mathrm{H}(n,p)\mathrm{H}\),

\[ g_1(E_n,T)=2\pi\int_{-1}^{x'_{\max}}\frac{d\sigma_{np}}{dx'}\,dx', \tag{4} \]

where

\[ \frac{d\sigma}{dx'} \]

is the differential cross section of \((n,p)\) scattering [3];

\(x'=\cos\Theta'\) is the cosine of the angle of the scattered neutron;

\(\Theta'_{\max}\) is the limiting neutron scattering angle at which the recoil proton has the threshold energy \(E_p(T)\) in the c.m. system.

\[ x'_{\max}=\{\gamma(E'_p+m_p)-[E_p(T)+m_p]\}/\gamma P'_p, \tag{5} \]

Fig. 1. Geometry of the crystal and directions of the detected neutrons.

where \(E'_p\) is the kinetic energy of the recoil proton;

\[ \beta=\sqrt{\frac{E_n}{2m+E_n}};\quad \gamma=\sqrt{\frac{2m+E_n}{2m}}; \]

\[ p'_p=\sqrt{\frac{2mE_n}{4}};\quad E'_p=\sqrt{\frac{2m(2m+E_n)}{2}}; \]

\(E_n\) and \(E'_n\) are the kinetic energy of the neutron before (after) scattering;
\(P'_p\) is the momentum of the recoil proton;
\(m=m_n\approx m_p\) is the mass of the proton and neutron for the second reaction channel.

For the second reaction channel \(C^{12}(n,\alpha)B^9_e\)

\[ g_2(E,T)=\int_{E_{\alpha\min}}^{E_{\alpha\max}}\frac{dE_\alpha}{2\beta\gamma p'} =\frac{E_{\alpha\max}-E_{\alpha\min}}{2\beta\gamma p'}, \tag{6} \]

where

\[ E_{\alpha\min}=E_\alpha(T), \]

\[ E_{\alpha\max}=\gamma E_\alpha+\beta\gamma p' + m_\alpha(\gamma-1). \]

For the third reaction channel \(C(n,n'3\alpha)\)

\[ g_3(E,T)=\int_0^1 f_{3\alpha}(E_n,T,\delta_n)N(\delta_n)d\delta_n, \tag{7} \]

where

\[ f_{3\alpha}(E_n,T,\delta_n)= 3\left[ 1-\frac{2\left(\frac{T}{Q}-0.046\right)}{0.007Q} \right] \]

is the fraction of detected neutrons due to the reaction \((n,n',3\alpha)\) for a fixed energy \(Q\); \(\delta_n=\dfrac{E'_n}{E'_{n\max}}\);

\(Q\) is the energy transferred to the compound nucleus.

In view of the fact that the excited nucleus \(B^{12}\) has a high density of low-lying levels, the spectral form given in [4] was used to calculate the contribution of this reaction, and the efficiency for the reaction \(C^{12}(n,p)B^{12}\) was determined from

\[ g_4(E_n,T)=\int_0^1 f_p(E,T,\delta_p)d\delta_p, \tag{8} \]

where

\[ f_p(E,T,\delta_p)=2\pi\int_{x'_{\min}}^{1}\frac{\partial^2\sigma_{np}}{\partial\delta\,\partial\Omega}\,dx' \]

is the fraction of neutrons registered through the reaction \(C^{12}(n,p)B^{12}\) at a fixed value of the parameter \(\delta_{p0}\);

\[ x'_{\min}=\cos\Theta_p \]

is the cosine of the angle at which the protons in the final state have an energy corresponding to the threshold energy \(E'_p\);

\[ \delta_p=\frac{E'_p}{E'_{p\max}} \]

is the parameter specifying the ratio of the threshold energy \(E'_p\) to the maximum possible energy for the given reaction;

\[ \frac{\partial^2\sigma}{\partial\delta\,\partial\Omega} \]

is the double differential cross section of the reaction \(C^{12}(n,p)B^{12}\), determined from [4].

The results of the calculated efficiencies for each reaction channel are shown in Fig. 2 as curves, each of which is labeled according to the type of reaction. The total efficiency of the neutron detector was determined by summing the efficiencies for the individual channels and is also shown in Fig. 2.

Fig. 2. Experimental and calculated values of the efficiencies of the neutron detector as a function of the neutron kinetic energy; I — experimental values; solid curve — calculated total efficiency; the remaining curves — calculated efficiency values for individual reaction channels.

The experimental estimate of the efficiency of the neutron detector was carried out in an experiment on the photoproduction of \(\pi^+\)-mesons on recoil neutrons, the energy and emission angle of which were specified by detecting the \(\pi^+\)-meson. The experimental setup and apparatus were the same as those used in the asymmetry measurement [5]. The bremsstrahlung beam was produced on a tantalum target \(0.4\) mm thick by electrons with an energy of \(600\) MeV. The \((\pi^+ n)\)-coincidences were measured at angles \(\Theta_\pi=90^\circ\) in the laboratory system and \(\Theta_n=30^\circ\) in the laboratory system at low photon energies in the range from \(210\) to \(240\) MeV.

Table 2

Table 3

Results of the theoretical calculation of the efficiency

The efficiency was calculated on the basis of comparing the results obtained with the data of Bonn [6] according to the formula

\[ \frac{d\sigma}{d\Omega_{\text{Bonn}}}=\frac{Y}{\varepsilon}, \tag{9} \]

where \(Y\) is the measured quantity.

Since two recording channels were used in this experiment, the calculation of \(\Delta\Omega\) was complex. The essence of the method of analytical calculation was that the effective recording surface of one of the channels (in this case, the neutron counter), or the solid angle of one of the recording channels, taking into account the reaction kinematics and the restrictions imposed in selecting events, was mapped into the other channel (the pion channel), cutting out in the solid angle of this channel the required solid angle of the two recorded channels, as shown in Fig. 3. In this case the required solid angle takes the form

\[ \Delta\Omega= \frac{ \displaystyle \int_{E_\gamma-\Delta E_\gamma}^{E_\gamma+\Delta E_\gamma} dE_\gamma \int_{V_M} dV \int_S \frac{\cos\delta(\rho_M)}{r_\pi^2(\rho_M)} \,dS(\theta_\pi,\theta_n,\rho_M,E_\gamma) }{ \Delta E_\gamma \Delta V_M }, \tag{10} \]

where

\(V_M,\ \rho_M,\ r_\pi,\ \delta(\rho_M),\ S,\ S_\pi,\ \theta_\pi\) and \(\theta_n\) are shown in Fig. 3.

Calculation of \(\Delta\Omega\) in this form is a typical problem for the Monte Carlo method. In the special case when \(\Theta_\pi=90^\circ\) in the c.m. system, this quantity was substantially simplified. The results of the direct estimate and of the analytical calculation showed good agreement within a 3% error. In calculating the efficiency, the value of the solid angle determined experimentally was used. The results are presented in Fig. 2 and in Tables 2 and 3.

Fig. 3. Experimental geometry.

REFERENCES

1. R. L. Kurz. Technical Report UCRL—11339, University of California Radiation Laboratory, Berkeley, 1964.

2. T. I. Goading. Nuclear Instrument and Methods, 7, 189 (1960).

3. P. Wilson. Nucleon-Nucleon Interactions. Moscow, “Mir,” 1965.

4. G. I. Frye, Z. Posen. Physical Review, 99, 1375, (1955).

5. V. M. Kuznetsov, O. I. Stukov, E. V. Reshenko, V. D. Eponeshnikov, V. N. Zabaev, A. P. Potylitsyn. Yadernaya fizika, 13, 1049 (1971).

6. D. Freytag, W. I. Schwille, R. I. Wedemeyer. Zeitschrift fur Physik, 185, 1 (1965).