Photoproduction of $\pi^-$ Mesons on Carbon $\mathrm{C}^{12}$

V. D. Epaneshnikov, V. M. Kuznetsov, O. I. Stukov

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

Introduction

The photoproduction of pions on free nucleons is currently being intensively studied and is fairly well known both theoretically and experimentally. On the other hand, experimental data on pion photoproduction on complex nuclei are insufficient. Moreover, reactions in which it has been possible to measure the integral photoproduction cross section are difficult for theoretical consideration because of the uncertainty of the excitation spectrum of the residual nucleus. Partial transitions, corresponding to individual definite levels of the residual nucleus, are more accessible to theoretical analysis. Data of this kind could serve as a serious touchstone for theory. The study of partial transitions, although the isolation of even one definite transition would require considerable improvement of the experimental technique, would make it possible to approach closely the construction of a quantitative theory of photoproduction on complex nuclei and the solution of a number of important problems [1]: clarifying the accuracy of the impulse approximation, refining information on the “elementary” amplitude of photoproduction on a nucleon, etc.

The measurement of the total probability of a partial transition in the present work is necessary for checking the theoretical prediction [2] and is the first attempt in this direction.

Brief Review of Theory and Experimental Data on the Photoproduction of Charged $\pi^-$ Mesons on Light Nuclei

An analysis of the experimental material accumulated up to the present time [3—7] qualitatively confirms the assumption, advanced as early as the 1950s, that the photoproduction process occurs on individual nucleons of the nucleus. There are a number of works [8, 9] in which the integral cross sections of several specific reactions of the type $\gamma + A \to \pi + B$ have been calculated under the assumption of volume and surface photoproduction of pions. However, when theoretical predictions are compared with experimental data, a large discrepancy between theory and experiment is observed, especially for the case of volume production; this is not surprising, since it is known that data concerning variations of total cross sections with energy and their dependence on $A$ can be used only for a qualitative test of the theory. A major difficulty in the quantitative interpretation of such data is—

there remains the uncertainty of the excitation spectrum of the residual nucleus, especially when its above-threshold states are involved.

The consideration of general questions in the theory of partial transitions in the photoproduction of charged \(\pi\)-mesons on light nuclei [2] is at present based on the following assumptions.

1. The photoproduction amplitude on a nucleus is constructed in the impulse approximation without taking into account terms depending on the momenta of the nucleon in the nucleus.

2. Absorption and scattering of mesons by the nucleus in the final state are not taken into account; for light nuclei this may be justified in a fairly wide range of \(\gamma\)-quantum energies near the photoproduction threshold, where the mean free path of the meson in nuclear matter is considerably larger than the nuclear radius [10].

The differential cross section for photoproduction on a nucleus with its transition from the ground state \(\psi_i\) to the final state \(\psi_f\) is expressed (in the c.m. system) through the transition \(T\)-matrix [2]:

\[ \frac{d\sigma}{d\mathbf{k}_\pi} = \frac{1}{(2\pi)^2 I}\, |T_{if}|^2\, \delta(E_i+\omega_\gamma-E_f-\omega_\pi). \tag{1} \]

Here \(\mathbf{k}_\pi,\ \omega_\pi\) and \(\mathbf{k}_\gamma,\ \omega_\gamma\) are the momentum and energy of the meson and of the \(\gamma\)-quantum;

\(E_i\) and \(E_f\) are the total energy of the nucleus in the initial and final states;

\(I\) is the flux density of the colliding particles:

\[ I = 1 + k_\gamma/E_i \quad (\hbar = C = 1). \]

In the impulse approximation the matrix element \(T_{if}\) is expressed through the \(t\)-matrix of photoproduction on a free nucleon and, after transformations, is written as

\[ T_{if} = 4\pi \sum_{\lambda l k \varkappa} i^l(-1)^{\varkappa} \left[ k^\lambda Y_l\left(\frac{\mathbf{q}}{q}\right) \right]^{\vec{k}}_{-\varkappa} \frac{\langle T_i M_i k\varkappa \mid T_f M_f\rangle}{(2T_f+1)^{1/2}} [\lambda l k]. \tag{2} \]

Here \(T_iM_i\) and \(T_fM_f\) are the values of the spins and their projections for the initial and final states of the nucleus;

\(\langle T_iM_i k\varkappa \mid T_fM_f\rangle\) is the Clebsch–Gordan coefficient;

\([\lambda l k]\) is the reduced matrix element of the operator [2].

The nuclear matrix elements entering formula (2) are characterized by three indices \(\lambda, k, l\), which determine the selection rules in partial transitions with respect to parity and total angular momentum, as well as to orbital and spin angular momenta: \(\Delta\pi=(-1)^l\), \(|T_i-T_f|\leq k\leq T_i+T_f\), etc. The presence in the matrix element of the operator \(\tau_\pm\) determines the selection rule with respect to isospin: \(\Delta T=0,\ \pm1\).

When the theory is applied to a real process, the reaction \(C^{12}+\gamma\to\pi^-+N^{12}\) occupies a special position. The nucleus \(N^{12}\) has only one bound state \((1^+)\) [11], so that from its \(\beta\)-decay a quite definite partial transition \(0^+\to1^+\) can be isolated. In such a transition only the nuclear matrix elements (101), (121) [2] are different from zero; therefore only the vector part of the photoproduction amplitude on the neutron enters the cross section. The nucleus \(C^{12}\) has been well studied; the intermediate-coupling model makes it possible to construct quite reliable wave functions of its states, which satisfactorily explain the usual experimental characteristics of this nucleus. This makes it possible to calculate, with a high degree of accuracy, the matrix elements of the transition \(C^{12}(0^+)\to N^{12}(1^+)\), and consequently also the total probability of formation of \(N^{12}\) in photoproduction. The total cross section of the partial transition \(0^+\to1^+\) as a function of the photon energy was calculated in [12], where, for comparison, the results of a calculation using the shell model with \(j\)–\(j\) coupling are given, as well as the magnitude of the spin-dependent part of the photoproduction cross section on a free nucleon.

The unique possibility of singling out the only partial transition in measuring the integral cross section was used by us in developing the experimental apparatus for studying the reaction \(C^{12}+\gamma\to\pi^-+N^{12}\) in the \(\gamma\)-beam of the “Sirius” synchrotron.

Measurement procedure

The difficulty of experimentally measuring integral cross sections of reactions of the type \(\gamma+A\to\pi+B\) consists in the fact that it is rarely possible to isolate the active nucleus \(B\) against the accompanying induced background activity.

The study carried out of the final product of the reaction \(C^{12}+\gamma\to\pi^-+N^{12}\), the nucleus \(N^{12}\), showed that this isotope has a very short half-life, but one differing by several orders of magnitude from the half-lives of all possible accompanying isotopes. It is precisely this feature that makes it possible to separate it from the background. Indeed, let us consider the main types of reactions when the \(\gamma\)-beam passes through the target:

\[ \begin{array}{lll} \gamma + C^{12} \to \pi^- + N^{12} & \tau = 0.011\ \text{s}.\\ & \pi^- + p + C^{11} & \tau = 1200\ \text{s}.\\ & \pi^- + p + p + B^{10} & \text{stable}\\ & \pi^+ + B^{12} & \beta^- \text{ decay } \tau = 0.027\ \text{s}\\ & \pi^+ + n + B^{11} & \text{stable} \end{array} \]

etc.

Photuclear reactions are also possible: \(C^{12}(\gamma,n)C^{11}\), \(C^{12}(\gamma,p)B^{11}\), \(C^{12}(\gamma,p,n)B^{10}\), \(C^{12}(\gamma,t)Li^{9}\), \(C^{12}(\gamma,\alpha)Be^{8}\), etc. The only source of a background that is difficult to separate may be nuclei undergoing positron decay [1,3].

\[ \begin{array}{ccc} C^{11} & C^{10} & B^{8}\\ \downarrow & \downarrow & \downarrow\\ \tau = 1200\ \text{s} & \tau = 19.1\ \text{s} & \tau = 0.7\ \text{s}\\ \beta_+ \sim 0.968\ \text{MeV} & \beta_+ \sim 2.2\ \text{MeV} & \beta_+ \sim 13.7\ \text{MeV}\\ \downarrow & \downarrow & \downarrow\\ B^{11}\sim 99.8\% & B^{10}\sim 98\% & Be^{8}\sim 100\% \end{array} \]

When considering the probability of photoproduction on \(C^{12}\) of these nuclei, it is clear that the contribution from \(C^{10}\) and \(B^{8}\) will be negligibly small, whereas \(C^{11}\), having a half-life shorter than the measurement time, will accumulate in the target and, owing to the large cross section of the reaction \(C^{12}(\gamma,n)C^{11}\), will make the decay of \(N^{12}\) nuclei indistinguishable. \(C^{11}\) can be separated only by using the difference between the decay spectra of \(N^{12}\) and \(C^{11}\).

On the basis of a study of the characteristics of \(N^{12}\) and possible sources of background, we chose an experimental method based on measuring the number of active \(N^{12}\) nuclei formed in the reaction \(C^{12}+\gamma\to\pi^-+N^{12}\). The positron emitted in the decay of \(N^{12}\) annihilated at rest into two 0.5 MeV \(\gamma\)-quanta, recorded in coincidence. To separate the background, a \(\beta\)-counter was used which recorded positrons only with energies greater than the maximum energy of the \(C^{11}\) decay spectrum. Since the half-life of \(N^{12}\) is 11 msec, the measurements were carried out directly in the beam after the instantaneous dumping of electrons onto the synchrotron target, in the intervals between radiation pulses.

For reliable identification of the process being singled out, during the measurement an amplitude analysis of the registered \(\gamma\)-quanta was performed and the decay curve of the active target nuclei was recorded.

Experimental equipment

The layout of the experiment is shown in Fig. 1. The target material was spectrally pure powdered carbon, pressed into an aluminum tube with a wall thickness of 2 mm. The dimensions of the carbon target, Ø 30 × 30 mm, were chosen with a view to obtaining a sufficiently large yield of N^12 (the calculated yield is of the order of 100 events per \(5 \cdot 10^7\) equiv. quanta) and small dimensions. Positrons from C^11 are absorbed in the aluminum tube.

Fig. 1. Layout of the experiment.

A β-counter, made on the basis of two plastic scintillators \(80 \times 80 \times 10\) mm, located on opposite sides of the target, is viewed by a single FEU-30 photomultiplier. Behind the scintillators, in front of the γ-detectors, aluminum absorbers 20 mm thick were installed, sufficient to stop the entire positron spectrum in the decay of N^12. The γ-detector, made on the basis of a NaI(Tl) crystal Ø \(100 \times 100\) mm, is viewed by an FEU-13 photomultiplier. Two γ-detectors with identical characteristics were used. The aperture of the NaI(Tl) crystal Ø \(100 \times 100\) mm subtends a large solid angle; 0.5 MeV γ-quanta lose all their energy in NaI and, consequently, are registered in the full-absorption peak. The FEU-13 retains good spectrometric characteristics under conditions of large loads. The signal to be analyzed is taken from the anode, and the timing signal from the dynode of the FEU-13. The amplitude resolutions of both detectors are of the order of 25% on the 0.5 MeV γ-quanta line from Na^22.

Measurements directly on the beam, in the intervals between beam-dump pulses, under conditions of large background loads, required the use of the fast–slow coincidence method for reliable selection of the reaction under study. For this purpose, the timing signals from the FEU-13 are amplified and fed to the fast γ–γ coincidence circuit CC1 (resolving time 30 nsec). The fast γ–γ coincidences are then brought into coincidence with pulses from β and are fed to control the linear gates LV1 and LV2, which pass pulses from the FEU-13 anodes to a 4096-channel amplitude analyzer (Tenzor). In this case one control pulse passes another pulse for analysis. The “Tenzor” operates in amplitude–timer mode. In this mode all 4096 channels are divided into several groups (in our case 16 groups of 256 channels), and registration is carried out in each group from the first to the last. The analysis time in one group is set (in our case 4.1 msec). Thus, by synchronizing the start of the “Tenzor” with the moment of electron dumping, it is possible to trace the decay of the active nuclei of the target.

Measurements and Experimental Results

A necessary and sufficient condition for reliable identification of the reaction \(C^{12}+\gamma\to \pi^-+N^{12}\) is the agreement of the period of the selected nuclei, active under electron bombardment, with the known half-life of \(N^{12}\). Figure 2 shows the decay curve of active target nuclei obtained at \(E_\gamma^{\max}=600\) MeV and with instantaneous dumping of the electrons onto the synchrotron target. The half-life obtained,

Fig. 2. Decay curve of the active target nuclei. \(E_\gamma=600\) MeV.

Fig. 3. Experimental data.

\(10.8\pm0.5\) msec, agrees satisfactorily with the expected value. Figure 3 presents the theoretical values of the total cross section of the partial transition using wave functions of the shell model with intermediate coupling (lower curve) [12]. For comparison, the results of a calculation using the shell model with \(j\)-\(j\) coupling are given.

The experimental values are in agreement with the theoretical predictions for the case of the shell model with intermediate coupling.

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