Measurement of differential cross sections for π+-meson production on hydrogen
Abstract Generated abstract
Differential cross sections for single photoproduction of positive pions on hydrogen were measured at a laboratory pion angle of 90 degrees using a 630 MeV bremsstrahlung photon beam and a polyethylene target. Hydrogen yields were obtained by subtracting carbon target measurements from polyethylene data, while pion identification and energy determination used a scintillation counter telescope, veto and threshold Cherenkov counters, and spark chambers with copper absorbers. The analysis applies corrections for telescope efficiency, pion nuclear absorption, decay in flight, multiple Coulomb scattering, and possible background contributions. The resulting cross sections are presented in comparison with Bonn measurements, with relatively large uncertainties attributed to the conventional difference method.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE,
ORDER OF THE OCTOBER REVOLUTION AND ORDER
OF THE RED BANNER OF LABOR,
named after S. M. Kirov
Volume 273 1975
MEASUREMENT OF DIFFERENTIAL CROSS SECTIONS FOR THE PRODUCTION OF \(\pi^+\)-MESONS ON HYDROGEN
V. M. KUZNETSOV, E. V. REPENKO, O. I. STUKOV
(Submitted by the Scientific and Technical Seminar of the High-Energy Laboratory, Research Institute of Nuclear Physics)
Differential cross sections for single photoproduction of \(\pi^+\)-mesons on hydrogen were measured using a polyethylene target. The measurements were carried out on a bremsstrahlung beam from a tantalum target \(0.4\) mm thick \((0.094\) radiation lengths). Electrons with an energy of \(630 \pm 20\) MeV were dumped onto the target for \(20\text{--}40\) ms at a frequency of \(1\) Hz. The bremsstrahlung beam was formed by two lead collimators \(25\) cm long, was cleaned by a magnetic field, struck the target, and was monitored by a quantameter. The diameter of the \(\gamma\)-beam on the cylindrical target was \(25\) mm.
A charged-particle telescope, surrounded by \(10\) cm thick lead shielding, was positioned at an angle of \(90^\circ\) to the direction of the photon beam.
The energy of the bremsstrahlung radiation was measured with a Wilson quantameter calibrated at the Kharkov 2 GeV linear accelerator. The quantameter constant for bremsstrahlung photons with maximum energy \(630\) MeV was \((3.86 \pm 0.077)\cdot 10^{-18}\) C/MeV. The measurements were carried out alternately with polyethylene and carbon targets. The meson yield from hydrogen was determined by the usual difference method [1], from the difference between the yields \(N_{\mathrm{CH}_2}\) from the polyethylene target and \(N_{\mathrm{C}}\), according to the formula \(N_{\mathrm{H}_2}=N_{\mathrm{CH}_2}-\xi N_{\mathrm{C}}\), where the coefficient \(\xi=0.58\) is the relative weight of the carbon-nucleus content in the polyethylene and carbon targets.
Fig. 1. Schematic of the recording setup: \(S_1, S_2, S_3, S_4\) — scintillation counters; \(\check{C}\) — Cherenkov counter; \(\mathrm{IK}_1, \mathrm{IK}_2\) — spark chambers; Cu — copper absorber.
The schematic of the recording setup is shown in Fig. 1. Photons with endpoint energy \(E_{\max}=630\) MeV passed through a target \(\varnothing\,50\times80\) mm and were monitored by a Wilson quantameter. Registration of \(\pi^+\)-mesons
Fig. 2. Curve of delayed triple coincidences.
Fig. 3. Curves of delayed anticoincidences obtained with the electron beam.
was carried out using a charged-particle telescope and two spark chambers, IK\(_1\) (with a wide gap of 100 mm, for determining the emission angle of the \(\pi^+\)-meson) and IK\(_2\) (a multigap chamber for determining the energy of the \(\pi^+\)-meson from its stopping in copper plates 2 mm thick). The solid angle \(\Delta\Omega\) was determined by the dimensions of the working region of the target and by the dimensions of counter \(S_1\). The flux of charged particles was selected by fast triple coincidences \((S_1,S_2,S_3)\). To reject the electron background, counter \(S_4\) and a threshold Cherenkov counter \(\check{C}\) [2] were provided. The proton background from hydrogen in this arrangement was considered negligible.
The curve of delayed triple coincidences \((S_1S_2S_3)\) is shown in Fig. 2. The voltages on the counters at a discriminator threshold of 0.2 V are given in Table 1.
Table 1
| Counters | \(S_1\) | \(S_2\) | \(S_3\) | \(S_4\) | \(\check{C}\) |
|---|---|---|---|---|---|
| \(U\) (V) | 2100 | 2080 | 2060 | 1700 | 2800 |
The signals from the veto circuit (Fig. 3), carrying information about the particle that stopped in IK\(_2\), were branched: to the PP-12 counting device, to trigger the spark chambers IK\(_1\) and IK\(_2\), and to trigger the recording camera RFK.
When setting up the telescope, particular attention was paid to its position in the space of the spark chamber IK\(_1\), from whose tracks the angles of emission of particles from the target were determined. IK\(_1\) was set to the direction of emission of \(\pi^+\)-mesons at an angle of \(90^\circ\) in the laboratory system with an accuracy of \(\pm 20'\). To tie the track to space, luminous fiducial marks were installed. The calculation of the differential cross sections was carried out according to the formula [3]:
\[ \frac{d\sigma}{d\Omega}(E_\gamma,\theta_\pi) = \frac{N_{\mathrm{H}_2}^{\circ}(E_\gamma,\theta_\pi)} {\Delta\Omega\;\dfrac{2N_A}{A_{\mathrm{CH}_2}}\;\rho_{\mathrm{CH}_2}\;\overline{l}\;\Delta N_{E_\gamma}(\Delta E_\gamma,E_\gamma)} , \tag{1} \]
where \(N_{\mathrm{H}_2}^{\circ}\) is the true counting rate of \(\pi^+\)-mesons from hydrogen at \(\Theta_\pi \pm \Delta\Theta_\pi\) and mean energy \(E_\gamma \pm \Delta E_\gamma\);
\[ \begin{aligned} \Delta\Omega &\text{ — solid angle;}\\ N_A &\text{ — Avogadro’s number;}\\ A_{\mathrm{CH}_2} &\text{ — atomic weight of polyethylene;}\\ \rho_{\mathrm{CH}_2} &\text{ — density of polyethylene;}\\ \overline{l} &\text{ — averaged target length;}\\ \Delta N_{E_\gamma}(\Delta E_\gamma,E_\gamma) &\text{ — number of photons with energy } E_\gamma \text{ in the interval } \Delta E_\gamma. \end{aligned} \]
The true counting rate is related to the measured counting rate by the following relation:
\[ N^{\circ}(R_i) = \frac{N(R_i)} {f_1 P_a(x_{i-1},R_i)P_s(x_{i-1},R_i)P_d(R_i)} - \frac{ f_2 \displaystyle\sum_{k=i+1}^{m} N(R_k)\,[P_a(x_{i-1},R_k)-P_a(x_i,R_k)]P_s(x_{i-1},R_k) } {f_1 P_a(x_{i-1},R_i)P_s(x_{i-1},R_i)P_d(R_i)} P_d(R_k)P_c(R_k-x_{i-1}); \tag{2} \]
where \(N(R_i)\) is the measured counting rate for the \(i\)-th plate of the spark chamber IK\(_2\);
\[ \begin{aligned} f_1 &\text{ — efficiency of the counter telescope;}\\ f_2 &\text{ — coefficient that takes into account how many times nuclear absorption in the plastic scintillator is smaller than in an equivalent amount of copper;}\\ R_i(R_k) &\text{ — mean range of a pion in the } i\text{-th }(k\text{-th}) \text{ plate }(\mathrm{g/cm^2});\\ x_i &\text{ — range of a pion in the } i\text{-th plate of IK}_2\;(\mathrm{g/cm^2});\\ P_a(x_{i-1},R_i) &\text{ — correction for nuclear absorption for a pion with mean range } R_i \text{ and absorber } x_{i-1};\\ P_d(R_i) &\text{ — correction for decay of a } \pi^+\text{-meson in flight, with range } R_i;\\ P_s(x_{i-1},R_i) &\text{ — correction for multiple Coulomb scattering for a pion with range } R_i \text{ and scatterer } x_{i-1};\\ P_c(R_k-x_{i-1}) &\text{ — correction for the probability that } \pi^+\text{-mesons with range }(R_k-x_{i-1}) \text{ can be registered by the Cherenkov threshold counter.} \end{aligned} \]
The first term in (2) is the counting rate taking into account the efficiency of the charged-particle telescope and corrections that account for the interaction of the \(\pi^+\)-meson with the medium. The second term takes into account the additional counting rate due to mesons with range \(R' \gg R_i\), but having a probability of stopping in the \(i\)-th plate of chamber IK\(_2\) due to nuclear absorption, i.e., of imitating \(\pi^+\)-mesons with range \(R_i\).
Let us consider the corrections to the measured counting rate: 1) the correction for nuclear absorption was calculated by the formula [3]:
\[ P_a(x,R)=\exp\left[-\frac{\rho N_A}{A}\int_0^x \sigma(R-\xi)\,d\xi\right], \tag{3} \]
where \(\rho=8.9\ \text{g/cm}^3\) is the density of copper;
\(N_A=6.02\cdot 10^{23}\) is Avogadro’s number;
\(A=63.54\) is the atomic weight of copper;
\(\sigma(R)\) are the absorption cross sections of \(\pi^+\)-mesons in copper as a function of thickness \(R\) (cm), taken from [4];
\(x\) (cm) is the final thickness of the absorber;
\(\xi\) is a variable.
2) The correction for \(\pi^+\)-meson decay in flight was calculated from the well-known formula for radioactive decay:
\[ P_d=e^{-\frac{0.693}{T^{1/2}}t}, \tag{4} \]
where \(T^{1/2}\) is the half-life of the \(\pi^+\)-meson, \((2.55\pm0.03)\cdot10^{-8}\ \text{s}\);
\(t\) is the flight time of the \(\pi^+\)-meson.
Taking into account the specific conditions of the passage of the \(\pi^+\)-meson through the target, counters, and absorber, formula (4) takes the form
\[ P_d=\exp\left[-\frac{0.693}{T^{1/2}}\left(\frac{l_1}{\beta_1 c}+\frac{l_2}{\beta_2 c}+\ldots+\frac{l_n}{\beta_n c}\right)\right], \tag{5} \]
where \(l_r\) is the distance from the target to the first counter, from the first counter to the second, etc., from the \(n-1\)-th to the \(n\)-th absorber,
\(\beta_r\) is the velocity of the \(\pi^+\)-meson in units of the speed of light for the \(r\)-th absorber.
3) The correction for Coulomb scattering, i.e., the probability of pions passing through the telescope of counters, is a function [5]:
\[ P_s=f(r_0',\rho_0'). \tag{6} \]
Here \(\rho_0'=\rho_0/R\);
\(\rho_0\) is the radius of the beam of \(\pi^+\)-mesons incident on \(S_3\);
\(R\) is the radius of counter \(S_3\).
The quantity \(r_0'\) is determined from the following expression:
\[ r_0'=\frac{r_0}{R}=\frac{\theta_s\left(l^2+l_x+\frac{x^2}{3}\right)^{1/2}}{R}, \tag{7} \]
where \(r_0\) is the rms scattering length in the plane perpendicular to the initial direction of the \(\pi^+\)-meson beam;
\[ \theta_s=\frac{E_s}{p\beta c}\sqrt{\frac{x}{x_0}}; \tag{8} \]
\(E_s=21\ \text{MeV}\) (see [6]);
\(\beta\) and \(p\) are the velocity and momentum of the \(\pi^+\)-meson;
\(x\) is the thickness of the scatterer;
\(x_0\) is the radiation length of the scatterer;
\(l\) is the distance between the scatterer and the recording counter.
A calculation of the quantity \(P_s\) is given in the works of Sternheimer [6]; Grushin and Leikin [7] considered the calculation of the correction \(P_s\) for large thickness \(x\). However, our case differed from those considered in [6, 7] in that the scatterers were located at different distances and had lar-
of thickness \(x\). In this connection, the correction for Coulomb scattering was calculated as follows:
\[ P_s=\prod_{\lambda=1}^{n} P_s^\lambda , \tag{9} \]
where \(P_s^\lambda\) is the correction for multiple scattering for a scatterer of small thickness.
All scatterers were divided into \(n\) sufficiently thin scatterers, for which the value \(P_s^\lambda\) was also calculated. Such a method is inconvenient for ordinary calculations, but it is simple when an electronic computer is used.
1) The correction for registration of a \(\pi^+\)-meson by the Cherenkov counter is
\[ P_c^{\vee}=1-\varepsilon_c^{\vee}, \tag{10} \]
where \(\varepsilon_c^{\vee}\) is the efficiency of the threshold Cherenkov counter for \(\pi^+\)-mesons.
\(P_c^{\vee}\) was taken equal to unity, since \(\varepsilon_c^{\vee}=0\) for 80 MeV [2].
2) The correction for the additional count from \(\mu\)-mesons produced as a result of the decay of \(\pi^+\)-mesons was automatically taken into account in selecting events in the spark chambers.
The results of the measured differential cross sections, in comparison with the data of Bonn [3], are shown in Fig. 4. The large errors in the cross sections are associated with the use of the ordinary difference-method technique.
Fig. 4. Results of measurements of differential cross sections for photoproduction of \(\pi^+\)-mesons on hydrogen.
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