Methodology for Analyzing an Experiment on Photoproduction of Neutral Mesons on Complex Nuclei
Abstract Generated abstract
This paper describes a computational and statistical methodology for processing experiments on neutral meson photoproduction in complex nuclei, where two decay gamma quanta are detected by Cherenkov total absorption spectrometers. It formulates the relation between measured yields and differential cross sections, then develops a Monte Carlo model of the experimental geometry, bremsstrahlung spectrum, nuclear Fermi motion, detector resolution, and registration efficiency. The authors also present procedures for defining admissible event regions in the two gamma energy plane, rejecting repeats and background, estimating meson spectra with maximum likelihood methods, and testing meson identification using truncated normal distributions. The resulting workflow provides estimates of reaction yield, efficiency corrections, and differential cross sections, and was applied to processing a photoproduction experiment.
Full Text
METHODology FOR PROCESSING AN EXPERIMENT ON THE PHOTOPRODUCTION OF NEUTRAL MESONS ON COMPLEX NUCLEI
S. M. BERSENEV, T. N. DUDKIN, V. N. EPONESHNIKOV, Yu. F. KRECHETOV,
N. I. SABLIN, V. A. TRYASUCHEV
(Presented by the scientific and technical seminar of the High-Energy Laboratory of the Research Institute of Nuclear Physics)
At present, in the High-Energy Laboratory, measurements are being made of the cross sections for the photoproduction of neutral mesons on complex nuclei. The reactions under study are:
\[ \gamma + \text{nucleus} \to \text{nucleus} + \pi^0_{2\gamma}. \]
With the aid of two Cherenkov total-absorption spectrometers, two \(\gamma\)-quanta from the decay of mesons are recorded. The energies of the decay \(\gamma\)-quanta, obtained in the form of a decimal code (the channel numbers of amplitude analyzers), are output to a digital printer, punched tape, or magnetic tape. A detailed description of the setup and the registration method is given in [1].
1. Simulation of the Experiment
The direct result of the experiment is the yield of the reaction \(N\), related to the differential cross section \(\dfrac{d\sigma}{d\Omega}(E,\Omega)\) by the relation:
\[ N = n \int_E \int_\Omega \frac{d\sigma}{d\Omega}(E,\Omega)\cdot \varepsilon(E,\Omega)\cdot \xi(E)\cdot dE\cdot d\Omega, \tag{1} \]
where \(n\) is the number of nuclei in the target per \(1\ \text{cm}^2\);
\(\varepsilon(E,\Omega)\) is the efficiency of registration of the process under study by the experimental setup;
\(\xi(E)\) is the bremsstrahlung spectrum of \(\gamma\)-quanta incident on the target;
\(E\) is the energy of the incident \(\gamma\)-quanta;
\(\Omega\) are the angular coordinates of the direction of meson emission.
The dependence \(\xi(E)\) was taken from tables [2]. In integral (1), the dependence \(\varepsilon(E,\Omega)\) remains unknown; it includes the specific conditions of the experiment (kinematics, target dimensions, solid angles of the spectrometers, etc.).
In experiments of this kind, the differential cross section is taken out from under the integral sign in (1) as averaged over the angular and energy resolutions of the setup, which are determined in the following way:
\[ R(\Omega)=\int_E \xi(E)\cdot \varepsilon(E,\Omega)\cdot dE , \tag{2} \]
\[ R(E)=\xi(E)\cdot \int_\Omega \varepsilon(E,\Omega)\cdot d\Omega . \tag{3} \]
Then the remaining integral
\[ \varepsilon=\int_{E\Omega}\xi(E)\cdot \varepsilon(E,\Omega)\cdot dE\cdot d\Omega \tag{4} \]
denotes the total probability of registering the photoproduction process for a given measurement geometry. Integrals (1), (2), (3) are usually calculated with the aid of a computer by the Monte Carlo method. In this case, a probabilistic mathematical model of the experiment is constructed, in which the specific experimental conditions are taken into account, beginning with the interaction of the incident \(\gamma\)-quanta with the target and ending with the registration of the \(\gamma\)-quanta from meson decay by Cherenkov spectrometers.
The model of the experiment is implemented in the form of a program written in the symbols of an \(\alpha\)-language. In the calculation, the momentum distribution of nucleons in the nucleus was taken into account. In [3] it is shown that the value of \(\varepsilon\) depends very little on the form of this distribution. Therefore, the simplest model of a degenerate Fermi gas was chosen:
\[ f(\bar p)= \begin{cases} \text{const.}, & P\leq P_f,\\ 0, & P>P_f, \end{cases} \tag{5} \]
where \(P_f\) is the limiting Fermi momentum.
When calculating the energy distribution of decay \(\gamma\)-quanta and mesons registered by the setup, allowance was made for the finite energy resolution of the spectrometers, which was determined experimentally [4].
In order to reduce the time required to solve the problem on a computer, an additional selection of meson registration events was made [10]. There is azimuthal symmetry of the experiment, and the registration probability will not change if both spectrometers are rotated by the same angle about the axis associated with the bremsstrahlung beam. This makes it possible to “install” several pairs of counters in the calculation scheme, and then the calculated value of the efficiency must be divided by the number of these pairs. In addition, when the spectrometer is rotated about the axis, its window describes an annular region. Therefore, the entry of the decay \(\gamma\)-quanta into this annular region was first checked. With such an additional selection, the counting time for the problem is reduced severalfold.
In addition to the integrals (2, 3, 4), the following quantities are calculated:
1) the mean emission angle of the mesons
\[ \overline{\cos\theta}= \frac{\int R(\cos\theta)\cdot \cos\theta\cdot d\cos\theta} {\int R(\cos\theta)\cdot d\cos\theta}, \tag{6} \]
2) the mean energy of the incident \(\gamma\)-quanta
\[ \overline E= \frac{\int R(E)\cdot E\cdot dE} {\int R(E)\cdot dE}; \tag{7} \]
3) the energy spectra of the decay \(\gamma\)-quanta, and the energy and mass distributions of the registered mesons with and without allowance for co-
...of the finite energy resolution of the spectrometers. The mass distribution was found from the formula:
\[ \mu=\sqrt{4\cdot \sin^2\frac{\alpha_{\rm cp}}{2}\cdot E_{\gamma 1}\cdot E_{\gamma 2}}, \tag{8} \]
where \(\mu\) is the meson mass,
\(E_{\gamma 1,2}\) are the energies of the decay \(\gamma\)-quanta,
\(\alpha_{\rm cp}\) is the average opening angle between the spectrometers.
The aim of the simulation is also to determine the admissible registration region in the plane \(E_{\gamma 1}0E_{\gamma 2}\), by which is meant the region in which, with a specified probability \(\beta\), events of the process under study are found. To construct the region, the assumption is made that the useful events in the plane \(E_{\gamma 1}0E_{\gamma 2}\) obey a normal distribution law; then the spectra \(N(E_{\gamma 1})\), \(N(E_{\gamma 2})\), \(N(E_{\eta,\pi^0})\) may be regarded as grouped samples from a normal law. The estimates of the parameters \(m\) and \(\sigma\) for the spectra were found by the maximum-likelihood method [5]. For this purpose, a program was written that implements the random-search method for finding the maximum of the likelihood function with respect to the parameters \(m\) and \(\sigma\):
\[ L(m,\sigma)=\sum_{i=1}^{k-1} N_i\cdot \log P_i, \tag{9} \]
\[ P_i=F(x_{i+1})-F(x_i)= \Phi\left(\frac{x_{i+1}-m}{\sigma}\right)- \Phi\left(\frac{x_i-m}{\sigma}\right)= \]
\[ =\frac{1}{\sqrt{2\pi}}\times \int_{\frac{x_i-m}{\sigma}}^{\frac{x_{i+1}-m}{\sigma}} e^{-\frac{t^2}{2}}\,dt; \]
\(N_i\) is the number of events falling in the interval \([x_i,x_{i+1})\);
\[ x=E_{\gamma 1},\ E_{\gamma 2},\ E_{\eta,\pi^0}. \]
In addition, the program finds the statistical errors of the parameters
\[ \theta=m,\sigma, \]
\[ \sigma(\theta)= \left(\sum_{i=1}^{k-1} N_i\cdot A\right)^{-1/2}, \]
\[ A=\sum_{i=1}^{k-1} P_i \left( \left. \frac{\partial \log P_i}{\partial \theta} \right|_{\substack{m=\hat m\\ \sigma=\hat\sigma}} \right)^2, \]
\(\hat m,\hat\sigma\) are the maximum-likelihood estimates. Having these estimates for the spectra \(N(E_{\gamma 1})\), \(N(E_{\gamma 2})\), \(N(E_{\eta,\pi^0})\), one can write the equation of the ellipse that bounds the admissible registration region:
\[ \left(\frac{E_{\gamma 1}-\hat m_{E_{\gamma 1}}}{\hat\sigma_{E_{\gamma 1}}}\right)^2+ \left(\frac{E_{\gamma 2}-\hat m_{E_{\gamma 2}}}{\hat\sigma_{E_{\gamma 2}}}\right)^2 -2\cdot \rho\cdot \frac{ \left(E_{\gamma 1}-\hat m_{E_{\gamma 1}}\right)\cdot \left(E_{\gamma 2}-\hat m_{E_{\gamma 2}}\right) }{ \hat\sigma_{E_{\gamma 1}}\cdot \hat\sigma_{E_{\gamma 2}} } = \]
\[ =2\cdot(1-\rho^2)\cdot \ln(1-\beta), \tag{10} \]
\[ \rho= \frac{1}{2\cdot \hat\sigma_{E_{\gamma 1}}\hat\sigma_{E_{\gamma 2}}} \left[ \hat\sigma^2_{E_{\eta,\pi^0}}- \hat\sigma^2_{E_{\gamma 1}}- \hat\sigma^2_{E_{\gamma 2}} \right]. \]
The last equality follows from the law of conservation of energy:
\[ E_{\eta,\pi^\circ}=E_{\gamma 1}+E_{\gamma 2} \tag{11} \]
and, consequently,
\[ \hat{\sigma}^{2}_{E_{\eta,\pi^\circ}} = \sigma^{2}_{E_{\gamma 1}} + \sigma^{2}_{E_{\gamma 2}} + 2\cdot \operatorname{cov}(E_{\gamma 1},E_{\gamma 2}). \]
Owing to the low-energy background, a cutoff condition may be imposed on the region:
\[ E_{\gamma 1}+E_{\gamma 2}>E_{\text{cutoff}}. \]
In this case the admissible registration region is bounded by the ellipse and the straight line
\[ E_{\gamma 1}+E_{\gamma 2}=E_{\text{cutoff}}, \]
II. Primary Processing
For this type of experiment, primary processing is naturally divided into the following stages.
- Recording the primary physical effects and converting them into the digital code of the given computer.
- Accounting for apparatus failures and background measurements.
- Processing for the purpose of identifying the registered process.
- Estimating the reaction yield and its variance.
To implement stages 1 and 2, a program has been written which groups the registered events into energy and mass spectra of mesons, using expressions (8), (11). Events \((E_{\gamma 1}, E_{\gamma 2})\) in the form of a sequence of codes
\[ P=\{(N_1,N_2)_i\} \]
are the program input information.
\(N_1, N_2\) are the channel numbers of the analyzers, related to the energy in the following way:
\[ E_{\gamma}=d_j\times N_j+b_j,\quad j=1,2, \]
where \(d_j, b_j\) are coefficients determined by calibration of the spectrometers.
The admissible registration region is specified by a condition approximating the ellipse (10) with the imposed cutoff:
\[ \underline{E}_{\gamma 1}<E_{\gamma 1}\leq \bar{E}_{\gamma 1} \wedge \underline{E}_{\gamma 2}\leq E_{\gamma 2}\leq \bar{E}_{\gamma 2} \wedge \underline{E}_{\eta,\pi^\circ}<E_{\eta,\pi^\circ}<\bar{E}_{\eta,\pi^\circ}. \tag{12} \]
The sequence of operation of the program is as follows:
1) from the sequence \(P\), repetitions are discarded; in this experiment a repetition is understood as the subsequence
\[ P'=\{(N'_1,N'_2)_i\}\subseteq P, \]
where \((N'_1,N'_2)_i=(N_1,N_2)_i\), if
\[ N_{1,i}=N_{1,i-1}\vee N_{2,i}=N_{2,i-1}; \]
2) events not satisfying (12) are discarded;
3) the spectra are formed
\[ N(E_{\gamma 1}),\; N(E_{\gamma 2}),\; N(E_{\eta,\pi^\circ}),\; N(\mu_{\eta,\pi^\circ}) \]
with account taken of background measurements.
Identification of neutral mesons in this type of experiment is expediently carried out by the meson energy. For this purpose the energy spectrum \(N(E_{\eta,\pi^\circ})\) is processed, which, when a cutoff is imposed on the registration region, has the form of a grouped sample.
from a truncated normal law. The mean value \(E_{\eta,\pi^\circ \mathrm{av}}\) is determined by finding the maximum of function (9). In this case
\[ F(x)= \begin{cases} 0, & \text{if } x<x_1,\\[6pt] \dfrac{ \Phi\!\left(\dfrac{x-m}{\sigma}\right) - \Phi\!\left(\dfrac{x_1-m}{\sigma}\right) }{ 1-\Phi\!\left(\dfrac{x_1-m}{\sigma}\right) }, & \text{if } x\leq x_1 . \end{cases} \]
where \(x_1=E_{\mathrm{truncation}}\).
The properties of estimates for this type of distribution have been investigated, and the problem of optimal grouping in order to reduce the variance of the mean has been solved [7]. To compute the optimal intervals, a program was written that implements the linearization method [6], whose rate of convergence does not depend on the number of intervals. For identification, the statistic [8] was used:
\[ Z= \frac{ E_{\eta,\pi^\circ \mathrm{av}}-\hat m_{E_{\eta,\pi^\circ}} }{ \sigma\left(E_{\eta,\pi^\circ \mathrm{av}}-\hat m_{E_{\eta,\pi^\circ}}\right) }, \]
which has the normal distribution
\[ N\!\left( 0,\, \sqrt{ \frac{\hat\sigma^2_{\eta,\pi^\circ}}{n} + \frac{\sigma^2}{\displaystyle\sum_{i=1}^{k-1} N_i} } \right); \]
here
\[ n \text{ is the number of events obtained in simulating the experiment;} \]
\[ \hat m_{E_{\eta,\pi^\circ}},\ \hat\sigma^2_{\eta,\pi^\circ} \text{ are estimates for the energy spectrum in the simulation,} \]
\[ E_{\eta,\pi^\circ \mathrm{av}} \text{ are estimates of the experimental energy spectrum.} \]
The critical region is specified by the relation
\[ |Z|>Z_q, \]
\(q\) is the significance level.
In the case of a positive result in testing the hypothesis, the quantity
\[ N=N'+\delta N, \]
where
\[ N'=\sum_{i=1}^{k-1} N_i, \]
\[ \delta N = N'\left\{ \left[ 1-\Phi\!\left( \frac{x_1-\hat m_{E_{\eta,\pi^\circ}}}{\hat\sigma} \right) \right]^{-1} -1 \right\} \]
determines, taking into account the accumulated dose and the efficiency, the value of the differential cross section on a nucleus
\[ \frac{d\sigma}{d\Omega} = \frac{N\cdot \varepsilon'}{n\cdot \varepsilon\cdot N_\gamma}. \]
In this formula
\(\varepsilon'\) is a coefficient associated with the absorption of decay and incident \(\gamma\)-quanta,
\(N_\gamma\) is the number of equivalent \(\gamma\)-quanta.
The error of the yield \(N\) is calculated as \(\Delta N=(N' + N_{\text{background}})^{1/2}\).
The described procedure was used in processing the experiment [9].
REFERENCES
-
G. N. Dudkin et al. Methodology of an experiment for recording neutral mesons. Proceedings of the VII Interuniversity Conference on Electron Accelerators. (Tomsk, 1968). Issue 4, Atomizdat, 1970.
-
A. S. Penfold, J. E. Leiss. Calculation of photoeffect cross sections. University of Illinois, 1958.
-
C. Bacci et al. Preprint, LNF-69/4, 1969.
-
G. N. Dudkin et al. Proceedings of the Research Institute of Nuclear Physics, Electronics, and Automation at the Tomsk Polytechnic Institute. Issue 2. Accelerators and radio electronics of the physics experiment. Atomizdat, 1972.
-
N. A. Bogin. Estimation of distribution parameters from grouped samples. Proceedings of the Mathematical Institute of the USSR Academy of Sciences, CXI, 1970.
-
I. N. Silin. Finding minima of functionals by the method of linearization. JINR Preprint D-B10, 1963.
-
S. M. Bersenev, N. I. Sablin. Simultaneous estimation of the parameters of a truncated normal law from a grouped sample. Abstracts of reports of the All-Union Conference “Development and Practical Application of Electron Accelerators.” Tomsk, TTU Publishing House, 1972.
-
I. V. Dunin-Barkovsky, N. V. Smirnov. Probability Theory and Mathematical Statistics in Engineering (General Part). Moscow, 1955.
-
G. N. Dudkin et al. Photoproduction of \(\pi^0\)-mesons on complex nuclei at \(E_{\max}=500, 700\) MeV. Abstracts of reports of the All-Union Conference “Nuclear Reactions at High Energies.” Moscow, 1972, Tbilisi, 1972.
-
S. P. Denisov. Dissertation. Moscow State University, 1963.