Theoretical Approach to the Energy Resolution of a Scintillation Spectrometer with Several Photodetectors

The exact mathematical description of processes at registration of incident particles by a scintillation spectrometer with several photodetectors allows receiving correct formulae for the mean value and the variance of the amplitude at the output of photodetectors, and for the covariance between signals of photodetectors. It was shown that using the relative covariance of signals allows the experimental determination of the covariance in the light yield and the Fano factor in a scintillator. The important advantage of the proposed method is its independence on the gains and the noises of photodetectors electronics.


INTRODUCTION
HERE are a number of processes in scintillation detectors that lead to output signal broadening in response to impinging monochromatic radiation. There are semiempirical formulae that account for the contribution of different factors to the energy resolution of scintillation detectors [1], [2], [3]. However, these formulae do not contain any information on relationship of these contributions with scintillator characteristics and characteristics of the processes that occur in a scintillation detector at registration of incident particles.
The correct formula for the variance of the output signal of a scintillation detector cannot be semiempirical. It must result strictly from the exact mathematical description of the processes that take place at registration of the incident particle by a scintillation detector. Only such formula has the predictive ability. Only in this case dependence of the energy resolution on characteristics of scintillation crystal and other parameters of the detector can be revealed, and the conditions at which the characteristics of the processes can be derived from the output signal. As the process of signal formation at the output of detector represents a random branching cascade process, the formalism of probability generating functions (PGF) is the most adequate for its formulation [4].
Our approach to formulation of the mathematical model of an incident particle registration by a scintillation detector is based on the approach developed in formulation of the Manuscript  theory of shower spectrometers in [5], and published in [6] and [7].

MATHEMATICAL MODEL OF A SCINTILLATION DETECTOR
Let us formulate the mathematical model of an incident particle registration by a scintillation detector with N photodetectors, which we number with the index n , ( , ) n N 1 . The process of an incident particle energy transformation into the output signals includes the following successive stages.
1. The stage of interaction of an incident particle with a scintillation crystal. Let divide the coordinate space into volume elements which characterize all possible distributions of secondary particles in the phase space elements after finishing all processes of downconversion of the energy of the incident particle into the energy of secondary particles that cannot produce electron-hole pairs in the scintillation crystal.
2. The stage of electron-hole pairs generation. The scintillation properties are strongly related to the crystal structure. The scintillation crystals have the property of semiconductor material, and generation of light photons is Theoretical Approach to the Energy Resolution of a Scintillation Spectrometer with Several Photodetectors Victor V. Samedov going via the generation of electron-hole pairs. Let s f ijk be the PGF of the process of electron-hole pairs generation by a secondary particle of the type from the element of the phase space ijk . Hereafter s is the auxiliary variable of the probability generating function. In a semiconductor crystal, only secondary particles with energies greater than the threshold energy k ij E min , which may depend on the point in the detector volume, and on the energy and moving direction of the secondary particle, can produce electronhole pairs. This fact can be accounted for by assuming that the mean value 6. The stage of a light photon emission by the luminescent center. Because of a random spatial distribution of luminescent centers in the host matrix, the energy levels of each luminescent center experience different Stark shifts arising from the electric field of the surrounding ions. Therefore, the number of light photons per elementary mode of radiation is less than unity, and we can consider that photons are emitted by the luminescent centers independently. In this case, the process of a photon emission by the luminescent center belonging to the volume element i V is binomial with the PGF From the probability generating function (10), the mean value and the variance of the sum signal take forms: In (12), is the variance of the output signal due to covariance of secondary particles in the phase space, where is the variance of the output signal due to fluctuations in the number of electron-hole pair. In (14), according to the Fano model [8], we assume that fluctuations in the number of electron-hole pairs are proportional to their average where ijk F is the Fano factor.
is the variance of the output signal due to fluctuations in the number of photoelectrons in photodetectors.
is the variance of the output signal due to fluctuations in the electronic gains of photodetectors.
is the variance due to electronic noise.
As the average number of electron-hole pairs is determined by the energy absorbed in the detector volume element, it is convenient to move from the particle distribution to the absorbed energy distribution as follows: Any detector is usually characterized by the energy resolution of the total absorption peak, when the total energy of the primary monoenergetic particle 0 E is absorbed in the detector volume where the Heaviside unit function, ( where D is the effective diffusion coefficient, and is the mean lifetime of carriers. The Taylor expansion of the coefficient (22) up to the second order on displacement r r has the form At averaging (25) over the probability density of diffusion (23), owing to isotropy the linear term vanishes, and the second order term comes to , , The expressions (21) (27) -(32) are the most general formulae for the average and the variance of a scintillation detector with several photodetectors and are the basis for various approximations. These expressions include the formulae for the average and the variance of any photodetector signal if we leave in the sum only one term for the given photodetector.

APPROXIMATION FOR LOW-ENERGY X-RAYS
For the following, we need approximation for low-energy X-rays, when we can ignore the process of multiple scattering of Xquantum and can consider that its energy is transmitted to the kinetic energy of a photoelectron at the point of interaction. If the volume of downconversion of the photoelectron energy into the energy of electron-hole pairs is small compared with the volume of the detector, we can consider that all electron-hole pairs are generated at the point of X-ray quantum interaction with the detector volume.
For uniform and isotropic scintillator with the only one radiative transition mode the expression (27) is simplified to where ( ) ( ) r a S E E is the standard notation of the probability of the luminescent center activation by one electron-hole pair that can depend only on the electron energy; Q is the standard notation of the quantum efficiency of the luminescence process defined as / 4 ( , , ) e l Q r is the probability of a photoelectron creation in the n -th photodetector by a photon emitted by the luminescent center at the point r of the scintillator.
In a uniform and isotropic scintillator e h e h ( , , ) ( ) r E E . For the case of the local absorption of the energy of an incident particle at the point c r of the scintillator, the differential density of absorbed energy in the detector volume is The averaging over of all possible combinations of secondary particles in the phase space is factorized, and the averaging over the spatial coordinate is due to variation of the X-ray quantum interaction point. In this case the mean value and the variance of the sum signal take forms is the relative variance of the photodetector signal caused by the covariance of the differential light yield. This term is due to dependence of the differential light yield on the energy of secondary particles in the course of downconversion of the energy of an incident particle. If the differential light yield does not depend on the energy of the secondary particles, then the total light yield from a scintillator linearly depends on energy of an incident particle, and this term is equal to zero.
If we define the average value of the probability of the luminescent center activation, the average value of the Fano factor, and the average energy of the electron-hole pair production by expressions