ON THE STATISTICS OF THE NUMBER OF QUANTA IN A BETATRON BREMSSTRAHLUNG BEAM PASSED THROUGH A PLANE BARRIER

V. A. VOROB’EV, T. P. TARASOV

(Recommended by the scientific seminar of the Research Institute of Electron Introscopy)

Introduction

A correct calculation of the operating parameters of radiation defectoscopes (for example, sensitivity) that use a betatron bremsstrahlung beam as the carrier of useful information is possible only when the statistical properties of the beam that has passed through the barrier under study are known. The statistics of the beam that has passed through a barrier are determined, on the one hand, by the statistical properties of the primary beam (before the barrier), and on the other, by transport processes. The microprocesses accompanying the transport of radiation through barriers are probabilistic in character [3], which must affect the fluctuations of the beam beyond the barrier. This article considers the influence of transport processes on the change in certain statistical characteristics of a bremsstrahlung beam beyond a barrier.

Model

A directed beam of bremsstrahlung irradiates the area \(F\) of a barrier and is described by the random field [4]

\[ N_0(\Gamma)=<N_0(\Gamma)>+N_1^0(\Gamma)+N_2^0(\Gamma), \tag{1} \]

which specifies the distribution of the numbers of quanta in the phase space \(\Gamma_0=F\times E\times \Omega\times T\), where \(\Gamma=(\mathbf r,E,\Omega,t)\) is the set of variables \(\in\Gamma\). \(N_1^0(\Gamma)\) and \(N_2^0(\Gamma)\) are statistically independent components with \(M[N_i^0(\Gamma)]=0\). The radiation that has passed through the barrier is also described by the field \(N(\Gamma)=<N(\Gamma)>+N'(\Gamma)\), where \(M[N'(\Gamma)]=0\). The process of radiation transport through the barrier is represented by the stochastic transformation \(T\), mapping the random field \(N_0(\Gamma)\) into \(N(\Gamma)\):

\[ N(\Gamma)=TN_0(\Gamma). \tag{2} \]

The stochastic operator \(T\) can be represented as the sum of a deterministic transformation \(<T>\) and a random transformation \(\Delta T\):

\[ T=<T>+\Delta T. \tag{3} \]

It is clear that \(<N(\Gamma)>=<T><N_0(\Gamma)>\), so that \(<T>\) is the solution operator of the kinetic equation of \(\gamma\)-quantum transport [3] in the geometry under consideration. The fields \(N_0(\Gamma)\) and \(N(\Gamma)\) are, strictly speaking, generalized random functions, but we shall consider it possible [1, 2] to carry out the discussion first for an ordinary field,

for which the number of photons \(\Delta N_{ijkl}\) located in the cell
\(\Delta \Gamma_{ijkl}=\Delta F_i \Delta E_j \Delta \Omega_k \Delta t_l\) of phase space is determined by

\[ \Delta N_{ijkl}=N(\Gamma_{ijkl})\Delta \Gamma;\quad \Gamma_{ijkl}\in \Delta\Gamma . \tag{4} \]

Assumptions

1. The statistical properties of the bremsstrahlung beam of the betatron irradiating the barrier are known. It is assumed [4] that \(N_0(\vec{\Gamma})\) is an inhomogeneous, time-stationary Gaussian generalized random field; 2) the microprocesses accompanying the passage of radiation through the barrier are additive, i.e., the interaction of each field quantum with the barrier is independent; 3) the operator \(T\), describing the transfer of radiation, does not depend on time. \(\langle T\rangle\) can physically be interpreted as the probabilities \(\tau_{ijk,pqr}\) with which a quantum of the cell \(\Delta\Gamma_{ijk'l}\) of the phase space describing the primary beam \(N_0\) passes into the cell \(\Delta\Gamma_{pqrl}\) of the phase space describing the radiation beyond the barrier \(N\); 4) fluctuations are considered small, i.e., deviations of the fields from their mean values with appreciable probability are at least an order of magnitude smaller than the mean values.

Covariance Function

Starting from expressions (1, 2, 3, 4), the deviation from the mean in a particular \(s\)-th realization for the number of quanta located in the cell \(\Delta\Gamma_{pqrl}\) of phase space can be represented as

\[ \Delta N'_{pqrl}=\langle T\rangle [\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}] +\Delta T\,[\langle \Delta N_{0ijkl}\rangle+\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}]. \tag{5} \]

Writing an analogous expression for \(\Delta\Gamma_{\alpha\mu\eta}\) and multiplying by (5), by averaging over all realizations \(s\) we obtain the covariance \(K^\Delta_{pqrl,\alpha\mu\eta}\):

\[ K^\Delta_{pqrl,\alpha\mu\eta} = M\{\Delta T\langle \Delta N_{0ijkl}\rangle \cdot \Delta T\langle \Delta N_{0\alpha\beta\gamma\delta}\rangle\} + \]

\[ +M\{\langle T\rangle [\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}] \cdot \langle T\rangle [\Delta N^0_{1\alpha\beta\gamma\delta} +\Delta N^0_{2\alpha\beta\gamma\delta}]\}. \tag{7} \]

Of the nine terms we neglect in (7) seven summands, since

\[ M\{\langle T\rangle [\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}] \cdot \Delta T\langle \Delta N_{0\alpha\beta\gamma\delta}\rangle\}=0, \]

\[ M\{\langle T\rangle [\Delta N^0_{1\alpha\beta\gamma\delta} +\Delta N^0_{2\alpha\beta\gamma\delta}] \cdot \Delta T\langle \Delta N_{0ijkl}\rangle\}=0, \]

\[ M\{\Delta T\langle \Delta N_{0ijkl}\rangle\cdot \Delta T[\Delta N^0_{1\alpha\beta\gamma\delta} +\Delta N^0_{2\alpha\beta\gamma\delta}]\}=0, \]

\[ M\{\Delta T\langle \Delta N_{0\alpha\beta\gamma\delta}\rangle\cdot \Delta T[\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}]\}=0 \]

— owing to the statistical independence between the fluctuations of the quanta in the primary beam \(N_0\) and the additive macroprocesses of the interaction of radiation with matter during transfer. The terms

\[ M\{\langle T\rangle [\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}] \cdot \Delta T[\Delta N^0_{1\alpha\beta\gamma\delta} +\Delta N^0_{2\alpha\beta\gamma\delta}]\}, \]

\[ M\{\langle T\rangle [\Delta N^0_{1\alpha\beta\gamma\delta} +\Delta N^0_{2\alpha\beta\gamma\delta}] \cdot \Delta T[\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}]\}, \]

\[ M\{\Delta T[\Delta N^0_{1ijkl}+\Delta N^0_{2ijkl}] \cdot \Delta T[\Delta N^0_{1\alpha\beta\gamma\delta} +\Delta N^0_{2\alpha\beta\gamma\delta}]\}. \]

we neglect, considering the fluctuations small (these terms are of second order of smallness). Thus, in (7) the first term is determined by the probabilistic character of the transfer (“fluctuations” of the operator describing radiation transfer for a deterministic primary beam), and the second by fluctuations in the beam of the primary radiation, considering

transfer processes deterministic. We now turn to revealing the content of the formal notation of expression (7).

By virtue of the additivity of the physical processes in the transfer, the operator \(T\) is additive; consequently, its deterministic part \(\langle T\rangle\) and random part \(\Delta T\) are linear. By virtue of its linearity, \(\Delta T\) can be represented by a matrix \((\Delta \tau)\), whose elements are random quantities \(\Delta \tau_{ijk,pqr}\), with \(M[\Delta \tau_{ijk,pqr}]=0\). Now the first term in (7) can be represented as:

\[ M\{\Delta T<\Delta N_{0ijkl}>\cdot \Delta T<\Delta N_{0\alpha\beta\gamma\delta}>\} = \]

\[ = M\left\{\sum_{ijk}\Delta\tau_{ijk,pqr}<\Delta N_{0ijkl}>\cdot \Delta\tau_{ijk,\varepsilon\mu\nu}<\Delta N_{0ijk\delta}>\right\}+ \tag{8} \]

\[ +2M\left\{\sum_{ijk}\sum_{\alpha\beta\gamma} \Delta\tau_{ijk,pqr}<\Delta N_{0ijkl}>\cdot \Delta\tau_{\alpha\beta\gamma,\varepsilon\mu\nu}<\Delta N_{0\alpha\beta\gamma\delta}>\right\} \]

\[ ijk\ne \alpha\beta\gamma, \]

where the last term in (8) is equal to zero, since

\[ M\{\Delta\tau_{ijk,pqr}\cdot \Delta\tau_{\alpha\beta\gamma,\varepsilon\mu\nu}\}=0,\quad ijk\ne \alpha\beta\gamma \]

— by virtue of the additivity of the microprocesses of interaction of quanta with the substance of the absorber, which entails the statistical independence of the transfer of any quantum from the cell \(\Delta\Gamma_{ikl}\) of the phase space of the primary beam into the cell \(\Delta\Gamma_{pcrl}\) of the phase space describing the radiation behind the absorber, from the transfer of a quantum, for example, from \(\Delta\Gamma_{\alpha\beta\gamma\delta}\) into \(\Delta\Gamma_{\varepsilon\mu\nu\eta}\). The expression

\[ M\{\Delta\tau_{ijk,pqr}<\Delta N_{0ijkl}>\cdot \Delta\tau_{ijk,\varepsilon\mu\nu}<\Delta N_{0ijk\delta}>\} \]

determines the covariance between the numbers of quanta \(\Delta N_{pqrl}\) and \(\Delta N_{\varepsilon\mu\nu\delta}\) in the cells \(\Delta_{pqrl}\) and \(\Delta\Gamma_{\varepsilon\mu\nu\delta}\), transferred from the cells \(\Delta I_{ijkl}\) and \(\Delta I_{ijk\delta}\), in which there are, respectively, \(<\Delta N_{0ijkl}>\) and \(<\Delta N_{0ijk\delta}>\) quanta, the transfer being carried out with probabilities \(\tau_{ijk,pqr}\) and \(\tau_{ijk,\varepsilon\mu\nu}\) (the operator \(\langle T\rangle\) does not depend on time). But in this case the random vector

\[ \begin{pmatrix} \Delta\tau_{ijk,pqr}<\Delta N_{0ijkl}>\\ \Delta\tau_{ijk,\varepsilon\mu\nu}<\Delta N_{0ijk\delta}> \end{pmatrix} \]

will be distributed according to a polynomial law; the covariance (8) is determined by [5]:

\[ M\{\Delta T<\Delta N_{0ijkl}>\cdot \Delta T<\Delta N_{0\alpha\beta\gamma\delta}>\} = \]

\[ \sum_{ijk}\left[<\Delta N_{0ijkl}><\Delta N_{0ijk\delta}>\right]^{1/2} \delta_{l\delta}\,[\tau_{ijk,pqr}(\delta_{pqr,\varepsilon\mu\nu}-\tau_{ijk,\varepsilon\mu\nu})]. \tag{9} \]

Writing the second term in (7) in the form

\[ M\{\langle T\rangle[\Delta N^{0}_{1ijkl}+\Delta N^{0}_{2ijkl}]\cdot \langle T\rangle[\Delta N^{0}_{1\alpha\beta\gamma\delta}+\Delta N^{0}_{2\alpha\beta\gamma\delta}]\} = \tag{10} \]

\[ = M[\langle T\rangle\Delta N^{0}_{1ijkl}\cdot \langle T\rangle\Delta N^{0}_{1\alpha\beta\gamma\delta}] +M[\langle T\rangle\Delta N^{0}_{2ijkl}\cdot \langle T\rangle\Delta N^{0}_{2\alpha\beta\gamma\delta}] + \]

\[ +M[\langle T\rangle\Delta N^{0}_{1ijkl}\cdot \langle T\rangle\Delta N^{0}_{2\alpha\beta\gamma\delta}] +M[\langle T\rangle\Delta N^{0}_{2ijkl}\cdot \langle T\rangle\Delta N^{0}_{1\alpha\beta\gamma\delta}], \]

it is easy to show that the last two terms vanish. Indeed, the operators \(M\) and \(\langle T\rangle\) are commutative and, moreover, in the second factor of each term on the right-hand side of (10), \(\langle T\rangle\) acts on primed indices; denoting the latter fact by \(\langle T\rangle'\) and taking into account the linearity of \(\langle T\rangle\), we obtain

\[ M\left[\langle T\rangle \Delta N^0_{1ijkl}\cdot \langle T\rangle \Delta N^0_{2\alpha\beta\gamma\delta}\right] = \langle T\rangle \langle T\rangle' M\left[\Delta N^0_{1ijkl}\cdot \Delta N^0_{2\alpha\beta\gamma\delta}\right]=0 \]

\[ M\left[\langle T\rangle \Delta N^0_{2ijkl}\cdot \langle T\rangle \Delta N^0_{1\alpha\beta\gamma\delta}\right] = \langle T\rangle \langle T\rangle' M\left[\Delta N^0_{20ijkl}\cdot \Delta N^0_{1\alpha\beta\gamma\delta}\right]=0, \]

since \(\Delta N^0_1\) and \(\Delta N^0_2\) are statistically independent.

From the above considerations, the expression for the first term on the right-hand side of expression (10) is evidently written as

\[ M\left[\langle T\rangle \Delta N^0_{1ijkl}\cdot \langle T\rangle \Delta N^0_{1\alpha\beta\gamma\delta}\right] = \langle T\rangle \langle T\rangle' K^{(1)}_{ijkl,\alpha\beta\gamma\delta} = \sum_{ijk}\sum_{\alpha\beta\gamma} \tau_{ijk,pqr}\cdot \tau_{\alpha\beta\gamma,\varepsilon\mu\nu} K^{(1)}_{ijkl,\alpha\beta\gamma\delta}, \tag{11} \]

where \(K^{(1)}_{ijkl,\alpha\beta\gamma\delta}\) is the covariance moment for \(\Delta N^0_1\) of the primary beam [4]; \(\tau_{ijk,pqr}\) is the probability of transfer of quanta from the cell \(\Delta \Gamma_{ijkl}\) to the cell \(\Delta \Gamma_{pqrl}\). Let us analyze the remaining term in (10). We have:

\[ M\left(\langle T\rangle \Delta N^0_{2ijkl}\cdot \langle T\rangle \Delta N^0_{2\alpha\beta\gamma\delta}\right) = \langle T\rangle \langle T\rangle' K^{(2)}_{ijkl,\alpha\beta\gamma\delta} = \sum_{ijk}\tau_{ijk,pqr}\cdot \tau_{ijk,\varepsilon\mu\nu}\cdot \delta_{l\delta} \left[\langle \Delta N_{0ijkl}\rangle \langle \Delta N_{0ijk\delta}\rangle\right]^{\frac12}. \tag{12} \]

Since the random variable \(\Delta N^0_{2ijkl}\) is distributed according to the Poisson law [4], and for it the covariance moment \(K^{(2)}\) has the form [5]:

\[ K^{(2)}_{ijkl,\alpha\beta\gamma\delta} = \left[\langle \Delta N_{0ijkl}\rangle\cdot \langle \Delta N_{0\alpha\beta\gamma\delta}\rangle\right]^{\frac12} \delta_{ijkl,\alpha\beta\gamma\delta}, \]

whence, after summation over \(\alpha\beta\gamma\delta\), the final expression in (12) follows. Here \(\langle \Delta N_{0ijkl}\rangle\) is the mean number of quanta in \(\Delta \Gamma_{ijkl}\). Combining expressions (9, 11, 12), after simplifications we obtain for (7) the following:

\[ K^\Delta_{pqrl,\varepsilon\mu\nu\delta} = K^{\Delta_1}_{pqrl,\varepsilon\mu\nu\delta} + K^{\Delta_2}_{pqrl,\varepsilon\mu\nu\delta}, \tag{13} \]

where

\[ K^{\Delta_1}_{pqrl,\varepsilon\mu\nu\delta} = \sum_{ijk}\sum_{\alpha\beta\gamma} \tau_{ijk,pqr}\cdot \tau_{\alpha\beta\gamma,\varepsilon\mu\nu}K^{(1)}_{ijkl,\alpha\beta\gamma\delta} \tag{13a} \]

\[ K^{\Delta_2}_{pqrl,\varepsilon\mu\nu\delta} = \sum_{ijk} \left[ \tau_{ijk,pqr}\langle \Delta N_{0ijkl}\rangle\cdot \tau_{ijk,\varepsilon\mu\nu}\langle \Delta N_{0ijk\delta}\rangle \right]^{\frac12} \delta_{pqrl,\varepsilon\mu\nu\delta} \tag{13b} \]

Here

\[ \tau_{ijk,pqr}\langle \Delta N_{0ijkl}\rangle = \langle \Delta N_{ijkl,pqrl}\rangle \]

is the mean number of quanta in the cell \(\Delta \Gamma_{pqrl}\), transferred from \(\Delta \Gamma_{ijkl}\). Summation over \(ijk\) in (13b) determines the complete mathematical expectations of the numbers of quanta in the cells \(\Delta \Gamma_{pqrl}\), so that

\[ K^{\Delta_2}_{pqrl,\varepsilon\mu\nu\delta} = \left[\langle \Delta N_{pqrl}\rangle\cdot \langle \Delta N_{\varepsilon\mu\nu\delta}\rangle\right]^{\frac12} \delta_{pqrl,\varepsilon\mu\nu\delta}, \tag{13c} \]

where

\[ \langle \Delta N_{pqrl}\rangle = \sum_{ijk}\tau_{ijk,pqr}\langle \Delta N_{0ijkl}\rangle . \]

All the reasoning carried out above referred to a simple field, for which \(\langle \Delta N_0\rangle=\langle N_0\rangle\Delta \Gamma\) and \(\langle \Delta N\rangle=\langle N\rangle\Delta \Gamma\). Let us denote: \(\Gamma_0\) is the phase space on which the distribution of the numbers of quanta for the primary beam is defined (\(\vec{\Gamma}_0\) is a vector \(\in \Gamma_0\)); \(\Gamma\) is the phase space for the beam beyond the barrier (\(\vec{\Gamma}\) is a vector \(\in \Gamma\)); then for a simple field we shall have:

\[ K^{(1)}_{ijkl,\alpha\beta\gamma\delta} = \Delta \Gamma^2 K_1(\Gamma_{0ijkl};\Gamma_{0\alpha\beta\gamma\delta}, \]

\[ K_1(\Gamma_{0ijkl};\Gamma_{0\alpha\beta\gamma\delta}) = M\{N^0_1(\Gamma_{0ijkl})\cdot N^0_1(\Gamma_{0\alpha\beta\gamma\delta})\}, \]

\[ K^\Delta_{pqrl,\epsilon\mu\nu\delta} = \Delta\Gamma \left[ \langle N(\Gamma_{pqrl})\rangle \cdot \langle N(\Gamma_{\epsilon\mu\nu\delta})\rangle \right]^{1/2} \delta_{pqrl,\epsilon\mu\nu\delta}, \]

i.e., expression (13) takes the form:

\[ K^\Delta_{pqrl,\epsilon\mu\nu\delta} = \Delta\Gamma^2 \left[ \sum_{ijk}\sum_{\alpha\beta\gamma} \tau_{ijk,pqr}\cdot\tau_{\alpha\beta\gamma,\epsilon\mu\nu} K_1(\Gamma_{0ijk};\Gamma_{0\alpha\beta\gamma}) + \right. \]

\[ \left. + \frac{1}{\Delta\Gamma} \left[ \langle N(\Gamma_{pqrl})\rangle \langle N(\Gamma_{\epsilon\mu\nu\delta})\rangle \right]^{1/2} \delta_{pqrl,\epsilon\mu\nu\delta} \right]. \tag{14} \]

Since \(N_0(\Gamma)\) and \(N(\Gamma)\), which describe the distributions of the numbers of quanta in phase spaces, are generalized random functions representable as limits of sequences of ordinary functions [2], and the covariance functions \(K_{N_0}\) and \(K_N\) are limits of the corresponding sequences of covariances, we obtain

\[ K_N(\Gamma_1\Gamma_2) = \lim_{\Delta\Gamma\to 0} \left[ \frac{1}{\Delta\Gamma^2}K^\Delta_{pqrl,\epsilon\mu\nu\delta} \right] = K_N^{(1)}(\Gamma_1\Gamma_2)+K_N^{(2)}(\Gamma_1\Gamma_2), \tag{15} \]

where

\[ K_N^{(1)}(\Gamma_1\Gamma_2) = \int_{\Gamma_0^1}\int_{\Gamma_0^1} T(\Gamma_{01}^1;\Gamma_{02}^1) T(\Gamma_{02}^1;\Gamma_2^1) K_1(\Gamma_{01}\Gamma_{02})\, d\Gamma_{01}^1 d\Gamma_{02}^1 \tag{15a} \]

\[ K_N^{(2)}(\Gamma_1\Gamma_2) = \left[ \langle N(\Gamma_1)\rangle \langle N(\Gamma_2)\rangle \right]^{1/2} \delta(\Gamma_1-\Gamma_2), \tag{15б} \]

\[ \langle N(\Gamma)\rangle = \int_{\Gamma_0^1} T(\Gamma_0^1;\Gamma^1)\, \langle N_0(\Gamma_0)\rangle\, d\Gamma_0^1. \tag{15в} \]

Here

\[ T(\Gamma_0^1;\Gamma') = \lim_{\Delta\Gamma\to 0} \left[ \frac{1}{\Delta\Gamma}\tau_{ijk,pqr} \right] \tag{15г} \]

is the Green’s function of the boundary-value problem of the kinetic equation for transport of quanta in the barrier geometry under consideration [3].

The integration in (15a, 15в) is performed over the subspace \(\Gamma_0^1=F\times E\times \Omega\subset\Gamma_0\), in which the operator \(\langle T\rangle\) acts. According to [4], the component \(K_1\) of the covariance function has the form:

\[ K_1(\Gamma_{01};\Gamma_{02}) = \sum_{s=1}^{5} C_s(\Gamma_{01}^1;\Gamma_{02}^1)K_s(t_1t_2). \]

Taking this into account, expression (15a) can be written as

\[ K_N^{(1)}(\Gamma_1\Gamma_2) = \sum_{s=1}^{5} G_s(\Gamma_1^1\Gamma_2^1)K_s(t_1t_2), \tag{15д} \]

where

\[ G_s(\Gamma_1^1\Gamma_2^1) = \int_{\Gamma_0^1}\int_{\Gamma_0^1} T(\Gamma_{01}^1;\Gamma_1^1) T(\Gamma_{02}^1;\Gamma_2^1) C_s(\Gamma_{01}^1;\Gamma_{02}^1) d\Gamma_{01}^1 d\Gamma_{02}^1. \]

It follows from (15) that the random component \(N'(\vec{\Gamma})\) of the field beyond the barrier, as in the case of \(N_0(\vec{\Gamma}_0)\) [4], can be represented as the sum of two uncorrelated components, one of which is a \(\delta\)-correlated process. Obviously, calculation of the covariance function of the radiation beam beyond the barrier is associated with finding the kernel \(T(\vec{\Gamma}_0^1;\Gamma')\) of transport; effective methods for the analytic solution of this problem do not yet exist; the principal results in this direction have been obtained by the Monte Carlo method.

Mean-square fluctuations

Owing to the \(\delta\)-correlatedness, \(N(\vec{\Gamma})\) can be given a physical meaning only as a random generalized function; moreover, for \(N(\Gamma)\) there exists no probability distribution at each \(\vec{\Gamma}\) (a definite point \(\vec r\), at any time \(t\)). But since in practical experiments we record the field \(N(\Gamma)\) by means of a physical instrument which, in measurement, inevitably performs averaging over some volume \(\Delta\vec{\Gamma}\) of phase space (because of the finite magnitude of the energy and angular resolving power, the finiteness of the aperture, and inertia), we shall always obtain a definite value for the probability distribution, not for \(N(\Gamma)\), but for some linear functional:

\[ \Phi(\varphi)=\int_{\Gamma}\varphi(\Gamma)N(\Gamma)\,d\Gamma, \]

which depends continuously on the physical characteristics of the measuring instrument \(\varphi(\Gamma)\).

The variance of the readings at the output of the measuring instrument is determined from the obvious expression

\[ D\Phi(\varphi)=\int_{\Gamma}\int_{\Gamma}\varphi(\Gamma_1)\varphi(\Gamma_2)K_N(\Gamma_1\Gamma_2)\,d\Gamma_1d\Gamma_2 \]

and will be a finite quantity.

As for the probability distribution of the readings of the measuring instrument, for a sufficiently large averaging interval \(\Delta\Gamma\), it will be close to Gaussian, as a linear transformation of \(N_0(\vec{\Gamma}_0)\) [4].

LITERATURE

1. I. M. Gel'fand, N. Ya. Vilenkin. Some applications of harmonic analysis. Issue 4, FM, 1961.
2. Functional Analysis. SMF. Edited by S. G. Krein. Nauka, 1964.
3. U. Fano, M. Berger, L. Spencer. Transport of gamma radiation. Atomizdat, 1963.
4. V. A. Vorob'ev, G. P. Tarasov. On fluctuations of the number of quanta in the beam of bremsstrahlung of a betatron. Izvestiya TPI, vol. 169.
5. V. M. Gol'danskii. Statistics of counts in the registration of nuclear radiations. Atomizdat, 1959.