Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 169　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1968

On Fluctuations in the Number of Quanta in a Betatron Bremsstrahlung Beam

V. A. Vorob’ev, G. P. Tarasov

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

Introduction

The use of bremsstrahlung beams in physical experiments, as well as the calculation of technical devices in which betatron bremsstrahlung is used (in particular, radiation flaw detectors and introscopes), requires knowledge of the statistical characteristics of the radiation, with a correct assessment of the accuracy of the experiment and of the sensitivity of the devices. However, the usual statistical methods for describing a bremsstrahlung beam should be applied with caution; in a rigorous formulation of the problem, the radiation beam is described by a generalized singular random field [1], so that a probability distribution for the function describing the beam at each particular point does not exist, although in practice it will almost always be obtained, since the measuring instrument with which we register the beam inevitably performs averaging over the energy spectrum, time, and area of registration. In this article, under certain restrictions mentioned below, the statistical characteristics of the beam are described.

Model of the Generation Process

A. We adopt the following model of the process under description. The flux of accelerated betatron electrons dumped onto a radiator target is a sequence of pulses of duration \(\tau \simeq (3—15)10^{-6}\) sec, following with frequency \(f\). Each pulse \(n'_k(E_e; t)\) has a definite shape in time \(t\) and a distribution of the number of electrons over energies \(E_e\), so that the total number of electrons in a bunch is determined by:

\[ n_k=\int_0^{\tau_k}\int_0^{E_{km}} n_k(E_e;\,t)\,dE_e\,dt. \tag{1} \]

The small temporal duration of the pulse \(\tau_k\) in comparison with the registration time intervals \(\Delta t\), which occur in most practical cases, and the insignificant energy spread of the electrons in a given pulse in comparison with the mean energy \(E_k\) in this pulse, make it possible to approximate the distributions \(n_k(E_e;\,t)\) by \(\delta\)-functions, i.e.

\[ n_k(E_e;\,t)=n_k\delta(E-E_k)\delta(t-t_k). \tag{2} \]

Thus, a particular \(s\)-th realization of the random process \(n(E_e;t)\) can be represented by the sequence:

\[ n^{(s)}(E_e;t)=\sum_{-\infty}^{\infty} n_k^{(s)}\delta\left(E_e-E_{ek}^{(s)}\right)\delta\left(t-t_k^{(s)}\right). \tag{3} \]

The set of sequences \(n(E_e;it)\) forms a certain functional space \(R(n)\), on which a probability distribution can be specified in the usual way.

B. A beam of bremsstrahlung quanta is described by specifying the distribution function of the number of quanta [6] over the elements \(d\Gamma=dF\,dE\,d\Omega\,dt\) of the phase space \(\Gamma\), formed by the direct product of the sets:

\(F(x,y)\)—the plane, \(\{E\}\)—the quantum energy, \(\{\Omega\}\)—the direction, \(\{t\}\)—the time. If registration of the beam is carried out in the plane \(F\), perpendicular to the beam axis and located at a distance \(r_0\) from the target, then it will be convenient for us to use the phase space \(\Gamma=E\times\Omega\times T\), since there is a one-to-one correspondence between \(d\Omega\) and \(dF\). Denoting the set of variables \((E;\vec O;t)\) by the vector \(\vec\Gamma\in\Gamma\), we shall write the \(s\)-th realization of the distribution function of the number of quanta in \(\Gamma\) as \(N^{(s)}(\vec\Gamma)\). The set of realizations \(N(\Gamma)\) forms the functional space \(R(N)\), whose elements are fields defined on \(\Gamma\). In \(R(N)\) a probability distribution can also be specified.

C. We can now represent the process of generation of a bremsstrahlung beam mathematically by the stochastic mapping \(T^\Pi\) of the space \(R(n)\) into the space \(R(N)\):

\[ R(N)=T^\Pi R(n). \tag{4} \]

The physical meaning of the stochastic operator \(T^\Pi\) is that, even when \(n\) is deterministic (\(n\) does not fluctuate), owing to the quantum nature of the radiation processes the radiation flux will fluctuate. Thus, fluctuations of the field \(N(\vec\Gamma)\) describing the bremsstrahlung beam are determined both by fluctuations of the flux of accelerated electrons \(n\) and by the quantum character of the braking processes in the target, which is described by \(T^\Pi\). As already mentioned, a rigorous description of the field \(N(\vec\Gamma)\) requires the apparatus of generalized random functions [1, 2]. We shall not use them, but note the following propositions: a) the generalized random field \(N(\Gamma)\) is the limit of a sequence of Hilbert random fields [2] (i.e., fields with bounded variance), the sequence of whose covariance functions contains a \(\delta\)-like sequence; b) a Hilbert random field can be approximated by simple fields (i.e., those taking a finite number of values—the constant value \(N_m\) on \(\Delta\Gamma_m\subset\Gamma\))

Assumptions

The probabilistic characteristics of \(N(\vec\Gamma)\) will be considered subject to the following restrictions:

1. The random process \(n(E_e,it)\), describing the flux of accelerated electrons incident on the target, is ergodic; 2) the frequency of acceleration cycles \(f(t)\) is a stationary random process; 3) the radiation processes are additive, i.e., each electron, when braking in the target, interacts with it independently of the other electrons of the bunch;

4) we neglect multiphoton processes in the target; 5) the conditional probability \(p_{ij}, E_e\) of emission of a \(\gamma\)-quantum with energy in the interval \(E_i \div E_i+\Delta E_i\), into the solid angle \(\Delta \Omega_j\), in the direction \(\vec{\Omega}_j\), under the condition that the decelerated electron had initial energy \(E_e \div E_e+dE_e\), is determined through the cross section \(\Sigma(E_e; E; \Omega)\) for bremsstrahlung yield [5]:

\[ p_{ij}(E_e)= \int_{\Delta E_i}\int_{\Delta \Omega_j} \frac{d\Sigma(E_e;E;\Omega)}{\partial E\,\partial\Omega}\, d\Omega\,dE \simeq \left. \frac{d\Sigma}{\partial E\,\partial\Omega} \right|_{\substack{E\in\Delta E_i\\ \Omega\in\Delta\Omega_j}} \cdot \Delta E_i\Delta\Omega_j . \tag{5} \]

(for thin radiators one may use Schiff’s formula); 6) the fluctuations are considered small, i.e., deviations from the mean values with appreciable probability are at least an order of magnitude smaller than the mean values.

According to expression (1), each realization \(N(\Gamma)\) may be represented as

\[ N()=\langle N(\Gamma)\rangle+\delta N(\Gamma) = [\langle T^n\rangle+\delta T](\langle n\rangle+\delta n)= \]

\[ \text{where}\quad = \langle T^n\rangle\langle n\rangle+ [\langle T^n\rangle\delta n+\delta n+\delta T\langle n\rangle+\delta T\delta n], \tag{6} \]

\[ \langle N(\Gamma)\rangle=\langle T^n\rangle\langle n\rangle \quad \text{is the mean value,} \]

\[ \delta N(\Gamma)=\langle T^n\rangle\delta n+\delta T\langle n\rangle+\delta T\delta n \quad \text{is the deviation from the mean.} \]

Taking into account the smallness of the fluctuations, we neglect the last term, and, owing to the additivity of the radiation processes in the target, averaging the square of \(\delta N\) over all realizations will contain two terms, since

\[ M(\delta T\langle n\rangle\cdot \delta n)=0. \tag{7} \]

Hence one may conclude that the field \(\vec{N}(\Gamma)\), describing the bremsstrahlung beam, is the sum of two uncorrelated components: one component is due to fluctuations of the electron flux, the other to the probabilistic nature of the processes in the target.

Covariance Function of the Field

Figure 1 gives a simplified picture of a segment of the \(s\)-th realization of the random process \(n_i(e;t)\). Here \(t_k\) are the starting times of the observation intervals, \(\Delta t_k\) are the durations of the corresponding intervals. Let the \(i\)-th pulse of accelerated electrons contain \(n_{kl}^{(s)}\) electrons with energy \(E_{kl}\) (\(k\) is the number of the time interval). Owing to the additivity of the interaction, the number of \(\gamma\)-quanta \(\Delta N_i^{(s)}(\Delta\Gamma_{ij})\) in the element \(\Delta\Gamma_{ijk}=\Delta E_i\Delta\Omega_j\Delta t_k\) of phase space from the \(i\)-th pulse is distributed according to the binomial law with conditional mean value:

\[ \overline{\Delta N_i^{(s)}(\Delta\Gamma_{ijk})} = n_{kl}^{(s)}p_{ij}(E_{kl}). \tag{8} \]

During the time \(\Delta t_k\) (the interval \(t_k \div t_k+\Delta t_k\)), the number of quanta \(\Delta N^{(s)}(\Delta\Gamma_{ijk})\) will be formed by the superposition of fields from a number of pulses that fit into \(\Delta t_k\), following with frequency

\[ f_k=f_0+\Delta f_k; \]

\[ \Delta N^{(s)}(\Delta\Gamma_{ijk}) = \sum_{l=1}^{f_k\Delta t_k} \Delta N_l^{(s)}(\Delta\Gamma_{ijk}) = \sum_{l=1}^{f_k\Delta t_k} \left[ \overline{\Delta N_l^{(s)}(\Delta\Gamma_{ijk})} + \delta\Delta N_l^{(s)}(\Delta\Gamma_{ijk}) \right]. \tag{9} \]

Since \(n_{kl}^{(s)}=\langle n\rangle+\Delta n_{kl}^{(s)}\) fluctuates from one acceleration cycle to another, and the energy of the accelerated electrons \(E_{kl}^{(s)}\) also fluctuates, the conditional mathematical expectation \(\Delta N_i^{(s)}(\Delta \Gamma_{ijk})\) will be a random quantity, which we shall represent as:

\[ \overline{\Delta N_i^{(s)}(\Delta \Gamma_{ijk})} = \left[\langle n\rangle+\Delta n_{kl}^{(s)}\right] \left[p_{ij}+\Delta p_{ij}\left(E_{kl}^{(s)}\right)\right], \tag{10} \]

where \(\langle n\rangle\) is the mean value of the number of accelerated electrons in a pulse, averaged over all realizations \(s\).

\(\Delta n_{kl}^{(s)}\) is the deviation from the mean in the \(i\)-th acceleration cycle over the interval \(\Delta t_k\) of realization \(s\).

\[ \Delta p_{ij}(E_{kl}) = \left. \frac{\partial p_{ij}(E_e)}{\partial E_e} \right|_{E_e=E_0} \cdot \Delta E_{kl} = a_{ij}\Delta E_{kl} \]

is the change in the probability of radiation into the element \(\Delta \Gamma_{ijk}\) in connection with the deviation of the energy \(E_{kl}\) from the mean \(E_0\).

Fig. 1. Model of a segment of the \(s\)-th realization of the sequence of pulses of accelerated electrons dumped onto the target

Taking this into account, the deviation from the mean, \(\delta \Delta N^{(s)}(\Delta \Gamma_{ijk})\), of the number of quanta in \(\Delta \Gamma_{ijk}\) is represented as:

\[ \delta \Delta N^{(s)}(\Delta \Gamma_{ijk}) = \langle n\rangle \sum_{l=1}^{f_0\Delta t_k} \Delta p_{ij}\left(E_{kl}^{(s)}\right) + p_{ij} \sum_{l=1}^{f_0\Delta t_k} \Delta n_{kl}^{(s)} + \]

\[ + \sum_{l=1}^{\Delta f_k\Delta t_k} \langle n\rangle p_{ij} + \sum_{l=1}^{0\Delta t_k} \delta \Delta N_i^{(s)}(\Delta \Gamma_{ijk}) + O. \tag{11} \]

Here it was assumed that

\[ \langle \Delta N(\Delta \Gamma_{ijk})\rangle = \langle n\rangle p_{ij} f_0\Delta t_k \tag{11a} \]

is the unconditional mathematical expectation of the number of quanta in \(\Delta \Gamma_{ijk}\); \(O\) denotes quantities of second order of smallness relative to the mean value.

Writing the expression for \(\delta \Delta N^{(s)}(\Delta \Gamma_{pqr})\) and multiplying it by (11), after averaging over realizations \(s\) (by virtue of the ergodicity of \(n()\)—over intervals \((\Delta t_k)\)), we find the covariance \(K_{ijk,pqr}^{3}\) between the random quantities: \(\Delta N(\Delta \Gamma_{ijk})\), the number of quanta in the cell \(\Delta \Gamma_{ijk}=\Delta E_i \Delta \Omega_j \Delta t_k\) of phase space, and \(\Delta N(\Delta \Gamma_{pqr})\), the number of quanta in \(\Delta \Gamma_{pqr}\), taking into account the assumptions (§ 3):

\[ \begin{aligned} K^\Delta_{ijk,pqr}={}& \langle n\rangle^2 a_{ij}a_{pq} \sum_{l=1}^{f_0\Delta t_k}\sum_{m=1}^{f_0\Delta t_r} K^E_{kl,rm} +p_{ij}p_{pq}\sum_{l=1}^{f_0\Delta t_k}\sum_{m=1}^{f_0\Delta t_r} K^n_{kl,rm} +{}\\ &+\langle n\rangle^2 p_{ij}p_{pq}\Delta t_k\Delta t_r K^f_{k,r} +\langle n\rangle p_{ij}a_{pq} \sum_{l=1}^{f_0\Delta t_k}\sum_{m=1}^{f_0\Delta t_r}K^{n,E}_{kl,rm} +{}\\ &+\langle n\rangle a_{ij}p_{pq} \sum_{l=1}^{f_0\Delta t_k}\sum_{m=1}^{f_0\Delta t_r}K^{E,n}_{kl,rm} +\langle n\rangle f_0 [p_{ij}\Delta t_k]^{1/2}[p_{pq}\Delta t_r]^{1/2}\delta_{ijk,pqr}, \end{aligned} \tag{12} \]

where

\[ \begin{aligned} K^E_{kl,rm}&=M[\Delta E_{kl}\cdot \Delta E_{rm}],& K^{n,E}_{kl,rm}&=M[\Delta n_{kl}\cdot \Delta E_{rm}],\\ K^n_{kl,rm}&=M[\Delta n_{kl}\cdot \Delta n_{rm}],& K^{E,n}_{kl,rm}&=M[\Delta E_{kl}\cdot \Delta n_{rm}],\\ K^f_{k,r}&=M[\Delta f_k\cdot \Delta f_r]. \end{aligned} \]

In calculating the component due to the statistical character of the radiation at the target, the binomial distribution was approximated by a Poisson distribution (taking into account: \(\langle n\rangle \simeq 10^{10};\ p_{ij}\ll 1\)). Let us replace the time-pulsed flux \(n_k(t)\) by an equivalent continuous \(n(t)\) so that, for any time interval

\[ \Delta t_k>\frac{1}{f_0}, \qquad \int_0^{\Delta t_k} n(t)\,dt=\sum_{i=1}^{f_0\Delta t_k} n_i \tag{13} \]

and the distribution of electrons over energies would be preserved (i.e., here we are speaking of the envelope \(n(t)\)); then we obtain:

\[ \begin{aligned} K^\Delta_{ijk,pqr}={}& \frac{\langle i\rangle^2}{e^2}a_{ij}a_{pq} \int_0^{\Delta t_k}\int_0^{\Delta t_r}K^E(t_1t_2)\,dt_1dt_2 +{}\\ &+p_{ij}p_{pq}\frac{1}{e^2} \int_0^{\Delta t_k}\int_0^{\Delta t_r}K^i(t_1t_2)\,dt_1dt_2 +\langle i\rangle p_{ij}a_{pq}\frac{1}{e^2} \int_0^{\Delta t_k}\int_0^{\Delta t_r}K^{i,E}(t_1t_2)\,dt_1dt_2 +{}\\ &+\langle i\rangle a_{ij}p_{pq}\frac{1}{e^2} \int_0^{\Delta t_k}\int_0^{\Delta t_r}K^{E,i}(t_1t_2)\,dt_1dt_2 +{}\\ &+\frac{\langle i\rangle^2}{f_0^2}p_{ij}p_{pq}\frac{1}{e^2} \int_0^{\Delta t_k}\int_0^{\Delta t_r}K^f(t_1t_2)\,dt_1dt_2 +{}\\ &+\frac{1}{e}\left[p_{ij}\int_0^{\Delta t_k}\langle i\rangle\,dt_1\right]^{1/2} \left[p_{pq}\int_0^{\Delta t_r}\langle i\rangle\,dt_2\right]^{1/2} \delta_{ijk,pqr}. \end{aligned} \tag{14} \]

This expression (11) determines the covariance between the number of quanta with energy \(E_i \div E+\Delta E_i\), propagating within the solid angle \(\Delta\Omega_j\), around \(\Omega_j\), during \(\Delta t_k\), and the number of quanta with energy \(E_p \div E_p+\Delta E_p\), propagating within the solid angle \(\Delta\Omega_q\), around \(\Omega_q\), during \(\Delta t_r\), i.e., for a simple random field \(N(\Gamma_{ijk})\), for which

\[ \Delta N(\Delta\Gamma_{ijk})=N(\Gamma_{ijk})\Delta\Gamma \]

and, correspondingly,

\[ K^{\Delta}_{ijk,pqr}=K[\Delta N(\Delta\Gamma_{ijk});\Delta N(\Delta\Gamma_{pqr})] =\Delta\Gamma^2 K[N(\vec{\Gamma}_{ijk});N(\vec{\Gamma}_{pqr})]. \tag{15} \]

It was mentioned above that the field \(N(\Gamma)\), describing the bremsstrahlung beam, is the limit of a sequence of simple fields, and its covariance function is the limit of the corresponding sequence of covariance functions of simple fields, i.e.,

\[ K_N(\Gamma_1\Gamma_2)=K[N(\Gamma_1);N(\Gamma_2)] =\lim_{\Delta\Gamma\to 0}\left[\frac{1}{\Delta\Gamma^2}K^{\Delta}_{ijk,pqr}\right]. \tag{16} \]

Taking into account \(\Delta\Gamma=\Delta E\Delta\Omega\Delta t=\Delta\Gamma'\Delta t\) from expressions (14 and 16), passing to the limit, we obtain

\[ K_N(\Gamma_1\Gamma_2)=\sum_{s=1}^{6} C_s^*(\Gamma'_1\Gamma'_2)K_s(t_1t_2), \tag{17} \]

where

\[ C_1^*(\Gamma'_1\Gamma'_2)=\frac{1}{e^2}\langle i\rangle^2 a^*(\Gamma'_1)a^*(\Gamma'_2) \qquad K_1(t_1t_2)=K^E(t_1t_2) \]

\[ C_2^*(\Gamma'_1\Gamma'_2)=\frac{1}{e^2} b^*(\Gamma'_1)b^*(\Gamma'_2) \qquad K_2(t_1t_2)=K^i(t_1t_2) \]

\[ C_3^*(\Gamma'_1\Gamma'_2)=\frac{1}{e^2}\langle i\rangle b^*(\Gamma'_1)a^*(\Gamma'_2) \qquad K_3(t_1t_2)=K^{i,E}(t_1t_2) \]

\[ C_4^*(\Gamma'_1\Gamma'_2)=\frac{1}{e^2}\langle i\rangle a^*(\Gamma'_1)b^*(\Gamma'_2) \qquad K_4(t_1t_2)=K^{E,i}(t_1t_2) \]

\[ C_5^*(\Gamma'_1\Gamma'_2)=\frac{1}{e^2 f_0^2}\langle i\rangle^2 b^*(\Gamma'_1)b^*(\Gamma'_2) \qquad K_5(t_1t_2)=K^f(t_1t_2) \]

\[ C_6^*(\Gamma'_1\Gamma'_2)=\frac{\langle i\rangle}{e}\,[b^*(\Gamma'_1)b^*(\Gamma'_2)]^{1/2} \delta(\Gamma'_1-\Gamma'_2) \qquad K_6(t_1t_2)=\delta(t_1-t_2). \]

Here \(e\) is the electron charge;

\(\langle i\rangle\) is the mean current from the target, under the condition that all electrons are stopped in it;

\(f_0\) is the mean repetition frequency of the acceleration cycles.

\[ b^*(\Gamma'_j)= \left. \frac{d\Sigma(E_e,E,\vec{\Omega})}{\partial E\partial\Omega} \right|_{\substack{E=E_j\\ \Omega=\Omega_j}} \qquad a^*(\Gamma'_j)= \left. \frac{\partial b^*(\Gamma'_j)}{\partial E_e} \right|_{E_e=E_0}, \tag{19} \]

the vector \(\Gamma'=(E,\Omega)\) lies on \(E\times\Omega\).

Passing from solid angles to irradiated areas by the known transformation \(d\Omega=\dfrac{1}{r_0^2}\,dF\), instead of \(N(E,\vec{\Omega},t)\) we shall write \(N_0(r,E,t)\). The covariance function of the field \(N_0(\Gamma)\), where \(\Gamma=(r,E,t)\), now has the form:

\[ K_{N_0}(\Gamma_1\Gamma_2)=\sum_{s=1}^{6} C_s(\Gamma'_1\Gamma'_2)K_s(t_1t_2), \tag{20} \]

where

\[ C_1(\Gamma'_1\Gamma'_2)=\frac{1}{e^2}\langle i\rangle^2 a(\Gamma'_1)a(\Gamma'_2) \qquad C_4(\Gamma'_1\Gamma'_2)=\frac{1}{e^2}\langle i\rangle a(\Gamma'_1)b(\Gamma'_2) \]

\[ C_2(\Gamma'_1\Gamma'_2)=\frac{1}{e^2} b(\Gamma'_1)b(\Gamma'_2) \qquad C_5(\Gamma'_1\Gamma'_2)=\frac{1}{e^2 f_0^2}\langle i\rangle^2 b(\Gamma'_1)b(\Gamma'_2) \]

\[ C_5(\Gamma'_1\Gamma'_2)=\frac{1}{e^2}\langle i\rangle b(\Gamma'_1)a(\Gamma'_2) \]

\[ C_6(\Gamma'_1\Gamma'_2)=\frac{1}{e}\langle i\rangle [b(\Gamma'_1)b(\Gamma'_2)]^{1/2}\delta(\Gamma'_1-\Gamma'_2). \]

Here

\[ a(\Gamma'_j)=\beta a^*(\Gamma'_j)\qquad b(\Gamma'_j)=\beta b^*(\Gamma'_j)\qquad \beta=\frac{1}{r_0^2}\Gamma''=(\mathbf r,E). \]

Mean-square fluctuations

The \(\delta\)-correlatedness of the random field \(N_0(\vec{\Gamma})\) compels us to understand it in the sense of a generalized function, since it is meaningless to speak of a probability distribution for \(N_0(\vec{\Gamma})\) at each particular point \(\vec{\Gamma}\), i.e., at a given point \(\vec r\) of the irradiated surface at a fixed instant of time \(t\), because the variance is infinite. However, any measuring instrument by means of which we register \(N_0(\vec{\Gamma})\) performs averaging over some volume of phase space \(\Delta\Gamma\): it has a definite energy resolution \(\Delta E\) and angular resolution \(\Delta\Omega\), a finite planar aperture \(\Delta F\), and an inertia \(\Delta T\). That is, instead of \(N_0(\vec{\Gamma})\), at the output of the meter we shall observe the quantity

\[ \Phi_0(\varphi)=\int_{\Gamma}\varphi(\Gamma)N_0(\Gamma)\,d\Gamma, \tag{22} \]

which is a functional determined by the characteristics \(\varphi(\vec{\Gamma})\) of the instruments. Let us calculate the variance \(D\Phi_0\) for the simplest case:

\[ \varphi(\Gamma)= \begin{cases} \dfrac{1}{\Delta\Gamma}, & \text{for } \Gamma\in\Delta\Gamma,\\[4pt] 0, & \text{for } \Gamma\notin\Delta\Gamma, \end{cases} \qquad \Delta\Gamma=\Delta F\Delta E\Delta T \]

Taking (20) into account:

\[ D\Phi_0(\varphi) = M\left\{\left[\int_{\Gamma}\varphi(\Gamma)\,[N_0(\Gamma)-\langle N_0(\Gamma)\rangle]\,d\Gamma\right]^2\right\} = \]

\[ = \iint_{\Gamma\Gamma}\varphi(\Gamma_1)\varphi(\Gamma_2)K_{N_0}(\Gamma_1\Gamma_2)\,d\Gamma_1d\Gamma_2 = \sum_{s=1}^{6} g_s k_s, \]

where

\[ g_s=\frac{1}{[\Delta\Gamma'']^2}\iint_{\Delta\Gamma''\,\Delta\Gamma''} C_s(\Gamma''_1\Gamma''_2)\,d\Gamma''_1d\Gamma''_2, \qquad \Delta\Gamma''=\Delta F\Delta E \]

\[ k_s=\frac{1}{\Delta T^2}\int_{0}^{\Delta T}\int_{0}^{\Delta T} K_s(t_1t_2)\,dt_1dt_2, \]

i.e., the variance will be finite, and the rms deviation will also be finite:

\[ \sigma_{\Phi_0}=\sqrt{D\Phi_0(\varphi)}, \]

but it will depend continuously on the characteristics of the measuring instrument.

The mean value of the field \(\langle N_0(\Gamma)\rangle\) is obtained from (11a) by passing to the limit, taking into account the transformation from solid angles to areas:

\[ N_0(\Gamma)=\lim_{\Delta\Gamma\to 0}\beta\left[\frac{1}{\Delta\Gamma}\langle n\rangle \rho_{ij} f_0 \Delta t_k\right]=\frac{1}{e}\langle i\rangle b(\Gamma''). \tag{23} \]

Probability distribution function

Although the probability distribution of the field \(N_0(\vec{\Gamma})\) does not exist at each point, it is nevertheless possible to speak of it in a generalized sense. Let \(\Delta\Gamma\) be an interval in phase space over which averaging is performed by the measuring device \(\varphi(\vec{\Gamma})\). Then, from each pulse \(n_k\), the distribution of \(\gamma\)-quanta in \(\Delta\Gamma\) will be Poissonian; it will also be Poissonian for a sequence of pulses fitting into the time-averaging interval \(\Delta T\), if the parameters of the electron beam \((n,E_e,f)\) are fixed. Since in reality these parameters fluctuate about their mean values, the Poisson distribution must be averaged, i.e.,

\[ p(\Delta N)=\int\!\!\int\!\!\int_{0}^{\infty} p(\Delta N/n;\,E_e;\,f)\,w(n;\,E_e;\,f)\,dn\,dE_e\,df, \tag{24} \]

where

\[ p(\Delta N/n;\,E_e;\,f)=\sum_{l=0}^{\infty} \frac{\langle \Delta N_{nEf}\rangle}{l!}\, e^{-\langle \Delta N_{nEf}\rangle}\, \delta(l-\Delta N) \]

is the probability density of the Poisson distribution;

\(w(n;\,E_e;\,f)\) is the joint distribution density of the parameters of the electron beam.

For betatrons of medium energies, the charge accelerated per cycle is \(n_k\simeq 10^{10}\) el.; if \(\Delta\Gamma\) is such that \(\langle \Delta N\rangle\) is sufficiently large (say, \(>10^2\) quanta), and \(w\) is concentrated near \((\langle n\rangle;\,E_0;\,f_0)\), then in (24) it is sufficient to retain only one term of the Edgeworth expansion, i.e., to obtain a Gaussian distribution for \(\Delta N\). But in this case the field itself, \(N_0(\Gamma)=\dfrac{\Delta N}{\Delta\Gamma}\), will also be normally distributed with the parameters defined above. Most practical calculations are explicitly or implicitly limited to adopting the Gaussian form of the distribution.

Conclusion

1. The random component of the function \(N_0(\vec{\Gamma})\) describing the distribution of quanta in phase space (i.e., after subtracting the mean value \(\langle N_0(\vec{\Gamma})\rangle\)) is the sum of two components:
   a) the component \(N_0'(\Gamma)\) is caused by fluctuations of the electron beam and is completely correlated in \(F\) and \(E\) for each \(t\) (this fact is used in differential schemes);

b) the component \(N_0''(\Gamma)\) is caused by the probabilistic character of the radiation in the target; it is “inhomogeneous white noise” with intensity

\[ \lambda(\Gamma)=\langle N_0(\vec{\Gamma})\rangle \]

1. The covariance function \(K_{N_0}(\vec{\Gamma}_1\vec{\Gamma}_2)\) has a separate temporal component. The temporal component \(K_s(t_1t_2)\) can effectively be determined experimentally; the space-energy compo-

component of the covariance function $\vec{C}_s(\vec{\Gamma}_1^{\,r}\,\vec{\Gamma}_2^{\,r})$ can be found by calculation.

REFERENCES

1. I. M. Gel'fand, N. Ya. Vilenkin. Some Applications of Harmonic Analysis. Issue 4, FM, 1961.

2. I. I. Gikhman, A. V. Skorokhod. Introduction to the Theory of Random Processes. Nauka, 1965.

3. V. M. Gol'danskii. Statistics of Counts in the Detection of Nuclear Radiations. Atomizdat, 1959.

4. V. S. Pugachev. Theory of Random Functions. FM, 1963.

5. S. V. Starodubtsev, A. M. Romanov. Interaction of $\gamma$-Radiation with Matter. Nauka, Uzbek SSR, T, part 1, 1964.

6. U. Fano, M. Berger, S. Spencer. Transport of $\gamma$-Radiation. Atomizdat, 1963.