ON THE INTERRELATION AND DEGREE OF INFLUENCE OF THE INPUT PARAMETERS OF A HIGH-CURRENT BETATRON WITH EXTERNAL INJECTION ON THE INTENSITY OF γ-RADIATION

M. S. Aleinikov, V. P. Ivanchenkov, V. A. Kochegurov, B. E. Tomilov

(Presented by the scientific seminar of the Scientific Research Institute of Nuclear Physics)

In the present work, estimates are given of the relations between the input parameters and the output quantity—the intensity of the γ-radiation of a betatron.

From theoretical considerations and from operating experience it is known [1] that the dependence of the intensity on such parameters as injection phase, inflection, voltage on the electromagnet winding, injection and inflection, currents in the electromagnet winding and the filament of the electron gun, has an extremal character. By selecting definite values of the parameters listed above, the maximum value of the γ-radiation intensity is obtained. It is necessary to estimate how substantial the relation is between the input parameters and the output quantity (the degree of correlation), and also to estimate the degree of interrelation among the input quantities.

Since it is known [1] that the static characteristics of the betatron are substantially nonlinear, it is impossible to estimate the degree of relation between the input parameters and the output quantity by means of the correlation coefficient [2].

Therefore the degree of relation was estimated by means of correlation ratios. The variation of the values of the quantity \(y\) is due to the variability of the quantities \(x_i\) related to it (where \(i = 1, 2, 3 \ldots\), etc.) and of a number of other factors influencing \(y\) and not depending on \(x_i\) [2]. Characterizing the variation of the quantity \(y\) by its variance, this statement may be represented analytically by the following formula [2]:

\[ \sigma_y^2 = \sum \sigma_{y1i}^2 + \sum \sigma_{y2i}^2, \tag{1} \]

where \(\sigma_y^2\) is the total variance of the quantity \(y\), i.e., the variance of the points of the correlation field relative to the line of mathematical expectation \(m_y\);

\[ \sigma_y^2 = M\left[(y - m_y)^2\right]; \tag{2} \]

\(\sigma_{y1i}^2\) is the variance of the points of the correlation field relative to the conditional mathematical expectation for the \(i\)-th parameter

\[ \sigma_{y1i}^2 = M\left[(y - m_{y/x_i})^2\right]; \tag{3} \]

\(\sigma_{y2i}^2\) is the variance of the regression curve relative to the mathematical expectation of the quantity \(y\)

\[ \sigma_{y2i}^2 = M\left[(m_{y/x_i} - m_y)^2\right]. \tag{4} \]

The correlation ratio between the quantities \(y\) and \(x_i\) is defined as the part of the total variability of the quantity \(y\) caused by a change in the value of the quantity \(x_i\):

\[ \eta_{y/x_i}=\sqrt{\frac{\sigma^2_{yxi}}{\sigma_y^2}} . \tag{5} \]

The conditional mathematical expectation of the output quantity (the intensity of \(\gamma\)-radiation) for specified values of the input quantities is calculated by the formula [3]:

\[ m_{y/x_i}=\sum_1^n y_k \cdot P\{y_k/x_{ik}\}, \tag{6} \]

where \(P\{y_k/x_{ik}\}\) is the conditional probability, i.e., the probability that the event \(y_i\) occurs under the condition that the event \(x_i\) has occurred.

The conditional probability is determined by formula (7) [3]

\[ P\{y_k/x_{ik}\}= \frac{P\{(y=y_k)\cdot n(x=x_{ik})\}}{P\{y/x_{ik}\}}, \tag{7} \]

where \(P\{(y=y_k)\cdot n(x=x_{ik})\}\) is the probability of the simultaneous existence of two events;
\(P\{y/x_{ik}\}\) is the unconditional probability.

To use formulas (6) and (7) for determining conditional mathematical expectations, correlation fields are constructed or tables are compiled from the measurement data. We took one-hour segments from three-hour realizations of the input and output quantities of the betatron. Having set a level of one of the input parameters, we find the number of intersections of this level with the realization of the given input quantity and the corresponding points for these intersections on the realization of the output quantity (Fig. 1). In doing so it is assumed that

Fig. 1

the other input parameters are unchanged. Then, plotting the corresponding values (levels) of the input quantity along the abscissa axis, and along...

on the ordinate axes the values of the output quantity corresponding to the points of intersection of the given level \(x_i\) with its realization; we construct correlation fields (Figs. 2, 3, 4, 5). Using expression (7), we determine the conditional probability. In the numerator of expression (7) is substituted the number of points of a definite value of the output quantity for a given

Fig. 2. Correlation field and conditional mathematical expectation (regression line) of the intensity of \(\gamma\)-radiation when the injection phase is varied.

value of the input parameter; in the denominator, the number of intersections of the corresponding level of the input quantity with its realization.

For example, in Fig. 2, for an injection-phase level equal to 46.8, 14 points are plotted: 2 at the \(\gamma\)-radiation intensity level equal to 1 relative unit; 6 points at 6 relative units; 1 at 7 units; 1 at 8 units; 2 at 8.5 units; 1 at 9 units; 1 at 10.5 units. Hence:

\[ P(y=1)\cdot n(x=46.8)=\frac{2}{N}, \]

where \(N\) is the number of intersections of all selected injection-phase levels with its time realization.

The unconditional probability is equal to:

\[ P\{y/x_{ik}=46{,}8\}=\frac{14}{N}. \]

Fig. 3. Correlation field and conditional mathematical expectation of the intensity of \(\gamma\)-radiation when changing the voltage on the electromagnet

The conditional probability for the intensity \(I=1\)

\[ P\{y_k=1/x_{ik}=46{,}8\}=\frac{2}{14}, \]

for \(I=7\)

\[ P\{y_k=7/x_{ik}=46{,}8\}=\frac{1}{14}, \]

for \(I=8\)

\[ P\{y_k=8/x_{ik}=46{,}8\}=\frac{1}{14} \]

and so on.

Using formula (6), we determine the conditional mathematical expectation at a given level of the input quantity. Thus, for the injection phase level 46.8 (Fig. 2),

\[ m_{y/x_{ik}=46{,}8} = 1\cdot\frac{2}{14} + 6\cdot\frac{6}{14} + 8\cdot\frac{1}{14} + 7\cdot\frac{1}{14} + 8{,}5\cdot\frac{2}{14} + 9\cdot\frac{1}{14} + \]

\[ +\,10{,}5\cdot\frac{1}{4}=6{,}4. \]

Thus, conditional mathematical expectations have been determined for various levels of the input quantities, and regression curves have been constructed (Figs. 2, 3, 4, 5).

Then, using expressions (2), (4), and (5), we determine \(\sigma_y^2\), \(\sigma_{y_s}^2\), and \(\eta\). As a result, for the hourly realizations we obtain:

Fig. 4. Correlation field and conditional mathematical expectation of the intensity of \(\gamma\)-radiation as the voltage on the inflector plates changes

1. Total variance of the output quantity (the intensity of \(\gamma\)-radiation)

\[ \sigma_y^2 = 14. \]

1. Variance of the output quantity relative to the regression line when the injection phase changes (all other quantities are assumed constant)

\[ \sigma^2_{y/\varphi_{\text{inj}}} = 8.14. \]

Correlation ratio \(\eta = 0.75\).

1. The same when the inflection phase changes

\[ \sigma^2_{y/\varphi_{\text{inf}}} = 1.314,\quad \eta = 0.31. \]

1. The same when the injection voltage changes

\[ \sigma^2_{y/U_{\text{inj}}} = 1.52;\quad \eta = 0.35. \]

1. The same when varying the injection voltage

\[ \sigma^2_{y/U_{\text{inj}}}=2.71;\quad \eta=0.44. \]

1. The same when varying the voltage on the electromagnet winding:

\[ \sigma^2_{y/U_M}=2.72;\quad \eta=0.44. \]

1. The same when varying the current in the electromagnet winding

\[ \sigma^2_{y/I_M}=0.842;\quad \eta=0.25. \]

1. The same when varying the cathode heating current of the gun

\[ \sigma^2_{y/I_{\text{h}}}=0.152;\quad \eta=0.105. \]

Fig. 5. Correlation field and conditional mathematical expectation of the intensity of γ-radiation when varying the injection voltage

It follows from the calculation given above that the injection phase has the greatest relationship with the output quantity; next, in decreasing order, come: the voltage on the electromagnet, the injection voltage, the current in the electromagnet winding, and, finally, the cathode heating current of the gun.

In an analogous manner, the correlation ratios for estimating the stochastic dependence between the input parameters were calculated. The correlation ratios have the following values: the injection phase and inflection phase are mutually correlated, \(\eta = 0.42\). The electromagnet current and voltage are completely correlated, \(\eta = 0.94\). For the filament current and the voltage on the electromagnet winding, \(\eta = 0.56\). For the injection and inflection voltages, \(\eta = 0.09\). For the injection voltage and injection phase, \(\eta = 0.08\). For the voltage on the electromagnet winding and the injection phase, \(\eta = 0.092\).

Thus, the injection voltage and phase, the voltage on the electromagnet winding, and the inflection voltage have a weak relationship and may be regarded as stochastically independent variables.

It is known [4, 5] that, in order to obtain a mathematical model of processes in an object by the method of regression analysis, the following conditions must be satisfied:

1. The output quantity (in our case, the intensity of \(\gamma\)-radiation) must be distributed according to the normal law, and its changes in time must have a stationary character.
2. The input quantities must be mutually independent.
3. The dependence between the input and output quantities (the static characteristic) must be approximated by functions of the form:

\[ f_1(x)=a_0+a_1x \quad \text{or} \quad f_2(x)=a_0+a_1x+a_2x^2. \]

Approximation by nonlinear functions is also possible,

\[ \left(\frac{1}{x};\ \ln x\ e^x\ \text{etc.}\right), \]

but in this case a new variable is introduced in order to preserve the above-indicated form of the approximating functions.

We have established that the output quantity of the betatron (the intensity of \(\gamma\)-radiation) has a normal distribution, and that its changes in time are stationary in character.

From consideration of the degree of influence of each parameter given above, the following parameters may be regarded as independent input variables:

1) injection phase,
2) inflection voltage,
3) voltage on the electromagnet winding,
4) injection voltage.

Since the remaining variables (inflection phase, current in the electromagnet winding, and filament current) have a weak relationship with the output quantity and their influence on the output parameter is insignificant, they are not taken into consideration further. Proceeding from theoretical ideas about the processes in the betatron and from the form of the conditional mathematical expectations, we choose approximating functions for the injection phase, the voltage on the electromagnet winding, and the injection voltage of the form:

\[ y=f(x_i)=a_{0i}+a_{1i}\cdot x_i+a_{2i}x_i^2, \tag{7} \]

and for the inflection voltage, of the form

\[ y=f(x)=\frac{1}{a_0+a_1x}. \tag{8} \]

From the analysis presented above, the following conclusions may be drawn:

1. The intensity of $\gamma$-radiation is determined primarily by four independent input parameters: the injection phase, the voltage on the electromagnet winding, the inflection voltage, and the injection voltage.

2. The following form of approximating function was chosen:

\[ f(x_i)=a_{0i}+a_{1i}x_i+a_{2i}x_i^2 \quad \text{and} \quad f(x)=\frac{1}{a_0+a_1x}, \]

on the basis of theoretical considerations concerning the acceleration process in the betatron and from the form of the functions of conditional mathematical expectations (regression lines in Figs. 2, 3, 4, 5).

1. The mathematical model of the betatron may have either the form of a quadratic polynomial [5]

\[ F=\sum_{1}^{4}\left(a_{0i}+a_{1i}x_i+a_{2i}x_i^2\right), \tag{9} \]

or the form of a product of quadratic polynomials [5]

\[ F=C\prod_{1}^{4}\left(a_{0i}+a_{1i}x_i+a_{2i}x_i^2\right). \tag{10} \]

REFERENCES

1. L. M. Anan’ev, A. A. Vorob’ev, V. I. Gorbunov. Induction Accelerator—Betatron. Atomizdat, 1956.

2. E. L. Inkovich. Statistical Methods in the Automation of Production. Energiya Publishing House, Moscow–Leningrad, 1964.

3. B. L. van der Waerden. Mathematical Statistics. IL, 1960.

4. A. Hald. Mathematical Statistics with Technical Applications. IL, Moscow, 1956.

5. V. M. Ordyncev. Mathematical Description of Automation Objects. Mashinostroenie, Moscow, 1965.