Determination of the Limiting Bandwidth of a Scanning Detector in an Inhomogeneous Radiation Field

V. A. Vorob’ev, A. N. Laferov

Presented by the joint seminar of the sectors
DSM and MRD of the Scientific Research Institute of EI

The question of determining the frequency spectrum at the output of a moving detector has been raised more than once in various works [1]. The difficulty lies in the complexity of analytically writing and calculating the function that describes the response of a detector with specified parameters to the inhomogeneity of the radiation field. Knowing this function, it is not difficult to determine the signal spectrum. In such a formulation of the problem, the spectrum will depend on the shape and dimensions of the collimator, on the velocity of the detector’s motion, on the distribution function of the inhomogeneities of the field under study, and on the sensitivity over the cross section of the detector collimator.

It is known that, for a given input signal, the frequency limits can be found from the modulus of the transfer function. Assuming that the transient processes in the system decay rapidly, determination of the transfer function reduces to determination of the spectral characteristic, that is, when \(RS = 0\), where \(S\) is the parameter of the transfer function.

Let us find the response of the detector to some distribution function \(\psi(x,y)\), bounded by the region of the collimator \(f(x)\), when the collimator moves in a static radiation field from \(0\) to \(\ell\). The detector response in general form may be written as

\[ \psi(\ell)=\int_{0}^{\ell}\int_{0}^{d_2} f(x\ell)\,\psi(x,y\ell)\,dx\,dy, \tag{1} \]

where the integration is carried out within the dimensions of the collimator \((d_1,d_2)\).

If we assume that the signal bounded by the detector field has a constant ordinate, then (1) takes the form

\[ \psi(\ell)=\int_{0}^{d} f(x)\,\psi(x)\,dx, \tag{2} \]

where \(f(\xi)\) is the weight of the distribution function \(\psi(x)\).

Taking into account that the integration is carried out within the dimensions of the collimator—

...of the collimator when the argument of the distribution function \(\varphi(x)\) changes in the interval \(0<x<\ell\), the detector response is determined by the “sliding” integral:

\[ \psi(x)=\int_{0}^{d} f(\tau)\,\varphi(x+\tau)\,d\tau, \tag{3} \]

where \(0<\tau<d\) for each \(x\) from the region \(0<x<\ell\).

This expression makes it possible to find the response of the filter for a known collimator configuration \(f(\tau)\) and distribution function \(\varphi(x)\).

Solving expression (3) can be simplified if certain restrictions are introduced:

a) the collimator has the form of a circle of diameter \(d\);

b) the detector sensitivity is uniform over the entire cross-sectional area.

Let us calculate \(\psi(x)\) for the case when \(\varphi(x)\) has the form of a rectangular pulse with amplitude \(A\) and duration \(\delta\).

Then

\[ f(\tau)=2\sqrt{\left(\frac{d}{2}\right)^2-\left(\tau-\frac{d}{2}\right)^2} \]

is the weight function (the envelope of the circular collimator), shifted relative to the origin to the right by \(\dfrac{d}{2}\),

\[ \varphi(x)= \begin{cases} A, & 0<x<\delta,\\ 0, & x>\delta, \end{cases} \]

a rectangular pulse of duration \(\delta\), shifted to the right relative to the origin. Substituting these expressions into (3), we obtain the tabular integral:

\[ \psi(x)=2\int_{0}^{d}\sqrt{\left(\frac{d}{2}\right)^2-\left(\tau-\frac{d}{2}\right)^2}\cdot A(x-\tau)\,d\tau. \tag{4} \]

Solution (4), taking into account that \(0<\tau<d\) for each \(x\) from the region \(0\div\delta\), is approximated with sufficient accuracy by a triangular symmetric pulse of duration \(2\delta\) and amplitude \(A\).

The spectral function of such a pulse has the form:

\[ S(\omega)=\frac{4A}{\delta}\,\omega^2\sin^2\frac{\delta}{2}\omega \quad [2]. \tag{5} \]

It is known that the frequency region bounded by the second zero of this function satisfies the majority of practical cases, since with...

in this case more than 95% of the energy of the spectrum is transmitted [2],

\[ \text{but}\qquad 2\delta=\frac{S}{V}=\frac{d}{2V}; \]

where \(d\) is the diameter of the collimator, \(V\) is the speed of motion of the collimator, then

\[ f=\frac{1}{2\delta}=\frac{2V}{d}. \]

Expression (5) is valid both for single pulses and for a periodic sequence of pulses.

Below is Table I of the limiting frequencies for the most frequently encountered scanning speeds and collimator sizes when measuring bremsstrahlung radiation by means of scintillation detectors.

Table I.

Limiting frequencies for various collimator sizes and detector speeds in the radiation field.

Literature

1. Nadavny O. I. Candidate’s dissertation. Tomsk, 1969.
2. Mason S., Zimmerman H. Electronic Circuits, Signals, and Systems. Moscow, 1963. “Foreign Literature.”