Radiation Detection and Measurement

Response Function and Energy Resolution

Nsignal, the response function should have a Gaussian shape, because N is typically a large number. Hence, we have FWHM = 2.35σ.

The response of many detectors is approximately linear, so that the average pulse amplitude is H = KN, where K is a proportionality constant. The standard deviation of the peak in the pulse height spectrum is then √(FWHM/2.35) = √(σ/2.35).

We would then calculate a limiting resolution R due only to statistical fluctuation in the number of charge carriers as √(FWHM/2.35) / (H/KN√N) = √(σ/2.35) / (KN√N).

Note that, in this limiting case, in order to achieve an energy resolution better than 1% one must have N greater than 55000. An ideal detector would have as many charge carriers generated per event as possible, so that this limiting resolution would be as small as possible.

Careful measurement of the energy resolution of some type of radiation detectors have shown that the achievable values of R can be lower by a factor as large.

as 3 or 4 than the minimum predicted by the statistical arguments given above. The Fano factor has been introduced in an attempt to quantify the departure from the pure Poisson statistics and is defined as observed variance N∈F = Poisson predicted variance( = N )

We thus obtain the following resolution limit √√2.35 K FN FR = =2.35 Statistical limit KN N

Although the Fano factor is substantially less than unity for semiconductor diode detectors and proportional counters, other types such as scintillation detectors appear to show a limiting resolution consistent with Poisson statistics and the Fano factor would, in these cases, be unity.

Any other source of fluctuations in the signal chain will combine with the inherent statistical fluctuations from the detector to give the overall energy resolution of the measuring system: 2 2 2 2 FWHM FWHM FWHM FWHM ...( ) =( ) +( ) +( ) + overall statistical noise drift.

Detection efficiency In general, the radiation entering the detector will not

deposit all its energy in the active volume of the detector itself. This is especially true for radiations such as gamma rays or neutrons, while it is less true for primary charged radiation, such as alpha and beta particles. In fact, while the former need to undergo a significant interaction before detection is possible, the second ones will typically form enough ion pairs along a small fraction of their path to ensure that the resulting pulse is large enough to be recorded. It then becomes necessary to have a precise figure for the detector efficiency to record pulses, in order to relate the number of pulse counted to the number of particles incident.

It is convenient to subdivide counting efficiencies into two classes: absolute and intrinsic. Absolute efficiencies are defined as number of pulses recorded ϵ = abs number of particles emitted by source and are dependent not only on detector properties but also on the details of the counting geometry (primarily the distance from the source).

intrinsic efficiency is defined as number of pulses recorded ϵ = int number of particles incident on the detector and no longer includes the solid angle subtended by the detector as an implicit factor. The two efficiencies are simply related for isotopic sources by 4πϵ = ϵint abs Ω where Ω is the solid angle of the detector seen from the actual source position. It is much more convenient to tabulate values of intrinsic rather than absolute efficiencies, because the geometric dependence is much milder for the former. The intrinsic efficiency usually depends on the detector material, the radiation energy and the physical thickness of the detector in the direction of the incident radiation.

Counting efficiencies are also categorized by the nature of the event recorded. If we accept all pulses from the detector, then it is appropriate to use total efficiencies. In practice, however, any measurement system always imposes a requirement that pulses be larger than some finite threshold.

level set to discriminate against very small pulses from electronic noise sources. The peak efficiency, instead, assumes that only those interactions that deposit the full energy of the incident radiation are counted.

The total and peak efficiencies are related by the peak-to-total ratio ϵ_peak = ϵ_total.

It is often preferable from an experimental standpoint to use only peak efficiencies, because the number of full energy events is not sensible to some perturbing effects such as scattering from surrounding objects or spurious noise.

5.3 DEAD TIME

In nearly all detector systems, there will be a minimum amount of time that must separate two events in order that they be recorded as two separate pulses. The minimum time separation in this sense is called dead time of the counting system. Because of the random nature of radioactive decay, there is always some probability that a true event will be lost because it occurs too quickly following a preceding event. These "dead time losses" can

Become rather severe when high counting rates are encountered, and any accurate counting measurements made under these conditions must include come correction for these losses..

Dead time behavior

Two models of dead time behavior of counting systems have come into common usage: paralyzable and nonparalyzable response. These models represents idealized behavior, one or the other of which often adequately resembles the response of a real counting system. The fundamental assumption are illustrated in figure #5. At the center, a time scale is shown on which six randomly spaced events are indicated. At the bottom is the corresponding dead time behavior of a nonparalyzable detector. A fixed time τ is assumed to follow each true event that occurs during the "live period" of the detectors; true events that occur during the dead period are lost. In contrast, the behavior of a paralyzable detector is shown in the top line of figure #5. The same dead time τ is assumed to follow each true.

interaction; true events that occurs during the dead period, however, although still not recorded as counts, are assumed to extend the dead time by another period τ following the lost event.

Figure #5: illustration of two assumed models of dead time behavior for radiation detectors

The two models predict the same first-order losses and differ only when true event rates are high. The behavior of real counting system will often be an intermediate between these extremes.

Let's now consider the response of a detector system to a steady-state source of radiation. We would like to obtain an expression for the true interaction rate, n, as a function of the measured rate, m, and the system dead time, τ, so that the appropriate corrections can be made.

In the nonparalyzable case, the fraction of all time that the detector is dead is given by mτ. Therefore, the rate at which true events are lost is nmτ. But because n-m is another expression for the rate of losses, n m nm- = τHence,