Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -1 Chapter General Properties of Radiation Detectors Ionizing radiation is ost coonly detected by the charge created when radiation interacts with the detector. The definition is, after all, ionizing radiation. Crudely, the efficiency with which a detector easures a particular type of radiation depends on the efficiency with which the radiation type creates charge within the detector. When charged particles like protons or electrons are incident on the detector, they continuously interact with atoic electrons of the detector aterial and produce electron-ion (or electron-hole) pairs along their tracks. This initial charge produced by radiation is ost iportant inforation and further processed by a proper signal processing electronics after collection. When neutral radiation fields like those consist of gaa-rays or neutrons are easured, a neutral particle interacts with the detector aterial and then the product secondary charged particle (electron, proton, alpha etc.) deposits its energy in the detector..1. Operation odes Assue that a radiation interaction with a detector created charge Q within the detector volue. As the collection process of charge Q progresses, a current signal i(t) is induced at the collection electrode. i(t) tc itdt () Q tc t t c : charge collection tie The detection event rate (or counting rate) is dependent on both radiation field and detector efficiency. If the event rate is not high, each detection event can be well separated and analyzed. i(t) Current flowing in detector tie
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology - A. Current ode A siple way of easuring a detector signal is current easureent. In this ode, a current eter is connected to the detector output. Since the current level is in pa or na, a precise eter is required. Given that the current eter has a response tie T, the observed current fro a sequence of events at tie t will be 1 i ( t) T t t T i( t' ) dt' The response tie is usually longer than the tie between individual detection events, so that an average current is recorded at a tie t. The current ode is used when event rates are very high, which akes a stable current. B. Pulse ode In the pulse ode, we preserve the inforation on energy and tiing of individual events, i.e. the inforation on the signal aplitude and tie of occurrence is preserved. The signal shape fro a radiation detector depends on the electronics to which the detector is connected as well as the detector response. Often, the input stage of the electronics is an RC circuit. Detector C R V(t) In the circuit, R represents the easuring circuit input resistance, C is the sued capacitance of the detector, the cable and the input capacitance of the preap. V(t) is the tie dependent voltage across the load resistor; V(t) is the signal that is produced. = RC is the tie constant of the easuring circuit. Fig..1. Assued output signal and signal voltage V(t).
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -3 There are two extrees of operation: sall RC ( << t c ) and large RC ( >> t c ). Circuits with large RC are coon in radiation spectroscopy systes, in which the energy distribution of the incident radiation is easured. Many of the detector types used in the 4R6 laboratory will have output pulses of roughly the shape described in Fig..1 when viewed on an oscilloscope. With the large RC circuits, the axiu voltage becoes V ax = Q / C, where, Q is the charge produced in the detector. If C is fixed and stable, V ax is directly proportional to Q. Therefore, easuring the pulse height V ax is equivalent to easureent of the charge (or deposited energy) produced by a radiation interaction... Energy resolution Fig... Exaples of good and poor energy resolutions. The energy resolution of a radiation detection syste is a ost iportant property when radiation spectroscopy is intended. Fig.. shows pulse height spectra fro two detection systes for the sae radiation source. When a onoenergetic source is easured and each syste produces the corresponding response of a siple peak, the syste with a good resolution gives a narrower peak width, which is beneficial for separating closely located peaks. In an ideal case, a delta function type resolution would be the best case, however, a real radiation spectroscopy syste always produces a finite energy width in the peak shape. The energy resolution depends on the type of the detector and the configuration of the noise filtering in pulse processing.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -4 Fig..3. Exaples of good and poor energy resolutions. When ono-energetic radiation is incident on a detector and the energy (or pulse height) spectru is easured, the peak produced by full-energy deposition is Gaussian as shown in Fig..3 if the nuber of counts is sufficient enough. The energy resolution of a peak can be expressed as the width of the peak. Thus, a good resolution eans narrower peak width. A conventional way of defining the width is Full Width at Half Maxiu (FWHM). Fro the definition of Gaussian function Ap G( E) exp ( E ) ( E ) where, A p : peak area or nuber of counts, : standard deviation, E : peak center (or the energy of the incident radiation), FWHM has a relation of FWHM =.355 with the standard deviation. One of the origins of the peak width is the statistical fluctuation in the nuber of charge carriers produced by radiation interaction. If this process follows a Poisson distribution, the standard deviation is the square root of the nuber of charge carriers. Since the nuber is proportional to the E, the peak width has a dependence of E and becoes broader as the incident radiation energy E increases. When a fractional instead of absolute value is required, the percent resolution can be defined as R 1 FWHM E For exaple, if a gaa-ray spectru shows a peak with kev FWHM at 133 kev, the percent resolution at this energy becoes.15 %..3. Detection efficiency Detection efficiency is another iportant property of a radiation detector. As the general eaning iplies, detection efficiency represents the probability of detection for a single radiation quantu. An accurate and precise calibration of detection efficiency is very iportant for quantitative easureent of an unknown radiation source and will be discussed in detail in the photon spectroetry chapter.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -5 An ideal radiation detection syste should have a high efficiency and a good energy resolution, which is hardly et in practical applications. Therefore, when you plan to set up a radiation detection syste, the detector type and aterial should be carefully chosen according to the priority of the easureent. A coproise is usually unavoidable. In general, detection efficiency is dependent on both radiation interaction and size of a detector. Charged particles (electron, proton, alpha) interact ore easily than neutral ones (X-ray, gaaray, neutron) and give high efficiencies. Theoretically, if we could create detectors with large enough volues, we could always detect 1 % of the particles incident on the detector. However, this is either ipractical or even ipossible, e.g. seiconductor crystals just cannot be grown large enough to be 1 % efficient for high energy photons. Neutrino detectors are already built in ineshafts. This is why the concept of detection efficiency was created. Not all particles can be detected, but if the proportion of detected particles is known, the nuber of particles can be calculated fro the nuber detected. Depending on the way of defining the nuber of radiation quanta, either absolute or intrinsic efficiency can be used. The definition of the absolute efficiency is Nuber of pulses det ected abs. Nuber of radiation quanta eitted fro source For exaple, if we have a radiation source eitting Y particles per second, C is the easured counting rate, and we know abs, we can calculate Y fro Y C abs [ particles / s] A shortcoing of the absolute efficiency is that it changes every tie when the detection geoetry is changed, for exaple, the source or the detector is relocated to a different position. To ake the efficiency alost independent of the detection geoetry, the intrinsic efficiency was introduced: int Nuber Nuber of pulses det ected of radiation quanta incident on det ector Now, the efficiency is defined per nuber of radiation quanta incident on the detector, so that the influence of the detection geoetry is uch relieved and the efficiency is nearly independent of the geoetry. However, at very short distances between detector and source, even the intrinsic efficiency ay vary significantly due to variation in the path length distribution and therefore, care ust be taken in this case. The intrinsic and absolute efficiencies can be related to each other by the probability of incidence on the detector. The solid angle is used to represent this probability. The definition of the solid angle is the fraction of a specific angular range (both polar and aziuthal angles) in 3- diensional space. There are two units used for the solid angle. The fractional solid angle akes a sphere (i.e. all direction) 1 while the steradian [sr] akes 4. The fractional solid angle has a
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -6 eaning of probability. Both solid angle units can be converted by ultiplying or dividing by 4. Practically, the steradian unit is ostly eployed. Thus, int can be converted into abs by abs int 4 when the solid angle is given in [sr]. Suppose an angular range is defined in ( 1, ), ( 1, ) intervals (: polar, : aziuthal angles). This angular range will ake a surface area of S on the surface of a sphere with radius r as S r 1 1 sin dd Since the solid angle is proportional to the surface area but has to be independent of the radius, it can be defined as S 4 sin d d d [ sr] 4r 1 1 1 1 in steradian. In general situation, the solid angle can be calculated by integrating the differential solid angle d over given angular intervals. As a siple exaple, suppose a point source and a cylindrical detector with radius a. The solid angle in this case becoes d sin dd (1 ) [ sr] d a When the distance d is close to, the solid angle becoes, which is equivalent to that of a heisphere. If the distance is uch longer than the radius a, the spherical surface area can be approxiated to the cross-sectional area a and the solid angle becoes S a [ sr] r r Efficiencies are also classified by the fraction of the energy deposited. Depending on interactions involved with each radiation particle, there are two possibilities of energy deposition: full or partial energies of the incident radiation. If we don t care how uch fraction of energy is deposited, i.e. accept all pulses fro the detector, the total efficiency is used. The full energy deposition events for a peak in the energy spectru as shown in Fig..3 and the peak efficiency is defined with full energy events. The detector efficiency should be specified according to both criteria. For exaple, the conventional forat used for gaa-ray detectors is the intrinsic peak efficiency.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -7.4. Dead tie Detector Preap Shaping ap MCA Fig..4. Pulse processing systes for radiation detectors. Typical pulse processing systes for radiation detectors are shown in Fig..4. The first one is for radiation spectroscopy while the second is a siple counting syste with a Geiger-Müller (GM) gas detector. In the first case, radiation is incident on the detector, where charge is created. A voltage pulse is passed to the aplifier where it is shaped and aplified. The pulse is then passed to the MultiChannel Anaylzer (MCA) where its height is digitized. All of these processes take tie. While one pulse is being processed, another event cannot be. The tie this takes is called the dead tie. The dead tie of a syste is the suation of all the processing ties of the different coponents - detector, aplifier, MCA. In the laboratory, you will easure the dead tie in a GM tube. This is a very siple syste, as shown above, and essentially you will be looking at the dead tie of the GM detector itself; the tie that the GM tube needs to process a pulse. If dead tie losses are not accounted for, this can lead to isleading results e.g. source activities will be underestiated. There are two odels for dead tie behavior: paralyzable (or extending) and non-paralyzable (or non-extending). These are idealized responses that predict extree behaviour. True systes, being a cobination of coponents, will often be soewhere between the two odels. Fig..5. Models of dead tie behavior.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -8 A. Non-paralyzable (non-extending) odel A fixed dead tie follows each event that occurs during the live period of the detector. Events that occur during the dead period are not recorded and have no effect on the syste. In Fig..5, we end up recording only 4 counts instead of 6 real events. Assue we have a detector syste with a steady state source e.g. an extreely long-lived radioisotope and let s define C n = true event rate, C = recorded count rate, = syste dead tie. In 1 s interval, we have C counts (each count with a tie width of ) and the event loss probability becoes C. Accordingly, the rate of the loss events is expressed as C nc. The rate of event loss is also C n C. Therefore, C C n C C nc C 1 1 n 1 C (or C ) C n B. Paralyzable (extending) odel In this odel, a fixed dead tie also follows each event during the live period of the detector. However, events that occur during the dead period, although not recorded, still create another fixed dead tie on the syste following the lost event. In Fig..5, we end up recording only 3 events rather than 6 true events. In extree cases, where the count rate is high, we can end up switching off the syste, as pulses and dead ties overlap and we record no events, hence the ter paralyzed. The dead periods are now not always of a fixed length, so the true event rate obtained in the non-paralizable odel is not effective here. Before deriving the dead tie, we need to investigate the distribution function for tie intervals between successive events. Fro statistics, the probability of the binoial distribution is given as n x n x P x! ( ) p (1 p ( n x)! x! ), where, n is the nuber of trials, p is the success probability per trial, x is the nuber of events occurred. The ean value and the variance of the distribution are np and = np(1- p), respectively. As an extree case, if the probability p is very sall, the binoial distribution reduces to x e P( x), x! which is the Poisson distribution. The Poisson distribution can be conveniently adopted to describe the tie interval distribution between successive events of a radiation detector if general requireents are satisfied. Assuing an event has occurred at tie t =, the differential probability that the next event will occur at t = t within a differential tie dt is Probability of next event Probability of no event Probability of an = taking place at t=t within dt in (, t) interval event during dt P 1 (t)dt P() C n dt'
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -9 Fro the Poisson distribution, P() becoes C nt' ( C nt' ) e C nt' P() e,! and accordingly, C ' P1 ( t' ) dt' C nt ne dt'. The probability of observing an interval larger than can be obtained by integrating P ( t' ) dt' e 1 C n The rate of such intervals is then obtained by ultiplying with the true rate C C C n ne There is no explicit solution for C n, it ust be solved iteratively to calculate C n fro easureents of C and. A plot of the observed rate C as a function of the true rate C n is given in Fig..6 for a dead tie of s. When the event rates are low, the two odels give the sae results, however, the trends are totally different in high rates. The nonparalyzable odel approach an asyptotic value of 1/. For paralyzable odel, a axiu value is fored at C n 1/ and then the observed rate decreases as the true rate increases. Thus, the observed rate can correspond to either a low true rate or a high rate as shown in the figure. In practice, any abiguity is solved by varying the count rate up or down and observing whether C increases or decreases. Since the dead tie odels are iperfect, the dead tie of a radiation detection syste is generally set at < %. Observed rate [s -1 ] 6 1/ 4 Dead tie: s Nonparalyzable Paralyzable 4 6 8 1 True rate [s -1 ] Fig..6. Observed rates as a function of the true rate for two dead tie odels.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -1.5. Liits of detection [4,5] A. Critical level, L C Suppose a radiation detector is given and we are easuring the nuber of detection events. If a radioactive source is positioned and we take a easureent for a certain counting tie, the detection syste will produce a nuber of counts (or detected events) C G. Since both radiation fro the source and background radiation contribute to the gross counts C G, the net count CN can be expressed as C N where C G C B CB is the average nuber of counts fro the background radiation. If there is no radioactive source or saple, the observed counts are fro the background radiation only. Suppose we take easureents a large nuber of ties with no source at a fixed counting tie. A series of background counts would be obtained and the net count C N will be distributed as shown in Fig..7. Since there is no source, the ean value of the net count is zero. The standard deviation of the distribution is as shown in the figure. Then we have the following question: how can we decide whether a easured net count fro an unknown saple close to zero is truly zero or is a true existing count fro a source? The critical level L C is the answer to this question and is defined as the count above which we can assue that a easured net count is eaningful. The critical level can be set at L C k as shown in Fig..7. In other words, if a easured CN is bigger than L C, we will conclude that the true detection events ay exist. Here, the constant k corresponds to the degree of confidence (or risk of istake, ) as in hypothesis testing. In our case, we don t care about the region of C N, so that one-sided testing is applied. For Risk of One-sided istake, degree of k confidence exaple, if the risk is set at.5 (or a confidence level of 95 %), the corresponding k value is 1.645 for a Gaussian distribution. Depending on how uch confidence level you want, k can be set differently..1 99. %.36.5 95. % 1.645.1 9. % 1.8 Not detected Detected (ay be) Not detected Detected (ay be) (will be) L C L D L c = k D k D Net counts Fig..7. Definition of critical level L C. Net counts Fig..8. Definition of detection liit L D.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -11 Fro the definition of the net count, the variance of the net count is represented as var( C ) var( C ) var( C ) C C C C N G Then the standard deviation can be obtained fro var( NC ) C B and the critical level becoes L k k C C B B G B N B B. Detection liit, L D The detection liit is the answer to the question of What is the iniu nuber of counts we can be sure of detection?. The critical level is not a strong indication of true detection as already discussed. If a easured count is exactly L C, we can only be sure of true detection in 5% of cases since the counts will be distributed syetrically relative to L C when any easureents are taken. Therefore, it is straightforward that the detection liit should be above L C. Suppose a radioactive saple produced a count exactly at the detection liit of the easureent syste with a standard deviation of D. If we would like to control the risk of not detecting this saple at, the detection liit L D is defined by L D LC k D k k D as shown in Fig..8. For convenience, if and are set at the sae risk level and accordingly, the constants k, k are sae, the detection liit is siplified to L D k k D The variance of the distribution D is D C C G B At the detection liit, the gross count C G is CG LD CB and the background count C B has a relation C B as already shown. With these relations, the standard deviation D is represented by D D L. Then, the detection liit equation becoes k D k k LD L k, D the solution of which gives
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -1 L D k k. In ost cases, the k ter is relatively uch saller and therefore, L D is finally siplified to L D k k C [ counts] B For exaple, if a risk of 5% (or 95% confidence level) is adopted, the detection liit of the syste becoes 4.65 CB. The Occupational Nuclear Medicine Group at McMaster has been developing in vivo analysis ethods for trace eleents in body and has eployed (.83 C B ). A conventional definition of 3 CB has been used in the radioanalytical nuclear cheistry counity. Since the detection liit is proportional to the square root of the background count, we should aintain the background counting rate as low as possible by adding appropriate shielding for the background radiation. The diension of the detection liit can be converted to radioactivity of the saple L D L [ Bq] D [ counts] 1 t [ s] p c abs e where, p e is the eission probability of the radiation per decay (note: for gaa-rays p e is generally lower than 1). If decay of radioactivity is negligible during the counting period and the background counting rate is constant, the detection liit in [Bq] is inversely proportional to t c. Therefore, we can iprove the detection liit by counting longer. References 1. G.F. Knoll, Radiation Detection and Measureent - 3 rd edition (Chapters 3, 4), John Wiley & Sons, 1999.. G. Gilore, J.D. Heingway, Practical Gaa-Ray Spectroetry, John Wiley & Sons, 1995. 3. K. Debertin, R.G. Heler, Gaa- and X-ray Spectroetry with Seiconductor Detectors, North- Holland, 1988. 4. L.A. Currie, Anal. Che. 4 (1968) 586. 5. L.A. Currie, Appl. Radiat. Isot. 61 (4) 145.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -13 Probles 1. Derive the Poisson distribution fro the binoial distribution.. For a Gaussian function, derive the full width at the fifth axiu in ters of the standard deviation. 3. Derive the solid angle forula for a point source and a cylindrical detector. 4. Suppose you counted the background radiation any ties without any radioactive saples, which gave a standard deviation of for the net counts. (a) Using the Gaussian probability density function 1 ( x ) G( x) exp( x ),write down a atheatical expression for the degree of confidence in the case the critical level is set at L c k. (b) Express the degree of confidence for k =1 using the copleentary error function (erfc). [ t erfc( x) e dt ] x Copute the degree of confidence using the data table. x erfc(x).1.888..777.3.671.4.57.5.48.6.396.7.3.8.58.9.3 1.157 5. A radiation detection syste showed the detection liit of counts for 1 hour counting at 95 % degree of confidence. (a) Find the counting rate of the background radiation. Risk of istake, One-sided degree of confidence k.1 99. %.36.5 95. % 1.645.1 9. % 1.8 (b) Find the detection liit for 3 hour counting at 9 % degree of confidence. 6. A radiation detection syste has a ean background counting rate of counts per second. A 3 in counting for an unknown radioactive saple led to the gross counts of 4, counts. (a) Copute the net count and its uncertainty. (b) Find the iniu counting tie required to reach 5% uncertainty for the net counts. 7. A 3 in counting for an unknown radioactive saple led to the gross counts of 4, counts. The background counts for the sae counting tie were 3,6 counts. (a) Copute the net count and its uncertainty. (b) Find the iniu counting tie (hours) required to reach 5% uncertainty for the net counts.
Med Phys 4RA3, 4RB3/6R3 Radioisotopes and Radiation Methodology -14 8. A radiation detection syste has an average counting rate of 5 counts per second. Supposing a detection event happened at tie t=. (a) Find the ean nuber of counts for an interval (, t 1 ). Find the probability of zero detection in this interval. (b) Find the probability that the next detection event will happen in the tie region (t 1, t ). Copute the probability for the case t 1 =.1 s and t =.5 s. 9. A point radioactive source was placed at 1 c fro a cylindrical detector with 1 s dead tie and the observed counting rate was.41 5 counts/in. (a) Using the non-extending (non-paralyzable) odel, find the fraction of the counting loss, i.e. lost events to true events ratio. (b) Estiate the observed counting rate when the source-to-detector distance is doubled. 1. A radiation detector undergoes a fixed dead tie of 1 s per each detection event. Using the C extending (paralyzable) dead tie odel n C C ne, (a) Find the percentage dead tie, i.e. dead rate to true rate ratio, for a true event rate of 3, counts/in. (b) Sketch C as a function of C n in the C has the sketched trend in this region. C n region of to 5 counts/s. Briefly explain the reason why 11. To deterine the dead tie of a radiation counting syste, a set of radioactive sources with various activities were produced and their counting rates were easured. Using the extending (i.e. paralyzable) dead tie odel, the following relation was built by fitting the experiental data: C ln( ).33. x ( C : easured counting rate [counts/s], x C n / C n, 1, C n : true interaction rate, x Find the dead tie. C n,1 : true interaction rate for the weakest source used) 1. A radiation detection syste showed the detection liit of 1 counts for an hour counting tie. The corresponding detection liit in Bq was 1 Bq. Write down the expected detection liits in [counts] and [Bq] for a four hour counting. Briefly explain the reason. 13. A point radioactive source was placed at 1 c fro a cylindrical detector with 1 s dead tie and the observed counting rate was.41 5 counts/in. Using the extending (paralyzable) odel, find the true interaction rate in [s -1 ] at which the observed counting rate is axiu. (1)