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General Properties of Radiation Detectors
Ho Kyung Kim
Pusan National University
Radiation Measurement Systems
Radiation spectroscopy
• Key idea
– For an event in a detector: the energy deposited Edep # of charge carriers Q the integrated current Itot the peak voltage Vmax
– Edep ~ Vmax or Itot
2
+ + + – – –
E0
Edep
Q
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SIMPLIFIED DETECTOR MODEL
• Imagine the interaction of a single ptl or quantum of radiation in a detector
– Very small interaction or stopping time (~ns in gases & ~ps in solids)
– => Q generation w/i the detector at time t = 0
– => Q collection (under E field) w/i (~ms in ion chambers & ~ns in semiconductors)
• Determined by the mobility of charge carriers & the avg. distance to collection electrodes
• Current flowing during charge collection time 𝑡𝑐
– 𝑖 𝑡 d𝑡𝑡𝑐0
= 𝑄
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When the irradiation rate is low
The time intervals btwn successive current pulses are randomly distributed because the arrival of radiation quanta is a random phenomenon governed by Poisson statistics
MODES OF DETECTOR OPERATION
1) Pulse mode
– Most common
– Record each individual quantum of radiation
• Record the time integral of each burst of current (Q ~ Edep) => radiation spectroscopy
• Record pulses above a low-level threshold regardless of the value of Q => pulse counting
– Impractical or impossible at very high event rates
2) Current mode
– In very high event rates
– Radiation dosimetry
3) Mean square voltage (MSV) mode (or Campbelling mode)
• Limited to some specialized applications (e.g., mixed radiation measurements)
• In reactor instrumentation
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Current mode
• Current ~ interaction rate avg. charge per interaction
– Average current 𝐼0 = 𝑟𝑄 = 𝑟𝐸
𝑊𝑞
• 𝐸 = avg. energy deposited per event
• 𝑊 = avg. energy required to produce a unit charge pair (e.g., e-ion pair)
• 𝑞 = 1.6 10-19 C
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𝐼 𝑡 =1
𝑇 𝑖 𝑡′ d𝑡′𝑡
𝑡−𝑇
• Statistical uncertainty = random fluctuations in the arrival time of the event
– 𝜎𝐼2(𝑡) =
1
𝑇 𝑖 𝑡′ − 𝐼0
2d𝑡′𝑡
𝑡−𝑇=1
𝑇 𝜎𝑖
2(𝑡′)d𝑡′𝑡
𝑡−𝑇
– 𝜎𝐼(𝑡) = 𝜎𝐼2(𝑡)
• Recall 𝜎𝑛 = 𝑛 = 𝑟𝑇, where 𝑛 = # of recorded events
• Then, the fractional std., 𝜎𝐼(𝑡)
𝐼0=𝜎𝑛
𝑛=
1
𝑟𝑇
– Note that this accounts for only the random fluctuations in pulse arrival time, but not for in pulse amplitude, because Q in each event is assumed to be constant
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Mean square voltage mode
• Block 𝐼0 (avg. value) and only pass 𝜎𝑖(𝑡) (fluctuating component), and then
compute 𝜎𝐼2(𝑡) (time averaging the squared amplitude of 𝜎𝑖(𝑡))
– 𝜎𝐼2(𝑡) =
𝐼0
𝑟𝑇
2=𝑟𝑄2
𝑇
• Proportional to 𝑟
• Proportional to the square of Q produced in each event
– Useful for mixed radiation environments
• Further weight the detector response in favor of the type of radiation giving the larger avg. Q per event
– e.g., neutron signal compared w/ smaller-amplitude gamma-ray signal
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Pulse mode
– 𝑅 = input resistance of a measuring circuit (usually a preamplifier)
– 𝐶 = equiv. capacitance of both the detector itself & the circuit (the cable & input cap. of premap.)
– The time constant of the measuring circuit, 𝜏 = 𝑅𝐶
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Case 1. Small RC ( << tc)
• 𝑉(𝑡) has a shape nearly identical to 𝑖(𝑡) produced w/i the detector
• Operated when high event rates or timing information is more important than accurate energy information
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Case 2. Large RC ( >> tc)
• Very little current flowing in 𝑅 during 𝑡𝑐, integrated on 𝐶, & then discharged thru 𝑅
• Leading edge of 𝑉(𝑡) is detector dependent & tailing edge circuit dependent:
– The pulse rise time, required for the signal pulse to reach its max. value, is determined by 𝑡𝑐 w/i the detector itself
• No properties of the external or load circuit influence
– The pulse decay time, required to restore 𝑉(𝑡) to zero, is determined only by 𝜏 of the load circuit
• Amplitude of signal pulse, 𝑉𝑚𝑎𝑥 =𝑄
𝐶
– Distribution of pulse amplitudes => distribution in energy of the incident radiation
– 𝐶 may change in the semiconductor diode detector, hence the proportionality btwn 𝑉𝑚𝑎𝑥 & 𝑄 breaks!
• Charge-sensitive preamplifier uses feedback to largely eliminate the dependence of the output amplitude on 𝐶 and restores proportionality to 𝑄
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PULSE HEIGHT SPECTRA
• Pulse amplitude (𝑉𝑚𝑎𝑥) distribution
– Variations in amplitudes
• Differences in the radiation energy
• Fluctuations in the inherent response of the detector to monoenergetic radiation
• Electronic noise
• How to display it?
– Differential pulse height distribution
• Abscissa
– A linear pulse amplitude in units of pulse amplitude [volts]
• Ordinate
– The diff’l number d𝑁 of pulses observed w/ an amplitude w/i the diff’l amplitude increment d𝐻, or d𝑁 d𝐻 , in units of inverse amplitude [volt-1]
– Integral pulse height distribution
• Ordinate
– # of pulses whose amplitude exceeds that of a given value of the abscissa H
– Always monotonically decreasing function
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# of pulses w/ amplitude btwn H1 & H2 = d𝑁
d𝐻d𝐻
𝐻2𝐻1
Tot. # of pulses
represented by the distri., 𝑁0 = d𝑁
d𝐻d𝐻
∞
0
Tot. area
The value at H = 0
Peak Local max. in slopes
Valley Local min. in slopes
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COUNTING CURVES AND PLATEAUS
• Pulse counting measurements
– Counting device w/ a fixed discrimination level, 𝐻𝑑
• Signal pulses must exceed 𝐻𝑑 to be registered by the counting circuit
• Variable 𝐻𝑑 to provide information about the amplitude distri. of the pulses
– e.g., Vary 𝐻𝑑 btwn 0 & H5
– Small drifts in 𝐻𝑑 during measurements => How to minimize this effect?
• Set 𝐻𝑑 at counting plateau in the integral distri. (or valley in the diff’l spectrum)
• Similarly, find the operating point (voltage or gain) of max. stability in counting curves
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Detector Amp. w/
adjustable gain Discriminator Counter
Set 𝐻𝑑
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ENERGY RESOLUTION
• Response function of a detector for an energy
– Its width reflects an amount of fluctuation from pulse to pulse even though the same E is deposited in a detector
• Determine the ability to resolve fine detail in the incident E => energy resolution
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The same area if the same # of pules are recorded
Monoenergy
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• Energy resolution
– FWHM (full width at half maximum) assuming negligible background or continuum
• In units of energy
• Common for good resolution systems
– 𝑅 =FWHM
𝐻0
• In units of percentage
• Common for poor resolution systems
• 𝑅 ≤ 1% for semiconductor diode detectors in alpha spectroscopy
• 𝑅 = 3 − 10% for scintillation detectors in gamma-ray spectroscopy
– As a rule of thumb, one should be able to resolve two energies that are separated by more than one value of the detector FWHM
FWHM R
Si detector for 5.49 MeV
20 keV 0.36%
NaI(Tl) detector for 0.662 MeV
45 keV 6.8%
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• Factors degrading the energy resolution
– Drift of the operating characteristics of a detector
– Random noise in the detector/instrumentation syst.
– Statistical noise arising from the discrete nature of the measured signal itself
• 𝑄 generated w/i a detector is not a continuous variable but instead represents a discrete # of charge carriers => subject to random fluctuation
– Ion pairs in ion chambers
– Electrons collected from the photocathode of PMTs
• Irreducible min. amount of fluctuation
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• Assuming the Poisson statistics => Gaussian function due to a large # of charge carriers 𝑁
– 𝐺 𝐻 =𝐴
𝜎 2𝜋𝑒𝑥𝑝 −
(𝐻−𝐻0)2
2𝜎2 with FWHM = 2.35𝜎
– If 𝐻0 = 𝐾𝑁 assuming the linear response of detectors, 𝜎 = 𝐾 𝑁 & FWHM = 2.35𝐾 𝑁
– 𝑅 Poisson limit =FWHM
𝐻0=2.35𝐾 𝑁
𝐾𝑁=2.35
𝑁
• Resolution improves as 𝑁 increases
• To achieve 𝑅 better than 1%, N should be 55,000
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Derive! (H.W. due the next class)
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• Achievable values for R can be lower by a factor of 3 or 4 than the Poisson limit
– Indicating that the charge-formation processes are not independent
– Tot. # of charge carriers cannot be described by simple Poisson statistics
• Fano factor
– Quantify the departure of the observed statistical fluctuations in the # of charge carriers from pure Poisson statistics
– 𝐹 =observed variance in 𝑁
Poisson predicted variance (=𝑁)
– 𝑅 statistical limit =2.35𝐾 𝑁 𝐹
𝐾𝑁= 2.35
𝐹
𝑁
– 𝐹 < 1 for semiconductor diode detectors & proportional counters
– 𝐹 ≈ 1 for scintillation detectors
• Total energy resolution
– (FWHM)overall2 = (FWHM)statistical
2 +(FWHM)noise2 +(FWHM)drift
2 +⋯
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Ideal response function
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Single-peak response function
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Continuum response function
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Peak + continuum response function
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Typical response function
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Impact of energy resolution on a spectrum
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DETECTION EFFICIENCY
• Absolute efficiency
– 𝜖abs =number of pulses recorded
number of radiation quanta emitted by source
– Dependent not only on detector properties but also on the details of the counting geometry (distance, solid angle …)
• Intrinsic efficiency
– 𝜖int =number of pulses recorded
number of radiation quanta incident on detector= 𝜖abs ∙
4𝜋
Ω
– 𝜖int ≤ 𝜖abs
– Does not include the solid angle subtended by the detector
– Dependent on the detector material (or composition) & thickness (or size & shape), and the type & energy of radiation
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• Total efficiency, 𝜖total
– The entire area under the measured spectrum
• Peak efficiency, 𝜖peak
– Consider only interactions that deposit the full energy of the incident radiation
– Not sensitive to some perturbing effects
• Scattering from surrounding objects
• Spurious noise
– Peak-to-total ratio: 𝑟 =𝜖peak
𝜖total≤ 1
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• Intrinsic peak efficiency, 𝜖ip
– 𝜖ip =number of pulses recorded in full energy peak region
number of radiation quanta incident on detector
– Consider 𝑁 events under the full-energy peak in the spectrum, assuming that the source emits radiation isotropically & that no attenuation takes place btwn the source & detector;
• 𝑆 = 𝑁4𝜋
𝜖ipΩ
• Solid angle [steradians]
– Ω = cos 𝛼
𝑟2d𝐴
𝐴
• = angle btwn the normal to the surface element & the source direction
• e.g., A point source located along the axis of a right circular cylindrical detector;
– Ω = 2𝜋 1 −𝑑
𝑑2+𝑎2
– Ω = 2𝜋 1 −𝑑
𝑑2+𝑎2≅
𝐴
𝑑2 for 𝑑 >> 𝑎
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Derive! (H.W. due the next class)
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Quiz
• Assuming that 𝜖ip = 35% & 𝑁 = 4321, find out the source strength referring to the
following geometry.
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3 cm 20 cm
DEAD TIME
• In all detector systems, there will be a min. amount of time that must separate two events in order that they be recorded as two separate pulses
– Due to the detector itself or the associated electronics
• Effects of the dead time
– In counting systems
• Loss of counts for late pulses (dead time loss)
– In spectroscopy systems
• Late counts are neglected & first count determines the height (signal loss)
• A pulse w/ a new height is created (pulse pileup)
• Dead time losses can distort the statistics of the recorded counts away from a true Poisson behavior
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Causes of dead time
• Recovery time of detector response
– e.g., GM counters
• Resolving time set by the threshold (LLD) of integral discriminator (~100 s)
– Paralyzable
• Pileup resolution time due to a wide pulse width
– e.g., Shaping amplifier (3–5 s for Ge detectors)
– Paralyzable
• Dead time of MCA
– Conversion time of ADC
– Memory storage time
– ~10 s
– Nonparalyzable
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LLD
Resolving time
Models for dead time behavior
• Paralyzable model: true events during the dead time extend the dead time by another period following the lost event
• Nonparalyzable model: true events during the dead time are lost
• Real counting systems will often display a behavior that is intermediate btwn these two models
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=> 3 counts
=> 4 counts
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Nonparalyzable model
• Define;
– 𝑛 = true interaction rate
– 𝑚 = recorded count rate
– 𝜏 = syst. dead time
• Intermediately,
– 𝑚𝜏 = fraction of all time that the detector is dead
• Count loss rate
– 𝑛 − 𝑚 = 𝑛𝑚𝜏
• Then, true rate
– 𝑛 =𝑚
1−𝑚𝜏
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Paralyzable model
• Dead period is not anymore fixed, instead note that 𝑚 is identical to the rate of occurrences of time intervals btwn true events that exceed 𝜏
• Recall the distri. of intervals btwn random events occurring at an average rate 𝑛 ;
– 𝑃1 𝑇 d𝑇 = 𝑛𝑒−𝑛𝑇d𝑇
• When a pulse is counted, the probability that the next pulse arrives later than the dead time 𝜏 can be obtained by integrating 𝑃1 𝑇 d𝑇 btwn 𝜏 & ∞;
– 𝑃2 𝜏 = 𝑃1 𝑇 d𝑇∞
𝜏= 𝑒−𝑛𝜏
• Then, the measured count rate = the true count rate prob. that the next pulse is counted
– 𝑚 = 𝑛 × 𝑃2 𝜏 = 𝑛𝑒−𝑛𝜏
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Comparison btwn two models
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At high rates (𝑛 ≫ 1), 𝑚 →1
𝜏 (∵ 𝑛 =
1
𝜏+1
𝑛
)
Virtually the same at low rates:
𝑚 =𝑛
1+𝑛𝜏≅ 𝑛 1 − 𝑛𝜏 for nonparalyzable
𝑚 = 𝑛𝑒−𝑛𝜏 ≅ 𝑛 1 − 𝑛𝜏 for paralyzable
Max. at the rate, 𝑛 =1
𝜏 Derive the max. value! (H.W. due the next class)
At very 𝑛, very few events are recorded because of a long extension of the dead period
Ambiguity either due to low rate or high rate???
Derive the max. value! (H.W. due the next class)
Methods of dead time measurement
• Single measurement of 𝑚 => 2 unknowns: 𝑛, 𝜏
• Two measurements of 𝑚1, 𝑚2 => 3 unknowns: 𝑛1, 𝑛2, 𝜏
– => Need to have a known relationship between 𝑛1 and 𝑛2
• Decaying source method
– 𝑛2 = 𝑛1𝑒−𝜆𝑡
• Two-source method
– 𝑛12 = 𝑛1 + 𝑛2
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Two-source method
– 𝑛12 − 𝑛𝑏 = 𝑛1 − 𝑛𝑏 + 𝑛2 − 𝑛𝑏 ⇒ 𝑛12 + 𝑛𝑏 = 𝑛1 + 𝑛2
• Assuming the nonparalyzable model,
–𝑚12
1−𝑚12𝜏+
𝑚𝑏
1−𝑚𝑏𝜏=
𝑚1
1−𝑚1𝜏+
𝑚2
1−𝑚2𝜏
• Solving for 𝜏;
– 𝜏 =𝑋 1− 1−𝑍
𝑌
• 𝑋 = 𝑚1𝑚2 −𝑚𝑏𝑚12
• 𝑌 = 𝑚1𝑚2 𝑚12 +𝑚𝑏 −𝑚𝑏𝑚12(𝑚1 +𝑚2)
• 𝑍 =𝑌(𝑚1+𝑚2−𝑚12−𝑚𝑏)
𝑋2
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Decaying source method
• Given, 𝑛 = 𝑛0𝑒−𝜆𝑡 + 𝑛𝑏 ≅ 𝑛0𝑒
−𝜆𝑡 (negligible bgnd)
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• Nonparalyzable model
– 𝑛 =𝑚
1−𝑚𝜏
– 𝑛0𝑒−𝜆𝑡 =
𝑚
1−𝑚𝜏⇒ 𝑚𝑒𝜆𝑡 = −𝑛0𝜏𝑚 + 𝑛0
• Paralyzable model
– 𝑚 = 𝑛𝑒−𝑛𝜏
– 𝜆𝑡 + ln𝑚 = −𝑛0𝜏𝑒−𝜆𝑡 + ln𝑛0
𝜏 =−slope
intercept
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Dead time losses from pulse sources
– e.g., X rays from linear accelerators
• If 𝜏 ≪ 𝑇 (steady-state source), the analysis discussed previously can be applied
• If 𝜏 < 𝑇, the most complicated situation (a small # of counts are registered)
• If 𝑇 < 𝜏 < 1
𝑓−𝑇, a single count per pulse at most
– Prob. of an observed count per source pulse = 𝑚
𝑓
– Avg. # of true events per source pulse = 𝑛
𝑓
– Prob. that at least one true event occurs per source pulse:
• 𝑃 > 0 = 1 − 𝑃 0 = 1 − 𝑒−𝑥 = 1 − 𝑒−𝑛 𝑓 =𝑚
𝑓 ⇒ 𝑚 = 𝑓 1 − 𝑒−𝑛 𝑓
• 𝑛 = 𝑓 ln𝑓
𝑓−𝑚≅
𝑚
1−𝑚 2𝑓 for 𝑚 ≪ 𝑓
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