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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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