General Properties of Radiation Detectors

Ho Kyung KimPusan National University

Radiation Measurement SystemsKnoll chap. 4

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

+ + +– – –E0

Edep

Q

2

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 𝑡𝑡𝑐𝑐

– ∫0𝑡𝑡𝑐𝑐 𝑖𝑖 𝑡𝑡 d𝑡𝑡 = 𝑄𝑄

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

3

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

4

Current mode

• Current ~ interaction rate × avg. charge per interaction

– Average current 𝐼𝐼0 = 𝑟𝑟𝑄𝑄 = 𝑟𝑟 𝐸𝐸𝑊𝑊𝑞𝑞

• 𝐸𝐸 = avg. energy deposited per event• 𝑊𝑊 = avg. energy required to produce a unit charge pair (e.g., e-ion pair)• 𝑞𝑞 = 1.6 × 10-19 C

𝐼𝐼 𝑡𝑡 =1𝑇𝑇�𝑡𝑡−𝑇𝑇

𝑡𝑡𝑖𝑖 𝑡𝑡′ d𝑡𝑡𝑡

5

• Statistical uncertainty = random fluctuations in the arrival time of the event

– 𝜎𝜎𝐼𝐼2(𝑡𝑡) = 1𝑇𝑇 ∫𝑡𝑡−𝑇𝑇

𝑡𝑡 𝑖𝑖 𝑡𝑡′ − 𝐼𝐼0 2d𝑡𝑡𝑡 = 1𝑇𝑇 ∫𝑡𝑡−𝑇𝑇

𝑡𝑡 𝜎𝜎𝑖𝑖2(𝑡𝑡𝑡)d𝑡𝑡𝑡

– 𝜎𝜎𝐼𝐼 (𝑡𝑡) = 𝜎𝜎𝐼𝐼2(𝑡𝑡)

• Recall 𝜎𝜎𝑛𝑛 = 𝑛𝑛 = 𝑟𝑟𝑇𝑇, where 𝑛𝑛 = # of recorded events

• Then, the fractional std., 𝜎𝜎𝐼𝐼 (𝑡𝑡)𝐼𝐼0

= 𝜎𝜎𝑛𝑛𝑛𝑛

= 1𝑟𝑟𝑇𝑇

– 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

6

Mean square voltage mode

• Block 𝐼𝐼0 (avg. value) and only pass 𝜎𝜎𝑖𝑖 (𝑡𝑡) (fluctuating component), and then compute 𝜎𝜎𝐼𝐼2(𝑡𝑡) (time averaging the squared amplitude of 𝜎𝜎𝑖𝑖 (𝑡𝑡))

– 𝜎𝜎𝐼𝐼2(𝑡𝑡) = 𝐼𝐼0𝑟𝑟𝑇𝑇

2= 𝑟𝑟𝑄𝑄2

𝑇𝑇

• Proportional to 𝑟𝑟• 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

7

Pulse mode

– 𝑅𝑅 = input resistance of a measuring circuit (usually a preamplifier)– 𝐶𝐶 = equiv. capacitance of both the detector itself & the circuit (the cable & input cap. of premap.)– The time constant of the measuring circuit, 𝜏𝜏 = 𝑅𝑅𝐶𝐶

8

Case 1. Small RC (τ << tc)

• 𝑉𝑉(𝑡𝑡) has a shape nearly identical to 𝑖𝑖(𝑡𝑡) produced w/i the detector

• Operated when high event rates or timing information is more important than accurate energy information

9

Case 2. Large RC (τ >> tc)

• Very little current flowing in 𝑅𝑅 during 𝑡𝑡𝑐𝑐, integrated on 𝐶𝐶, & then discharged thru 𝑅𝑅

• Leading edge of 𝑉𝑉(𝑡𝑡) is detector dependent & tailing edge circuit dependent:– The pulse rise time, required for the signal pulse to reach its max. value, is determined by 𝑡𝑡𝑐𝑐 w/i

the detector itself• No properties of the external or load circuit influence

– The pulse decay time, required to restore 𝑉𝑉(𝑡𝑡) to zero, is determined only by 𝜏𝜏 of the load circuit

• Amplitude of signal pulse, 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑄𝑄𝐶𝐶

– Distribution of pulse amplitudes => distribution in energy of the incident radiation– 𝐶𝐶 may change in the semiconductor diode detector, hence the proportionality btwn 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 & 𝑄𝑄

breaks!• Charge-sensitive preamplifier uses feedback to largely eliminate the dependence of the

output amplitude on 𝐶𝐶 and restores proportionality to 𝑄𝑄

10

PULSE HEIGHT SPECTRA

• Pulse amplitude (𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚) 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 d𝑁𝑁 of pulses observed w/ an amplitude w/i the diff’l amplitude

increment d𝐻𝐻, or ⁄d𝑁𝑁 d𝐻𝐻, 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

11

# of pulses w/ amplitude btwn H1 & H2 = ∫𝐻𝐻1𝐻𝐻2 d𝑁𝑁

d𝐻𝐻d𝐻𝐻

Tot. # of pulsesrepresented by the distri., 𝑁𝑁0 = ∫0

∞ d𝑁𝑁d𝐻𝐻

d𝐻𝐻

Tot. area

The value at H = 0

Peak ⇔ Local max. in slopes

Valley ⇔ Local min. in slopes

12

COUNTING CURVES AND PLATEAUS

• Pulse counting measurements

– Counting device w/ a fixed discrimination level, 𝐻𝐻𝑑𝑑• Signal pulses must exceed 𝐻𝐻𝑑𝑑 to be registered by the

counting circuit• Variable 𝐻𝐻𝑑𝑑 to provide information about the amplitude

distri. of the pulses– e.g., Vary 𝐻𝐻𝑑𝑑 btwn 0 & H5

– Small drifts in 𝐻𝐻𝑑𝑑 during measurements => How to minimize this effect?

• Set 𝐻𝐻𝑑𝑑 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

Detector Amp. w/ adjustable gain

Discriminator Counter

Set 𝐻𝐻𝑑𝑑

13

14

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

The same area if the same # of pules are recorded

Monoenergy

15

• Energy resolution– FWHM (full width at half maximum) assuming

negligible background or continuum• In units of energy• Common for good resolution systems

– 𝑅𝑅 = FWHM𝐻𝐻0

• In units of percentage• Common for poor resolution systems• 𝑅𝑅 ≤ 1% for semiconductor diode detectors in

alpha spectroscopy• 𝑅𝑅 = 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 detectorfor 5.49 MeV α 20 keV 0.36%

NaI(Tl) detector for 0.662 MeV γ 45 keV 6.8%

16

• 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

• 𝑄𝑄 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

17

• Assuming the Poisson statistics => Gaussian function due to a large # of charge carriers 𝑁𝑁

– 𝐺𝐺 𝐻𝐻 = 𝐴𝐴𝜎𝜎 2𝜋𝜋

𝑒𝑒𝑒𝑒𝑒𝑒 − (𝐻𝐻−𝐻𝐻0)2

2𝜎𝜎2with FWHM = 2.35𝜎𝜎

– If 𝐻𝐻0 = 𝐾𝐾𝑁𝑁 assuming the linear response of detectors, 𝜎𝜎 = 𝐾𝐾 𝑁𝑁 & FWHM = 2.35𝐾𝐾 𝑁𝑁

– |𝑅𝑅 Poisson limit = FWHM𝐻𝐻0

= 2.35𝐾𝐾 𝑁𝑁𝐾𝐾𝑁𝑁

= 2.35𝑁𝑁

• Resolution improves as 𝑁𝑁 increases• To achieve 𝑅𝑅 better than 1%, N should be ≥ 55,000

Derive! (H.W. due the next class)

18

• 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

– 𝐹𝐹 = observed variance in 𝑁𝑁Poisson predicted variance (=𝑁𝑁)

– |𝑅𝑅 statistical limit = 2.35𝐾𝐾 𝑁𝑁 𝐹𝐹𝐾𝐾𝑁𝑁

= 2.35 𝐹𝐹𝑁𝑁

– 𝐹𝐹 < 1 for semiconductor diode detectors & proportional counters– 𝐹𝐹 ≈ 1 for scintillation detectors

• Total energy resolution– (FWHM)overall2 = (FWHM)statistical2 +(FWHM)noise2 +(FWHM)drift

2 +⋯

19

Ideal response function

20

Single-peak response function

21

Continuum response function

22

Peak + continuum response function

23

Typical response function

24

Impact of energy resolution on a spectrum

25

DETECTION EFFICIENCY

• Absolute efficiency

– 𝜖𝜖abs = number of pulses recordednumber 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

– 𝜖𝜖int = number of pulses recordednumber of radiation quanta incident on detector

= 𝜖𝜖abs �4𝜋𝜋Ω

– 𝜖𝜖int ≤ 𝜖𝜖abs– 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

26

• Total efficiency, 𝜖𝜖total– The entire area under the measured spectrum

• Peak efficiency, 𝜖𝜖peak– 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: 𝑟𝑟 =𝜖𝜖peak𝜖𝜖total

≤ 1

27

• Intrinsic peak efficiency, 𝜖𝜖ip– 𝜖𝜖ip = number of pulses recorded in full energy peak region

number of radiation quanta incident on detector

– Consider 𝑁𝑁 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;

• 𝑆𝑆 = 𝑁𝑁 4𝜋𝜋𝜖𝜖ipΩ

• Solid angle [steradians]

– Ω = ∫𝐴𝐴cos 𝛼𝛼𝑟𝑟2

d𝐴𝐴

• α = 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 = 2𝜋𝜋 1 − 𝑑𝑑𝑑𝑑2+𝑚𝑚2

– Ω = 2𝜋𝜋 1 − 𝑑𝑑𝑑𝑑2+𝑚𝑚2

≅ 𝐴𝐴𝑑𝑑2

for 𝑑𝑑 >> 𝑎𝑎

Derive! (H.W. due the next class)

28

Quiz

• Assuming that 𝜖𝜖ip = 35% & 𝑁𝑁 = 4321, find out the source strength referring to the following geometry.

3 cm20 cm

29

DEAD TIME

• In all detector systems, there will be a min. amount of time that must separate two eventsin 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

30

Causes of dead time

• Recovery time of detector response– e.g., GM counters

• Resolving time set by the threshold (LLD) of integral discriminator (~100 µs)

– Paralyzable

• Pileup resolution time due to a wide pulse width– e.g., Shaping amplifier (3–5 µs for Ge detectors)– Paralyzable

• Dead time of MCA– Conversion time of ADC– Memory storage time– ~10 µs– Nonparalyzable

LLD

Resolving time

31

Models for dead time behavior

• Paralyzable model: true events during the dead time extend the dead time by another period τ 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

=> 3 counts

=> 4 counts

32

Nonparalyzable model

• Define;– 𝑛𝑛 = true interaction rate– 𝑚𝑚 = recorded count rate– 𝜏𝜏 = syst. dead time

• Intermediately,– 𝑚𝑚𝜏𝜏 = fraction of all time that the detector is dead

• Count loss rate– 𝑛𝑛 − 𝑚𝑚 = 𝑛𝑛𝑚𝑚𝜏𝜏

• Then, true rate

– 𝑛𝑛 = 𝑚𝑚1−𝑚𝑚𝜏𝜏

33

Paralyzable model

• Dead period is not anymore fixed, instead note that 𝑚𝑚 is identical to the rate of occurrences of time intervals btwn true events that exceed 𝜏𝜏

• Recall the distri. of intervals btwn random events occurring at an average rate 𝑛𝑛;– 𝑃𝑃1 𝑇𝑇 d𝑇𝑇 = 𝑛𝑛𝑒𝑒−𝑛𝑛𝑇𝑇d𝑇𝑇

• When a pulse is counted, the probability that the next pulse arrives later than the dead time 𝜏𝜏 can be obtained by integrating 𝑃𝑃1 𝑇𝑇 d𝑇𝑇 btwn 𝜏𝜏 & ∞;

– 𝑃𝑃2 𝜏𝜏 = ∫𝜏𝜏∞𝑃𝑃1 𝑇𝑇 d𝑇𝑇 = 𝑒𝑒−𝑛𝑛𝜏𝜏

• Then, the measured count rate = the true count rate × prob. that the next pulse is counted

– 𝑚𝑚 = 𝑛𝑛 × 𝑃𝑃2 𝜏𝜏 = 𝑛𝑛𝑒𝑒−𝑛𝑛𝜏𝜏

34

Comparison btwn two models

At high rates (𝑛𝑛 ≫ 1), 𝑚𝑚 → 1𝜏𝜏

(∵ 𝑚𝑚 = 1𝜏𝜏+1𝑛𝑛

)

Virtually the same at low rates:𝑚𝑚 = 𝑛𝑛

1+𝑛𝑛𝜏𝜏≅ 𝑛𝑛 1 − 𝑛𝑛𝜏𝜏 for nonparalyzable

𝑚𝑚 = 𝑛𝑛𝑒𝑒−𝑛𝑛𝜏𝜏 ≅ 𝑛𝑛 1 − 𝑛𝑛𝜏𝜏 for paralyzable

Max. at the rate, 𝑛𝑛 = 1𝜏𝜏

Derive the max. value! (H.W. due the next class)

At very high 𝑛𝑛, 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)

35

Methods of dead time measurement

• Single measurement of 𝑚𝑚 => 2 unknowns: 𝑛𝑛, 𝜏𝜏

• Two measurements of 𝑚𝑚1,𝑚𝑚2 => 3 unknowns: 𝑛𝑛1,𝑛𝑛2, 𝜏𝜏– => Need to have a known relationship between 𝑛𝑛1 and 𝑛𝑛2

• Decaying source method– 𝑛𝑛2 = 𝑛𝑛1𝑒𝑒−𝜆𝜆𝑡𝑡

• Two-source method– 𝑛𝑛12 = 𝑛𝑛1 + 𝑛𝑛2

36

Two-source method

– 𝑛𝑛12 − 𝑛𝑛𝑏𝑏 = 𝑛𝑛1 − 𝑛𝑛𝑏𝑏 + 𝑛𝑛2 − 𝑛𝑛𝑏𝑏 ⇒ 𝑛𝑛12 + 𝑛𝑛𝑏𝑏 = 𝑛𝑛1 + 𝑛𝑛2

• Assuming the nonparalyzable model,

– 𝑚𝑚121−𝑚𝑚12𝜏𝜏

+ 𝑚𝑚𝑏𝑏1−𝑚𝑚𝑏𝑏𝜏𝜏

= 𝑚𝑚11−𝑚𝑚1𝜏𝜏

+ 𝑚𝑚21−𝑚𝑚2𝜏𝜏

• Solving for 𝜏𝜏;

– 𝜏𝜏 = 𝑋𝑋 1− 1−𝑍𝑍𝑌𝑌

• 𝑋𝑋 = 𝑚𝑚1𝑚𝑚2 −𝑚𝑚𝑏𝑏𝑚𝑚12

• 𝑌𝑌 = 𝑚𝑚1𝑚𝑚2 𝑚𝑚12 + 𝑚𝑚𝑏𝑏 − 𝑚𝑚𝑏𝑏𝑚𝑚12(𝑚𝑚1 + 𝑚𝑚2)

• 𝑍𝑍 = 𝑌𝑌(𝑚𝑚1+𝑚𝑚2−𝑚𝑚12−𝑚𝑚𝑏𝑏)𝑋𝑋2

37

Decaying source method

• Given, 𝑛𝑛 = 𝑛𝑛0𝑒𝑒−𝜆𝜆𝑡𝑡 + 𝑛𝑛𝑏𝑏 ≅ 𝑛𝑛0𝑒𝑒−𝜆𝜆𝑡𝑡 (negligible bgnd)

• Nonparalyzable model

– 𝑛𝑛 = 𝑚𝑚1−𝑚𝑚𝜏𝜏

– 𝑛𝑛0𝑒𝑒−𝜆𝜆𝑡𝑡 = 𝑚𝑚1−𝑚𝑚𝜏𝜏

⇒ 𝑚𝑚𝑒𝑒𝜆𝜆𝑡𝑡 = −𝑛𝑛0𝜏𝜏𝑚𝑚 + 𝑛𝑛0

• Paralyzable model

– 𝑚𝑚 = 𝑛𝑛𝑒𝑒−𝑛𝑛𝜏𝜏

– 𝜆𝜆𝑡𝑡 + ln𝑚𝑚 = −𝑛𝑛0𝜏𝜏𝑒𝑒−𝜆𝜆𝑡𝑡 + ln𝑛𝑛0

𝜏𝜏 =−slope

intercept

38

Dead time losses from pulse sources

– e.g., X rays from linear accelerators

• If 𝜏𝜏 ≪ 𝑇𝑇 (steady-state source), the analysis discussed previously can be applied• If 𝜏𝜏 < 𝑇𝑇, the most complicated situation (a small # of counts are registered)• If 𝑇𝑇 < 𝜏𝜏 < 1

𝑓𝑓−𝑇𝑇, a single count per pulse at most

– Prob. of an observed count per source pulse = 𝑚𝑚𝑓𝑓

– Avg. # of true events per source pulse = 𝑛𝑛𝑓𝑓

– Prob. that at least one true event occurs per source pulse:

• 𝑃𝑃 > 0 = 1 − 𝑃𝑃 0 = 1 − 𝑒𝑒−�̅�𝑚 = 1 − 𝑒𝑒− ⁄𝑛𝑛 𝑓𝑓 = 𝑚𝑚𝑓𝑓⇒ 𝑚𝑚 = 𝑓𝑓 1 − 𝑒𝑒− ⁄𝑛𝑛 𝑓𝑓

• 𝑛𝑛 = 𝑓𝑓 ln 𝑓𝑓𝑓𝑓−𝑚𝑚

≅ 𝑚𝑚1− ⁄𝑚𝑚 2𝑓𝑓

for 𝑚𝑚 ≪ 𝑓𝑓

39