Gas Detectors

Ho Kyung KimPusan National University

Radiation Measurement SystemsKnoll chaps. 5, 6, & 7

• Ionization chambers– Current or pulse mode operation (most commonly current mode operation)

• Proportional counters– Introduced in the late 1940s– Pulse mode operation– Can be used for the spectroscopy of low-energy x-ray

• Geiger-Mueller (GM) counters (or Geiger tubes)– Introduced in 1928– Pulse mode operation– Qcol ~ 109 – 1010 (~ V)

• Not need for external amplification– Lack of E information & large dead time

• Only applicable to low counting rates

2

Regions of operation of gas-filled detectors

Insufficient E-field to prevent recombination of the generated ion pairs(Qcol < Qgen)

Suppressed recombination region; Operation mode of ionization chambers

Threshold field at which gas multiplication begins

Linear gas multiplication(Qcol ∝ Qgen);Operation mode of proportional counters

Nonlinear gas multiplicationdue to (+)ive space charge

Self-limiting Geiger discharge(chain reaction of avalanche processes)=> the same amplitude regardless of Ngen

3

Ionization Chambers

Ionization process in gases

• Ion pairs are generated by direct interaction with the incident radiation or by secondary processes (“delta rays”)

• Avg. E required to create an ion pair (defined as the W-value [typ. 25–35 eV/ip]) > Min. ionization E for the least tightly bound electron shells (= 10–25 eV): see Table 5.1

– 𝑊𝑊 = ∆𝐸𝐸𝑁𝑁𝑔𝑔𝑔𝑔𝑔𝑔

– A weak function of gas species, radiation types & energy

– Fano factor 𝐹𝐹 < 1 => fluctuations in Ngen < 𝑁𝑁𝑔𝑔𝑔𝑔𝑔𝑔 (Poisson limit)

5

• Diffusion, charge transfer & recombination

– Diffusion of ion pairs due to the thermal motion (e.g., the m.f.p. of atoms or molecules of the gas = 10-6–10-8 m)

• Gaussian spreading w/ 𝜎𝜎 = 2𝐷𝐷𝐷𝐷

– Recombination btwn (+)ive & (-)ive ions is ×10 larger than that btwn (+)ive ions & free electrons

• Columnar (or initial) recombination– In a column along the track of the ionizing

particle (e.g., HCP)• Volume recombination

– Remote sites (due to drift or diffusion) from the track (e.g., electrons, gammas)

– Dependent on rates

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Charge migration & collection

• Drift of charge carriers under E-field

– 𝑣𝑣 = 𝜇𝜇𝜇𝜇𝑝𝑝

• 𝜀𝜀 = E-field strength [V/m]• 𝑝𝑝 = gas pressure [atm]• 𝜇𝜇 = mobility [m2 atm/V s]

– Ions: 1–1.5×10-4 m2 atm/V s for ions– Electrons: ×103 higher than ions

• 𝑣𝑣 = drift velocity [m/s]– ~1 m/s at 1 atm & 104 V/m for ions

• Transit time in a detector w/ a cm– Ions: ~10 ms– Electrons: ~µs

– Electrons diffuse during the drift motion• ~mm during few µs ("charge spreading")

Electron drift velocity

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• Ionization current

Due to recombination

Ion saturation=> dc ion chamber

Bulk recombination increases w/ irradiation rate

Another loss in the saturation current is due to the diffusion: negligible for ions but significant for electrons

8

Design and operation of dc ion chambers

• In general,– To avoid signal loss due to electron diffusion, use gases (e.g., air) that have high electron

attachment coefficient (negative ions rather than electrons)– To increase the sensitivity, elevate gas pressures

• Guard rings– To reduce the effects of insulator leakage

• Need 1016 Ω to keep the leakage level below 1% of Iionization = 10-12 A at 100 V– Most of V appears across the outer insulator, for which the resulting Ileak does not contribute to

the measured Iionization

Ileak cannot flow

o

×

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• Measurement of ion current– DC electrometer: measuring I by sensing Vdrop across

a series R (typ. 109 – 1012 Ω)• Due to the DC coupling, sensitive to any drifts

or changes in component values

– Dynamic-capacitor or vibrating reed electrometer• Oscillating the capacitance to induce an AC

voltage

– Δ𝑉𝑉 = 𝐼𝐼 𝑅𝑅𝐶𝐶Δ𝐶𝐶

– Integration of ionization current over a finite period of time

• Initially set to V0 and disconnect, and then measuring V at a later time;

– Δ𝑉𝑉 = Δ𝑄𝑄𝐶𝐶

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Radiation dose measurement w/ ion chambers

• “Exposure” includes the ionization created by all the secondary electrons generated by x-ray or gamma-ray irradiation

– How to measure all the ionization created along their tracks? Note the range in air of secondary electrons can be several meters!

– => Use the principle of compensation: • Q created outside the test vol. from 2ndry

electrons that were formed w/i the vol. (①+②) = Q created w/i test vol. from 2ndry electrons formed in the surrounding air (③+④)

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• Free-air ionization chamber– Exposure measurements for gamma-ray

energies < 150 keV

• Cavity chambers– Gamma-ray exposure measurements at

higher energies– Small vol. of air is surrounded by an air-

equivalent solid mat’l• Similar 2ndry electron yield & rate of

electron E loss per unit mass of air• Mat’l having a similar Z w/ air (e.g., Al

or plastics)

Air-equiv. walls should be thicker than the max. 2ndary electron range => electronic equilibrium in which the flux of 2ndary electrons leaving the inner surface of the wall becomes indep. of the wall thickness

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• Absorbed dose– Bragg-Gray principle

• The absorbed dose 𝐷𝐷𝑚𝑚 in a given mat’l can be deduced from the ionization produced in a small gas-filled cavity w/i that mat’l

• 𝐷𝐷𝑚𝑚 = 𝑊𝑊𝑆𝑆𝑚𝑚𝑃𝑃– 𝑊𝑊 = avg. E loss per ion pair formed in the gas– 𝑆𝑆𝑚𝑚 = rel. mass stopping power (energy loss per unit density) of the mat’l to that of the

gas– 𝑃𝑃 = # ion pairs per unit mass formed in the gas

– “Tissue-equiv.” ion chambers• Wall mat'l w/ an elemental composition similar to that of tissue

– Extrapolation chambers

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Applications of DC ion chambers

• (Portable) Radiation survey instruments– Read the saturated ion current by using a battery-powered electrometer circuit

• Condenser R-meter or pocket chamber– Measure the total integrated ionization charge

14

• Radiation source calibrators– Long-term stability (± 0.1% over several years) eliminating frequent recalibration– Is = ~10-13 A for 1 µCi of 60Co

15

• Measurement of radioactive gases

– Fill the radioactive gas as a detector gas mat’l

– 𝐼𝐼 =𝐸𝐸𝛼𝛼𝑔𝑔𝑊𝑊

• 𝐼𝐼 = ionization current (A)• 𝐸𝐸 = avg. E deposited in the gas per integration (eV)• 𝛼𝛼 = tot. activity (Bq)• 𝑊𝑊 = avg. E deposited per ion pair in the gas (eV)

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Pulse mode operation

– Initially 𝑉𝑉0 across the ion chamber– Drift motion of charges under the E-field gives rise to induced charges on the electrodes of the

ion chambers– Voltage drop from 𝑉𝑉0 across the load resistance 𝑅𝑅– Return back to the equilibrium conditions (𝑉𝑉0) on a time scale determined by 𝑅𝑅𝐶𝐶

• For several cm;– 𝐷𝐷𝑐𝑐 = ~µs for electrons– 𝐷𝐷𝑐𝑐 = ~ms for ions (either positive or negative)– 𝑅𝑅𝐶𝐶 should be > ms for the full collection of charges

» Limited to very low pulse rates

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• Assumptions

– Parallel-plate geometry (const. E-field strength 𝜀𝜀 = 𝑉𝑉𝑑𝑑

or uniformly spaced equipotentials)

– Gas spacing 𝑑𝑑 << electrode length/width (neglecting edge effects)– Negligible electron-ion recombination– Generation of ion pairs at 𝑥𝑥– 𝜏𝜏 = 𝑅𝑅𝐶𝐶 → ∞

𝑉𝑉0 − 𝑉𝑉𝑐𝑐𝑐 − 𝑉𝑉𝑅𝑅 = 0

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– Signal pulse 𝑉𝑉𝑅𝑅 ~ a linear rise w/ time during drifting of charges– Note that the induced charge results only from the motion of the charge carriers w/i the chamber

vol. and does not require their collection at either electrode

originalstoredenergy

energyabsorbedby ions

energyabsorbed

by electrons

remainingstoredenergy

= + +

12𝐶𝐶𝑉𝑉0

2 𝑛𝑛0𝑒𝑒𝜀𝜀𝑣𝑣+𝐷𝐷 𝑛𝑛0𝑒𝑒𝜀𝜀𝑣𝑣−𝐷𝐷12𝐶𝐶𝑉𝑉𝑐𝑐𝑐

2= + +

12𝐶𝐶 𝑉𝑉02 − 𝑉𝑉𝑐𝑐𝑐2 = 𝑛𝑛0𝑒𝑒𝜀𝜀(𝑣𝑣+ + 𝑣𝑣−)𝐷𝐷

12𝐶𝐶 𝑉𝑉0 + 𝑉𝑉𝑐𝑐𝑐 𝑉𝑉0 − 𝑉𝑉𝑐𝑐𝑐 = 𝑛𝑛0𝑒𝑒

𝑉𝑉𝑐𝑐𝑐𝑑𝑑 (𝑣𝑣+ + 𝑣𝑣−)𝐷𝐷

𝑉𝑉𝑅𝑅 =𝑛𝑛0𝑒𝑒𝑑𝑑𝐶𝐶 (𝑣𝑣+ + 𝑣𝑣−)𝐷𝐷

𝑉𝑉𝑅𝑅 = 𝑉𝑉0 − 𝑉𝑉𝑐𝑐𝑐𝑉𝑉0 + 𝑉𝑉𝑐𝑐𝑐 ≅ 2𝑉𝑉0

𝑉𝑉𝑐𝑐𝑐𝑑𝑑 ≅

𝑉𝑉0𝑑𝑑

𝑄𝑄+∆𝜑𝜑 𝑄𝑄−∆𝜑𝜑

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– After 𝐷𝐷− = 𝑥𝑥𝑣𝑣−

• 𝑉𝑉𝑅𝑅 = 𝑔𝑔0𝑔𝑔𝑑𝑑𝐶𝐶

(𝑣𝑣+𝐷𝐷 + 𝑥𝑥)

– After 𝐷𝐷+ = 𝑑𝑑−𝑥𝑥𝑣𝑣+

• 𝑉𝑉𝑅𝑅 = 𝑔𝑔0𝑔𝑔𝑑𝑑𝐶𝐶

𝑑𝑑 − 𝑥𝑥 + 𝑥𝑥 = 𝑔𝑔0𝑔𝑔𝐶𝐶

– 𝑉𝑉𝑚𝑚𝑚𝑚𝑥𝑥 = 𝑔𝑔0𝑔𝑔𝐶𝐶

for 𝑅𝑅𝐶𝐶 ≫ 𝐷𝐷+

• Independent of the position at which ion pairs are formed w/i the chamber

– In electron-sensitive operation (neglecting ion drift)

• |𝑉𝑉 𝑔𝑔𝑒𝑒𝑔𝑔𝑐𝑐 = 𝑔𝑔0𝑔𝑔𝐶𝐶 𝑥𝑥𝑑𝑑

• Dependent on the position 𝑥𝑥

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• The gridded ion chamber– Frisch grid transparent to electrons– Remove the dependence of the pulse

amplitude on position of interaction in electron-sensitive ion chambers

– Consider the signal ONLY from the drift of electrons

– 0 < 𝐷𝐷− < 𝑦𝑦𝑣𝑣−

• 𝑉𝑉𝑅𝑅 = 0

– 𝑦𝑦𝑣𝑣−

< 𝐷𝐷− < 𝑑𝑑𝑣𝑣−

• 𝑉𝑉𝑅𝑅 = 𝑔𝑔0𝑔𝑔𝑑𝑑𝐶𝐶

𝑣𝑣−𝐷𝐷

– 𝐷𝐷− > 𝑑𝑑𝑣𝑣−

• 𝑉𝑉𝑅𝑅 = 𝑉𝑉𝑚𝑚𝑚𝑚𝑥𝑥 = 𝑔𝑔0𝑔𝑔𝐶𝐶

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• Pulse amplitude

– e.g., For a 1-MeV charged ptl

• 𝑛𝑛0 = 𝐸𝐸0𝑊𝑊≅ 106 eV

35 eVip

= 2.86 × 104

– Note that typically 𝑛𝑛0 will be smaller by as much as a factor of 100 because of the low 𝑑𝑑𝐸𝐸/𝑑𝑑𝑥𝑥 in gases

– Limited to the pulse mode operation

• 𝑉𝑉𝑚𝑚𝑚𝑚𝑥𝑥 = (2.86×104)(1.6×10−19 C)10−10 F

= 4.58 × 10−5 V

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• Statistical limit to energy resolution

– e.g., Measurement of α w/ 5.5 MeV using a gas detector w/ W-value of 30 eV & a Fano factor of 0.15

• 𝑛𝑛0 = 𝐸𝐸𝑑𝑑𝑊𝑊≅ 5.5×106 eV

30 eVip

= 1.83 × 105 ion pairs

• 𝜎𝜎𝑔𝑔0 = 𝐹𝐹𝑛𝑛0 = 0.15(1.83 × 105) = 2.75 × 104 = 166

• FWHM 𝑛𝑛0 = 2.35𝜎𝜎𝑔𝑔0 = 390

• FWHM 𝐸𝐸 = 2.35𝜎𝜎𝑔𝑔0 𝑊𝑊 = 390 30 eV = 11.7 keV

• 𝑅𝑅 =2.35𝜎𝜎𝑔𝑔0𝑊𝑊

𝐸𝐸𝑑𝑑= 0.213%

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• Charged particle spectroscopy

– Free to arbitrary size & geometry– Can control gas pressure to tailor the stopping power or the thickness of vol.– Hard to radiation damage

– The Frisch-gridded ion chamber for the low-level α ptl spectroscopy• Requiring a low-noise preamplifier & avoidance of mechanical vibrations (to minimize the

modulation of capacitance)• FWHM of 35–45 keV for 5-MeV α ptl’s (reported up to FWHM of 11.5 keV)

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Proportional Counters

Gas multiplication

• Avalanche formation

– For E-field > 106 V/m, electrons undergo collisions w/ other neutral gas molecules and create secondary ionization in the form of cascade

– Townsend avalanche: d𝑔𝑔𝑔𝑔

= 𝛼𝛼d𝑥𝑥

• 𝛼𝛼 = first Townsend coefficient– Const. for a spatially const. field (as in parallel plate

geometry)– 𝑛𝑛 𝑥𝑥 = 𝑛𝑛(0)𝑒𝑒𝛼𝛼𝑥𝑥

• Charge amplification w/i a detector itself reduces the demands on external amplifiers

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• Geometry

– Cylindrical geometry

• 𝜀𝜀 𝑟𝑟 = 𝑉𝑉𝑟𝑟 ln ⁄𝑏𝑏 𝑚𝑚

– 𝑎𝑎 = anode wire radius– 𝑏𝑏 = cathode inner radius– e.g., 2000 V, 𝑎𝑎 = 0.008 cm, b = 1.0 cm

» 5.18 × 106 V/m @ anode» Equiv. to parallel plate => 51,800 V for 1-cm gap

• When the region of gas multiplication is confined to a very small vol. compared w/ the tot. vol. (i.e., vicinity of the anode wire), each electron undergoes the same multiplication process regardless of its original position of formation, and the multiplication factor will be the same for all original ion pairs

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– Electron drift velocity

• 𝑣𝑣 = 𝜇𝜇 𝜇𝜇𝑝𝑝

= 𝜇𝜇𝑉𝑉𝑝𝑝 ln ⁄𝑏𝑏 𝑚𝑚

1𝑟𝑟

– Time required to the drift

• 𝐷𝐷 = ∫𝑚𝑚𝑟𝑟 d𝑣𝑣𝑣𝑣

= 𝑝𝑝 ln ⁄𝑏𝑏 𝑚𝑚2𝜇𝜇𝑉𝑉

𝑟𝑟2

– Typically ~10 µs or more

An avalanche triggered by a single electron as simulated by a Monte Carlo calculation

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Design features of proportional counters

• Sealed tubes

• Windowless flow counters

No gas multiplication

(φ0.25 mm)

Anode wire (φ0.025 mm)

“Pancake” proportional counter2π gas flow proportional counter

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• Fill gases– Gases not exhibiting an appreciable electron attachment coeff.– Quench gas

• Polyatomic stabilizing additive (e.g., methane) to absorb the visible or UV photons from excited gas molecules that cause:

– A loss of proportionality; spurious pulses; increased dead time; reduced spatial resolution

– Argon, Ar 90%/CH4 10% (P-10) – most common– Krypton, Xenon – high efficient gamma-ray detection– Hydrocarbon gases (methane, ethylene) – not a considerable stopping

power– BF3, 3He – thermal neutron detection– CH4 64.4%/CO2 32.4%/N2 3.2% – dosimetry– Penning effect

• Further ionization of the additive due to a collision btwn the metastable excited atom & a neutral additive atom

• Reduce the W-value & improve the energy resolution

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Proportional counter performance

• Gas multiplication factor

– 𝑄𝑄 = 𝑛𝑛0𝑒𝑒𝑒𝑒• 𝑛𝑛0 = initial ion pairs• 𝑒𝑒 = avg. gas multiplication factor

– For a cylindrical geometry

• ln𝑒𝑒 = ∫𝑚𝑚𝑟𝑟𝑐𝑐 𝛼𝛼 𝑟𝑟 d𝑟𝑟 = ∫𝜇𝜇(𝑚𝑚)

𝜇𝜇(𝑟𝑟𝑐𝑐)𝛼𝛼 𝜀𝜀 𝜕𝜕𝑟𝑟𝜕𝜕𝜇𝜇

d𝜀𝜀 = 𝑉𝑉ln( ⁄𝑏𝑏 𝑚𝑚)∫𝜇𝜇(𝑚𝑚)

𝜇𝜇(𝑟𝑟𝑐𝑐) 𝛼𝛼 𝜇𝜇𝜇𝜇

d𝜇𝜇𝜇𝜇

– Assuming 𝛼𝛼 ∝ 𝜀𝜀

• ln𝑒𝑒 = 𝑉𝑉ln( ⁄𝑏𝑏 𝑚𝑚)

ln 2∆𝑉𝑉

ln 𝑉𝑉𝑝𝑝𝑚𝑚 ln( ⁄𝑏𝑏 𝑚𝑚)

− ln𝐾𝐾

– ∆𝑉𝑉 = the potential difference thru which an electron moves btwn successive ionization events

– 𝐾𝐾 = min. value of 𝜇𝜇𝑝𝑝

below which multiplication cannot occur

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⇒ Diethorn plotSee also Table 6.1

Slope = (ln2/∆V)Intercept = (-ln2)(lnK)/∆V

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• Space charge effects– Slow positive ions around the anode wire– Distortion (or reduction) in the E-field

• Reduced pulse amplitude => nonlinearity• Fluctuations in the reduced pulse amplitudes => loss of energy resolution

– Self-induced space-charge effect (when the gas gain is high)• Positive ions during a given avalanche => alteration in E-field => reduction in # of electrons in

further stages of the same avalanche• Depends on the gas gain & detector geometry• Not depends on the event rate

– General space-charge effect• Cumulative effect of positive ions from many diff’t avalanches• Important at lower gas gain• More serious with increasing event rate

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• Energy resolution

– 𝑒𝑒 = 1𝑔𝑔0∑𝑖𝑖=1𝑔𝑔0 𝐴𝐴𝑖𝑖 ≡ 𝐴𝐴 avg. multiplication factor

• 𝐴𝐴 = e– multiplication factor of an avalanche triggered by a single e–

• 𝑛𝑛0 = # of ion pairs or # of avalanches

– Recall 𝑄𝑄 = 𝑛𝑛0𝑒𝑒𝑒𝑒

– Rel. variance in Q:𝜎𝜎𝑄𝑄𝑄𝑄

2=

𝜎𝜎𝑔𝑔0𝑔𝑔0

2+ 𝜎𝜎𝑀𝑀

𝑀𝑀

2=

𝜎𝜎𝑔𝑔0𝑔𝑔0

2+ 1

𝑔𝑔0

𝜎𝜎𝐴𝐴𝐴𝐴

2

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• Variations in the number of ion pairs, 𝑛𝑛0

– Introducing the Fano factor

• 𝜎𝜎𝑔𝑔02 = 𝐹𝐹𝑛𝑛0

•𝜎𝜎𝑔𝑔0𝑔𝑔0

2= 𝐹𝐹

𝑔𝑔0

• Typically, 0.05 ≤ 𝐹𝐹 ≤ 0.20 (see Table 6.2)

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• Variations in single-electron avalanches, 𝐴𝐴– Assuming that the prob. of ionization by an e– is independent of its previous history, the expected

distri. in # of e– produced in a given avalanche follows the Furry distri.

• 𝑃𝑃 𝐴𝐴 = (1− ⁄1 𝐴𝐴)𝐴𝐴−1

𝐴𝐴≅ 𝑔𝑔 ⁄−𝐴𝐴 𝐴𝐴

𝐴𝐴

• Then, we have 𝜎𝜎𝐴𝐴𝐴𝐴

2= 1

– In strong E-fields, the prob. of ionization by an e– depends on the previous history and the distri. follows a Polya distri.

• 𝑃𝑃 𝐴𝐴 = 𝐴𝐴(1+𝜃𝜃)𝐴𝐴

𝜃𝜃exp −𝐴𝐴(1+𝜃𝜃)

𝐴𝐴

– 0 < 𝜃𝜃 < 1

• Then, we have 𝜎𝜎𝐴𝐴𝐴𝐴

2= 1

𝐴𝐴+ 𝑏𝑏 ≅ 𝑏𝑏

– 𝑏𝑏 = 11+𝜃𝜃

, typically 0.4 ≤ 𝑏𝑏 ≤ 0.7

If 𝐴𝐴 ≥ 50 or 100

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• Overall statistical limit

– Recall𝜎𝜎𝑄𝑄𝑄𝑄

2=

𝜎𝜎𝑔𝑔0𝑔𝑔0

2+ 1

𝑔𝑔0

𝜎𝜎𝐴𝐴𝐴𝐴

2= 1

𝑔𝑔0(𝐹𝐹 + 𝑏𝑏)

– Since 𝑛𝑛0 = ⁄𝐸𝐸 𝑊𝑊,𝜎𝜎𝑄𝑄𝑄𝑄

= 𝑊𝑊(𝐹𝐹+𝑏𝑏)𝐸𝐸

• Note that the term 𝑊𝑊(𝐹𝐹 + 𝑏𝑏) is constant for a given fill gas (see Table 6.2)

• The statistical limit of E resolution of a proportional counter ∝ 𝐸𝐸−1

– Ex) W = 35 eV, F = 0.2, b = 0.61

• For 10 keV, 𝑅𝑅 = FWHM𝐻𝐻0

= 2.35𝜎𝜎𝑄𝑄𝑄𝑄

= 2.35 𝑊𝑊(𝐹𝐹+𝑏𝑏)𝐸𝐸

= 12.5%

• For 100 keV, 𝑅𝑅 = 3.9%

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• Other factors affecting energy resolution– Electronic noise (negligible)– Geometric nonuniformities

• Uniformity & smoothness of the anode wire– Variations in operating parameters (see Table 6.3)

• Gas purity• Gas pressure• Stability of high voltage

– Count-rate effect– Aging

𝑊𝑊(𝐹𝐹 + 𝑏𝑏)

Anode wire w/ improved uniformityP-10, 1 atm

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• Signal pulse

Generation of ion pairs (~ns) Drift of electrons (~few µs)

Onset of avalanche (< µs)Drift of (-) avalanches (< µs)

Drift of (+) avalanches (~few µs),producing most of the output pulse amplitude

39

– Starting from the energy absorbed by the motion of a positive charge thru a difference in E-potential in cylindrical geometry:

• d𝐸𝐸 = −𝑄𝑄d𝜑𝜑

• d𝐸𝐸d𝑟𝑟

= 𝑄𝑄𝜀𝜀 𝑟𝑟 = 𝑄𝑄 𝑉𝑉0𝑟𝑟 ln ⁄𝑏𝑏 𝑚𝑚

from 𝜀𝜀 𝑟𝑟 = d𝜑𝜑(𝑟𝑟)d𝑟𝑟

& 𝜀𝜀 𝑟𝑟 = 𝑉𝑉𝑟𝑟 ln ⁄𝑏𝑏 𝑚𝑚

– Assuming 𝑛𝑛0 electrons & ions in an avalanche at a fixed distance 𝜌𝜌 from the surface of the anode wire and 𝑄𝑄 = 𝑛𝑛0𝑒𝑒:

• E absorbed by the ions: 𝐸𝐸+ = ∫𝑚𝑚+𝜌𝜌𝑏𝑏 d𝐸𝐸

d𝑟𝑟d𝑟𝑟 = 𝑄𝑄𝑉𝑉0

ln ⁄𝑏𝑏 𝑚𝑚 ∫𝑚𝑚+𝜌𝜌𝑏𝑏 d𝑟𝑟

𝑟𝑟= 𝑄𝑄𝑉𝑉0

ln ⁄𝑏𝑏 𝑚𝑚ln 𝑏𝑏

𝑚𝑚+𝜌𝜌

• E absorbed by the e–’s: 𝐸𝐸− = − 𝑄𝑄𝑉𝑉0ln ⁄𝑏𝑏 𝑚𝑚 ∫𝑚𝑚+𝜌𝜌

𝑏𝑏 d𝑟𝑟𝑟𝑟

= 𝑄𝑄𝑉𝑉0ln ⁄𝑏𝑏 𝑚𝑚

ln 𝑚𝑚+𝜌𝜌𝑏𝑏

• Total E: ∆𝐸𝐸 = 𝐸𝐸+ + 𝐸𝐸− = 𝑄𝑄𝑉𝑉0ln ⁄𝑏𝑏 𝑚𝑚

ln 𝑏𝑏𝑚𝑚+𝜌𝜌

𝑚𝑚+𝜌𝜌𝑏𝑏

= 𝑄𝑄𝑉𝑉0

40

– From the energy conservation:

• 12𝐶𝐶𝑉𝑉𝑐𝑐𝑐2 = 1

2𝐶𝐶𝑉𝑉02 − ∆𝐸𝐸

– Assuming 𝑉𝑉𝑐𝑐𝑐 + 𝑉𝑉0 ≅ 2𝑉𝑉0 & substituting 𝑉𝑉𝑅𝑅 = 𝑉𝑉0 − 𝑉𝑉𝑐𝑐𝑐:

• 𝑉𝑉𝑅𝑅 = ∆𝐸𝐸𝐶𝐶𝑉𝑉0

= 𝑄𝑄𝐶𝐶

– Max. pulse amplitude if RC >> the ion collection time

– Note that 𝐸𝐸−

𝐸𝐸+= ln ⁄(𝑚𝑚+𝜌𝜌) 𝑚𝑚

ln ⁄𝑏𝑏 (𝑚𝑚+𝜌𝜌)= 1.9% for a = 0.025 mm, b = 10 mm & ρ = 0.003 mm

• The positive ion drift dominates the pulse formation• We may neglect the electron contribution in the pulse formation

41

– Putting the ion drift velocity 𝑣𝑣+ 𝑟𝑟 = 𝜇𝜇𝜇𝜇(𝑟𝑟)𝑝𝑝

= 𝜇𝜇𝑝𝑝

𝑉𝑉0ln ⁄𝑏𝑏 𝑚𝑚

1𝑟𝑟

into the law of motion ∫𝑚𝑚𝑟𝑟(𝑡𝑡) d𝑟𝑟

𝑣𝑣+ 𝑟𝑟= ∫0

𝑡𝑡 d𝐷𝐷

• Time-dependent position of the ions: 𝑟𝑟 𝐷𝐷 = 2 𝜇𝜇𝑝𝑝

𝑉𝑉0ln ⁄𝑏𝑏 𝑚𝑚

𝐷𝐷 + 𝑎𝑎2⁄1 2

• Time required to collect the ions: 𝐷𝐷+ = 𝑏𝑏2−𝑚𝑚2 𝑝𝑝 ln ⁄𝑏𝑏 𝑚𝑚2𝜇𝜇𝑉𝑉0

~ several hundred µs

– Time-dep. signal pulse: 𝑉𝑉𝑅𝑅 𝐷𝐷 = 𝑄𝑄𝐶𝐶

1ln ⁄𝑏𝑏 𝑚𝑚

ln 2𝜇𝜇𝑉𝑉0𝑚𝑚2𝑝𝑝ln ⁄𝑏𝑏 𝑚𝑚

𝐷𝐷 + 1⁄1 2

• |𝐷𝐷 half amplitude = 𝑚𝑚𝑚𝑚+𝑏𝑏

𝐷𝐷+

– The half-amplitude point is reached after only 0.25% of the full ion drift time for a = 0.025 mm & b = 10 mm, which corresponds to a fraction of µs

• 𝑟𝑟 |𝐷𝐷 half amplitude = 𝑎𝑎𝑏𝑏– The ions moves 0.475 mm from the wire surface for the above case, where the E-field

is down to 5% of its surface value (= ⁄𝑎𝑎 𝑏𝑏)• Fast leading edge of the pulse is followed by a much slower rise corresponding to the drift of

the ions thru the lower-field regions found at larger radial distance

42

– Additional spread in the rise time due to the ion pairs produced along the track of the incident radiation that covers a range of radii (i.e., All ion pairs are not formed at a fixed position!)

– Ballistic deficit: signal loss due to a finite shaping time to remove the slow component of the drift of the ions (compared to the pulse amplitude achieved by an infinite time constant)

• Spurious pulses– Avalanche produced by a photoelectron emitted from the cathode wall due to the photoelectric

interaction of photons emitted by excited atoms produced in the primary avalanches– Field emission of electrons at the cathode wall due to the positive ions

Initial ionization at a single radius

Uniform ionization along a dia.

Leading edges of output pulses

43

Detection efficiency & counting curves

• Selection of the operating voltage– Recall a plateau on the resulting rate vs. voltage curve

• Alpha counting– Monoenergetic charge particles whose range is less than

the dimensions of the chamber– Windowless flow counters rather than sealed tubes

whose window may result in energy loss

• Beta counting– The beta particle range greatly exceeds the chamber

dimension– Signal is proportional to small fraction of the energy lost

in the gas– 4π flow counter

44

• X-ray & gamma-ray sources– E should be low enough to interact w/ reasonable efficiency in the

counter gas– Low-E x-rays

• A few hundred ion pairs• Not enough compared w/ the equiv. noise charge at the input

of typical low-noise preamplifiers• Require additional “internal” gains• Escape peaks

– Characteristic x-rays (e.g., Kα = 2.97, 12.6, & 29.7 keV for Ar, Kr, & Xe, respectively)

• Absorption of characteristic x-rays from the entrance window or wall of the counters

– Low-Z beryllium or aluminum (50–250 µm)

5.08-cm thickness

45

Variants of the proportional counter design

• Tissue equiv. proportional counter

• Parallel plate avalanche counters

• Position-sensitive proportional counters

46

• Multiwire proportional counters

47

48

49

• Gas proportional scintillation counters

50

• Micropattern gas detectors

– Microstrip (A. Oed, 1988)/microgap (R. Bellazzini, 1995) gas chamber

Microdischarges Full breakdown

51

52

– Gas electron multiplier (F. Sauli, 1997)

53

9-keV absorption radiography of a bat (30 × 60 mm2)

54

– Micromegas (micro-mesh gaseous structure) (Y. Giomataris, 1996)

– Resistive plate chambers

55

Geiger-Mueller Counters

• Unlike proportional counters, at a critical value of E-field, each avalanche can itself trigger one more avalanches and self-propagating chain reaction results in G-M counters, yielding exponentially growing number of avalanches (~109–1010 ion pairs => ~V) w/i a very short time

• Once this Geiger discharge reaches a certain size, collective effects of all the individual avalanches terminate the chain reaction (this limiting point is always reached after about the same number of avalanches have been created)

• All pulses from a Geiger tube are of the same amplitude regardless of the number of original ion pairs that initiated the process

• Only used as a simple counter

57

The Geiger discharge

• Conditions– Higher M = 106–108 (Note that M = 102–104 in proportional counters)– 𝑛𝑛0′ 𝑝𝑝 ≥ 1

• 𝑛𝑛0′ = # of excited molecules formed in a avalanche• 𝑝𝑝 = prob. of photoelectric absorption of any given photon (visible or UV from excited

molecules) w/ other atoms or the cathode wall– Time for the development of the Geiger discharge = a few µs

• Buildup of positive ion space charge eventually terminates the Geiger discharge– Terminates after developing about the same total charge regardless of # of original ion pairs (the

same size of output pulses)

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Fill gases

• Noble gases: Helium, Argon• Usually operated at less than or equal to the atmospheric press

• Quenching– To prevent the possibility of excessive multiple pulsing due to secondary discharges initiated by

electrons produced when positive ions neutralize at the cathode surface– External quenching

• Reducing the high voltage, for a fixed time after each pulse, to a value that is too low to support further gas multiplication

• Quenching time > transit time of positive ions to the cathode (a few hundred µs) + transit time of free electrons (~µs)

• External resistor: limited to very low counting rates

~108 Ω, hence RC ~ms

59

– Internal quenching• Adding a second component (quench gas) having a lower ionization potential and more

complex molecular structures w/ a concentration of 5–10% to the primary fill gas• While the quench gas absorbs UV photons in proportional counters, it prevents multiple

pulsing thru the mechanism of charge transfer collisions in G-M counters• Positive ions transfer charges to quench gas molecules, and the excess energy of quench gas

is used in disassociation (rather than ionization) when neutralized at the cathode• Organic molecules: ethyl alcohol, ethyl formate

– Limited lifetime: ~109 counts• Halogen molecules: chlorine or bromine

– Infinite lifetime because halogen molecules recombines after disassociation

60

Time behavior

• Pulse profile– A fast rise on the order of µs (the motion of the ions thru high-

field region)– A much slower rise time due to slow drift of ions thru lower-

field region

• Dead time– Period btwn the initial pulse and the time at which a second

Geiger discharge, regardless of its size, can be developed (50–100 µs)

• Resolving time– The elapsed time required to develop a second discharge that

exceeds a certain level of amplitude

• Recovery time– Time interval required for the tube to return to its original state

capable of producing a second pulse of full amplitude

61

The Geiger counting plateau

62

Design features

63

Counting efficiency

64

Time-to-first-count method

65

G-M survey meters

66