Coincidence in 22Na - Department of Physics · 2019-02-15 · of Gamma Ray Spectroscopy. There are...

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γ -γ Coincidence in 22 Na Experiment GGC University of Florida — Department of Physics PHY4803L — Advanced Physics Laboratory Objective You will learn how to use two NaI scintilla- tion detectors with specialized electronic mod- ules to measure and analyze the coincident emission of gamma ray pairs created in the positron-electron annihilation process. You will also learn about “square root statistics” for counting experiments, and about the at- tenuation of gamma rays as they pass through lead. References Jerome L. Duggan, Laboratory Investigations in Nuclear Science, (The Nucleus, Oak Ridge, TN 1988). A. C. Melissinos, Experiments in Modern Physics, (Academic Press, New York, NY 1966). Positron Emission Tomography article on Wikipedia Annihilation Radiation 22 Na radioactively decays to an excited state of 22 Ne either by emission of a positron (90% probability) or by electron capture (10% prob- ability). The excited 22 Ne nucleus decays with a mean life 3 ×10 -12 s to the ground state with the emission of a 1.274 MeV gamma. Figure 1: The decay of 22 Na proceeds by positron β + emission (90%) or electron cap- ture (10%) to produce an excited state of 22 Ne which decays by emission of a 1.274 MeV gamma. The positrons are emitted with a range of ki- netic energies up to about 0.5 MeV. They lose this energy quickly (10 -9 s) in the material surrounding the source and, when they reach atomic (eV) energies, capture an electron to form positronium—a hydrogen-like “atom.” The positronium decays (with a lifetime on the order of 10 -10 s) by annihilation of the e + and e - into two gammas. By energy conserva- tion, the energy of the gammas must equal the GGC 1

Transcript of Coincidence in 22Na - Department of Physics · 2019-02-15 · of Gamma Ray Spectroscopy. There are...

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γ-γ Coincidence in 22Na

Experiment GGC

University of Florida — Department of PhysicsPHY4803L — Advanced Physics Laboratory

Objective

You will learn how to use two NaI scintilla-tion detectors with specialized electronic mod-ules to measure and analyze the coincidentemission of gamma ray pairs created in thepositron-electron annihilation process. Youwill also learn about “square root statistics”for counting experiments, and about the at-tenuation of gamma rays as they pass throughlead.

References

Jerome L. Duggan, Laboratory Investigationsin Nuclear Science, (The Nucleus, OakRidge, TN 1988).

A. C. Melissinos, Experiments in ModernPhysics, (Academic Press, New York, NY1966).

Positron Emission Tomography article onWikipedia

Annihilation Radiation

22Na radioactively decays to an excited stateof 22Ne either by emission of a positron (90%probability) or by electron capture (10% prob-ability). The excited 22Ne nucleus decays witha mean life 3×10−12 s to the ground state withthe emission of a 1.274 MeV gamma.

Figure 1: The decay of 22Na proceeds bypositron β+ emission (90%) or electron cap-ture (10%) to produce an excited state of22Ne which decays by emission of a 1.274 MeVgamma.

The positrons are emitted with a range of ki-netic energies up to about 0.5 MeV. They losethis energy quickly (10−9 s) in the materialsurrounding the source and, when they reachatomic (eV) energies, capture an electron toform positronium—a hydrogen-like “atom.”The positronium decays (with a lifetime onthe order of 10−10 s) by annihilation of the e+

and e− into two gammas. By energy conserva-tion, the energy of the gammas must equal the

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energy (including the rest mass energy) of thepositronium and, by momentum conservation,the net momentum of the two gammas mustequal the initial momentum of the positron-ium.

The magnitude of the photon momentum isgiven by

pγ =Eγc

(1)

In the rest frame of the positronium, there isno initial momentum and thus the two an-nihilation gammas must be of opposite mo-mentum in that frame. Consequently, the twophotons must be equal in energy and prop-agate in opposite directions. Since the ini-tial energy of the positronium (neglecting thebinding energy of a few eV) is simply therest mass energy of an electron and positron(0.511 MeV each), each gamma will have anenergy Eγ = 0.511 MeV.

In the laboratory frame, the positroniumwill be moving with a range of kinetic ener-gies up to a few eV (typical energy of elec-trons with which it forms). Then, dependingon the direction of the initial positronium mo-mentum relative to the gamma emission di-rection, the transformation to the lab framegives gamma energies that might differ from0.511 MeV and/or produce gammas that arenot emitted exactly 180◦ apart.

Exercise 1 Assume the positronium is mov-ing with a kinetic energy of 10 eV in the labframe. Determine the lab-frame energy of eachgamma if they are emitted in the rest frameparallel to the direction of motion. Determinethe lab frame angle between the gammas if theyare emitted in the rest frame perpendicular tothe direction of motion.

The previous exercise demonstrates that inthe lab frame the two annihilation gammas areemitted within a degree or less of 180◦ from

one another. The main objective of this ex-periment is to verify this behavior.

Apparatus

Read up in Experiment GA on the principlesof Gamma Ray Spectroscopy. There are twoscintillation detectors for this experiment, oneto detect each of the annihilation gammas.Each detector consists of an integral NaI scin-tillation crystal, photomultiplier tube (PMT),and a preamplifier. Each preamplifier outputis connected to a linear amplifier with variablegain. The amplifier output consists of analogpulses—voltage pulses having a duration of afew microseconds and having a height or am-plitude proportional to the gamma ray energydeposited in the scintillator. Read up aboutthe pulse height analyzer (PHA)—an elec-tronic module inside the computer that sorts,records, and histograms the analog pulses ac-cording to their pulse heights. Learn aboutthe spectrum of analog pulse heights expectedfrom a monoenergetic gamma source; the pho-topeak, Compton plateau, and backscatterpeak are features you will need to be able torecognize in the pulse height spectrum.

Figure 2 shows how the detectors and sourcewill be arranged for the experiment. Sim-ply stated, the experiment will detect coinci-dences. Our definition of a coincidence will bethe near simultaneous detection of a gammain the 0.511 MeV photopeak in each detec-tor. The rate of these coincidences will be highwhen the detectors are on opposite sides of the22Na source and will decrease as the detectorsare moved away from this orientation. Thetheory section will describe a model predict-ing the coincidence rate, but before presentingthis model, we next describe how a coincidencewill be detected.

The electronics for the coincidence measure-ments is shown in Fig. 6.

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γ-γ Coincidence in 22Na GGC 3

Figure 2: The two detectors for coincidencemeasurements have lead (Pb) shields withapertures in front of scintillation crystals.

Of fundamental importance for this experi-ment is the PHA’s gate input. The gate is alogic input (0, 5 V) used with the analog pulseinput. When the gate input is used, the PHAonly processes those pulses that occur whilethe gate is high (at a 5 V level); it ignoresthose pulses occurring while the gate is low(at a 0 V level). To detect coincidences, thePHA will be set up to process analog pulsesfrom the movable detector with its gate inputderived from the analog pulses of the fixed de-tector. As described next, the Single ChannelAnalyzer and the Gate and Delay Generatorwill be used to create the gating signal andwith the Delay Amplifier will reliably overlapit with the analog pulses.

The analog pulses from the fixed detectorwill be used to make the gating signal. To doso they are first processed by a Single Chan-nel Analyzer (SCA), which is used to pickout those pulses that are in the 0.511 MeVphotopeak. The SCA has a “window” from Eto E + ∆E with both the lower limit E andthe window width ∆E user adjustable. TheSCA window is actually a voltage window but

is labeled as an energy in anticipation of itsuse for gamma ray energy analysis. The SCA“looks at” its analog pulse input and “decides”whether the pulse height is within the window.If it is, the SCA puts out a short 5 V logicpulse about 0.5 µs long. E and ∆E will beset so that the logic pulse is produced only foranalog pulses in the 0.511 MeV photopeak.

The logic pulses from the SCA are routedto the Gate and Delay Generator. Thismodule puts out a 5 V pulse that lasts afew microseconds after the arrival of the SCAlogic pulse which initiated it. This rectangu-lar pulse is called the gating signal and will berouted to the gate input of the PHA.

Before going to the PHA, the analog pulsesfrom the movable detector are sent throughthe Delay Amplifier, a module which out-puts these analog pulses with little or nochange in their shape or height, but delayedfrom the input pulses by some fixed amount.This delay is needed because the SCA and theGate and Delay Generator take some mini-mum time to create the gating signal. With-out the delay, the gating signal would be toolate to overlap with the analog pulses. Exam-ine Fig. 3 to see the timing relationships.

The name singles will be used to describeeither those gammas which result in the pro-duction of an SCA logic pulse (fixed detectoror SCA singles), or those gammas that resultin a count in the 0.511 MeV photopeak of thePHA spectrum with gating off (movable detec-tor or PHA singles). With the SCA properlyset, both types of singles are gammas detectedin the 0.511 MeV photopeak. We can now de-fine more precisely that coincidences will referto those gammas in 0.511 MeV photopeak ofthe PHA spectrum with gating on, i.e., in co-incidence with 0.511 MeV gammas from thefixed detector. (The Procedure section will de-scribe exactly how the modules are set up.)

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Figure 3: Relative timing of the various pulses.

Theory

The nuclear decay rate (number of nuclear de-cays per second) is Γn = α · 3.7 × 1010 de-cays/sec/curie, where α is the source activ-ity in curies. Ninety percent of the time,the nuclear decay proceeds by β+ emission,and these always annihilate with an electronto produce two 0.511 MeV gammas. Thusthe rate of emission of 0.511 MeV gammas is0.9 ·2 ·Γn = 1.8Γn. These gammas are emitteduniformly (in oppositely directed pairs) fromthe source. Thus at a distance R from the

source they are spread out over an area 4πR2

and the flux Φ (number per unit area per sec-ond) will be

Φ =1.8Γn4πR2

(2)

The geometry of the source and detectors isshown in Fig. 2. The source is about 2 mmin size located near the middle of the disk andthe annihilation events occur within a few mil-limeters of the source. One detector will befixed and the other will be rotated about thesource. Both use 1′′×1′′ cylindrical NaI scintil-lators. In front of each detector there is a leadshield with a small aperture of radius ra drilledthrough its center. The source-aperture dis-tance will be measured and is given the symbolR. The apertures, the scintillators and theirrelationship to the source and each other arean important factor in determining the ratesat which gammas are detected.

To calculate the true rate at which gammasare striking the face of the scintillator, the ef-fect of the lead aperture needs to be taken intoaccount. We call the effective aperture radiusra and the effective scintillator radius rs. Theterm “effective” is used because we will nottake into account the full three dimensionalnature of the shielding aperture or of the scin-tillator. We will determine the rate of passageof gammas as the rate passing through circlesof the effective radii assuming both are a dis-tance R from the source.

The fraction of 0.511 MeV gammas whichget through the lead shielding will also need tobe specified. It is expected to be on the orderof 20% and will be expressed by the symbol κ.

With these considerations, the rate Q of0.511 MeV gammas striking the face of thescintillator can be calculated.

Q = Φ[Aa + κ(As − Aa)] (3)

where the area of the aperture is Aa = πr2a,and the area of the scintillator is As = πr2s .

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γ-γ Coincidence in 22Na GGC 5

Figure 4: With the assumption of 180◦ gammapairs, real coincidences only occur when themovable scintillator detects a gamma in the c-flux—outgoing gammas for which the oppositemember of the pair is detected in the fixedscintillator.

Thus As−Aa is the annular area of the detec-tor shielded by lead.

For a variety of reasons, SCA or PHA sin-gles will occur for only a fraction f of the0.511 MeV gammas entering the correspond-ing scintillation crystal. f is called the pho-topeak efficiency and depends on crystal size,SCA or PHA settings and source/detector ge-ometry. The singles rates are thus suppressed(from the value Q) by the factor f , and thesingles rates S will be given by

Sf,m = ff,m ∗Q (4)

where we have allowed for the possibility thatthe photopeak efficiency may be different forthe fixed (subscript f) and movable (subscriptm) detectors.

Coincidence rate

With the assumption that 0.511 MeV gam-mas are always produced in pairs emitted 180◦

apart, we may model the detection of singles in

the fixed detector as creating a flux of possiblecoincidence gammas moving in the oppositedirection. We will call this c-flux (for coinci-dence possible flux) and use the symbol Φ′ forit. The situation is as shown in Fig. 4.

The c-flux is not the same as the raw flux of0.511 MeV gammas from the source. It is notuniform; it extends only into the region oppo-site the fixed detector from the source. Withboth detectors at the same distance R fromthe source, the c-flux will be the product ofthe raw flux at that distance and the probabil-ity that the fixed detector will register a countin the photopeak. In the region opposite theaperture, it is reduced in intensity from theraw flux by the photopeak efficiency.

Φ′center = Φ · ff (5)

In the annular region it is further reduced bythe attenuation due to the Pb shielding.

Φ′annulus = Φ · κff (6)

It will help in the analysis to remodel thesec-flux regions as the sum of two complete cir-cles of c-flux: a higher c-flux extending over acircular area of radius ra

Φ′a = Φ(1− κ)ff (7)

and a lower c-flux extending over a circulararea of radius rs

Φ′s = Φκff (8)

The sum of these c-fluxes give the same resultas the two c-fluxes Φ′center and Φ′annulus.

Keep in mind the c-flux is a real flux ofgammas radiating out from the source oppo-site the fixed detector. It is modeled based onthe assumption that the 0.511 MeV gammasare always created in 180◦ pairs. The c-fluxis simply a selected fraction of the total flux,for which the opposite gamma of the pair is

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detected in the fixed detector. Consequently,any of the c-flux gammas detected in the mov-able detector’s photopeak will be counted ascoincidences.

Because of the movable detector’s photo-peak efficiency and shielding aperture, it willonly count a fraction of the c-flux gammas.Over its aperture area, the detection proba-bility is fm; in its annular (shielded) area, theprobability is lower—κfm.

Again, this situation is more easily modeledas the sum of two full circular detection re-gions: one of (scintillator) radius rs with alower detection probability κfm and anotherof (aperture) radius ra with a higher detectionprobability (1 − κ)fm. The efficiency of thesum of these two detection regions will be thesame as that of circular and annular regionsdiscussed in the previous paragraph.

Real coincidences should only occur whenthere is some overlap of the movable detectorinto the region of the c-flux. If the fixed andmovable detectors are very far from being op-posite one another, there will be no overlapand no chance of real coincidences. As themovable detector is brought in to the c-flux,real coincidences just start to occur when itsshielded area starts touching the circular areacontaining the lower c-flux. As the overlaparea grows, the coincidence rate increases inproportion to that area. In fact, the coinci-dence rate would be the product of the overlaparea, the magnitude of the c-flux in that areaand the movable scintillator’s detection proba-bility for that area. Of course, as there beginsto be overlap with the higher c-flux areas andhigher detection probability regions, these willalso need to be taken into account. The waythe c-flux and movable detector efficiency wereeach modeled as a sum of two complete circu-lar regions, there will be four possible overlap-ping areas, and the coincidence rate will bethe sum of four such products.

Figure 5: The overlap area A′ (hatched) de-pends on the radii r1 and r2 of the circles andthe separation δ between their centers. Thehalf-sector angles φ1 and φ2, found from thelaw of cosines, is useful in expressing the areaformula.

The overlap areas between the various cir-cles involved can be calculated from the ge-ometry of the apparatus. They will dependon the radii of the two circles involved andon the separation δ between the centers of thecircles. We will assume the aperture and scin-tillator circles for each detector have commoncenters so that all overlap areas for a givenposition of the movable detector will have thesame value for δ.

Exercise 2 With reference to Fig. 5, showthat the hatched overlap area A′ of two cir-cles of radii r1 and r2 (where r1 > r2) is given

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γ-γ Coincidence in 22Na GGC 7

by

for δ < r1 − r2 :

A′ = πr22

for r1 − r2 < δ < r1 + r2 :

A′ = r21(φ1 − 1

2sin 2φ1

)+

r22(φ2 − 1

2sin 2φ2

)for δ > r1 + r2 :

A′ = 0

(9)

where the half-sector angles φ1, φ2 can be ob-tained from

r21 = r22 + δ2 − 2r2δ cosφ2 (10)

r22 = r21 + δ2 − 2r1δ cosφ1

Components of the coincidence rate C cannow be calculated as the product of the c-flux,the overlap area, and the detection probabil-ity.

Lower c-flux area with lower detection prob-ability area

Css = Φ′sκfmA′ss (11)

= fmffΦκ2A′ss

Lower c-flux area with higher detection prob-ability area

Csa = Φ′s(1− κ)fmA′sa (12)

= fmffΦκ(1− κ)A′sa

Higher c-flux area with lower detection prob-ability area

Cas = Φ′aκfmA′as (13)

= fmffΦ(1− κ)κA′as

Higher c-flux area with higher detectionprobability area

Caa = Φ′a(1− κ)fmA′aa (14)

= fmffΦ(1− κ)2A′aa

In each case, the doubly subscripted area rep-resents the overlap area of two circles of thecorresponding two radii when their centers areoffset by the amount δ. Of course, the netcoincidence rate will be the sum of the fourcomponents above.

C = Css + 2Csa + Caa (15)

where use has been made of the fact thatA′as = A′sa.

Exercise 3 The coincidence rate would begreatest when the movable detector is centeredover the c-flux pattern, i.e., for δ = 0. Showthat the maximum coincidence rate reduces tothe expected result

Cmax = Φfmff [Aa + κ2(As − Aa)] (16)

Also explain why this is the expected result.

Equations 4 and 16, respectively, predict thesingles rates Sm and Sf and the maximum co-incidence rate Cmax based on the (presumedunknown) flux Φ and photopeak efficienciesfm and ff and the (presumed known) geom-etry factors Aa and As and attenuation fac-tor κ. The next exercise asks you to invertthese equations to predict the flux Φ and thegeometric mean of the photopeak efficienciesfrom the known geometry and attenuation fac-tor and your actual measured rates for Sm, Sfand Cmax. Of course, you can then use thedetermination of Φ with Eq. 2 and anotherknown geometry factor (4πR2) and a theoryfactor (1.8) to determine the actual source ac-tivity Γn.

Exercise 4 Show that the raw flux Φ can beexpressed

Φ =SmSfCmax

[Aa + κ2(As − Aa)

(Aa + κ(As − Aa))2

](17)

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and that the geometric mean of the photopeakefficiencies can be expressed

√fmff =

Cmax√SmSf

Aa + κ(As − Aa)Aa + κ2(As − Aa)

(18)

Random Coincidences

Even when SCA and PHA singles arrive ran-domly, there is a finite probability that onewill arrive in each detector near enough in timeto be counted as a coincidence. The averagerate at which these random coincidences canbe expected to occur is easily calculated anddepends on the singles rates and the width τof the gating signal.

If SCA pulses arrive at a rate RS, the prod-uct τRS is the fraction of the time f that thePHA gate will be open. For example, considerthe case where τ is 4 µs and RS = 500/s. Onaverage, every second, there will be 500 SCApulses and thus, during that second, the PHAgate will be open for 2000 µs. The fractionof the time the gating window is open is thusf = 2000 µs/s = 0.002.

If PHA-singles are occurring randomly at arate RP , the rate Rrc at which they should bedetected arriving in the gating window (andthus count as coincidences) is Rrc = fRP .Thus random coincidences can be expected tooccur at the rate

Rrc = τRSRP (19)

Procedures

Safety

The 22Na source used in this experiment issealed in a plastic disk and can be handledwithout special equipment. It is a radiationsource, however, so do not keep it close to yourbody for an extended time . Always handle the

lead sheets and lead apertures using gloves ora paper towel and wash your hands afterward.Do not eat or drink in the laboratory.

There are many electronic modules used inthis experiment. If you are unsure how touse them or just want to get more informa-tion about them, please consult the manu-als. Reread the Apparatus section and referto Fig. 6 for an overview of the connections.

Setup

1. Measure and record the relevant geo-metric quantities such as R (the source-detector distance), ra (the aperture ra-dius), and d (the lead shielding thick-ness).

2. The source will be placed on a bracketthat is mounted to a 3-axis positioner.The positioner is mounted on the fixeddetector arm and used for precisely cen-tering the source between the detectors.Adjust the positioning knobs to get thepositioner near the middle of its range inall three directions. Tape down the 22Nasource on the positioning bracket care-fully centering it over the wooden cylinderabout which the detectors rotate. Beforedoing a more precise positioning in a laterstep, do a coarse “eyeball” check and ad-just the positioner if needed. You are try-ing to get the bottom center of the sourcedisk (close to where the source was de-posited) to be centered over the woodencylinder and at the correct height to be onthe centerline between the detector aper-tures. If you can discern that the disk isnot centered, re-position the disk and oradjust any of the positioning knobs.

3. Position the fixed detector at the 0◦ set-ting on the table and position the mov-able detector opposite it, at 180◦. As its

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Figure 6: Schematic of the apparatus for the experiment.

name implies, the fixed detector will notbe moved during this experiment.

4. The HV supply has dual outputs and sup-plies voltage to both detector PM tubes.Make sure you use the special coax cablebetween the HV supply and the detectorHV inputs. The connectors on these com-ponents are called MHV (on the detec-tors) and SHV (on the HV supply) andthe special coax cables have correspond-ing mixed connectors. The BNC connec-tors found on most coax cables in the lab-oratory and usable for all other connec-tions in this experiment are not designedto handle the high voltages. The highvoltage should be set for positive polar-ity and 1000 V. The voltage reading onthe HV supply may not indicate the setvoltage. Trust the rotary dials used to setthe voltage.

5. Connect the detector preamp outputs to

the Linear Amplifier inputs. Set the Lin-ear Amplifiers to direct, unipolar output.Set the coarse gain to x4 and the fine toabout x5.

6. Put a BNC tee on the SCA input andon the oscilloscope channel 1. Connectthe fixed detector’s Linear Amplifier out-put to the SCA input, then to the os-cilloscope channel 1, and then to thePHA analog terminal labeled INPUT. ThePHA is the small electronics box pluggedinto one of the computer’s USB ports.With the oscilloscope triggering properly,pulses of various sizes, but with similarshape should be visible. A large frac-tion of the pulses should arise from the0.511 MeV annihilation gammas; on theoscilloscope they should have nearly thesame size and be quite a bit more intensethan pulses of other sizes. Adjust theamplifier coarse and fine gain controls so

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Figure 7: A region of interest set over apeak. The software provides the FWHM, thepeak position, the gross area and the net area(hatched).

that the 0.511 MeV annihilation gammas(the more intense group of traces) pro-duce pulse heights around 3 V.

7. Start the Maestro software. SelectAcquire|ADC Setup and then set the Con-version Gain to 1024 and the Gating toOff. Select Acquire|Preset Limits and clearany entries in the dialog box that ap-pears. Select Calculate|Calibration and De-stroy Calibration. Select Acquire|Clear andthen Acquire|Start. The conversion gain of1024 implies that a 10 V analog pulse willbe sorted into channel 1024, with smallerpulse heights sorted into correspondinglylower channels in the spectrum. Thus,the spectrum now coming in should showthe 0.511 MeV photopeak at about 3 V(near channel 300 in the spectrum), anda weaker 1.274 MeV photopeak around7.5 V (near channel 750). Adjust the Lin-ear Amplifier gain if the peaks are too faroff the suggested channel locations.

8. Acquire the PHA spectrum for the fixeddetector with good statistics (until itlooks smooth) and save it.

9. Check the detector resolution for the0.511 MeV and 1.274 MeV peaks. To doso, determine the Full Width at Half theMaximum height of the peak (abbrevi-ated FWHM). The ratio of this width tothe center position of the photopeak is thedetector resolution. The FWHM and thepeak position must be measured in thesame units, e.g., MeV, volts, or channels.Thus, measure the number of channelsacross the peak at half the peak heightand divide by the position of the center ofthe peak in channels. This is easily donewith the Maestro software. Set a RegionOf Interest (ROI) using the Maestro soft-ware to cover the 0.511 MeV photopeak;include the whole peak with a few chan-nels out in the shoulders to either sideof the peak. Click in the region to getthe FWHM and the center channel of thepeak. For the 0.511 MeV peak, the resolu-tion should be around 10-15%, and some-what better for the 1.274 MeV peak.

10. The ROI information also gives the grossand net counts in the peak. The grosscount is simply the sum of all counts ineach channel of the ROI. The net countis the count above a baseline at a heightequal to the average of the first and lastthree points of the ROI. See Fig. 7. Thetrue counts in the 0.511 MeV photopeakwould be the net count; the gross countwould include, for example, counts in theCompton plateau for the 1.274 MeV gam-mas. Determine the singles rate Sf fromthe net count and the live time.

In the next few steps the electronics will beset up to make coincidence measurements ofthe fixed detector’s analog pulses with a gat-ing signal also generated from the fixed de-tector, i.e., coincidences of the fixed detector’spulses with themselves. As you will see, this

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will make it easier to set the timing properlyand it will make it easy to set the SCA win-dow. When finished, all that will be necessaryto get true coincidences is to switch the PHAinput to the movable detector’s analog pulses.

11. Put a tee on the input of the Delay Am-plifier. Disconnect the Linear Amplifieroutput/SCA input at the scope channel 1tee and reconnect it to the Delay Ampli-fier input. Connect the Delay Amplifieroutput to the scope channel 1 (and thenon to the PHA analog input). Connectthe scope channel 2 to the Delay Ampli-fier input. Put in a total of 4 µsec of de-lay on the Delay Amplifier and note howthe output is delayed relative to the in-put with virtually no change in the pulseshape or height. If you do not see an out-put, jiggle the time delay switches on theDelay Amplifier. There seems to be an in-termittency problem with these switches.

12. Take and save a PHA spectrum with goodstatistics for these delayed pulses. Com-pare it with the one for the direct (unde-layed) pulses taken in Step 8 to see if theDelay Amplifier has any effect.

13. Set the SCA Mode switch to the middleposition (∆E, 0 to 10 V), set E = 1.5 V(middle dial) and ∆E = 3 V (top dial).Connect the SCA output logic pulses tothe Gate and Delay Generator.

14. On the Gate and Delay Generator, setthe Amplitude dial to about 5 V (mid-way on the dial). Set the Delay Rangeswitch to 0.1-1.1 µs and set the Delay dialto the minimum (fully counterclockwise).Set the Width dial fully clockwise (about5 µs). Put a BNC tee on the scope chan-nel 2 and connect the Gate and DelayGenerator output to it and then on to the

PHA GATE input. Set the scope triggerfor channel 2 (the rising edge of the gatesignal).

15. Next, you will overlap the (delayed) ana-log pulses and the gating signal. The 4 µsdelay on the Delay Amplifier should workwell and need not be changed. For propergating, the gate must occur before thepeak of the analog pulses and extend for0.5 µs beyond the peak. Because of tim-ing jitter, and because making the widthof the gate bigger than is absolutely nec-essary is not very detrimental to the ex-periment, a gate width of around 5 µs isrecommended. Measure and record thiswidth. Set the Delay to give a good over-lap of the gate and the analog pulses.

The fixed detector’s (delayed) analog pulsesare now being processed by the PHA with thegate generated from these same (undelayed)pulses when they are within the SCA window.Next, the SCA will be adjusted while a spec-trum is acquired with the PHA.

16. Select Acquire|ADC Setup and then set theGating to Coincidence. The software oftenfails to use the gating signal (even when itis properly set to Coincidence mode) andprocesses all pulses as if gating is set toOff. Turning the gating to Off, and thenback to Coincidence seems to fix the prob-lem. If you are unsure whether gating isworking, it is wise to go through this pro-cedure just to be sure. It seems to onlyhappens when you start the Maestro soft-ware, so you should only have to do itonce.

Clear and restart the PHA. The spec-trum should now show only peak heightsfrom 1.5 to 4.5 V, i.e., only channels fromaround 150 to 450 should now be receiv-ing counts. Adjust the SCA E and ∆E,

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clearing and restarting the PHA spec-trum each time, until only the pulses inthe 0.511 MeV photopeak are processed.Clearing the data is most convenientlydone by the keyboard combination Alt|3.Adjust the SCA so the PHA spectrum in-cludes all channels over the entire peak; itis better to include a few channels outsidethe peak than to miss any in the peak.

17. Acquire this gated spectrum with goodstatistics (until it looks smooth) and saveit. Set a Region Of Interest (ROI) usingthe Maestro software to cover all SCA-gated channels. Record the ROI grosscounts. Connect a scaler to the SCA out-put, measure the SCA singles rate, andcheck that the ROI gross count rate iswithin a percent or so of the SCA singlesrate.

18. Switch the Delay Amplifier input fromthe Linear Amplifier for the fixed detec-tor to the Linear Amplifier for the mov-able detector. Now the movable detec-tor’s analog pulses go to the Delay Am-plifier and on to the scope channel 1 andthe PHA analog input. All connectionsending at the PHA gate input should bethe same; the gate should still be gener-ated from the fixed detector.

19. Turn the PHA gating off and start ac-quiring a new spectrum. Adjust the Lin-ear Amplifier gain to get the 0.511 MeVphotopeak around channel 300 and the1.274 MeV photopeak around channel750. Set an ROI over the 0.511 MeV pho-topeak; again get into the shoulders.

20. Turn gating on and start a new PHAspectrum. If the detectors are 180◦ op-posite one another relative to the 22Nasource, the spectrum should come in quite

a bit more slowly because these counts aremostly all real coincidences arising fromthe annihilation gamma pairs.

21. Move the movable detector to 90◦ andrestart a PHA spectrum. Note how thespectrum now comes in much more slowly.The counts coming in now are not true co-incidences; they are random coincidences.

Before taking coincidence measurementsversus the angular position of the movable de-tector, there is one more adjustment to make.In order to maximize the rate at which coin-cidences will occur, the small spot where the22Na nuclei are located must be positioned onthe centerline between the two detectors.

22. Do not put any markings on thesource disk. Carefully position the mov-able detector at the 180◦ position and thefixed detector at the 0◦ position.

23. Select Acquire|Preset Limits and set theLive Time to around 60 s. Start a PHAspectrum with gating on. Set an ROI overthe 0.511 MeV photopeak and its Comp-ton plateau and record coincidence counts(area in the ROI) as you translate the diskin the plane bisecting the detectors. Dothis one direction at a time. You will haveto keep track of the number of turns onthe positioner from where it started. Theknobs have scratch marks to help. Beforestarting, you should rotate the scratchmarks to some convenient angle, such asstraight up or along the centerline. Aboutone turn between measurements over arange of ±6 turns should be as precise asneeded. Plot the results, locate the max-imum, and then set the positioner therebefore moving on to the second transversepositioner. Check the positioning on thefirst axis again to see if the maximum has

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γ-γ Coincidence in 22Na GGC 13

moved. If necessary, recheck the position-ing along the second axis also. For thesesecond checks, the range can be reducedto ±2 turns.

The coincidence rate doesn’t decreasevery much if you move the source longitu-dinally a few millimeters off-center alongthe axis between the detectors. Just eye-balling the center of the disk to the cen-ter of the wooden cylinder is good enoughalong this dimension.

The apparatus is now ready to make coinci-dence measurements.

24. Turn gating off and take a non-coincidence spectrum for the movable de-tector with good statistics and save it.Check the resolution for this detector.Record the gross and net counts in the0.511 MeV photopeak. Determine thesingles rate Sm from the net count andthe live time.

25. Turn gating on. With the source alignedand giving maximum coincidences, takea spectrum with good statistics and saveit. (An hour or more may be necessary.)Set an ROI over the 0.511 MeV photo-peak and record the gross and net countsin the ROI. Determine the maximum ex-perimental coincidence rate. (This is notCmax.)

26. Position the movable detector near the90◦ mark and restart a PHA spectrum.Take a random coincidence spectrumwith good statistics and save it. (Anovernight run is definitely necessary.) De-termine the random coincidence rate Rrc.

27. Answer C.Q. 3 and 4.

28. Position the movable detector for max-imum coincidences. Set the preset live

time to get at least 3000 coincidencesat the maximum coincidence rate. Thisshould take no longer than 5 min per an-gle.

29. Make a data table for coincidences vs. theangle reading for the movable detector.Measure to both sides of the maximumuntil you get into the region where thereare only random coincidences. Measure-ments every one degree seem to work well.Graph the coincidence count rate vs. de-tector angle.

CHECKPOINT: The procedure shouldbe completed through the previous step,including the graph of coincidence countrate vs. angle.

Mini-Experiments

“Square root statistics”

If the number of detected gammas N (coinci-dences or singles) is measured over some timeinterval and the measurement is repeated overand over again under unchanging experimen-tal conditions, the result is a sample of N -values. Because of the random nature of nu-clear decay and gamma detection, the valuesof N will not all be the same. For an infi-nite number of samples, the values for N arepredicted to occur with probabilities governedby the Poisson distribution, several features ofwhich are discussed next.

A Poisson distribution is characterized by asingle number—its mean µ (which would bethe mean of an infinite sample of N -values).The standard deviation σ of a Poisson distri-bution (or of an infinite sample) is given byσ =

õ. One goal of this mini-experiment is

to verify that the standard deviation of a sam-ple of N -values is reasonably well-described bythe square root of the sample average. Lastly,

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the shape of the Poisson distribution (for µgreater than 30 or so), is very close to that ofthe Gaussian of the same mean and standarddeviation.

When only a single measurement is madeand a count value N is obtained, that valuebecomes the best estimate of the true expectedaverage µ. Furthermore, knowing that themeasured N is a sample from a Poisson dis-tribution, one may then estimate its standarddeviation, i.e., the uncertainty of the measuredN , as the

√N . This kind of measurement is

often said to obey “square root statistics.” Incalculations or fits using measured N values(greater than 30 or so), one can reliably ap-ply techniques relying on the Gaussian distri-bution, using

√N for the uncertainty in any

measured N .

30. With gating off, adjust the preset livetime so that the PHA singles count willbe between 1000 and 2000. Take twentyseparate singles counts without changingthe experimental conditions.

31. Answer Comprehension Question 1

Gamma ray attenuation

The transmission of monoenergetic gammasthrough shielding of thickness z follows an ex-ponential decay law

S(z) = S0e−µz (20)

where the decay constant µ depends on theshielding material and the energy of thegamma. Thus the value of κ discussed in thetheory section is expected to be κ = e−µd

where d is the thickness of the lead apertures.To determine the value of µ (and thus κ)

experimentally, lead sheets of known thicknessare available for placement in front of one ofthe detectors and then the count rate in the

0.511 and 1.274 MeV photopeaks can be mea-sured.

32. Set two ROI’s; one over the 0.511 MeVphotopeak and one over the 1.274 MeVphotopeak. Place various thickness oflead sheets in front of the movable detec-tor (leaving the original aperture in place)and record the photopeak count rates S(net counts in each ROI divided by thelive time) vs. the thickness of the leadsheets.

33. Fit the data for each photopeak to Eq. 20.(Add a constant to the equation to takeinto account background counts.) Thedifferent energy gammas may have differ-ent fitting parameters. Remember to usesquare root statistics for the data pointuncertainties.

34. Answer Comprehension Question 2.

Analysis of the Coincidence Data

35. Fit your coincidence counts vs. angle toEq. 15 plus a constant B to represent ran-dom coincidences. Including as parame-ters in your fit R, ra, rs, b, θ0, B, κ, andan overall multiplicative constant repre-senting the quantity fmffΦ. Rememberto use square root statistics for the datapoint uncertainties. The next few para-graphs describe how to construct the pre-diction column for measured coincidencerate as a function of the angle setting, θ.

Despite your best efforts, the c-flux patternmay still be somewhat above or below theplane of the movable detector. In addition,because of the finite size of the active area, theoverlap cannot be perfect for all source points.The average overlap area might be reasonablywell modeled by Eq. 15 with δ given by

δ =√b2 + [R(θ − θ0)]2 (21)

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γ-γ Coincidence in 22Na GGC 15

where θ0 describes the horizontal angle wherethe best case overlap is achieved, and b is an ef-fective offset describing the imperfect overlapat θ = θ0. Remember to convert all angularmeasures to radians before using them in anyformulae.

Set aside a spreadsheet area for the fittingparameters. Populate those cells with reason-able guesses for their values. For b, start withthe value 0.1 cm.

Set up columns for θ, the measured coinci-dence counts N , and the live time t. Create acolumn for the coincidence count rate Cmeas

(=N/t) and another for its uncertainty σCmeas

(=SQRT(N)/t). Enter your data in a consis-tent (monotonic) order for θ.

Use your values of θ with the cells for R, band θ0 to construct a column for δ accordingto Eq. 21.

To make columns for the three overlapareas, the IF(LOGICAL EXPRESSION, TRUEVALUE EXPRESSION, FALSE VALUE EX-PRESSION) Excel function will be used to takecare of the various cases given in Eq. 9. TheIF function returns values based on one or theother of the two latter expressions based onthe value of the first expression (which mustevaluate to a true or false value).

What follows are equivalent spreadsheettranslations of Eq. 9 with intermediatecolumns for the half angles as obtained fromEq. 11. For A′sa the circle radii r1 and r2 andthe half angles φ1 and φ2 in those equations be-come rs and ra and φs and φa below. For A′aa,the radii r1 and r2 are both equal to ra andthe half angles φ1 and φ2 are both equal andtaken as φa below. For A′ss, the radii r1 andr2 are both equal to rs and the half angles φ1

and φ2 are both equal and taken as φs below.Non-spreadsheet notations used below are (a)variable names are used while the spreadsheetneeds cell addresses for the variables, (b) im-plied multiplication is shown while the spread-

sheet needs the specific ∗ symbol, (c) squaringrequires the ˆ symbol.

A column for the overlap area A′sa ismost easily constructed using two intermedi-ary columns. Create a column for φa using

= IF(δ > rs − ra, IF(δ < rs + ra, (22)

ACOS((r2a + δ2 − r2s)/(2raδ)), 0),PI())

and create a column for φs using

= IF(δ > rs − ra, IF(δ < rs + ra, (23)

ACOS((r2s + δ2 − r2a)/(2rsδ)), 0), 0)

Then, create a column for A′sa using

= r2a(φa − 0.5 SIN(2φa)) + (24)

r2s(φs − 0.5 SIN(2φs))

The conditional φ-values will give the properarea for all cases of Eq. 9.

Columns for the two overlap areas A′aa andA′ss are most easily constructed using one in-termediary columns for each area. The cal-culation is illustrated for A′aa with the trans-lation to A′ss obtained by changing subscriptsfrom a to s. Create a column for φa using

= IF(δ < 2ra,ACOS(δ/(2ra)), 0) (25)

Then, create a column for A′aa using

A′aa = 2r2a(φa − 0.5 SIN(2φa)) (26)

Construct a column for each of the termsCss, 2Csa, and Caa from Eqs. 12-15 and onemore for their sum plus the random coinci-dence rate — the predicted coincidence rateCpred: = Css + 2Csa + Caa +B.

Construct a column for the contribution ofeach point to the total χ2

=(Cmeas − Cpred)2

σ2Cmeas

(27)

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Then construct a cell for the sum of all thesecontributions using the spreadsheet SUM()function. Finally, perform the fit by havingthe Excel Solver vary the fitting parametersto minimize this sum. You can eliminate baddata points from consideration in the fit bytyping a zero in the cell where its χ2 contribu-tion would have been calculated. Rememberto exclude R from the Solver variable list.

The fitting parameters, (in particular, b andra) may interact in the fit. Both may simul-taneously change with only a modest changein the shape of the overlap function. Con-sequently, relatively good data (high coinci-dence counts) are needed to fit both parame-ters. Watch out for this and, if necessary, fix(instead of fitting) the value of ra to its mea-sured value.

Comprehension Questions

1. For the mini-experiment on square rootstatistics: (a) Calculate the sample meanand standard deviation of the twenty trialvalues of N and compare with the predic-tion based on square root statistics. (b)Show that square root statistics are the-oretically self consistent. That is, if N1,N2, ... have σNi

=√Ni, show that prop-

agation of errors dictates that their sumNT =

∑Ni will have σNT

=√NT . (c)

Show how cube root statistics would notbe self consistent.

2. For the mini-experiment on the attenu-ation of gammas by lead: (a) Use yourfitted µ values and the density ρ =11.3 g/cm3 of lead to determine the massattenuation coefficients µ/ρ at the two en-ergies. (b) Compare your results with ref-erence values, e.g., those from the graphin the auxiliary material for this experi-ment of mass attenuation coefficients for

various elements in K.Z. Morgan and J.E.Turner, eds., Principles of Radiation Pro-tection. (c) Determine the fraction κof 0.511 MeV gammas predicted to getthrough the thickness for the lead aper-ture shielding.

3. Print, compare, and explain the spectrataken in Steps 8, 12, 17, and 24-26.

4. Compare the measured random coinci-dence rate (coincidence rate at large an-gles) with the value predicted by Eq. 19.Discuss any significant discrepancy.

5. Discuss the results of the fit to the coin-cidence data. In particular, check if anyof the parameters are unrealistic. Reworkthe fit until all fitting parameters are rea-sonable.

6. Use the results of the fit to the coin-cidence data to determine Cmax. Next,use the fit results, Cmax, and the singlesrates Sm and Sf (the net photopeak countrates as measured in Procedure Steps 10and 24) to determine (a) the raw flux Φat a distance R from the detector and (b)the geometric mean of the photopeak effi-

ciencies√fmff . Determine the source ac-

tivity α from Φ and the distance R usingEq. 2. Use your value for α and the knowninitial activity of the source (10µC) to de-termine the age of the source.

7. What does a non-zero value of θ0 indi-cate? How well do your results “prove”that the gammas are emitted 180◦ apart?Suppose the positronium “atoms” decaywhile moving in random directions withan average, non-zero lab frame kineticenergy, Eav. Assuming Eav was largeenough to have an effect, how might theshape of the graph of measured counts vs.angle change? Would it become broader,

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γ-γ Coincidence in 22Na GGC 17

covering a wider range of angles? Wouldits center shift to a new angle? Approx-imately (order of magnitude) how largewould Eav have to be to have an observ-able effect on the data?

February 15, 2019