The Quantum Mechanics of Alpha Decay

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uraninite (∼325g of nearly pure UO 2 ). After severaldays (comparable to the mean life of radon so that itis in equilibrium with its precursors) of exposure to theemanations of the mineral, the charcoal carries a load ofadsorbed radon which can be transferred to liquid scintillator.The jar is split into two halves, labeled ripe andunripe, please use the former!The maximum activity of radon atoms in equilibriumNwith one mole of uranium isτ disintegrations s−1 ,U 238where N = 6.03 x 10 23 is Avogadro’s number, and τ U 238= 6.5 x 10 9 yr is the mean life of uranium. Thus we findthat the maximum activity we can expect to find in thecharcoal to be 2.9 x 10 6 s −1 (∼70 µCi). If this is dividedamont 10 packets of charcoal, then the maximum activityof each is 2.9 x 10 5 s −1 , which is more than enough forthese experiments.You will determine the mean life of Po 218 by measuringthe rapdid growth (minutes) of its radioactivity in afreshly prepared sample of Rn 222 , and the mean life ofRn 222 from the slow decay products. You will have towork rapidly at the beginning of this measurement in ordernot to miss the early stages in the build up of thePo 218 activity. Study the following procedure before youbegin. Make up a data table in your lab book torecord the times (starts and durations), the numberof counts, and your comments. Practice usingthe Trump-PCI MCA so that you can record successiveenergy spectra for various “preset” time intervals. Youmust also learn how to use the ROI (region of interest)feature to display the total number of counts recordedbetween two different selected amplitudes in a spectrum,as well as the number with background subtracted.The “sum” signals from the Beckman scintillationcounter (ignore for now the “coin” signals) should be feddirectly to the signal input of the MCA located on theback of the computer. The amplifier should be set fora negative inpt with fast pulse shaping and a gain thatspreads the full spectrum over about 2/3 of the range ofthe display. You may need to set the lower level discriminator(LLD) at 2 or 3% to suppress the counting of verysmall pulses (which may be rf interference).Detailed Procedure for Rn 222 and Po 218Make up a solution of radon in liquid scintillator hermeticallysealed in a glass vial in which the alpha-decayactivity of Rn and its shorter-lived decay products canbe assessed from time to time over the next several dayswith the Beckman scintillation spectrometer. Place in aglass vial a radon-loaded charcoal packet extracted fromthe “radon farm” jar in the lead box in the hood. Pumpin 10 cc of liquid scintillator from the reservoir. Cap thevial and rock it gently for ∼ten seconds. Remove thepacket and discard it in the waste jar. To prevent falseresults due to radon leaking fromthe vials, it is essentialthat they be hermetically sealed. You can improve theseal by applying vacuum grease to the rim of the vialand capping it tightly so the radon does not leak outduring the several days required for the measurements.Lower the vial into the counting chamber and immediatelystart a sequence of measurements, appropriatelyspaced in time so as to define accurately the build up ofthe Po 218 activity. The following is a suggested sequence:1. Record three 30-second spectra with the shortestpossible intervals between, making note of thestarting times of each measurement.2. After the third spectrum, wait one minute and takeanother 30 second spectrum. Repeat with intervalsof 2, 4, and 8 minutes between the starts of successiveaccumulatins, each time setting the preset durationof the measurement to be half the time sincethe start of the last measurement (e.g. to 60 secafter the 2-minute wait, 120 sec after the 4-minutewait, etc.) These increased integration times willgive you better statistical accuracy as you approachthe asymptotic rate.The eight spectra should reveal the growth of the ∼6MeV alpha particle peak from Po 218 as a shoulder on thepeak due to the alpha particles from Rn 222 . You shouldalso begin to see the beginnings of a peak at 7.6 MeV dueto alpha particles from the decay of the ultra short-livedPo 214 .Measure on the MCA and plot, as a function of time,the intensity of the combined (and unresolvable) 6MeVand the 5.5 MeV alpha-particle groups. To do this youwill have to set up identical ROI’s in each spectrum withthe lower level set at the lower boundary of the alphaparticle peak to as to exclude as much as possible, thecounts due to the beta decays, and the upper limit setin the valley between the 6 and 7.6 MeV peaks so asto exclude as efficiently as possible the alpha decays ofPo 214 whose rate will continue to grow and confuse theresult. The MCA will give you both the “gross area”(total number of counts) and the “net area” (the numberof counts above a straight-line drawn from just to theleft boundary of the ROI to the right boundary of theROI (see the Trump-PCI manual for details). The areaunder the straight line is attributable mostly to the betadecayevents which continue to grow in rate due to theaccumulation of Pb 214 and Bi 214 long after the rate ofalpha decays from Po 218 has reached its maximum rate.Think about your preliminary results and how youmight obtain improved data on your second run. Makea practice run to familiarize yourself with the procedureand to make any needed adjustments of the electronics.Make a second run for a good set of data. Each teamshould attempt to get at least two good sets of data.After you have completed the comparatively rapid seriesof measurements required for a determination of the
Po 218 mean life, and extracted with the ROI feature ofthe MCA a good measure of the unresolved activities ofthe Rn 222 and Po 218 , you can turn your attention to themuch more gradual process of the decay of the parentnuclide itself, i.e. Rn 222 . Since the mean life of radonis several days, you will have to follow the decay of yoursample for several days or a week to get accurate results.Here you must overcome three problems. The first isthat the beta activities of Pb 210 and Bi 210 and the alphaactivity of Po 214 will continue to grow for severalhours before they reach the equilibrium condition of beingequal to the acitivities of Rn 222 and Po 218 . Sinceyou cannot isolate the counts due to Rn 222 , any rate youmeasure will be influenced by these other activities. Thebest thing to do, therefore, is to wait until all the activitiesare nearly equal (a couple of hours). Then anyreproducible ROI will give you a rate proportional to theRn 222 activity from which you can determine the meanlife. The second problem is to define the ROI in termsof features of the spectrum that you can identify fromday to day as you return to the setup after it has beenaltered by other users. The upper limit can simply beset somewhat above the obvious upper limit of the entirespectrum. The lower limit can be set just below the lowerlimit of the distribution of alpha pulses, i.e. where thealpha peak just starts to blend into the plateau of betaraycounts. The third problem is the possible leakage ofthe radon gas from the vial.Detailed Analysis for Rn 222 and Po 218Given the follwing facts:1. The mean life of Po 218 is much shorter than thatof Rn 222 ,2. The mean lives of the Po 218 decay products thatlead to the production of alpha-active Po 214 arelong compared to the mean life of Po 218 ,3. Some Po 218 may be dissolved in the liquid scintillatoralong with Rn 222 ,one can show that the measured counting rate r(t) isrepresented to a good approximation by a function ofthe formwhere a, b and τ are constants. Thusr(t) = a − be −t/τ (13)ln[a − r(t)] = ln b − t/τ (14)The constant a is the total counting rate that is approachedasymptotically by the two alpha activities attimes much longer than the mean life of Po 218 , but stillmuch less than the mean life of Rn 222 . It can be estimatedfrom an examination of a plot of r vs. t. A plotof ln[a − r(t)] vs. t during the first few minutes shouldbe nearly a straight line with a slope of −1/τ where τ isthe mean life of Po 218 . (Note: as r approaches a, statisticalfluctuations will cause wild gyrations or worse in thevalues of ln a − r, so the most useful information aboutτ must be derived from the early part of the rising curveof activity.Additional questions:1. Describe how the pulse height spectrum changeswith time?2. Identify the part of the pulse height spectrum dueto the beta decays of Pb 214 and Bi 214 ?3. What is the effect of the accumulating radionuclidePo 210 on your results?After initiating the Rn 222 measurements, return tothe silicon detector experiment and measure the channelnumbers of all the peaks of the one–hour spectrum.With the aid of the chart of the nuclides (located onthe wall above the experiment), your preparatory questionanswers and [9], identify the peaks and explain theirshapes. Do you see evidence of the famous high–energyalphas from Po 212 ? If so, estimate their energies and thefraction of the decays that give rise to them.• Try to explain any discrepancies between the spectrumof pulse heights you have observed and theirexpected energies (e.g. consider the effect of athin coating of protective material or dust over thesource or the detector.)• In light of the likely age of the uranite source andthe opportunity it has had to achieve equilibriumamong the quantities of its radioactive decay products,try to explain the inequalities you may see inthe intensities of the various alpha particle groups.At the end of the first lab session, seal your can withthe stopper/detector and connect your high–voltage boxbetween the can and the shield of the BNC connectorwith the appropriate polarity to draw the polonium ionsto the detector. This will build up a supply of radioactivenuclides in the gas in the can and on the surface of thedetector for analysis at your next session. Note the timein your lab notebook.DAYS 2 AND 3: DETERMINATION OF THEMEAN LIVES OF SEVERAL NUCLIDESIn preparation for your second session devise a plan forthe determination of the mean lives of as many nuclidesas possible with data that you will be able to obtain fromthe setup used in the first session. Base your plans onthe possibilities of measurement implied by equations 11
Page 1 and 2: The Quantum Mechanics of Alpha Deca
Page 3 and 4: and proceeding by alpha decays, pro
Page 5 and 6: EXPERIMENTSr B (t) = B(t)/τ B ≈
Page 13 and 14: HVINalpha signalOUTSilicon detector
Page 15: FIG. 7: The Three Naturally Occurri

The Quantum Mechanics of Alpha Decay

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