Table of Contents
The positron is the antiparticle of the electron; it has the same mass, but opposite electrical charge. When a positron is created (usually in a radioactive decay), it can lose kinetic energy through Coulomb interactions and eventually bind with an electron to form positronium. This state is short-lived, however, and the two constituent particles will annihilate, converting their mass into energy which escapes in the form of photons (gammas). In addition to the conservation of energy, this annihilation process conserves other quantities such as linear momentum, angular momentum, parity, etc. This lab will study the angular distribution of the resulting annihilation gammas and establish the two-photon decay process always results in gammas emitted 180 degrees relative to each other.
| * 1References |
| * 22 Theory |
| * 33 Procedure |
| * 3.13.1 Apparatus and setup |
| * 3.1.13.1.1 Apparatus |
| * 3.1.23.1.2 Making connections |
| * 3.1.33.1.3 Setting the PMT high voltage |
| * 3.1.43.1.4 Looking at signals on the oscilloscope |
| * 3.23.2 Setting the discriminators |
| * 3.2.13.2.1 Obtain a Na-22 spectrum |
| * 3.2.23.2.2 Gating the PHA |
| * 3.2.33.2.3 Selecting the 511 keV peak |
| * 3.33.3 Time calibration |
| * 3.43.4 Data collection |
| * 3.53.5 Accidental coincidence rate |
| * 44 Analysis |
| * 5Rubric |
| * 5.1Winter quarter |
References
| [1] A.C. Melissinos, //Experiments in Modern Physics 1st Ed//., Academic Press, pp. 412-429, (1966); A.C. Melissinos, J. Napolitano, Experiments in Modern Physics 2nd Ed., Academic Press, pp. 409-421, (2003). |
This textbook describes the experiments done in this lab and includes analysis of student data.
1 Goals
In this experiment you will study the the angular correlation of the gammas produced in the two-photon decay of positronium. Specifically, your goals for this experiment include the following:
- to understand how the conservation of different quantities in the annihilation of positronium determines the properties of the resulting emitted photons;
- to wire-up and understand the logic behind the electronics required to measure a two-fold detector coincidence, including the following:
- to use a constant fraction discriminator and to understand gating to select different features of a detector spectrum, and
- to use a time-to-amplitude converter to measure time differences and to establish coincidence criteria; and
- to model the geometric overlap of your two detectors in order to compare the measured angular distribution of count rates to the theoretical one.
2 Theory
Sodium-22 decays to an excited state of neon-22 by either electron capture or by positron emission. The neon later decays to its ground state by the emission of a 1.27 MeV gamma. (See Fig. 1.)
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Figure 1: Nuclear decay scheme for sodium-22. (Source: C. Michael Lederer, Jack M. Hollander, and Isadore Perlman, Table of Isotopes, 6th Edition, John Wiley & Sons, 1967.)
The emitted positrons are slowed down and are captured by electrons in the source to form an electron-positron bound state called positronium, a hydrogen-like “atom.” The ground state of positronium, which has a binding energy of 6.8 eV, has two possible configurations depending on the relative orientation of the electron and positron spins. The state with anti-parallel spins has net spin equal to 0, and is variously referred to as the singlet state, _para-positronium, _or, in spectroscopic notation, the state 1S0. The state with parallel spins has net spin equal to 1, and is referred to as the triplet state, _ortho-positronium, _or the state 3S1.
NOTEBOOK: Photons have integer spin of 1. Given that para-positronium (singlet state) has a net spin of 0 and ortho-positronium (triplet state) a net spin of 1, explain why para-positronium can decay to two photons (but not one or three), while ortho-positronium can decay to three (but not one or two) photons.
The para- and ortho- states are both short-lived, but decay (by means of electron-positron annihilation) with very different lifetimes. Para-positronium has a mean lifetime of about 10-10 s and decays to two photons. (In general, decays to 2_n_ photons are allowed (for integer _n ≥1), though the two photons process is by far the most likely.) Ortho-positronium has a mean lifetime of about 10-7 s and decays to three photons. (In general, decays to 2_n+1 photons are allowed (for integer _n _≥1), though the three photons process is by far the most likely.) If each spin state is equally populated, there will be three times as many positronium “atoms” in the triplet state as in the singlet state. Therefore, one would naively expect three times as many 3γ decays as 2γ decays.
However, that is not what is observed! Another process contributes to the actual outcome: a photon may be absorbed by the electron in positronium, causing the electron to flip its spin, thus changing the state of the positronium from ortho- to para-, or vice versa. The lifetime for the spin flip process is about 10-9 s – long compared to the para- lifetime, but short compared to the ortho- lifetime. Thus, para-positronium decays (via two photons) before a spin flip can occur, whereas spin flips occur before ortho-positronium has time to decay (via three photons), converting the ortho- to para-. Therefore, the two photon decay is much more likely.
Since the rest masses _m_0 of the electron and positron are converted to energy in the annihilation process, each of the resulting two photons has energy E = _m_0_c_2 = 511 keV, and are created simultaneously.
3 Procedure
3.1 Apparatus and setup
3.1.1 Apparatus
Since our model for electron-positron annihilation assumes the simultaneous creation of two 511 keV photons, we wish to set up our detection and counting system to look for coincident events at 511 keV. We will count rate for those events as a function of angle between the two detectors. The apparatus is illustrated in Fig. 2.
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Figure 2: Apparatus for measuring the angular correlation of pair-annihilation gammas.
[A] - _Radioactive source: _ You are provided with a Na22 rod source. (The source activity at time of creation is given on the safety label.) The active material is a small pellet, sealed in the dark-colored tip of a plastic rod. Sodium-22 emits positrons which ultimately annihilate with electrons in the source, producing gammas.
[B] – NaI and photomultiplier tubes (PMTs) : The NaI+PMT detectors are mounted in carriages. One detector is fixed in position (defined as _θ _= 0°) and the other is free to move around a circular track. Energetic particles from the source [A] which strike the NaI generate electrical pulses in the PMT. The total charge of the pulse is proportional to the energy of the incident particle. The bases of these PMTs have been wired to take negative high voltage. We can look at the signals coming from both the final dynode and the anode of the PMTs. For more information, see NaI Detector Physics and Pulse Height Spectra. | [C] - Constant fraction (CF) discriminators: The output pulses from the detectors are sent to the input of the CF discriminators. The CF discriminators generate an output pulse only if the voltage of the input pulse is within an upper and lower limit set by the user. [D] - _Analog time delay: _ This box uses coaxial cable of different lengths to delay pulses by different times. For reference, a cable of length 0.6 foot delays pulses by about 1 ns. Delay times of up to 60 ns are set by toggle switches. [E] - _Time to Amplitude Converter (TAC): The TAC has s_tart and stop inputs. A start pulse initiates a linearly increasing voltage ramp in the TAC. The ramp stops increasing when either a pulse arrives at the stop input, or a preset time limit (range) is exceeded. If a stop pulse arrives before the range is reached, the TAC generates an output pulse whose voltage is proportional to the time between the start and stop pulses. If the range is exceeded the TAC resets itself and produces no output pulse. | [F] - Pulse Height Analyzer (PHA): The PHA integrates the charge in each pulse (proportional to energy of the gamma) and displays a histogram of pulse charges on a computer. For more detailed information, see NaI Detector Physics and Pulse Height Spectra. | [G] - Scalers: The scaler module counts input pulses for a time set by the accompanying timer. Note that the timer sets times in the format N x 10_M_-1 seconds, where N is the number on the first dial, and M is the number on the second. (Note the “-1” in the exponent). [H] - Amplifier: The amplifier is used to amplify dynode pulses from the PMTs. [I] - High Voltage (HV) Supply: Each PMT is powered by its own external HV supply. The PMTs in this experiment can be safely run up to -2500 V. 3.1.2 Making connectionsThe schematic wiring diagram is shown in Fig. 3.
{ Use coaxial cable to connect the PMT anodes to the inputs of the CF discriminators. The upper and lower level thresholds of the CF discriminators must be set so that only pulses from 511 keV photons will generate an output pulse. In this way we can reject unwanted background events, such as the 1.27 MeV photons from the neon-22 decay. One output from each CF discriminator is sent to a scaler so that the rate of 511 keV events can be measured for each detector. These rates are often referred to as the singles rates. A second output from one of the CF discriminators (it does not matter which) is connected to the start input of the TAC. A second output from the other CF discriminator is sent through the delay box and then into the stop input of the TAC. Thus, 511 keV photons which hit the two detectors simultaneously will produce start and stop pulses at the TAC separated by a time equal to the added delay on the stop channel. 3.1.3 Setting the PMT high voltageTurn on the high voltage (HV) supplies as follows:
Verify that the HV supplies for each PMT are set as follows:
IMPORTANT: Leave the PMT high voltages at the values given above for the entire experiment. They have been chosen to optimize the singles and coincident rates. 3.1.4 Looking at signals on the oscilloscopeIn order to understand what each element of the electronic circuit does, it helps to look at input and output signals on the oscilloscope, both individually and simultaneously with other, related signals. With a Na-22 rod source mounted in the center of the table, start by looking at the anode and dynode pulses from one of the PMTs on the scope at the same time. (Place the anode pulse in channel 1 and the dynode pulse on channel 2 and trigger on either of the pulses.) Use a 50 Ω terminator at each scope input.
Continue through the electronics chain, sketching and commenting on what you see. When possible, compare the timing and voltage of input signals to the time and voltage of output signals. Notice which circuit components produce uniform output pulses and which produce outputs that vary in voltage or time.
3.2 Setting the discriminators3.2.1 Obtain a Na-22 spectrumConnect the dynode of one of the PMTs to the amplifier. Connect the output of the amplifier to the direct input of the PHA. Use the SpecTech UCX software on the computer to collect a pulse height spectrum of the dynode pulses. Adjust the gain of the amplifier until the spectrum shows peaks for both the 511 keV and the 1.27 MeV gammas expected from the decay of Na-22. You should see the full energy peak and Compton shelf associated with each of the two energies. For best resolution, adjust the gains so that the 1.27 MeV peak is near the right end of the PHA display. The idea is to get the full pulse height spectrum displayed on the PHA so that you can determine which part of the spectrum represents the 511 keV photons.
3.2.2 Gating the PHAIn the next section we wish to have visual feedback to select the relevant portion of the spectrum. We will make use the PHA gate input to set the criterion for which pulses will be analyzed. A pulse arriving at the direct input of the PHA will be analyzed only if another pulse arrives simultaneously at the gate input. We will use the SCA output from the CF discriminator as a gate pulse for the PHA. Whenever the CF discriminator is triggered by a pulse from the PMT anode, it will generate an SCA pulse which will “open” the gate of the PHA. The amplified dynode pulse from the same PMT is connected to the direct input of the PHA. We want the dynode pulse to arrive at the PHA input during the time interval in which the corresponding SCA pulse is activating the gate input. To check this timing, connect the dynode output of one of the PMTs to an amplifier and the output of the amplifier to one channel of the scope. Locate the CF discriminator which is connected to the anode of the same PMT. Connect the SCA out (the cable coming from the rear of CF discriminator) to the other channel of the scope. The dynode pulse should overlap in time with the SCA pulse.
3.2.3 Selecting the 511 keV peakWe wish to count only the relevant 511 keV annihilation photons. This can be accomplished by adjusting the upper and lower levels on the CF discriminators so that only pulses in the 511 keV photopeak portion of the spectrum will be accepted. The gating scheme, shown in Figure 4, gives visual feedback on where the levels are set.
{ For one of the detectors (it doesn't matter which), make the connections shown in Fig. 4. On the discriminator, gently adjust the upper level threshold to its maximum setting and the lower level threshold to its minimum setting, thus allowing the maximum range of pulse sizes to be accepted by the discriminators and generate an output pulse. The PHA will now only count pulses on the direct input which arrive when there is also a pulse present on the gate input. As the upper and lower level discriminators are adjusted, only pulses which pass the discriminator settings will generate an SCA pulse to gate the PHA. As such, the PHA will only collect and display dynode pulses for events which pass the CF discriminators. Start collecting data on the PHA. With the upper and lower level thresholds on the CF discriminator fully open you should see the full spectrum of the Na-22 source, including the full energy peaks of both the 511 keV and 1.27 MeV gammas. Now adjust the upper and lower level thresholds on the CF discriminator until the 511 keV full energy peak is the main remaining feature on the PHA. Note that the threshold controls do not produce a sharp cut-off in the spectrum, but instead are quite “mushy”. (In fact, the upper and lower discriminators are not even independent, but depend loosely on each other.) Try to adjust the thresholds to remove the low energy noise and the higher energy (1.27 MeV) peak. WARNING: Bracketing too closely on the 511 keV photopeak may reduce the overall rate at which discriminator output pulses appear. It is preferable to let a little bit of the rest of the spectrum on either side through.
Repeat this procedure for the other PMT. Once you have adjusted the windows on both discriminators, use the scalers to observe the singles rates of the output pulses. The two rates should be similar and should be several thousand counts per second. If one is smaller than the other (or if both are less than 1000 per second), readjust the threshold levels to make the rates more equal.
3.3 Time calibrationIn order to calibrate the output of the TAC we will use the output of a single CF discriminator to generate both a start pulse, and a delayed stop pulse. Configure the electronics as shown in Figure 5. Place a BNC “Tee” on the delay box input. Connect the output of the discriminator to one side of the Tee and the TAC Start input to the other side of the Tee. Now the same pulse, from the CF discriminator, both starts and stops the TAC. The only difference between the start and stop pulses is the amount of delay added to the stop pulse. Connect the output of the TAC to the direct input of the PHA. Make sure to disconnect the gate input on the PHA.
{ Start the PHA and observe the effect of the delay box toggle switches. Disconnect the inputs to the start and stop inputs on the TAC and look at them simultaneously on the scope. Use the cursors on the scope to directly measure the time delay between the start and stop pulses for a range of delay settings on the delay box. Now reconnect the start and stop pulses to the TAC and use these settings to calibrate the x-axis of the PHA in units of time.
3.4 Data collectionReconnect the electronics according to Fig. 3 and set the delay box to a value which corresponds to a peak near the center of the visible range. Check to make sure that both detectors are equidistant from the source.
You will now collect count rates as a function of detector angle. Coincident events in the two detectors will produce TAC output peaked at the delay time set above, and the rate can be computed from the number of counts falling in that peak. Collect a TAC output spectrum for a range of detector angles, θ, around 180°.
<blockquote> NOTEBOOK: Measure the full-width half-maximum (FWHM) of one of your TAC output plots and call this your “resolving time”. Based on this value, would you be able to distinguish between the following two cases? </HTML>
Hint: Consider the average positronium lifetime given above in the “Theory” section to determine how far apart in time the events of case (2) will likely be. </blockquote></HTML> 3.5 Accidental coincidence rateIn coincidence counting experiments there will be accidental coincidences caused either by two uncorrelated events entering different detectors at the same time or by a single event which simultaneously causes a signal in both detectors at the same time (e.g. a photon which scatters from one detector into the other, thereby depositing energy in both). In this experiment, we likely have both situations and this can lead to a background count rate that is present at all angles. While this rate may not be constant as a function of angle, we can at least make an estimate by measuring the coincidence rate at an angle far from 180º where we expect no true coincidences.
You do not need to subtract the background rate from your coincidence rates measured above. Instead, you will use this background rate when you compare your data to a predictive model in the analysis below. 4 AnalysisTo first order, we can estimate how the coincidence rate will fall as a function of the angle between the detectors by looking at the overlap of the fields of view of the two detectors. We make the following assumptions:
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