In this experiment, you will study Compton scattering of the back-to-back photons produced by the annihilation of positronium in order to collect evidence in support of the theory of conservation of parity. In particular, your goals for this experiment include the following:

• to understand why the conservation of parity leads to pairs of photons with orthogonal polarizations;
• to understand how the polarization-dependence of Compton scattering leads to an excess of coincident scatters in perpendicular directions when compared to parallel directions;
• to familiarize yourself with how to make coincidence measurements between multiple detectors and in particular to further familiarize yourself with the electronics of counting experiments;
• to get a handle on the origin of accidental coincidences and to make both theoretical calculations and actual measurements of these rates;
• to collect coincident measurements in 16 permutations of parallel/perpendicular detector orientation;
• to compute the ratio of average coincident rate for the perpendicular orientation versus the parallel orientation; and
• to compare this result to the predicted ratio after taking into account the geometry of the experimental apparatus.

The positron is the anti-particle equivalent of the electron; the two have identical mass and spin, but opposite charges. If a positron is slowed down enough by Coulomb interactions, it can pair up with an electron to form a bound hydrogen-like system called positronium. Once formed, though, the positronium is short lived as the two constituents will eventually come close enough to each other to annihilate and convert their mass into energy in the form of light. Depending on the spin state of the electron and positron at the time of annihilation, positronium will most likely produce either two or three photons. Positronium in a triplet state (ortho-positronium) has a total spin of s = 1 and has a mean lifetime of 142 ns; it will decay preferentially into three photons. Positronium in the singlet state (para-positronium) has a total spin s = 0 and has a mean lifetime of 125 ps; it decays preferentially into two photons.

NOTEBOOK:  Why is decay to one photon not allowed? Why does the total spin of the positronium before decay control the number of photons produced? What quantity is conserved by this process?

In this experiment, we will look exclusively at the decay of para-positronium resulting in two photons. Conservation of energy and momentum require that each photon have an energy equal to the rest mass energy of the electron and that the two photons be emitted back-to-back. Additionally, conservation of a quantity called parity predicts that the two photons will be preferentially plane polarized orthogonal to one another. By measuring the correlation of the two photon polarizations, the conservation of parity in electromagnetic interactions can be tested.

Here we give a brief overview of parity and its role in the decay of positronium. For more complete details see Ref. [1].

Parity describes how a system behaves under an inversion of the coordinate system, for example

{ ${/download/attachments/131006890/eqn_1.png?version=2&modificationDate=1439389852000&api=v2}$

(1)

where { ${/download/attachments/131006890/r_arrow.png?version=1&modificationDate=1439224064000&api=v2}$ is a position vector. The parity operator P, when applied to a wave function, has eigenvalues of ±1. In the case of even parity (P = +1), this means

{ ${/download/attachments/131006890/eqn_2.png?version=1&modificationDate=1439224211000&api=v2}$

(2)

In the case of odd parity (P = -1), this means

{ ${/download/attachments/131006890/eqn_3.png?version=1&modificationDate=1439224217000&api=v2}$

(3)

Since the equations of electrodynamics are invariant under the inversion of the coordinate system, parity is a conserved quantity in electromagnetic interactions. Both para- and ortho-positronium have odd parity before the decay; therefore, the parity of the system after the decay should also be odd. 

The wave function for a two photon state with momenta { ${/download/attachments/131006890/k_arrow.png?version=2&modificationDate=1439224289000&api=v2}$ and { ${/download/attachments/131006890/k_arrow_minus.png?version=1&modificationDate=1439225101000&api=v2}$ (photons emitted back-to-back), photon frequency ω (equal energy), and exhibiting odd parity can be shown to be

{ ${/download/attachments/131006890/eqn_4.png?version=2&modificationDate=1461609876000&api=v2}$

(4)

where { ${/download/attachments/131006890/r_12_arrow.png?version=1&modificationDate=1439225141000&api=v2}$, and { ${/download/attachments/131006890/epsilon_1.png?version=1&modificationDate=1439225149000&api=v2}$ and { ${/download/attachments/131006890/epsilon_2.png?version=1&modificationDate=1439225155000&api=v2}$ are the photon polarizations. Although neither photon is produced in a well-defined polarization state, the expectation value can be calculated for observing the two photons in a state with a well-defined relative angle between the two polarizations. This expectation value is a maximum when the planes of polarization are orthogonal. Thus, for the decay of para-positronium one expects to observe a greater number of photon pairs with orthogonal polarizations than pairs with parallel polarizations. It is this prediction that we will test in this experiment.

Our positronium source is 22Na. Sodium-22 decays by positron emission to 22Ne as shown in Fig. 1. The positron rapidly slows down and comes to rest in the source material where it can pair up with an electron to form positronium. The positronium then decays into either two or three photons as described earlier.

{ ${/download/attachments/131006890/Na-22.png?version=1&modificationDate=1439225301000&api=v2}$ Figure 1: Sodium-22 decay scheme (Source: James E. Martin, Physics for Radiation Protection, Wiley-VCH, 2006.)

{ ${/download/attachments/131006890/Figure_2.png?version=1&modificationDate=1439225490000&api=v2}$ Figure 2: Sodium-22 source housing and photomultiplier tube arrangement

Two NaI coupled PMTs (let us call them P1 and P2) are mounted as shown in Fig. 2. P1 and P2 can be rotated about the x-axis, as defined by the photon pairs which emerge from the lead box. These PMTs are shielded from directly detecting radiation from the source. However some of the photons emitted by the source will undergo an approximately 90º Compton scatter in the aluminum rods and be scattered into the NaI scintillator of P1 or P2. The observation of time coincident pulses from P1 and P2 can then be correlated with the emission of back-to-back gammas from the source which each happened to scatter from the Al rods into the associated NaI scintillators. We will make use of the polarization dependence of the Compton scattering effect to observe the correlations of the two photon polarizations produced by the decay of positronium.

Some of the coincidences will be due to random chance detections of uncorrelated photons. As such it will be important to demonstrate that the observed coincidence rate is statistically significant compared to the expected rate of random chance coincidences.

The most commonly stated form of the Compton scattering formula relates the energy of the incident photon E to the energy of the scattered photon energy E' through the scattering angle θ as

{ ${/download/attachments/131006890/eqn_5.png?version=1&modificationDate=1439225926000&api=v2}$

(5)

where me is the electron mass and c is the speed of light. However, in order to infer the polarizations of the emitted photons, we must make use of use of the more complete Compton scattering formula, the Klein-Nishina formula for the differential cross section,

{ ${/download/attachments/131006890/eqn_6.png?version=1&modificationDate=1439226068000&api=v2}$

(6)

where d_σ_/d_Ω_ is the differential cross section (i.e., the differential “probability” for scattering into an infinitesimal solid angle d_Ω,)_ e is the electron charge and η is the angle between the incident photon polarization and the scattering plane (i.e. the plane formed by the paths of the incident and scattered photon). In our geometry θ ≈ 90º, so the cross section is maximized when η = 90º or 270º. Put another way, the cross section for scattering is highest when the plane of scattering is perpendicular to the plane of polarization.

How does this information help us? If the Klein-Nishina formula suggests that photons preferentially Compton scatter perpendicular to the plane of their polarization axis, and parity conservation requires the two photons of the decay to have orthogonal polarization, then we should observe a higher rate of scattering in perpendicular rather than parallel directions.  Let us study coincident detections in P1 and P2 and compare rates when the two are orthogonal (Φ = 0º or 180º) to when they are parallel (Φ = 90º or 270°).

The logic for this experiment is straightforward as shown in Fig. 3.

{ ${/download/attachments/131006890/Fig_3.png?version=1&modificationDate=1439238890000&api=v2}$ Figure 3: Logical flow of the signal processing chain

• First, any pulses that are not large enough to have been caused by a 90º Compton scattered gamma are rejected. This step gets rid of the large number of spurious noise pulses common to photomultiplier tubes.
• Second, a counter (scaler) is incremented each time a PMT pulse in P1 (large enough to exceed the voltage threshold) occurs simultaneously with a PMT pulse in P2. (In our case, “simultaneous” means within a 50 ns window.) These pulses are the signal we want to measure.
• Finally, another scaler is incremented each time a pulse from P1 is coincident with (that is, arrives within the 50 ns window of) a delayed pulse from P2. By choosing the delay time to be large enough (we use approximately 100 ns) we ensure that no true coincidences are measured; therefore, any simultaneous pulses detected in this way must be random. This count rate represents the background rate of accidental coincidences.

This logic is implemented by a series of electronics modules as illustrated in Fig. 4. 

{ ${/download/attachments/131006890/Parity_pos_schematic_v4.png?version=5&modificationDate=1459793105000&api=v2}$\\

Figure 4: Electronics wiring diagram

• The signal from each PMT goes into a discriminator which rejects pulses smaller than a set value. A NIM logic pulse output (with amplitude V = -700 mV) is produced for each input pulse above the threshold. 
• The discriminator output pulses from P1 and P2 go into a second discriminator, for reasons which will be described below. The outputs from these 2nd-level discriminators then go into one channel of a coincidence unit labeled P1•P2 in Fig. 4. The coincidence unit produces an output pulse only when it sees simultaneous (overlapping within 50 ns) input pulses from P1 and P2. The output of the P1•P2 coincidence then goes to a scaler.
• A second set of outputs, from the 2nd-level P1 and P2 discriminators – with the P2 output sent through a delay box of approximately 100 ns – go to a second channel of the coincidence unit labeled P1•P2d. The output of the P1•P2d coincidence then goes to a scaler.
• All of the scalers are controlled by the scaler timer. This module is used to simultaneously open and close the counting gates for all of the scalers. When in the “timer” mode, it can be used to run for preset time intervals of up to 8000 seconds.

The high voltage for P1 and P2 is provided by two individual high voltage power supplies. Verify that the outputs of the high voltage supplies are correctly cabled to the appropriate PMT and check that the signal cables are of equal lengths. Turn on the high voltage and turn on power to the rack which feeds all of the electronics. The suggested starting values for the PMTs are as follows:

PMT #1: V = -1800 V

PMT #2: V = -1800 V

NOTEBOOK: Record the starting high voltage values in your notebook.

Much of your preliminary work for this experiment is in verifying the electronics setup and understanding the signals and logic at each point. Work through this slowly and make notes at each step about what you observe and why. Listed below are the most important points to record in your notebook, but include as much information as you can as you explore the setup. These notes will help you when writing your report.

• Begin by placing the sodium-22 rod source in the lead collimator. Make sure that the binder clip is attached in order to prevent it from falling too low into the box.

CAUTION:  This source is fairly hot – with a strength on the order of millicuries as opposed to microcuries like smaller button sources – and should therefore always be inside the lead collimator box or inside its storage canister so that it is properly shielded. 

• Verify that all of the connections among the NIM modules are as shown in Fig. 4.
• Check that the discriminator thresholds are set to 30 mV. Use a digital voltmeter at the test points on the discriminator front panel to measure the threshold settings. Note that the test point voltage is 10 times greater than the actual threshold voltage.

NOTEBOOK: Record the measured discriminator threshold values. Alert a TA or staff member if the values are not approximately 30 mV as described.

NOTEBOOK: For each PMT, sketch the pulses (to scale) and record the typical pulse widths in your lab notebook.

• Note that for P1 and P2 the pulses are long enough that if their discriminator output pulse widths are set to a small value (say 25 ns), the discriminator would fire several times as the ragged pulse repeatedly moved above and below threshold. You can now see why the discriminator output width must be set to (the maximum value of) 1 μs in order to get only one output pulse from one input pulse. You may find that even with this longer output width, some pulses produce more than one output.
• The outputs of these two “1st-level” discriminators go into two “2nd-level” discriminators whose purpose is to scale the final output to a standard 25 ns width. The outputs of these 2nd-level discriminators go into the Lecroy coincidence unit, and the output of that goes to a scaler.
• Put the P1 signal into a tee connected to one channel of the scope, and run a cable from the tee to the P1 discriminator input. Put the discriminator output into the other channel of the scope. Trigger the scope on the discriminator output and look at the input signal to measure the threshold setting on the discriminator.
• Now look at the 25 ns long discriminator output pulses of P1 and P2 which go to the P1*P2 coincidence module on the scope. Trigger on P1 and look at P2 to verify that the discriminator outputs are simultaneous to within a couple of nanoseconds. Note that the point of this exercise is to verify that simultaneous inputs into the P1 and P2 discriminators will generate 25ns long output pulses which arrive simultaneously at the inputs to the coincidence module.  

NOTEBOOK: Make sketches (to scale) in your notebook of the discriminator outputs after both the first and second level discriminations. Note the widths and voltages and verify that they are as expected. 

• Now look at the output of the P1•P2 coincidence unit on the scope in conjunction with one of the P1 or P2 discriminator outputs.

Reset the scalers and take a short run (100 sec) to measure the count rates of P1, P2, and P1•P2. If the singles rates of P1 and P2 are R1 and R2, the accidental coincidence rate will be given by

where τ = _w_1 + _w_2 -2_w_0 is the resolving time, where w1 and w2 are the widths of the two discriminator output pulses, and where w0 is the minimum overlap required to fire the coincidence unit.

NOTEBOOK:  Calculate the accidental coincidence rate assuming that w0 ~ 2 ns.

This accidental coincidence rate is experimentally measured by running two other outputs of the P1 and P2 discriminators into a second coincidence unit, but with the P2 signal delayed by 100 ns. This delay eliminates all genuine coincidences, but does not alter the accidental (random) coincidence rate. This rate is measured by the P1•P2d scaler.

NOTEBOOK:  Does the measured accidental rate agree with your calculation? 

The rate of genuine photon coincidences is measured by taking the difference between the P1•P2 and P1•P2d scalers. This subtraction is necessary to eliminate the accidental coincidence background.

Take data at _Φ_1 = 0º (Fig. 5) and _Φ_2  = 0º, 90º, 180º, and 270º. Run for as long as is needed to get sufficiently low uncertainties on your count rates. Repeat the measurements with _Φ_1 =  90º, 180º, and 270º. This gives a total of 16 configurations. It is necessary to take data over all permutations of the four detector angles because of the sensitivity of PMTs to magnetic fields. Simply changing the orientation of one of the PMTs in the Earth’s local magnetic field will have a significant effect on the counting rates. By averaging data taken over all permutations of parallel and perpendicular PMT orientations, the effect of the Earth’s magnetic field will be averaged out.

NOTEBOOK: At each pair of angles, record the counts on the four discriminator outputs, the coincidence output and the delayed coincidence (i.e. accidental) output. Record the time for each run and compute the corrected coincidence rate (i.e. the rate with accidental rate subtracted) along with uncertainties.

{ ${/download/attachments/131006890/Detector_angles.png?version=1&modificationDate=1442932271000&api=v2}$ Figure 5.  Recommended PMT angles as viewed looking into the PIG through the Al scatterer.  Note that the angles are chosen to lie along the diagonals because the PMT's are too long to fit directly underneath the scatterer.

The correlation in the polarizations of the two decay photons should be evident from the difference in averaged coincident rates when the PMTs are parallel as compared to when they are perpendicular. The ratio of the count rates for these two cases can be used to make a comparison with the expected symmetry level of the experiment.

NOTEBOOK: Average together the eight perpendicular coincidence rates (with background subtracted) and average together the eight parallel coincidence rates (with background subtracted). Compute the ratio of the two averages. By propagating uncertainties properly, can you show that this ratio exceeds 1? Put another way, can you show that you conclusively see more coincidences in the perpendicular orientation than the parallel one?

Since this measurement is a convolution of the expected perpendicular polarization with the dependence of the azimuthal efficiency of Compton scattering on polarization, the calculation of the magnitude of the count rate ratio described above is complicated. It is further affected by the fact that the detectors have finite size and therefore cover a range of angles. This is a general problem for any experiment which revolves around count rates. The acceptance angle of the experiment must be made large enough to allow a reasonable count rate, but at the same time must not be so large as to obscure the angular dependence being investigated. In this experiment there are two detector angular extents that we need to consider as shown in Fig. 6.

{ ${/download/attachments/131006890/Acceptance_angles.png?version=2&modificationDate=1460471286000&api=v2}$ Figure 6: Detector acceptance angles

The first is clearly the finite extent of the detector in the azimuthal angular direction. The other is the angular extent of the scatter measurement parallel to the gamma-ray beam. This affects the ability of the Compton scatter to respond to polarization. A calculation of the expected asymmetry in this experiment as a function of these angular resolutions is shown in Fig. 7 (compiled from Ref. [2]).

NOTEBOOK and REPORT: Sketch the apparatus and note the dimensions. From these values, compute the two angles referred to above, θ and _φ, _with uncertainties.

{ ${/download/attachments/131006890/Fig_6.png?version=1&modificationDate=1439315173000&api=v2}$ Figure 7: Expected count asymmetry ratio: [90º, 270º] / [0º, 180º]. Note that both θ and Φ are the half-angles of the cones illustrated in figure 6. (Compiled from Ref. [2].)

This incorporates the effects of the finite detector sizes and their effect on the ratio. For example, if the effective angular acceptance half-angle of P1 and P2 along the beam direction is 30° and perpendicular to the beam (azimuthal direction) is 20º then the expected ratio for the coincidence rate for perpendicular versus parallel scatters is ~1.8.

REPORT: Use your values for the two angles to determine the expected count asymmetry ratio. Is your measured ratio consistent with the expected ratio? Note that you will have uncertainty in both your computed ratio and in the estimated expectation value from the plot above.

When writing your report, consult the rubric and notes below for the appropriate quarter.