Polarization Correlation in the Gamma- Gamma Decay of Positronium Bin LI Department of Physics & Astronomy, University of Pittsburgh, PA 56, U.S.A April 5, Introduction Positronium is an unstable bound state of a positron and an electron. It is very like hydrogen atom, except that a positron replaces the proton. The ground state of positronium can be either a singlet S or triplet state 3 S, where the angular momentum number L=, and these two states are corresponding to spin zero and spin one respectively. Due to the conservation principles of linear momentum, angular momentum, parity and charge conjugation, we know the singlet decays through annihilation into a pair of gamma particles, and the triplet decays into three gamma particles. Theoretical calculation shows that the decay rate of singlet is three orders of magnitude fasters than that of triplet state. So most of the decaying gamma rays come from singlet state annihilation and they will exist as two back-back gamma particles with same magnitude but vice-visa momentums. Now, we begin to discuss about the polarizations of the two gamma particles. In order to satisfy the conservation laws mentioned previously, both of the two back-back gamma particles must be RHC (right hand circulated) or LHC (left hand circulated). Meanwhile, we know the parity of singlet state of positronium is (-) L+ =(-) + =-, so the two-photon state should also be negative parity, we can write it as following: ψ = R R > L L >, where R R > is the uncoupled states of R > and R > ; same with L L >. We are supposed to set up a counter that only accepts the lights with x-polarization on one side of our source ----- X >, and another counter that only accepts the lights with y-polarization on the other side ----- Y >. So we can calculation the amplitude that ψ will be in the state of X Y > since we know the RHC gamma is X > + i Y > X > + i Y > R > = or R > =, and the LHC gamma is L >= X > i Y > or = L > X > i Y >.
So from < X Y ψ > = < > < > i i X Y RR X Y L L = ( ) =i, we get the possibility to obtain one photon with x-polarization on one side and another photon with y-polarization on the other side is unit ( i =). Similarly, we can also get the possibility to obtain both of these two photons with x-polarizations ( or both with y-polarizations) is < X X ψ > =. Since the directions of x-axis and y-axis are arbitrary, the only constraint is that they are perpendicular, so more generally, we get that the polarizations of the two detectable correlated back-back gamma particles should be orthogonal. Method In our experiment, we use Na as irradiation source. As Na decays into Ne, a positron will be emitted. And this positron forms positronium when it becomes bound to an electron. Most of positroniums are in singlet state, and they will decay into two backback gamma rays. Now we use Compton Scattering to measure the relative polarizations of the two photons since the signal rate is proportional to the scattering cross section, which is strongly dependent on the polarizations of incident gamma particle and outgoing gamma particle. From reference, we have the formula of deferential cross section for Compton Scattering: d e = ( dω mc k' ) ( ) k ε ε, If we consider =, k and ε are the momentum and polarization of the incident h gamma ray, k' and ε are the momentum and polarization of the scattering gamma ray. From the point of classical electrodynamics, we can estimate the differential scattering cross section as the following: (omit) k n In this diagram, and are the propagation directions of the incident wave and scattering wave respectively, so these two vectors build up a plane called scattering plane. ε and ε are two special choices for the possible polarization of the two outgoing ε k wave after the scattering, where is in the plane containing and (scattering plane), ε is perpendicular to it. Since both cases ε and ε are perpendicular to, in term of unit vectors parallel to the coordinate axes, we have: ε = cosθ ( e cosφ + e sinφ) e x y z sinθ n n
ε = e x sinφ + e y cosφ If we assume that the out-going wave is unpolarized (this might be a reasonable assumption for circular polarized wave), we will get: () For an incident linearly polarized wave with polarization parallel to x-axis, the angular distribution summed over final polarizations is: cos θ cos φ + sin φ. () For the incident wave with polarization on the y-direction, the over all angular distribution for final polarization is: cos θ sin φ + cos φ In order to see the apparent difference due to the variation of the polarizations of the gamma particles, we choose θ = 9. That is to say, the propagation direction of incident momentum and that of the scattering momentum are perpendicular. So we will put our detector for the out-going gamma particle at some place perpendicular to the incident direction with some azimuthal orientation. The following figure shows how our detecting system works. We use the combination of two detectors. () Parallel Configuration ()Diagonal Configuration
Everything is the same with the previous configure for parallel case except that the relative orientations of the two detectors are not parallel but perpendicular and they are still kept orthogonal to the incident directions respectively. Now we know, for both of these two cases, θ = 9. Using Compton Scattering formula: = ( cosθ ) and the energy conservation for Positronium k k mec annihilated to two back-back gamma particles with same magnitude momentum: k kc = mec, we can easily get d e =. So we have = ( ) sinφ for the incident k dω mc d e wave with x-polarization; while = ( ) cosφ for the incident wave with y- dω mc polarization. If assuming that spherical angle of the surface area of detector due to the scattering center is small, we can calculate d at an approximate constant θ 9. So dω π e we obtain the scattering cross section at that orientation is = d φ( ) sinφ, or mc π φ( e = d ) cos mc φ. So the signal rate (due to cross section) is totally depended on the polarizations of the incident gamma rays. (Since here these two correlated gamma particles are circular polarized, we can only change the conformation of our detectors to see this effect.) Now we are going to estimate the expected counting rates of the two different detectors configurations. From the discussion in Introduction, we know the polarizations of the two correlated back-back gamma particles should be perpendicular with each other. So for the case ----- parallel configuration: If the polarization of one incident gamma particle is along the x-direction, the polarization of the other one must be the y-direction (or just exchange their polarizations). For the other case ----- orthogonal detecting configuration: If the polarization of one incident gamma particle is along x-direction, in order to satisfy the condition that the polarizations of these two particles are perpendicular, the polarization of the other one must be also the its x-direction, or both of them are on their own y-directions. So for the first case ----- parallel configuration, we have: π e () = α * dϕ[sin ϕ * cos ϕ*], where α = ( mc For the second case ----- perpendicular configuration, we have: π ϕ () = α * dϕ[sin ϕ *sin ] ), it s a constant.
After calculation, we get = 3, so the ratio of the expected coincident counting rate of the second configuration to the first configuration is about 3. But this is only true for the ideal case: A). Solid angles from the detector surface to the scattering centers on both sides of the source are small and exactly same ----- so θ 9 for all the detected outgoing scattering gamma rays. This requires the surface of our detector is small, and is pretty far from the scattering center, but it couldn t be too far away since that will reduce the counting rate greatly! B). The solid angles for the two configurations are same. It requires that when we change the detecting configuration from one to the other, we should keep the distances from the scattering center to the detectors are identical with the previous case. C). The Compton Scattering here is single scattering (just scatters from free electron once). This is pretty true if the thickness of our scattering material (here is aluminum) is not two big. D). The detecting system and the electronic instruments can work consistently. For point D), it is very hard to say for our experiment set-up. For point C), it is approximately true. And we try our best to satisfy the requirement B), but we still need to consider the correction for A). This is due to that the count rate is partially contributed by the signals scattering at an angle not equal to 9, it will make the experimental result of less than 3. The precise calculation requires us to measure the surface of the PMT detector and the distance away from scattering center, then do integral for both θ and φ. From previous experiment record, we can estimate. Experiment Set-Up and Procedures The following are all the instruments involved in our experiment. G: Sample Holder with irradiation source inside. Na P, P : Photo Multiplier Tubes as detectors for outgoing gamma particles.
H, H: High voltage for the Photo multipliers P and P respectively. S, S: Aluminum Pieces as Compton Scattering center for gamma rays. A: Model 776, Sixteen Channel Amplifier. Q: Model 74, QUAD Linear. D: Model 7, Six Channel Discriminator. C: Model 465, Coincidence Unit. C`: Digital Counter. O: Oscilloscope. H HHH S S H P G P A Q D C C O ch ch In this experiment, our goal is to measure the coincidence counting rates when the two photo multipliers P, P are in parallel configuration and in perpendicular configuration. So we have to figure out the best working conditions of our system and meanwhile find out a method to estimate and get the background for our measurements. Here, I want to give a summary of our set up. The high voltage is about 9 Volts, which is the suggested value from the manual. P, P can change the scattering light
signal into electronic signal, and then send it to amplifier. The amplifications for the signals coming out from both tubes are same ----- approximate. The Instrument Q was used to adjust the offset of the signals, we made it very closed to zero. Discriminator can transform the incoming signals into the square wave fronts with width at some magnitude if the amplitudes of the incoming signals are beyond the threshold value. A good threshold value can differentiate the signals from the huge amount of noises. Our threshold voltages for both of the two channels are about 5 mv. We can also adjust the width of the Discriminator and use oscilloscope to watch the change of shape of the square wave front and measure it. Here, our widths are. us and.5 us for the two different channels. Coincidence Module has several different options, it can read the signals from channel one, signals from channel two and coincidence signals of channel A and channel B. The counter here is used to count the number of invents in some finite time scale. After successful setting-up and adjustment, we can begin to do measurement. Experimental Data As we know, backgrounds are very important in this experiment since our signal rate is very low. And the background rates come from cosmology irradiation, the rate of randomly overlapping coincidences, the fluctuation and inconsistence of our electronic instruments, etc. First, we try to check the consistence of our instruments for measurement. So, we removed the source Na away from our detectors and measured bare-counting-rates. We got the following results: Channel : 564/5mins 48/min Channel : 46/5mins 844/min Coincidence: 7/5mins 4/min So, we can see even there is no source at all, the coincidence rate is still very high, beyond /min. Knowing this information, then we can measure the backgrounds (keeping the irradiation source but removing the aluminum scattering targets) and the coincidence rates for the two configurations-------parallel and diagonal. In order to reduce the accidental deviations and measure our data as precisely as possible, we extended our measurement time to one day (4 Hrs). But before doing that, we can measure the single channel rates for both two cases in short time scale since the high rate of single channel will reduce the accidental fluctuation greatly. Parallel Configuration:
Channel : 53636/5mins Channel : 39/5mins 377/min 645/min Background: 64/8.5Hrs Coincidence: 4388/4.43Hrs 3.46/min 8.3/min Diagonal Configuration: Channel : 83755/6mins Channel : 455/6mins 366/min 379/min Background: 3477/.75Hrs Coincidence: 4766/4.83Hrs 6.6/min 3./min The width values that we set in the two channels of our discriminator are: Channel : T =. us Channel : T =.3 us We can use it to evaluate the background rates due to the randomly overlapping coincidence by referring to the formula R= ν ν T + ). ( T Configuration Expectation Background Due to randomly overlapping Experimental Background Rates Coincidence Rates Refined Coincidence Rates (Real correlated signal) Parallel 4.3/min 3.46/min 8.3/min 4.77/min Diagonal 7.83/min 6.6/min 3./min 5.4/min Discussion and Conclusion
From the Data table, we see either for the parallel configuration or the diagonal configuration, the experimental background rates are very close to the calculated background rates due to random overlapping signals. So we know the main part of background of gamma-gamma coincident signals is the random overlapping signals. But unfortunately, our experimental value of is far from expected value, (here it is only.3), this is because here the background rate is very high, it almost accounts for 7% of the total signals, so how could we estimate it will give us reasonable results at this condition. We have two possible ways to improve our experimental result: () Use a much stronger decaying source, which will give us higher coincidence rate. () Improve the electronic detecting circuit, and make the ratio of signal to noise increasing.