Department of Physics, Middlebury College | 1992-93 |

Modern Physics Laboratory |

The positron has positive charge +e, spin 1/2, and the same mass m

_{e}as the electron. The positron is emitted in the^{+}decay of radioactive nuclei such as^{22}Na; however, it was first identified by C.D. Anderson in 1932 in cosmic ray photographs.When a positron interacts with ordinary materials, it rapidly associates itself with one of the electrons of the material and forms a bound system called positronium. In less than 10

^{-7}s the positron and electron annihilate to produce two gamma rays in the charge-conserving reaction

e ^{+}+ e^{-}--> 2 .The energy E

_{}of each gamma ray can be computed using conservation of total relativistic energy for a positron and electron initially at rest, that is

2 m _{e}c^{2}= 2 E_{}so that

E _{e}= m _{e}c^{2}= 0.511 MeV. The emission of two gamma rays of the same energy is required by linear momentum conservation. To see this, assume that initially the positronium system is at rest and the total initial linear momentum is zero. Linear momentum conservation cannot permit pair annihilation into a

singlephoton since that photon would have to carry energy 2m_{e}c^{2}and a non-zero linear momentum 2m_{e}c required by the photon energy-momentum condition p_{}= E_{}/c. Therefore, if linear momentum is to be conserved, at least two photons must be emitted. It is easy to show that if two photons conserve energy and linear momentum they must be emitted with the same energy E_{}= m_{e}c^{2}and must travel in exactly opposite directions.^{1}It should be noted that three or more photons can simultaneously conserve electric charge, energy and linear momentum, but that the probability for multiphoton processes is very small relative to the two photon process observed in this experiment.Precise measurements of the exact collinearity of the two annihilation gamma rays can be used to test relativistic four-momentum conservation. The detector arrangement to be used in this experiment is shown in Fig. 1. Positron decays of

^{22}Na in a solid source provide correlated, pair annihilation gamma rays for detection by the two NaI(T) detectors. One NaI(T) detector remains fixed while the other is free to rotate by angle about a vertical axis through the radioactive source. An electronic coincidence circuit produces a pulse each time a gamma ray is detected simultaneously in the two detectors. When the coincidence counting rate is measured as a function of the angle of the movable detector, one obtains data such as that in Fig. 2.^{2}One clearly observes a maximum coincidence counting rate for = 180^{o}, as expected.Two perhaps unexpected features are also apparent in Fig. 2. (1) The counting rate peak in Fig. 2 has considerable angular breadth and, at first, this might seem to suggest that the two annihilation gamma rays are not emitted in exactly opposite directions as required by relativistic four-momentum conservation. In this experiment you will show that the observed angular breadth of the coincidence peak is entirely due to the rather large, finite solid angles subtended by both NaI(T) detectors and is not due to a breakdown of our most important conservation law. If each detector subtends a half-angle as shown in Fig. 1, then the movable detector can detect perfectly collinear annihilation events throughout the angular range from = 180

^{o}- 2 to = 180^{o}+ 2. The angular breadth of the coincidence curve of Fig. 2 can be made smaller by moving the detectors farther from the^{22}Na source, thereby reducing .(2) At = 90

^{o}or 270^{o}one finds that the coincidence counting rate is not exactly zero. This results from the finite resolving time of the electronic coincidence circuit, that is, the electronics cannot determine that two signals are exactly simultaneous, but only that they both fell into the same time "window" of some very small, but finite duration. Thus, it is possible for the two detectors to detect gamma rays from entirely separate annihilation events and have the two gamma rays arrive accidentally within the time window of the coincidence circuit. The resulting coincidences are called accidental or random coincidences and they are present to some degree in every nuclear or atomic coincidence measurement.An electronics diagram for the pair annihilation experiment is given in Fig. 3. Low-level, -10 to -100 mV pulses from the NaI(T) detectors are amplified in separate sections of an ORTEC 535 Quad Fast Amplifier. Amplified signals are sent to two channels of an ORTEC T105/N Dual Discriminator where low level noise pulses are rejected. The -1 V, 5 ns full width at half maximum (FWHM) discriminator pulses from the fixed detector and the movable detector are separately routed to the Start and Stop inputs of an ORTEC 567 Time-to-Amplitude Converter (TAC)/Single Channel Analyzer (SCA), respectively.

The TAC unit produces a square pulse with a magnitude that is directly proportional to the time difference between the arrival of the Start pulse and the arrival of the Stop pulse. In this experiment, the TAC Range and Multiplier controls are set to give a full range of 200 ns. This means that when a Stop pulse arrives 200 ns after a Start pulse, a TAC output pulse of +10 V magnitude is created. If a Stop pulse arrives more than 200 ns after a Start pulse, no TAC output is generated and the TAC is recycled to await the arrival of the next Start pulse. Since the two annihilation gamma rays are created simultaneously, it is necessary to delay electronically the arrival of the Stop signal from the movable detector by at least 10 to 20 ns to insure that the TAC does not produce pulses too close to 0 V. The appropriate delay may be obtained by inserting a low-loss coaxial cable of 10 to 20 ft length between the discriminator output and the TAC Stop input.

In Pulse Height Analysis (PHA) mode, the Quantum 8 multichannel analyzer (MCA) accepts 0 to +10 volt pulses from its Direct input (on rear panel) and maps them into channels 0 through 1023. When the number of counts versus channel number is displayed, one sees a time spectrum such as that shown in Fig. 4, where full scale along the horizontal axis corresponds to a Start-Stop time difference of 200 ns. The TAC peak has been placed near 130 ns by choosing an appropriate length of coaxial cable. When the time delay and time jitter due to the coaxial cable and other electronic components are properly accounted for, the 5 ns FWHM of the TAC peak establishes an upper limit of 5 - 10 ns on any lack of simultaneity in the emission of the two annihilation gamma rays.

The ORTEC 567 TAC/SCA unit contains an SCA that can be used to window the annihilation gamma ray TAC peak and eliminate counting of random coincidences lying far from the main TAC peak. Pulses from the SCA-windowed peak can be sent to an audio amplifier and loudspeaker so that experimental parameters can be adjusted by listening directly to the gamma-gamma coincidences.

(1) With the Tektronix 2465A CT oscilloscope, examine the various signals used in the gamma-gamma coincidence circuit. Photograph the signals observed on the oscilloscope at the following points in the coincidence circuit:

For (a), (b), and (c), a 50 terminator must be provided at the oscilloscope input if the signals are to be properly displayed. Your photographs should be affixed securely to pages of your laboratory notebook and be accompanied by labels identifying the signals seen in the various oscilloscope traces.

- (a) photomultiplier outputs,
- (b) fast amplifier outputs,
- (c) discriminator outputs,
- (d) TAC/SCA outputs.
(2) Place a stack of three 1 µCi

^{22}Na sources on the stand at the center of the pair annihilation apparatus. Adjust the two NaI(T) detectors so that the front metal surfaces of both of the NaI(T) detectors are a distance D = 25 cm from the radioactive sources. Be sure that the distance D does not change during the experiment. Arrange all cables and equipment on the experimental table so that the movable detector can be rotated freely.(3) Use the ORTEC 776 Counter/Timer to measure the number of gamma-gamma coincidence counts obtained in five minutes for ten different angles . Most of the angles should be chosen near = 180

^{o}so that an accurate profile of the coincidence peak is obtained, as in Fig. 2. One of the ten data points should be taken as near to 180^{o}as you can determine. One data point should be at = 90^{o}where you expect only accidental coincidences. Record the width, t, of the SCA time window that you apply to the TAC output.(4) At = 180

^{o}, determine R_{Start}and R_{Stop}, the counting rates at the TAC Start and Stop inputs, respectively. To do this, connect the Start and Stop signals to the ORTEC 776 Counter/Timer, and count each one for one minute. Convert the measured values for R_{Start}and R_{Stop}to counting rates in counts per second (Hz).

Your goal is to provide a quantitative justification of the finite solid angle and random coincidence effects seen in your experimental data. You will not be able to compute these effects exactly; however, you will be able to make well-reasoned estimates of them.

(1) Use the region of interest (ROI) feature of the Q8 program to obtain the number of coincidence counts in five minutes for the ten angles you chose. For consistency, you must apply the same ROI Start and End channels to all ten TAC spectra. Plot your data points, with error bars, in a well-labeled figure. Draw a smooth line through the data points.

(2) Assume that the NaI(T) detector occupies the entire 1 1/2" dia x 1/2" deep cylindrical volume at the front end of the NaI(T) /PMT detector assembly. (You are neglecting the thickness of the thin metal shroud that surrounds the NaI(T) crystal.) Assume that all parts of the NaI(T) crystal have equal efficiency for the detection of annihilation gamma rays. Treat the stack of radioactive sources as a single point source.

Make a full scale, top view drawing of your detection geometry, showing (a) the position of the source, (b) the position and top view cross section (1 1/2" x 1/2" rectangle) of the fixed detector, and (c) the circle along which the movable detector moves. From an index card, cut out a rectangle (1 1/2" x 1/2") corresponding to the cross section of the movable detector. Move this rectangle around the circle of your drawing, taking care to orient it correctly. Use a ruler passing through the source point to help you decide whether or not "back-to-back" gamma-gamma coincidences can be detected for a given setting of the angle of the movable detector.

Suppose that when = 180

^{o}the coincidence counting rate is measured to be R_{o}. Use your scale model to develop answers to the following three questions.

- (a) Over what range of angles do you expect a coincidence counting rate greater than 3/4 R
_{o}?- (b) Over what range of angles do you expect a coincidence counting rate greater than 1/2 R
_{o}?- (c) Over what range of angles do you expect a coincidence counting rate greater than 1/4 R
_{o}?As you answer these questions, outline your reasoning in your laboratory notebook. Remember that the NaI(T) volume is actually a cylinder, not a rectangle. Indicate the three angular ranges (a), (b), and (c) in the figure containing your ten data points.

(3)

Random coincidences. If the signals presented to the Start and Stop inputs of the TAC have rates R_{Start}and R_{Stop}, respectively, and the time window of the TAC is t, then the rate of random outputs of the TAC, R_{TAC random}, is given by

R _{TAC random}= R_{Start}R_{Stop}t. (1)Eq. (1) is derived under the assumption that the input pulses presented to the Start and Stop inputs of the TAC are both randomly distributed in time.

Use your values of R

_{Start}and R_{Stop}from step (4) in the procedure section to compute R_{TAC random}from Eq. (1). Does your value of R_{TAC random}computed from Eq. (1) agree with the TAC counting rate you measured at = 90^{o}, where you expected only accidental coincidences?

1. R. Eisberg and R. Resnick,

2. T. G. Ferris,