# Polarization Correlation Of Annihilation Radiation

## (Optional Extension: Angular Correlation) Caltech Senior Physics Laboratory Experiment 14 September 2000

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N.B. EXERCISES No. 1, 2, and 3 are required BEFORE starting the experiment.

The early 1930's saw explosive progress in the efforts to unravel the structure of the nucleus, as well as the application of the new quantum mechanics. Until 1932, the nucleus was believed to contain only electrons and protons. That year, James Chadwick (Cavendish Laboratory) reported the discovery of a new component, the neutron. In 1933, Carl Anderson and Seth Neddermeyer (California Institute of Technology) discovered the positive electron, or positron, and the phenomenon of pair production. P. A. M. Dirac, et al, developed quantum mechanical theories that correctly explained many experimental phenomena, and predicted features soon to be confirmed experimentally.

The ability to fabricate artificial isotopes permitted hitherto impractical experiments. The polarized nature of the products of positron/electron interactions (annihilation) was first observed by C.S. Wu and I. Shaknov (Columbia University, 1950) using 64Cu and modern scintillation techniques. Independent calculations by M. H. Pirenne (1944), J. A. Wheeler (1946), and A. Ore with J. L. Powell (1949) [I, II] predicted that positrons would interact with electrons to produce a substance (named positronium) with two unstable states. M. Deutsch (Massachusetts Institute of Technology, 1951), working with 22Na, provided experimental evidence that positronium existed and the multiple decay mechanisms were as predicted. The experiment here allows you to examine the polarization correlation of the photons produced by positron annihilation, as well as gain experience in coping with time-coincidence measurement techniques.

## Theory:

When the atom [X] with a nucleus of Z protons and N neutrons is more massive than its nearest neighbor [Y] which has (Z - 1) protons and (N + 1) neutrons, it will either emit a positron-neutrino pair, or attract one of its inner atomic electrons into the nucleus (electron capture) and then emit a neutrino. The processes are quite similar to b-decay and are written:

When a positron emitter (22Na) is surrounded by any material, the positrons will lose energy by ionization-loss processes and, when nearly at rest, form a bound structure with an electron called positronium. The energy levels are similar to those of hydrogen, but with a unique "Rydberg". If the positron/electron spins are parallel, a 13S1 state (ortho-positronium) will be formed. When the spins are anti-parallel, a 11S0 state (para-positronium) is formed. Because the collision processes that produce these states cause them to be equally populated, there will be three times as many of the triplet (3S1) state as the singlet (1S0). Additional interactions between the meta-stable ortho-positronium and its environment induce transitions (quenching) to the lower energy 1S0 state in 10-9 sec. Nearly all the detected electron-positron annihilation emissions will thus occur from the spin-zero state (t ~ 10-10 sec), producing two 511 keV (E = mc2) photons emitted in opposite directions.

If one photon exhibits an X-polarization, its mate always shows a Y-polarization, i.e., the planes of their polarizations must be perpendicular to each other. This may be confirmed experimentally by utilizing the feature that Compton-scattering cross sections for polarized photons are significantly greater for scattering into the plane at right angles to the E-vector of the incident photon, i.e., 90° to the direction of polarization. The optical analog of the scattering material is the polarizing filter. The Klein-Nishina formula shows the scattering cross section, sigma, is proportional to:

with k0 and k representing the momenta of the incident and the scattered photons respectively, q is the angle of scattering, and f is the angle between the plane of scattering and the E-vector of the incident photons. At q = 90°, annihilation radiation scattering is five times stronger for f = 90° than for f = 0°, thus providing an effective g-ray polarization analyzer. (The ultimate analyzing power is somewhat higher and is seen at q = 82°.)

Figure 1. The geometry of the polarization correlation experiment.

Consider the source-scatterer-detector configuration of Figure 1. The 22Na source of annihilation radiation is collimated by the Pb cylinder, so that the two quanta emitted up and down may scatter at A1 and A2 into detectors D1 and D2, respectively, with q = 90°.

The randomly polarized "up" beam can be resolved into incoherent linearly polarized components parallel and perpendicular to the page. Therefore, the "down" beam must be polarized perpendicular and parallel to the page, i.e., "down" beam components are cross correlated with the "up" beam polarization.

The probability of coincident counts in both D1 and D2 may now be added up for the two polarizations, with a probability assigned, of 1/6 for the f = 0° scattering and 5/6 for f = 90°. Thus the count rate in the pictured geometry (detector axes parallel) is:

If the detector D2 is now rotated out of the page (i.e., 90° about k0), the count rate becomes:

The anisotropy ratio is Nperp / N// = 2.6. When the same calculation is carried out for the maximum Compton analyzing power at q = 82°, the anisotropy is found to be 2.85.

The anisotropy ratios computed above are for ideal geometry, i.e., a point source, infinitesimal size scatterers, zero variation of solid angles, and 100% detector efficiency. In practice, none of these conditions can be achieved. The selection of the scattering material involves tradeoffs between the largest possible Compton scattering cross section, while simultaneously presenting the lowest possible Photoelectric interaction cross section. This requires choosing a low-Z material for a small Photoelectric cross section, while increasing size to achieve an acceptable Compton yield. Detectors must be thick enough for high detection efficiency at the highest scattered energy, with a large enough frontal area to produce an acceptable counting rate at practical detector-scatterer separation when using a safe and economically feasible source. They should, at the same time, be thin enough to minimize response to unwanted higher-energy background. These compromises produce significant variations in source-scatterer solid angles, as well as an appreciable spread in both q and f. The net result is a substantial reduction in the experimentally observable anisotropy. Some calculations for finite geometry have been made by Snyder, et al, Phys. Rev., 73, 440 (1948), with some modifications for practical geometries. Some additional factors to consider when analyzing the data from a real experiment are: double scattering, variations in detection efficiency with energy and angle, accidental random (chance) coincidences, contributions of sources of "background", and dead time.

## The Experiment

A cross section of the apparatus is shown in Figure 2. The positron emitter used here is 22 Na (900 µCi ± 15%, on 1 February 1987) embedded in solid plastic and then sandwiched between two thin aluminum discs to insure that very nearly all of the positrons are stopped near the site of their emission, with minimal attenuation of 511 keV g-rays. (See a complete decay scheme diagram , and other data describing the 22Na source.) The "up" and "down" beams are defined by conical lead collimators, which are in turn surrounded by 5 cm of lead to minimize the exposure of the experimenter (and the detectors) to radiation. [The dose rate close to the apparatus is quite high and should be carefully checked.] The scattering cylinders (Aluminum) are symmetrically mounted on the beam axis. The two detectors are mounted on rotating arms that also permit radial adjustment of the distance between the detectors and the scattering cylinders.

Figure 2. Cross-section view of the Source/Scatterers/Detectors.

### Time Coincidence Measurements:

The process to be examined produces two correlated back-to-back g-rays that can be separated from unwanted radiation by testing for both amplitude and the time relationship between them. Ideally, only the 511 keV annihilation photon pairs will be correlated in time, i.e., are simultaneously emitted and scattered into the detectors. If only those detector signals that are "simultaneous", i.e., are produced within a very small time interval, are selected for amplitude analysis, it will be possible to reject many (but not all) spurious events of similar amplitude. The system to accomplish this is shown schematically in Figure 3. A single positron source provides the correlated g-rays to be examined by the two detectors. The amplitudes of the detector

Figure 3. Wave forms of an "Overlap" Coincidence Analyzing System.

pulses are then tested by the Single Channel Analyzers (SCA). These generate a logic signal if the analog input exceeds the Lower Level Discriminator (LLD) setting, but is less than the Upper Level Discriminator (ULD) setting. These logic signals are then applied to an "overlap" type Coincidence Analyzer that will in turn produce a Gate signal output whenever the inputs "overlap", or coincide in time. This Gate tells the Multichannel Analyzer (MCA) that the corresponding analog signal applied to its input is to be accepted for processing.

The spectra produced are not "pure". A significant number of counts are uncorrelated events that fall within the amplitude/time window of the electronics. Accidental/random/chance coincidences are unavoidable in this (or any) timing experiment, be they generated by two uncorrelated 511 keV quanta or by uncorrelated emissions at other energies ("background"). The number of "accidentals" is estimated by measurement of the singles rates (r1, r2), i.e., the count rate in each channel, and the time resolution (t) of the coincidence analyzer. The accidental coincidence rate for a 2-fold coincidence measurement is then given by:

Rrandom = 2 tr1 r2.

References: (6) p. 308-314, (5) p. 692-698.

This is the rate that will be subtracted from the raw data. The value calculated should be carefully compared to the values measured using other methods described below, and under Data Acquisition . Carefully consider the shortcomings of each of the techniques. To this end you should consider the total emission from the 22Na source and the NaI(Tl) detectors efficiency at the appropriate energies.

Background is usually experiment specific, but may have a significant environmental component. Experiment specific background may be from uncorrelated source emissions that reach the detectors within the resolving time "window" by paths other than scattering off the analyzing cylinders. Remember that the source also emits 1.27 MeV g-rays in time coincidence with the 511 keV annihilation quanta, and that these are not completely stopped by the modest lead shielding available. Also, a 1.27 MeV g-ray scattered off a cylinder could produce a detector pulse within the energy window, and in time coincidence with a single scattered 511 keV photon, or even another 1.27 MeV photon. Because it is not feasible (safe) to remove the 22Na source from the apparatus, you will have to devise ways to realistically estimate the total effect of all of the "accidental" and background components. A possible scheme would be to remove one of the scatterers, and take a counting run. Another method is to remove both detectors from close proximity to the 22Na source, then place separate sources in front of each detector, being careful that each source/detector pair does not "see" the other.

Another possible source of error may be the "dead time" of the various elements of the electronic chain. This effect may be estimated by considering the average count rate seen by the device and the width of the pulses involved. The dead time (count rate) specifications of the amplifiers, SCA's, Coincidence Analyzer, and MCA must be examined to determine if this effect will be of significance for this experiment.

References: (5) p. 96-102.

## Instrumentation And Experimental Techniques

The electronics system consists of general purpose NIM (Nuclear Instrumentation Modules) units. The complicating feature is that there are two interdependent channels, with the data from each to be analyzed for both amplitude and time information, and then analyzed once again for the information that each contains about the other. The setup and adjustment procedures are not complicated, but must be performed in the correct order to obtain meaningful results. Refer to Figures 3 and 4 .

### NaI(Tl) Scintillation Detectors:

The NaI(Tl) crystals on the two integrated units are 3.18 cm diam x 1.27 cm thick, mounted on 3.81 cm diameter photomultipliers (PMT). The crystal dimensions provide the highest detection efficiency for scattered 511 keV photons while presenting the minimum interaction cross section for the 1.27 MeV emissions that take a direct path through the modest Pb shielding between the source and detectors. Each detector is enclosed in a Pb housing to further reduce "background". N.B. note that these Pb housings restrict the effective frontal area of both detectors. The two PMT dynode chains have significantly different gains at the same HV. The PMT preamplifiers are set to produce approximately (~5%) the same output amplitude for the same energy of g - ray input when both operate from the same high voltage power supply. This scheme is economical, simplifies system setup, and tends to equalize the differences in transit times through the two channels.

### Linear Amplifier:

This is a low noise active Gaussian filter unit with a maximum gain of 640. Peak amplitude is reached at 1.5 µs for a unipolar output, and 1.1 µs for the bipolar output, with a bipolar zero-crossover point at 2.5 µs. Output non-linearity is ~0.15% from 0 to l0 V with saturation at about 11.5 V.

### Constant-Fraction Timing Single-Channel Analyzer:

The unit analyzes Positive input pulses of 50 mV - l0 V maximum (unipolar preferred). Simultaneous positive and negative outputs are produced: NIM standard pulses of +5 V into 50 W, with ~20 ns rise time, and 500 ns width; plus a NIM/FAST negative pulse of ~ - 800 mV into 50 W, with ~ 5 ns rise time and ~ 20 ns width. Three operational modes provided are: INTegral, NORmal, and WINdow. INTegral mode produces an output whenever an input pulse exceeds the Lower Level Discriminator (LLD) setting [the Upper Level Discriminator (ULD) is disabled]. NORmal mode produces an output only when the input exceeds the LLD setting, but is below the Upper Level Discriminator setting. LLD and ULD settings are independent, each providing a range of 50 mV to 10 V. If they should be "crossed" ALL outputs are disabled! The WINdow mode is not relevant to this experiment. Accurate timing data are obtained by use of the Constant Fraction technique. If the input pulse falls between the Lower Level and Upper Level discriminator settings, the output circuit will be triggered at the 50% point on the input pulse trailing edge. Outputs appear at the front panel connectors at this time + the time set on the Delay control. Time shift (walk) vs. input amplitude is < 2 ns for a 10:1 dynamic range and < 4 ns for a 50:1 dynamic range. Pulse pair resolving time for the + 5 V output is the sum of the output pulse width + Delay control setting + 200 ns.

### Multipurpose Coincidence Unit:

The TC 404A logic signal processor determines the time relationship between as many as five input signals and generates an output if user-selected criteria are met. Each channel requires a + input > 1 V with a width > 50 ns. The output coincidence gate is a rectangular wave form with rise and fall times of < 50 ns, > 4 ms width, and an amplitude of 8 V open circuit, or 4 V into 50 W. Each input channel has three operational modes: OUT (bypass), COINcidence, and ANTIcoincidence. These describe the relationship between that channel's input and any other active channel that will produce an output pulse. The time coincidence "window" can be continuously varied between 50 ns and 250 µs. A test jack allows accurate measurement of a gate width equal to the resolving time. The number of active channels is set with a five position switch. The COINcidence mode produces output pulses when input pulses overlap by at least the length of the resolving time gate. The ANTIcoincidence mode produces output pulses only when the two (or more) inputs do not overlap. The leading edge of the output gate appears at a time equal to the resolving time t + 500 ns after the appearance of the leading edge of the first input pulse. The module dead time persists for 300 ns after the end of the output pulse.

### Delay Amplifier:

This is an analog unity gain amplifier whose output can be delayed with respect to its input by means of switch - selected delay lines. The delay interval may be any combination of 0.25, 0.5, 1.0, 1.0, and 2.0 ms ± 5%. The minimum delay is 150 ns. Input range is + 0.1 - 10 V with the output polarity user selected. The output may be attenuated by lx, 2x, or 3.3x. Integral nonlinearity is < 0.05%, with variations in gain vs delay of < 4%.

### Multichannel Analyzer (MCA):

A Tracor - Northern TN-7200 Multichannel Analyzer (MCA) will be used. This 4096-channel device accepts a full scale analog input of + 8 V, with user selectable coincidence gating logic. This unit is the same as used for Experiments 12 , 13 , 15 , and 16 . It will be useful to review any of those experiments you have completed. The new feature to be utilized in this experiment is the Coincidence Gating input that allows for analysis and storage of data from only those analog pulses that have the correct time relationship to a gating signal, also derived from the same electronics system. The coincidence gate input must be a TTL pulse ( < + 2 V), with a leading edge that precedes the analog pulse peak by > 500 ns, and with a width of at least 1 µs . The Coincidence Gate input connector, and its enabling switch, are on the TN-7200 rear panel.

The ultimate interconnection of the modules is shown in Figure 4 . Remove the lower preamplifier-detector assembly (No.2) from its cradle, place it in the long aluminum channel and rest them on the top plate in line with the upper preamplifier-detector assembly (No.l) (set them corner to corner for stability and safety) . Place the 22Na Calibration Source (a lead housing 3" dia. X 4" long and painted RED) on the short aluminum cradle midway between the two NaI(Tl) detector crystals. Rotate the upper scattering cylinder off the centerline. Set the HV for the photomultipliers (PMT) at + 600 V. Disconnect the Coincidence analyzer OUTPUT Gate cable at the TC404A panel.

Figure 4. The electronics system for time/amplitude analysis.

Set amplifier No.l for a positive bipolar output and set the gain to put the 511 keV full energy peak near + 4 V, using the MCA default Conversion Gain (Suggestion: 1/4 memory with a minimum of 512 channels displayed). Set the MCA (Rear Panel) analog input coupling switches to DIRECT and PASSIVE, and the Coincidence Gate mode switch to ANTI. [Be sure that the Coincidence Gate cable is disconnected at the TC404A output.] Set the gain for amplifier No.2 to closely match the pulse height of No. 1 (use a different memory group). The Coarse gain control positions on the two amplifiers should be the same with some differences in the Fine Gain control positions. (Recall that the two preamps have been adjusted to produce nearly identical output amplitudes [in spite of different PMT gains] when both PMT's are operated at the same high voltage.)

It will be convenient to make the TC 812 Pulse-Height dial direct reading (full scale = 1000 keV), by pre-setting the 10-turn Pulse Height dial to 511, selecting the appropriate fixed attenuators, and then carefully adjusting the TC 812 front-panel CALib. trim pot (Use the small tan plastic tool) to position the output at the center of the 511-keV full-energy peak for Channel 1. Feed the Pulser Attenuated Output to the preamplifier Test inputs, using the scheme shown in Fig. 4 . Set the MCA Cursor to the center of the 511 keV full-energy peak for No. 1. Turn off the high voltage to eliminate detector pulses, and select an empty memory group. Adjust the Pulser output to match the Cursor position. Set the Channel 1 Single Channel Analyzer (SCA) to NORmal mode, the LLD to 0.020, the ULD at full scale, and Delay at minimum. Trigger the CRO from the TC 812 Direct Output, and connect the CRO CH 1 input to the + output of SCA No.l. Set the TC 812 Pulser for a 50 keV output, and adjust the LLD of SCA No. 1 for a flickering CRO trace. This indicates that the SCA is set to a level that allows only some of the Pulser outputs to generate a standard SCA output. The Channel 1 singles Scaler will count at a 60 Hz rate when the LLD is set below the pulse amplitude, at a lower rate when it only partially matches the pulse amplitude, and stops completely when the LLD is set high. The scaler provides a quantitative measure of the display seen on the CRO. Repeat the Pulser calibration procedure for Channel 2 (REMEMBER, the PMT preamplifier gains differ), but set this channel for a LLD at 100 keV and upper level to 350 keV. (The LLD and ULD settings are not sacred, but only reasonable suggested starting values.)

The next step is to adjust the SCA's for proper time overlap of pulses from the two channels. Connect the + outputs of both SCA's to two of the TC 404A Multipurpose Coincidence inputs. Set these for COINCidence, Coincidence Requirement = 2, with a 50 ns Resolving Time. Connect the two 10x CRO probes to the appropriate input test jacks and trigger the CRO sweep EXTernally from the TC 812 pulser Direct Output.. Set the CRO for 5 V/div on each channel, with ALT vertical mode. Reset the TC 812 pulser for an output that will be accepted simultaneously by both SCA's. You will see the outputs in their true time relationship, with one slightly leading the other. Change the CRO vertical mode to ADD. The two pulses are summed, with the regions that are in time coincidence at twice the height of those that are not. Adjust the Delay control on the SCA of the "fast" channel for precise overlap, keeping the Delay controls as close to minimum setting as possible. Check that both Singles scalers and the Coincidence scaler all count at the same rate, i.e., 60 Hz. The final adjustment requires that the detectors/PMT's be part of the system. Turn off the Pulser and trigger the CRO INTERNALLY from Channel 1. Turn on the High voltage. The CRO display will show a bright trace for the CH. 1 SCA output pulses (Amplitude = +1), and a dimmer flickering trace above it (amplitude = +2) for the parts of the pulses from CH. 2 that are in time coincidence with CH. 1. Adjust the appropriate SCA output Delay for precise overlap of the pulses.

### Delay Amplifier (Gate/Analog Pulse Overlap) Adjustment:

Connect the Amplifier No. 1 output to the Delay Amplifier input, and the CRO Ch.1 10x input probe to the Delay amplifier output. Preset the Delay Amplifier for minimum delay (all buttons OUT). Connect the CRO Ch. 2 10x input probe to the TC 404A + output. Trigger the CRO INTERNALLY from the leading edge of the Coincidence Gate (Ch.2) to display the time relationship between the linear amplifier analog pulses and the Coincidence Gate wave form. It will be helpful to temporarily set the TC 404A for Coincidence Requirement = 1. This will provide a much higher coincidence rate and brighter CRO display. Adjust this time relationship (Delay Amp switches) to that required by the MCA for proper operation in its Coincidence mode. Connect the Coincidence Gating signal to the MCA Coincidence input on its rear panel. First store a spectrum with the Gate OFF, then with it ON. There will be a noticeable drop in the rate at which channels are addressed, i.e., dead time will drop. Make small adjustments in the Delay Amplifier switches to maximize this lower count rate. Be sure to reset the TC 404A Coincidence Requirement = 2.

### Data Acquisition:

Remount the No.2 NaI assembly on the lower track, position each of the detectors the same distance from the scatterers. Accurately measure diameters and distances so that solid angles may be computed. Then proceed to measure the anisotropy. At least several hours are needed for each position, || and perpendicular . If the laboratory schedule permits, it is strongly advised that you take overnight run(s). Obtain data to accurately determine the accidental coincidence rates, as well as a quantitative determination of the effect of the several sources of "background". Take a random-coincidence spectrum by increasing the SCA delay setting for the channel No.2 mixer channel by > 0.5 µs to completely destroy any pulse overlap in the coincidence unit. You can double check this last by first restoring the Delay control settings, and then separating the detectors as much as possible, place a separate source in front of each, shielding each source from the "other" detector, and take an "accidental coincidence" spectrum from two truly uncorrelated sources (suggest 137Cs).

References: (5) p. 687-698, (6) p. 308-314, (7) p. 791-793.

## References

1. C. S. Wu and I. Shaknov, "The Angular Correlation of Scattered Annihilation Radiation," (137 KB) Phys. Rev., 77, 136, (1950). The first experiment of this type using scintillation counters.
2. H. S. Snyder, S. Pasternack, and J. Hornbostel, "Angular Correlation of Scattered Annihilation Radiation," (1.20 MB) Phys. Rev., 73 , 440 (1948). Useful theoretical calculations, including geometrical effects.
3. M. Deutsch and S. Berko, Alpha-, Beta-, And Gamma-Ray Spectroscopy, K. Siegbahn, Ed., (North-Holland Pub. Co., 1965), p. 1583. Review article regarding positronium, with extensive references to the literature (1963).
4. M. Deutsch, "Annihilation of Positrons," (2.83 MB) Progress in Nuclear Physics , 3 , 131, (1963). Another useful survey article regarding the annihilation of positrons.
5. G. F. Knoll, Radiation Detection And Measurement, (John Wiley & Sons, 1979).
6. N. Tsoulfanidis, Measurement And Detection Of Radiation, (McGraw-Hill, 1983, 1995).
7. R. D. Evans, The Atomic Nucleus, (McGraw-Hill, 1955).
8. R. P. Feynman, R. B. Leighton, and M. Sands, Lectures In Physics (CLAS), (Addison-Wesley Publishing Company, 1965), III, section 18-3.
9. A. C. Melissinos, Experiments In Modern Physics (CLAS), (Academic Press, 1966), Ch. 6, Section 3.1 .
10. W. R. Leo, Techniques for Nuclear and Particle Physics Experiments: A How-to Approach (CLAS), 2nd. rev. ed., (Springer-Verlag, 1994).
Reprints of the papers listed above are available at the experiment, with additional copies bound up for overnight signout.

## 22Na Source - Production And Characteristics:

A common method of production involves use of an accelerator to bombard Mg with high energy deuterons, producing Na and an a particle. Isotopic purity is enhanced by careful control of the accelerator, followed by meticulous chemical procedures.

Decay scheme for 22Na.

# Angular Correlation

In many quantum systems a great deal of information about the underlying dynamics can be extracted from knowledge of charge and current densities. One method commonly used in both particle and nuclear physics for determining electromagnetic transition matrix elements, spins of quantum levels, and the multipolarity of transitions between levels is " g-ray directional angular correlation ". By this we mean observing the intensity of a particular g-ray as a function of the angle between the spin axis of the nucleus and its direction of emission. In the general case, we have a "particle" (nucleus or hadron) in an initial state with spin (Ji) and parity ( p i ) denoted by Ji p i, with 2Ji + 1 magnetic substates undergoing a transition to a final state by Jf p f, with 2Jf + 1 substates. If the initial state is unaligned, then all the substates are equally populated, i.e., the probability of being in a given substate mi is p(mi) = l/(2Ji + 1), since we now have the constraint $sum$ p(mi) = 1. If the initial state is polarized the substate populations are not equal. We call the state "aligned" if the substate populations are not equal but p(mi) = p(-mi).

Many techniques are available to align the initial state, such as particle capture, particle emission, g - ray emission, with usually only one being suitable for the particular system under investigation. We will consider the case where the initial state is aligned through g - ray emission, a technique known as g - g directional angular correlation.

## Theory:

We will discuss transitions in the 60Co nucleus. The level scheme of 60Co is shown in Figure 1. The upper state (J p = 4+) is populated from b - decay of the parent nucleus. Since the electron is not observed, the 4+ state has no preferred spatial orientation. and hence is unaligned.

Figure 1. Decay scheme for 60Co.

The transition to the first excited state (J p = 2+) is via an E2 electric quadrupole (L p = 2 + ) transition. Observation of this g - ray [call it (1)] along some axis (z-axis) thereby gives the system a spatial orientation with the 2 + state aligned along this axis. The substate populations of the 2 + state are not equal, however p(2) = p(-2), p(l) = p(-l).

Let us now calculate the population parameters (density matrix) of the 2 + state, p(m2) from Clebsch-Gordan coefficients. It can be shown that the m-state population of J = 2 reached by the E2 g - ray (1) emitted from an unpolarized state J = 4, and measured along the axis of observation is given by:

This summation is over all the possible m quantum numbers of the (unpolarized) initial state (m 1 = ±4, ±3, ±2, ±1, 0) and over the two directions of circular polarization (t) of the g - ray. The distribution of the g - ray (2) from the second transition ( 2 + - 0 + ) will therefore be spatially correlated with the direction of the first g - ray. W(q) is the angular correlation between the first and second g - rays, q being the angle between the two g - ray detectors. We can then write:

where Pk(cosq) are the Legendre polynomials of order k. The Qk = Qk(l) Qk(2) is the product of the "efficiency functions" for the detectors that depend upon the source-detector geometry and detector crystal size. In the perfect case of point detectors with unit efficiency they become Qk = 1. The Ak = Ak(l) Ak(2) are the coefficients that contain the physics discussed previously and can be written in terms of Clebsch-Gordan coefficients as:

The first terms in the series are defined to be Q0 = A0 = 1. The angular correlation between the two g - rays in this case is uniquely determined with no free parameters. This in general is not true for an arbitrary cascade. When a transition can proceed by more than one angular momentum channel, the population parameters depend upon the relative amount of each multipole. In principle, one could calculate this from nuclear-physics models and determine the distribution, but in practice, the reverse happens. The angular correlation enables the relative contribution from each multipole to be extracted.

You will examine the nuclei of 22Na and 60Co. Study the level diagrams for each, identify a measurable cascade, and measure its angular correlation.

You will use exactly the same electronics as for the Polarization measurements, but with larger NaI(Tl) detectors (1 3/4" D x 1 1/2" H) mounted on the Angular Correlation Goniometer. Ask the T. A. for help in connecting them up safely and correctly.

Use a 22Na source to set amplifier gains, the SCA Lower Level and Upper Level Discriminators, and to determine the "0" of the goniometer. You will then need to modify the Linear Amplifier gain settings and reset the SCA Lower Level and Upper Level Discriminators to be appropriate for the 60Co source. Carefully consider, and discuss with the T.A., the best settings for the windows in both channels.

The accidental coincidence rate (determined by the source strength, solid angle, detector efficiency, singles rates, and the resolving time) must be kept small with respect to the true coincidence rate. If N = disintegrations/unit time of the source , wl and w2 the fraction of solid angle subtended by the detectors, el and e2 the efficiencies of the detectors, the "singles" rates will be

.

When two cascade g - rays are spatially uncorrelated, or only weakly correlated, the coincidence rate is

The "accidental" rate will be

where dt is the resolving time of the coincidence circuit. The ratio of "accidental" to true coincidences is then Ra / Rc = N dt. See Reference (1) p. 414-415, and Reference (2) p. 597. Note that increasing source strength to improve counting statistics may quickly become counterproductive. This would not be the case for the Polarization experiment where the g - rays are strongly correlated both in space and time.

Take data at 15° intervals from 180° to 90°, or at intervals you deem appropriate. Determine accidental coincidence rates, by methods similar to those used for the Annihilation exercise, and apply appropriate corrections to the data.

## Questions

1. Derive and draw the angular correlation of the two g-rays emitted during the transition in the following hypothetical nucleus:

Write the angular correlation in terms of Legendre polynomials.

2. The first nucleon resonance D(1232) has spin and parity Jp( d ) = 3/2+. The d+,0 can electromagnetically decay to a proton or a neutron, each with Jp (p,n) = l/2 +. What is the highest multipolarity of radiation that can be emitted? In the quark model the transition is purely a quark spin-flip, in this case what is the multipolarity of the emitted radiation?

3. Given a 1 µCi source of 60Co, at 10 cm. from each NaI(Tl) detector (1 3/4 " diam. x 1 1/2 " thick), what is the ratio of true coincidences to random coincidences if the time resolution of the system is 10 µs, 1 µs? What are the ratios at 20 cm? Explain.

4. If you measure the angular correlation between two g - rays that have W(q) = aP0 (cos q) + bP2 (cos q), what distance from the source should you place 1 3/4" x 1 1/2 " NaI(Tl) crystals, and at what angular intervals should you measure the g - ray intensity?

5. Why are only even Legendre polynomials included in the formula for W(q)? [Physical Explanation.]

6. If the coincidence gate is open for 50 ns, what is the effective resolving time of the system?

## References

1. A. C. Melissinos, Experiments In Modern Physics (CLAS), (Academic Press, 1966), 9, Section 3.
2. K. Siegbahn, Ed., Beta and Gamma Ray Spectroscopy, 2nd Edition, Vol. 2, (No. Holland Publishing Co., 1955) Copy of relevant material available in laboratory and Experiment Reference Binder.
3. H. J. Rose and D. M. Brink, Rev. Mod. Phys., 39, 306, (1967). Available in the laboratory and in the Experiment Reference Binder. This article is about 40 pages long, so it will be awhile until it gets scanned.
4. Ortec Application Note, Experiments in Nuclear Science , AN 34, 2nd Edition.
5. Canberra Manual for Nuclear Sciences, 1983.
6. R. P. Feynman, R. B. Leighton, and M. Sands, Lectures In Physics (CLAS), III , 18-5.
7. M. Deutsch, "Evidence for the Formation of Positronium in Gases," (390 KB) Phys. Rev., 82 , 455 (1951).
8. C. D. Anderson, "The Positive Electron," (983 KB) Phys. Rev., 43 , 491 (1933). This paper discusses the initial discovery of the positron made in Guggenheim Laboratory.
9. J. A. Wheeler, "Polyelectrons," (1.89 MB) Ann. New York Acad. Sci., 48 , 219 (1946).
10. A. Ore and J. L. Powell, "Three-Photon Annihilation of an Electron-Positron Pair," (699 KB) Phys. Rev., 75 , 1696 (1949).
11. A. Ore and J. L. Powell, "Three-Photon Annihilation of an Electron-Positron Pair," (261 KB) Phys. Rev., 75 , 1963 (1949).

Last updated 10 August, 2000. Broken links and bibliography updated Tuesday, July 25, 2000.