Angular Correlation of Gamma Rays from Cobalt 60

Tanner Lange, Carrie Pfeifer

University of Minnesota – School of Physics and Astronomy

Minneapolis, MN 55455

Abstract

The angular correlation of successive γ-rays emitted from a single nucleus in the decay chain of 60Co was studied using two scintillation detectors. The experimental data was compared to the angular correlation functions derived by Hamilton to confirm the j=4 2 0 quadrupole-quadrupole transition of the excited 60Ni nucleus. The experimentally determined coefficients in the angular correlation function were determined to be a1 = 0.114 ± 0.041 and a2 = 0.061 ± 0.039. These values are 0.27 and 0.49 sigma away respectively from the predicted values of 0.125 and 0.0417 and are consistent with theory.

Introduction and Theory

When studying emission spectrums of nuclear decay processes, much attention is often spent on determining the energy of the photons released when the excited states decay. From this, one can determine the energy of the excited states involved. Another property of interest is the spin of the excited states. One method to measure this would be to apply a magnetic field to induce Zeeman splitting of the energy levels. However, a very large field would be necessary to observe the effect. A simpler method that applies to certain decay processes can be used due to the phenomenon of angular correlation of γ-rays.

When an excited nucleus emits two successive photons, the directions of the photons are correlated. For an excited state that decays to ground through an intermediate state, the spins of the excited states affect the distribution of the photons [1], resulting in an anisotropic distribution. Due the random orientation of the atoms, the distribution of the first photon is isotropic. However, detection of the first photon defines the z-direction. The probability of the intermediate state being in any one of the possible mz states is no longer the same as without the z-axis defined. This produces a similar effect to changing the populations of the states with an external field [3]. A quantum mechanical treatment of the problem accounting for conservation of angular momentum was done by Hamilton [2]. The results are summarized in the angular correlation function:

W(θ) = 1 + Σ ai cos2i(θ) (1)

θ is the angle between the emitted photons, summation is from 1 to 2l where 2l is the lowest multipole of the two radiations (dipole, quadrupole), and W(θ) gives the relative probability per unit solid angle of detecting the second photon at an angle θ to the first. By measuring the distribution of coincident γ-rays, one can compare to predicted distribution functions for possible spin state transitions. Brady and Deutsch tabulated the coefficients ai for various j transitions and combinations of multipole radiation predicted by Hamilton. These values are reproduced in figure 1.

Figure 1. The theoretically predicted coefficients for the angular correlation function (1). Table from [3]

The nucleus of interest here is 60Co. The energy diagram in figure 2 outlines the decay chain. 60Co decays through β- emission into an excited state of 60Ni. The excited state decays through an intermediate state by emitting a 1.1 and a 1.3 MeV photon. The lifetime of the intermediate state is 8 x 10-13 seconds [1]. This makes it favorable for angular correlation studies since the nucleus does not appreciably rotate between emission of the two photons. The largest contribution to such disturbance is Larmor precession, which has been observed to have a period on the order of 10-9 seconds [3]. Based on work done by previous groups [1][3], 60Co is expected to follow a j=4 2 0 quadrupole-quadrupole transition. Referencing figure 1, the expected angular correlation function is:

W(θ) = 1/8 cos2(θ) + 1/24 cos4(θ) (2)

The angular correlation function is defined to be one at 90 degrees, so the coincidence counting rate Γ(θ) can be predicted relative to the counting rate observed at 90 degrees in the following manner:

Γ(θ) = Γ90 * W(θ) (3)

By constructing a coincidence circuit to detect both photons from a single decay and observing the counts/second at various angles, the correlation function can be measured.

Figure 2. The energy diagram for 60Co. The excited state of the 60Ni nucleus emits a 1.1 and a correlated 1.3 MeV photon. Diagram from [1]

Experimental Setup

Figure 3 depicts the setup. Two NaI scintillation detectors with photomultiplier tubes (PMTs) were used to detect the γ-rays. A coincident circuit was constructed to measure only pairs of γ-rays originating from the same nucleus. The detectors were placed 13 cm from the source on mounts with rotating arms. The mounts locked at 13 cm, so any difference in distance between different angle measurements was minimized. Coincidence measurements were taken for angles between 90 and 180 degrees.

Figure 3. Two scintillation detectors set 13 cm. from the 60Co source. The angle between the detectors was varied between 90 and 180 degrees. Original figure.

The signal from the detectors was a negative pulse whose amplitude was proportional to the energy deposited in the scintillating medium. The signal from Detector B in figure 4 below was sent to a Canberra Model 2020 Spectroscopy Amplifier. The amplifier integrates the incoming signal and outputs a positive pulse. The shape of the pulse could be adjusted, and a shaping constant of 1 μ-sec was used. The amplitude was still proportional to energy, so the signal could be sent to an Ortec Easy MCA multi-channel analyzer to observe the spectrum.

The signal from Detector A was used to gate the MCA. A Tennelec TC 247 amplifier was used. An Ortec 850 single-channel analyzer was used to create a logic pulse whenever a 1.1 MeV photon was detected. This pulse was widened using a Lecroy Dual Gate Generator. This signal enabled the MCA. A model 128L Linear Fan-Out was used in the calibration process.

Figure 4. Schematic of the coincidence circuit. The spectrum from Detector B is observed on the MCA only when Detector A observes a 1.1 MeV photon. Original Figure.

Calibration

With the coincidence circuit constructed, the gain on the amplifiers was adjusted so that the spectrum was spread out over the range of the MCA. Other photon sources of known energies were used to calibrate the relationship between channel and energy. The signal from Detector A was sent through the fan out which sent the same signal through both arms of the circuit in figure 4. With the SCA window set to trigger on a large range of energies, the delay and widener could be adjusted by observing the signals on an oscilloscope. The widener was set to output a 5.3 μsec pulse. The length of the pulse was necessary to meet the specifications on the MCA; the end of the enable pulse needed to occur at least 0.5 μsec after the peak of the observed signal. The delay was calibrated so that the signal from the second amplifier was entirely enclosed in the enable signal. This was achieved using ~200 feet of BNC cable. With the timing calibrated, the window on the SCA could be set. The upper and lower thresholds were adjusted until the spectrum observed on the gated MCA showed only the 1.1 MeV peak. With the calibration complete, Detector B was re-connected as seen in figure 4. We make the assumption that any time difference between Detector A and Detector B is negligible. The same length of cable was used between the detectors and the coincidence circuit, and any difference should be on the order of nanoseconds, which is well within the tolerances of the 5.3 μsec gate.

Data Acquisition and Analysis

The distance from the detectors to the source was 13.0 ± 0.2 cm. Due to the locking mechanism, this distance did not change as the angle was adjusted. Coincidence measurements were taken at 9 angles between 90 and 180 degrees. The measurement at 140 degrees was taken twice due to a labeling mistake, resulting in 10 measurements. The uncertainty on each angle measurement was 1 degree. Each data run lasted at least 2.8 days to minimize statistical uncertainty and overcome the weak activity of the available source. After each data run, an un-gated spectrum was taken for approximately 7 to 10 minutes. The observed MCA spectrums for an un-gated and gated Detector B are shown below in figures 5 and 6 respectively. Here the channel is proportional to energy and the vertical axis is the number of counts normalized in time. By determining the number of counts in the 1.3 MeV peak in the gated spectrums (figure 6) and plotting vs. angle, equation 3 can be mapped.

Figure 5. An un-gated spectrum observed on Detector B. The 1.1 MeV and 1.3 MeV peaks are clear. Original figure.

Figure 6. The gated spectrum observed on Detector B when Detector A observes a 1.1 MeV photon. Original figure.

In order to define the channel ranges to integrate over, both peaks in each un-gated spectrum (figure 5) were fitted to Lorentzian peaks with a linear term in Origin. For the purpose of finding the peak, a linear background approximation is enough. The peak locations and FWHM values were averaged for all 10 plots, and the channel range was defined to include channels within 1.3 times the HWHM on either side of the peak. This was done to include as many counts as possible while excluding counts beyond the actual peak. These channel ranges were used for all spectrums in the rest of the analysis.

The number of counts in each 1.3 MeV peak was found by integrating over the channel range found above. These values were then normalized in time. The uncertainty was statistical: square root of the number of counts. Due to the long period of time over which data was taken, it was necessary to take account of the decay of the source. The mean life of 60Co is τ = 2777.5 days. The strength of the source was reduced by a factor of exp(-t/τ), where t was the number of days since the start of the first measurement. Each measurement was adjusted by multiplying the measured count rate by exp(t/τ).

By making the substitution x = cos2θ in equations 2 and 3, the data could be fit to a 2nd order polynomial. This was done using a least squares program developed by Kurt Wick [4]. The result is an equation in the form Γ = a + bx + cx2. The fit is shown below in figure 7. By comparing with equation 3, it is clear that to make a comparison with W(θ) (equation 2) the coefficients b and c must be divided by Γ90, the coincident count rate observed at 90 degrees. It is clear from the small 1.1 MeV peak in the gated spectrum in figure 6 that a certain amount of the un-gated spectrum, or “background,” was observed. The 1.3 MeV peak contains a certain number of non-coincident counts. This count rate of non-coincident 1.3 MeV photons is constant for all angles. The MCA is triggered by Detector A seeing a 1.1 MeV photon, and the rate of this observation is the same regardless of the position of Detector B. Thus the MCA is enabled the same fraction of the total time at all angles, the same amount of the background spectrum is allowed in, and the rate of non-coincident photons is constant.

Any constant offset is absorbed into the a value of the fit. Thus: a = Γ90 + α, where α is the constant count rate of non-coincident 1.3 MeV photons. To determine α, we analyze the un-gated spectrums. It is not clear where the x-intercept of the linear Compton background is, so a linear term applied to the whole 1.1 MeV peak may overestimate the number of Compton scattered counts sitting under the peak. From figure 2 we see that there is the same number of 1.1 and 1.3 MeV photons. In figure 5, the 1.1 MeV peak is larger due to Compton scattering of the 1.3 MeV photons. The amount of Compton photons sitting under the 1.1 peak is proportional to the number of counts in the 1.3 peak. We subtract the counts in the 1.3 peak from the 1.1 peak to get the number of Compton photons. We then take the ratio of these Compton counts to 1.3 counts and average for all un-gated spectrums.

For each gated spectrum, we multiply the number of counts in the 1.3 peak by the ratio of Compton to 1.3. This gives the number of Compton photons sitting under the 1.1 peak of the gated spectrum. We subtract this Compton from the total number in the 1.1 channel range, leaving the number of counts above Compton scattering. This value is normalized in time, and this counts/sec value is averaged for all gated spectrums. Appealing to the argument that there are the same number of 1.1 and 1.3 MeV photons, this averaged value is the counts/sec of non-coincidence 1.3 MeV photons, or α. Using the fit value a and the value for α just determined, Γ90 was calculated. The fit coefficients b and c were divided by Γ90, producing the coefficients of the cos2θ and cos4θ terms in equation 2.

Results

The least squares fit resulted in a chi squared of 6.05 with 7 degrees of freedom. The probability to exceed this chi squared is 0.53, indicating the functional form fit the data well. Using the analysis methods outlined above, the experimental value of the a1 coefficient was 0.114 ± 0.041. The experimental value of the a2 coefficient was 0.061 ± 0.039. These values were 0.27 and 0.49 sigma away from the predicted values of 1/8th and 1/24th. A plot of the best fit with these coefficients is in figure 7. Significantly, a value of 0 for the a2 coefficient would be 1.6 sigma below the experimental value. This result is significantly better than that achieved by a previous study which was unable to bound the coefficient away from 0 [1]. Alternatively, fitting to a cos2θ function produces a coefficient of 0.175 ±.012, which does not fit any of the possible correlation functions in Figure 1.

Figure 7. The resulting best fit of the experimental data. The coefficients of the cos2 and cos4 terms is equation 2 were 0.114 ± 0.041 and 0.061 ± 0.039 respectively. The chi squared was 6.05 with 7 degrees of freedom. Original figure.

Conclusion

Angular correlation of successive γ-rays was used to study the spin of the intermediate states in the decay chain of 60Co. The experimental data is shown to be consistent with the j = 4 2 0 quadrupole-quadrupole cascade predicted by Hamilton [2]. Bounds are placed on the coefficients such that the a1 coefficient must be 0.114 ± 0.041 and the a2 coefficient must be 0.061 ± 0.039. The greatest limitation in these bounds is the limited number of measurements taken due to the long time required. A stronger source and/or more time should reduce uncertainties.

References

[1] C. D. Muhlberger, Experiment IX: Angular Correlation of Gamma Rays, WWW Document,

(http://pages.physics.cornell.edu/~cmuhlberger/documents/phys405-paper.pdf).

[2] D. R. Hamilton, Phys. Rev. 58, 122 (1940).

[3] E. L. Brady and M. Deutsch, Phys. Rev. 78, 558 (1950).

[4] Kurt Wick, University of Minnesota, LSQ2D Least Square Fitting spreadsheet.