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POLARIZATION OF RADIATION PRODUCED DURING POSITRON–ELECTRON PAIR ANNIHILATION
According to the theory of annihilation of a positron–electron pair, the planes of polarization of the two quanta produced in annihilation and flying apart in opposite directions should be mutually perpendicular. The communications reviewed here¹˒² are devoted to an experimental verification of this prediction, which follows necessarily from the law of conservation of angular momentum. In both investigations the measurement method was almost identical, differing only in details.
The source of radiation was the annihilation of slow positrons (0.66 MeV) emitted by Cu⁶⁴ (half-life 12.8 hours), prepared by irradiating copper with deuterons in a cyclotron. The photons produced had an energy of 0.51 MeV. Hanna² carried out measurements with 13 different specimens of the source, having an initial activity of about \(3 \cdot 10^8\) positrons per second and giving the observed annihilation radiation for 36 hours after irradiation. Bleuler and Bradt¹ used four sources of different intensity.
Fig. 1. Schematic of the apparatus for measuring the polarization of annihilation radiation¹.
Fig. 2.
The source \(P\), enclosed in an aluminum tube, was placed at the center of a collimator pierced through with a lead block that played the role of the collimator. (In the apparatus of Bleuler and Bradt the channel had a diameter of 9.5 mm, Fig. 1.) The photons flying out in opposite directions, after passing through the channel, entered the scatterers \(S_1\) and \(S_2\), made of aluminum¹˒² or copper² and serving as analyzers. (In Hanna’s work the scatterers were 25 mm long and 15 mm in diameter.)
As is known, the probability of Compton scattering in a given direction depends substantially not only on the scattering angle \(\theta\), but also on the angle \(\varphi\) formed by the plane of scattering with the plane of polarization of the radiation being scattered. If from the source \(P\) (Fig. 2) two photons simultaneously fly out in opposite directions, are then scattered in \(S_1\) and \(S_2\), and if their planes of polarization are mutually perpendicular, then the probability of simultaneous detection of the scattered photons within the solid angles \(d\Omega_1\) and \(d\Omega_2\) in the directions \(\theta_1, \varphi_1\) and \(\theta_2, \varphi_2\), where \(\varphi_1\) and \(\varphi_2\) are measured from an arbitrary plane that includes the direction of flight of the photons from the source, is given by the expression
\[ W(\theta_1,\varphi_1,\theta_2,\varphi_2)\,d\Omega_1\,d\Omega_2 = k\left\{ \frac{\left[(1-\cos\theta_1)^3+2\right]\left[(1-\cos\theta_2)^3+2\right]} {(2-\cos\theta_1)^3(2-\cos\theta_2)^3} - \frac{\sin^2\theta_1\sin^2\theta_2\cos 2(\varphi_1-\varphi_2)} {(2-\cos\theta_1)^2(2-\cos\theta_2)^2} \right\} d\Omega_1\,d\Omega_2 . \]
$k$ is a constant depending on the intensity of the source, the scattering cross sections of the scatterers $S_1$ and $S_2$, and the parameters of the apparatus.
In the case when the planes of polarization of the scattered photons are not perpendicular to one another, this relation will not hold. Thus, verification of this relation by counting the number of pairs of photons undergoing simultaneous scattering in an apparatus of the described type within the given solid angles $\Omega_1$ and $\Omega_2$ for given $\theta_1$ and $\theta_2$, but different $\varphi_1$ and $\varphi_2$, is at the same time a check of the mutual perpendicularity of the planes of polarization of the photons arising in the process of annihilation of a pair.
Geiger counters were used as photon counters in both works. The solid angles $\Omega_1$ and $\Omega_2$ were cut out by lead diaphragms; Hanna used double diaphragms, cutting out a wide and a narrow solid angle. Bleuler and Bradt’s apparatus contained two counters—one on each side of the channel—which could rotate around the axis of the channel (Fig. 1). Measurements were made at $\varphi_2-\varphi_1=0^\circ$, $90^\circ$, $180^\circ$, and $270^\circ$, with the positions of both counters being interchanged. In Hanna’s apparatus, paired counters were placed on both sides of the channel, set at the angles $\varphi_1$, $\varphi_1+\pi$, $\varphi_2$, and $\varphi_2+\pi$, one of the pairs being rotatable about the axis of the channel relative to the other pair, so that $\varphi_2-\varphi_1=0^\circ$ or $90^\circ$. Hanna had $\theta_1=\theta_2=-90^\circ$; Bleuler and Bradt had $\theta_1=\theta_2=82^\circ$, since, as a consequence of absorption in the scatterers, the mean scattering angle proved to be somewhat less than $90^\circ$, in agreement with theory.
Corrections were introduced for the change in the source intensity due to decay and for accidental coincidences.
In Hanna’s experiments, about 70% of the total number of counts of each pair of counters was due to photons scattered in $S_1$ or $S_2$. The number of coincidences connected with the scattering of annihilation radiation was approximately—
Table I
| Number of counts per minute | |
|---|---|
| Mean number of single counts in the absence of a scatterer | 3000 |
| Mean number of single counts in the presence of a scatterer | 5370 |
| Probable number of coincidences ($T=1.2\cdot10^{-7}$ sec.) | 0.117 |
| Actual number of coincidences for $\varphi_2-\varphi_1=90^\circ$ or $270^\circ$ ($W_{\perp}$) | 0.152 |
| Actual number of coincidences for $\varphi_2-\varphi_1=0^\circ$ or $180^\circ$ ($W_{\parallel}$) | 0.073 |
| Ratio | $\displaystyle \frac{W_{\perp}}{W_{\parallel}}=2.1\pm0.64.$ |
0.5 per minute. About 3 coincidences per minute were caused by cosmic rays.
The measurements of Bleuler and Bradt are characterized by the following table, referring to one of their measurement series of 16 hours’ duration (Table I).
The mean values
\[ \frac{W_{\perp}}{W_{\parallel}}, \]
obtained for various measurement series, are collected in Table II.
Table II
| Authors | Bleuler and Bradt | Hanna | Hanna | Hanna |
|---|---|---|---|---|
| Scattering material | aluminum | aluminum | brass | brass |
| Lead diaphragms | ? | narrow | narrow | wide |
| $\dfrac{W_{\perp}}{W_{\parallel}}$ Theory | 1.7 | 1.86 | 1.82 | 1.55 |
| $\dfrac{W_{\perp}}{W_{\parallel}}$ Experiment | $1.94 \pm 0.37$ | $1.51 \pm 0.10$ | $1.31 \pm 0.17$ | $1.39 \pm 0.07$ |
Thus, the fact of a relative rotation of the polarization planes of two photons arising in pair annihilation, close to $90^\circ$, has been established beyond doubt. In Hanna’s opinion, the systematic discrepancy from theory observed in his experiments cannot be explained by fluctuations. On the other hand, the influence of secondary scattering is not excluded, since for scattering in aluminum the discrepancy with theory proves to be smaller than for scattering in brass. In this connection Hanna considers further experiments necessary in order to establish how real this discrepancy is and to what extent it can be attributed to an instrumental effect. As regards the results of Bleuler and Bradt, which agree well with the predictions of the theory, Hanna finds that the errors indicated within the possible limits make them compatible with his results.
G. Rozenberg
CITED LITERATURE
- E. Bleuler and H. L. Bradt, Phys. Rev. 73, 1398 (1948).
- R. C. Hanna, Nature 162, 332 (1948).