Experiment 7. POSITRON ANNIHILATION

Positrons are created by beta decay or pair production. The typical energy of positrons emitted during beta (+) decay is ~1 MeV. Positrons incident on a solid penetrate, losing energy by excitation and ionization of atoms, and slow down to thermal energies in ~1 picosecond. After thermalization they annihilate in the solid with electrons, their antiparticles. In fair approximation, the probability for annihilation per unit time (or the inverse of the mean lifetime) is proportional to the electron density at the location of the positron. Usually positron annihilation leads to two gamma-rays that carry away the energy and momentum of the e⁺e^- pair. Since kinetic energies and momenta of the electron and positron are small after thermalization of the positron, the gamma-rays are emitted in opposite directions with energies of 0.511 MeV in good approximation .

In this experiment you will first confirm the emission of the two annihilation radiations in time-coincidence. Then, you will measure the angular correlation of the radiations and verify that the gammas are emitted quite precisely in opposite directions, within the precision of measurement methods used. In an advanced experiment with much greater angular resolution, one can observe a slight degree of non-collinearity from which one can determine the Fermi energy of condcution electrons in copper or aluminum.

Equipment: Time coincidence spectrometer consisting of two movable scintillation detectors, a coincidence circuit, and a scaler; collimating lead bricks; ²²Na source.

Readings: Positron annihilation (refer to an introductory text); time-coincidence techniques (general), sect. 9.1, 9.2; angular correlations, sect. 9.3.1, 9.3.2, 9.3.3.

Key concepts: Annihilation process, angular correlation, resolving time, true and accidental coincidences.

10.1 Predicting the width of the angular distribution of annihilation radiation (do prior to measurement)

Positrons are emitted during decay of ²²Na nuclei in a continuous beta spectrum that has an end-point energy of 0.542 MeV. After injection in a solid and thermization, they usually avoid ion cores of atoms in the solid due to coulomb repulsion and annihilate with conduction electrons (metals) or weakly-bound valence electrons (insulators).

In this experiment you will measure the angular correlation in two-photon decay. To prepare for the measurements, you should estimate the expected width of the angular distribution. First, estimate the avarage energy for a conduction electron in a metal using a simple model such as that of a box containing free, noninteracting fermions. Second, estimate the average energy of the positron at the time of annihilation in the two limiting cases:

a. the e⁺ annihilates before losing any kinetic energy, so that it has an energy equal to a significant fraction of 0.542 MeV.

b. the e⁺ thermalizes (k_BT~ 0.025 eV), and then annihilates with one of the conduction or valence electrons.

Then, for two-photon decay, estimate the width of the angular distribution about a relative angle of emission of 180^o under each limiting case. Your measurements should demonstrate which limiting case, if either, is consistent with experiment.

10.2 Accidental coincidences and the resolving time

In Section 3 below, you will monitor time-coincidences of annihilation radiation by detecting them in two detectors. In addition to "true" coincidences between the two gamma-rays emitted in a single annihilation event, there will be "accidental" coincidences which arise when emissions fromdifferent annihilation events are detected. A coincidence (true or accidental) is registered when radiations are detected in the two detectors within an experimental resolving time T_res. The resolving time depends on the design of the electronic coincidence circuit and signal pulse widths. In chapter 9 (cf. eq. 2.1 and text) it is shown that the accidental coincidence counting rate R_acc is given in terms of the "singles" counting rates R_i in detectors 1 and 2 and the resolving time Tres by R_acc= R₁R₂T_res. Once one knows the value of T_res, one can calculate the accidental counting rate from the singles counting rates and then subtract R_acc from the total rate R_tot to obtain the true coincidence rate R_true: R_true= R_tot- R_acc.

To measure T_res place sources near each detector which emit no time-coincident radiations, such as ¹³⁷Cs, and shield detectors to avoid counter-to-counter scattering. In this way only accidental coincidences should be detected. Make sets of measurements of R_acc, R₁ and R₂ by varying the source-detector distances. Plot R_acc versus the product R₁R₂ and determine T_res from the slope.

(Note that the "singles" counting rates in each detector can be measured by connecting the output of the detector to both inputs of the coincidence circuit.)

10.3 Angular correlation measurement

Set up two NaI scintillation detectors on the optical bench on opposite sides of the positron source and collimate the angles to obtain an angular resolution of a few degrees. Rotate one detector about the source and determine the coincidence and singles counting rates as a function of detector-detector angle. Make lots of measurements near 180^o. Obtain sufficient statistics to determine the width and shape of the angular distribution. Determine the accidental coincidence counting rate at each angle and subtract it from the total coincidence rate to obtain the true coincidence counting rate. This curve represents the angular correlation of annihilation radiation and is your principal experimental result.

Interpret the observed shape and width in terms of the angular resolution function which you would expect for your geometry. Is the experimental distribution function consistent with the radiations being emitted at angles of precisely 180^o? Or is the experimental distribution broader than the angular resolution function, and by how much? Estimate an upper limit for the maximum momentum of annihilating e⁺e^- pairs.

Questions and Considerations

a. What effect should unequal width collimating slits have on the shape and width of the angular resolution function?

b. The source is not necessarily a point source. What effects might this have on the measurements?

c. Only low resolution measurements were made in this experiment. If the precedure in part 3 is repeated at much higher angular resolution using prediction from part 1, one can measure the distribution of momenta of annihilating e⁺e^- pair.