Angular Correlation of Electron Positron Annihilation Radiation

In the majority of annihilation events, two gamma quanta are created that are, in the reference frame of the electron-positron pair, emitted in exactly opposite directions.

In the laboratory frame, there is a small angular deviation from collinearity, which is caused by the momentum of the electron.

Hence, measuring the angular correlation of the annihilation radiation yields information about the momentum distribution of the electrons in the solid.

The border between occupied and unoccupied momentum states, the Fermi surface, is arguably the most significant feature of the electronic structure and has a strong influence on the solid's properties.

A measurement on a free electron gas for example would give a positive intensity for momenta

Measuring the momentum of one electronic state gives a distribution of momenta which are all separated by reciprocal lattice vectors.

Hence, an ACAR measurement on a solid with completely filled bands (i.e. on an insulator) gives a continuous distribution.

An ACAR measurement on a metal has discontinuities where bands cross the Fermi level in all Brillouin zones in reciprocal space.

Since positrons that are created by beta decay possess a longitudinal spin polarization it is possible to investigate the spin-resolved electronic structure of magnetic materials.

[3] ACAR has several advantages and disadvantages compared to other, more well known techniques for the investigation of the electronic structure like ARPES and quantum oscillation: ACAR requires neither low temperatures, high magnetic fields or UHV conditions.

However, ACAR is reliant on defect free samples as vacancy concentrations of up to 10−6 per atom can efficiently trap positrons and distort the measurement.

Therefore, the underlying physical observable is often called 'two photon momentum density' (TPMD) or

can be expressed as the squared absolute value of the Fourier transform of the multi-particle wave function

of all the electron and the positron in the solid: As it is not possible to imagine or compute the multi-particle wave function

[note 2] There exist sophisticated enhancement models to describe the electron-positron correlations,[4] but in the following it will be assumed that

A very illustrative form of the TPMD can be obtained by the use of the Fourier coefficients for the wave function product

If one assumes that the overlap of the electron and the positron wave function is constant for the same band

is folded back into the first Brillouin zone, the resulting density is flat except at the Fermi momentum.

When a positron is implanted into a solid it will quickly lose all its kinetic energy and annihilate with an electron.

By this process two gamma quanta with 511 keV each are created which are in the reference frame of the electron positron pair emitted in exactly anti-parallel directions.

In the laboratory frame, however, there is a Doppler shift from 511 keV and an angular deviation from collinearity.

In ACAR position sensitive detectors, gamma cameras or multi wire proportional chambers, are used.

In order to get a high angular resolution of 1×10−3 rad and better, the detectors have to be set up at distances between 16 and 20 m from each other.

Another way to evaluate ACAR spectra is by a quantitative comparison with ab initio calculations.

[7] In the early years, ACAR was mainly used to investigate the physics of the electron-positron annihilation process.

[8][9][10] Otto Klemperer could show with his angular correlation setup that the electron-positron pairs annihilate mainly into two gamma quanta which are emitted anti-parallel.

[9] In the 1950s, it was realized that by measuring the deviation from collinearity of the annihilation radiation information about the electronic structure of a solid can be obtained.

[note 4] A measurement with a long slit setup yields a 1D projection of the electron momentum density

The development of two-dimensional gamma cameras and multi wire proportional chambers in the 1970s and early 1980s led to the setting up of the first 2D-ACAR spectrometer.

An important early example of the use of spin-polarized 2D-ACAR is the proof of half metallicity in the half-Heusler alloy NiMnSb.