In a PET scanner a positron emitted in the patient slows down and annihilates with an…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime · Relativistic dynamics and fields in flat spacetime

Relativity and Gravitation, Ch Mass as energy: binding, defect and annihilation

Problem

In a PET scanner a positron emitted in the patient slows down and annihilates with an electron in the tissue. Because the electron-positron pair is still moving very slightly when it annihilates, the two photons that emerge are not exactly back to back, and this limits the sharpness of the image. Take the pair, at the instant of annihilation, to have in the scanner frame a total energy

E = 1.022\ \mathrm{MeV}

(its kinetic energy, a few eV, lies below the four figures kept here) and a total momentum of magnitude

p

with

pc = 2.0\ \mathrm{keV}

. The annihilation produces photons only, and the two photons carry equal energies. Take

m_ec^2 = 0.5110\ \mathrm{MeV}

. ((a)) Evaluate the invariant mass of the pair in

\mathrm{MeV}/c^2

and use it to show that the annihilation cannot produce a single photon. ((b)) Find the angle between the two photon directions, expressed as the number of degrees by which it falls short of

180^\circ

. ((c)) The annihilation happens at the centre of a detector ring of radius

0.42\ \mathrm{m}

. The scanner assumes the annihilation lies on the straight line joining the two detectors that fired. By what perpendicular distance, in mm, does that line miss the actual annihilation point?

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