Positronium

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Positronium (Ps) is a quasi-stable system consisting of an electron and its anti-particle, a positron, bound together into an "exotic atom". The orbit of the two particles and the set of energy levels is similar to that of the hydrogen atom (electron and proton). However, because of the different reduced mass, the frequencies associated with the spectral lines are less than half of those of the corresponding hydrogen lines.

1 Details
2 Energy levels
3 References
4 See also
5 External links

[edit] Details

Positronium is unstable, with a half-life of at most about 10⁻⁷ seconds (100 nanoseconds). The particles "spiral" closer to each other (although this actually takes place in quantized steps of decreasing radius), until their existence is terminated by electron-positron annihilation. At annihilation, gamma rays are produced. Any number of gamma rays (greater than one) can be produced with a total energy of 1022 keV (since both particles have mass of 511 keV), though natural reactions usually produce two or three rays, depending on the spin of the particles. A single photon decay is forbidden by relativistic momentum conservation. Up to five annihilation gamma rays have been observed in laboratory experiments, confirming the predictions of quantum electrodynamics to very high order.

Positronium, like hydrogen, can have different configurations; the ground state can be a singlet state with antiparallel spins (S = 0, M_s = 0) known as parapositronium (term symbol ¹S₀). The singlet state lasts about 10⁻¹⁰ seconds (mean lifetime = 125 ps). The other case, a triplet state, with spins parallel (S = 1, M_s = −1, 0, 1) is known as orthopositronium (term symbol ³S₁). The triplet state lives about 10⁻⁷ seconds (mean lifetime = 140 ns).

[edit] Energy levels

See Bohr model for a derivation of the equation for energy levels.

The similarity between positronium and hydrogen extends even to the equation that gives a rough estimate of the energy levels. The energy levels are different between the two because of a different value for the mass, m*, used in the energy equation:

$E_n = \frac{-m^* q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,$

See Electron energy levels in hydrogen for a derivation.

q e

is the charge magnitude of the electron (same as the positron)

h

is Planck's constant

ε 0

is the electric constant (otherwise known as the permittivity of free space) and finally

m *

is the reduced mass

The reduced mass in this case is:

$m^* = {{m_e m_p} \over {m_e + m_p}} = \frac{m_e^2}{2m_e} = \frac{m_e}{2}$

where

m e

and

m p

are, respectively, the mass of the electron and the positron - which are the same.

Thus, for positronium, its reduced mass only differs from the rest mass of the electron by a factor of 2. This causes the energy leves to also roughly be half of what they are for the hydrogen atom.

So finally, the energy levels of positronium are given by

$E_n = \frac{1}{2} \frac{-m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} = \frac{-6.8 \ \mathrm{eV}}{n^2} \,$

The lowest energy level of positronium (n = 1) is −6.8 electron volts (eV). The next highest energy level (n = 2) is −1.7 eV. Note that the negative sign implies a bound state.

[edit] References

[edit] See also

Quantum electrodynamics
electron	positron	photon
self-energy	vacuum polarization	vertex function
Gupta-Bleuler formalism	ξ gauge	Ward identities
Compton scattering	Bhabha scattering	Moeller scattering
anomalous magnetic dipole moment
positronium
bremsstrahlung