Positronium energy level

Pachucki, K. and Karshenboim, G. (1998). Complete results for positronium energy levels at order ma6. Phys. Rev. Lett. 80 2101-2104. 

Here we intend to summarize several calculations on precision positronium physics which we have recently completed. We will describe the calculation of recoil corrections to positronium nS energy levels and the hyperfine splitting at 0(ma6), (D(ma7 In2 a) corrections to positronium energy levels and second order corrections to parapositronium lifetime. Additional details of those projects can be found in the original papers [1,2,3,4]. For the summary of the experimental situation, we refer to Ref. [11]. 

At the moment, it is not possible to compute ma7 corrections completely. Nevertheless, the leading logarithmic correction, 0(ma7In2 a), to positronium energy levels can be computed. This provides an estimate of higher order effects and hence of the uncertainty in the current theoretical prediction. In what follows we briefly describe our calculation of these corrections [3],... 

Table 1. Contributions to the positronium energy levels in units of cR.oo- In order to keep the size of the table as small as possible the following contributions which must be added to all energy levels have been omitted the lowest order contribution — and the contribution with the Bethe logarithm — lnfco(n, L)...

Positronium can exist in the two spin states, S = 0, 1. The singlet state (5 = 0), in which the electron and positron spins are antiparallel, is termed para-positronium (para-Ps), whereas the triplet state (5 = 1) is termed ortho-positronium (ortho-Ps). The spin state has a significant influence on the energy level structure of the positronium, and also on its lifetime against self-annihilation. 

Fig. 7.6. The energy levels of ground state positronium in a magnetic field.

Karplus, R. and Klein, A. (1952). Electrodynamic displacement of atomic energy levels. III. The hyperfine structure of positronium. Phys. Rev. 87 848-858. 

We will review here experimental tests of quantum electrodynamics (QED) and relativistic bound-state formalism in the positron-electron (e+,e ) system, positronium (Ps). Ps is an attractive atom for such tests because it is purely leptonic (i.e. without the complicating effects of nuclear structure as in normal atoms), and because the e and e+ are antiparticles, and thus the unique effects of annihilation (decay into photons) on the real and imaginary (related to decay) energy levels of Ps can be tested to high precision. In addition, positronium constitutes an equal-mass, two-body system in which recoil effects are very important. 

Abstract. We review our recent results on higher order corrections in positronium physics. We discuss a calculation of the recoil 0(ma6) corrections to the hyperfine splitting [1] and energy levels of a positronium atom [2], 0(ma7 In2 a) contributions to the positronium S-wave energy levels [3] and Ola2) radiative corrections to the parapositronium decay rate [4],... 

The Rydberg constant is the scale factor that connects all theoretical calculations and experimental measurements of energy levels in any system involving electrons. This includes all atoms, molecules and condensed matter. In simple systems, such as hydrogen, positronium, muonium, and possibly helium, the theoretical accuracy is comparable to that of experiments. In this case, experimenters can be said to measure the Rydberg constant, if not to test theory. 

Fig. 2 Lowest energy level structure of positronium showing the measured energy level splitting.

Abstract. We have collected all known theoretical contributions to the energy levels of positronium and present a complete listing for the states ra = 1, 2 and 3. We give the explicit dependence of the energy levels on the quantum numbers n, L, S and J up to the order Rood In the next higher order RccOi only the contributions to S- and P-states are completely known. The annihilation rates of para- and ortho-positronium are completely listed up to the orders Poo a and PooCt , respectively. We compare calculated values of energy levels and annihilation rates with experimentally observed quantities. 

If, however, the process takes place in excited states near the band edge, then the momentum of the particle and of the hole are almost zero, and they experience a mutual Coulomb attraction. The problem is essentially the same as that of the H atom or that of the energy levels of positronium the electron and hole possess discrete energy levels given by the equation ... 

The energy level diagram of the n=l and n=2 states of positronium is shown in Fig. 1. By now the hyperfine structure interval (really of order fine structure) in the ground n=l state (or the S, to So interval), the fine structure intervals in the n=2 state, and the 1S-2S interval have been measured with high precision. 

Figure 2. Zeeman energy levels of positronium in its ground state.

A further term, which has no analogue in hydrogen, arises in the fine structure of positronium. This comes from the possibility of virtual annihilation and re-creation of the electron-positron pair. A virtual process is one in which energy is not conserved. Real annihilation limits the lifetimes of the bound states and broadens the energy levels (section 12.6). Virtual annihilation and re-creation shift the levels. It is essentially a quantum-electrodynamic interaction. The energy operator for the double process of annihilation and re-creation is different from zero only if the particles coincide, and have their spins parallel. There exists, therefore, in the triplet states, a term proportional to y 2(0). It is important only in 3S1 states, and is of the same order of magnitude as the Fermi spin-spin interaction. Humbach [65] has given an interpretation of this annihi-... 

Ito, Y., Electron and positron energy levels and hole sizes in positronium formation in polymers. Mater. Sci. Forum, 175-178, 627-634 (1995). 

A positronium atom (Pos) resembles a hydrogen atom. It has a positron, instead of a proton, as the nucleus, with an electron moving around it (or vice versa, as the two particles have the same mass). According to the Bohr s model, the energy levels and the radii of the orbits for a hydrogen atom are ... 

There are yet other kinds of "cherished beliefs" which are occasionally interesting to call into question. An obvious case is at hand for leptonic atoms (especially positronium) where one s confidence in the calculability of its energy levels is deservedly very high but clearly worth questioning. 

Turning to component II, if the initial kinetic energy of the ortho-positronium is above E then it can undergo inelastic conversion by excitation of the Ag level of O2. However, Figure 7.15 shows that as the O2 pressure increases so the peak of component II increases at the expense of the tail (at energies > 1 eV). This tail is due to the... 

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