Positronic states

Introduction of negative energy (positron) states. The coupling between the electronic and positronic states introduce a small component in the eleetronic wave funetion. The result is that the shape of the orbitals change, relativistic orbitals, for example, do not have nodes. 

To effectively form a bound antiproton-positron state starting from free particles, excess energy and momentum has to be carried away by a third particle. Various schemes for producing antihydrogen have been proposed and discussed in some detail [21,22,23,24,25,26,27], with the first mention of the possible production of antihydrogen in traps by Dehmelt and co-workers [28],... 

Equation (3.125) is the required transformed Hamiltonian, and we see that in the representation in which (3 is diagonal, the Dirac equation decomposes into uncoupled equations for the upper and lower components of the wave function, i.e. for electron and positron wave functions. Setting (3 equal to +1 gives the positive energy (electron) states, whilst (3 equals -1 gives the negative energy (positron) states. 

Simultaneous measurement of positron lifetime and the momentum of the annihilating pair can give information on thermalisation and transitions between positron states (and hence on chemical reactions of positrons or Ps). The most recent version uses MeV positron beams [35]. A full description of AMOC can be found elsewhere in this volume. 

The energy distribution spectra collected by DB can be resolved into a sum of Gaussians, each representative of a positron state [3], with intensities, iP, and full widths at half maximum (fwhm), Tj. The subscripts are the same as for PALS. The DB intensities can be quantitatively correlated to those from PALS. Because p-Ps annihilates in an intrinsic mode, one has T T3 < r2. The DB results are conveniently presented in the form of the global fwhm of the energy distribution spectra. Similar treatments can be made, with a better resolution, in angular correlation (AC) experiments. 

In case of electron scavenging (and no Ps lifetime quenching, as is true for both Cl" and Tl+), no other positron states are present than free e+ and Ps then, the intensities from PALS and from DB are the same. The p-Ps and o-Ps intensities are expected to decrease so that the fwhm of the DB spectra should increase with solute concentration (the narrow components are suppressed). The variations of fwhm with C can be completely calculated, knowing the intensities Ij from PALS and the Tj previously established for a given solvent. This is illustrated by the solid line in Figure 4 for Tl+ this ion, as expected from its high solvated electron scavenging rate constant, is thus shown to suppress Ps formation by electron capture. 

By contrast, the variation of fwhm with C(C1") shows a decrease (Figure 4.4). This can only occur if a fourth positron state, not present in the case of... 

The calculation of PAES intensities largely reduces to the calculation core annihilation probabilities for positrons in the surface state [11]. This follows from the fact that almost all of the core hole excitations of the outer cores relax via Auger emission and that almost all of the positrons incident at low energies become trapped in a surface state before annihilation. First-principles calculations of the positron states and positron annihilation characteristics at metal and semiconductor surfaces are based on a treatment of a positron as a single charged particle trapped in a "correlation well" in the proximity of surface atoms. The calculations were performed within a modified superimposed-atom method using the corrugated-mirror model of Nieminen and Puska [12]. 

The two quantities which can be observed when an individual positron annihilates in condensed matter are the positron age r, which is the time interval between implantation and annihilation of the positron, and the momentum p of the annihilating positron-electron pair. Time-resolved information on the evolution of positron states is obtained by correlated measurements of the individual positron lifetime (= positron age) and the momentum of the annihilating positron-electron pair (Age-Momentum Correlation, AMOC). AMOC measurements are an extremely powerful tool for the study of reactions involving positrons. It not only provides the information obtainable from the two constituent measurements but allows us to follow directly, in the time domain, changes in the e+e momentum distribution of a positron state (cf. Sect. 1). 

AMOC allows time-dependent observations of the occupations and transitions of different positron states tagged by their characteristic Doppler broadening. Chemical reactions of positronium have been studied by beam-based AMOC as well as bound states between positrons (e+) and halide ions (cf. Sect. 2). 

In order to take advantage of the full information contained in the AMOC data we use a two-dimensional fitting procedure A two-dimensional model function representing the number of counts as a function of positron age and energy of the annihilation quanta is fitted to the raw AMOC relief without prior data reduction. On the age axis, each positron state is represented by an exponential decay function convoluted with the time resolution function of... 

Tj =1/A,i Lifetime (inverse of annihilation rate) of the positron state i... 

In 1974, O.E. Mogensen and V.P. Shantarovich concluded from ACAR measurements on aqueous solutions of sodium chloride [15] that in this system a fourth positron state (in addition to p-Ps, free positrons and o-Ps) was formed, which they identified as an e+Cl bound state. The existence of such a bound state found support in several Doppler broadening and ACAR investigations on aqueous and non-aqueous solutions of halides and pseudohalides [16-20]. Since the commonly used expression bound state of positrons may lead to confusion with positrons bound in Ps, we refer to this fourth state as positron molecules and characterize it by the subscript M. 

The lifetimes of the positron-molecule states rM (see Table 14.2) are considerably longer than that of the free positrons (rc+= 400 ps). rM is longest for PsF, which has the smallest number of electrons in its shell, and, hence, the lowest electron density, and decreases with increasing number of electrons. Seeger and Banhart [25] give an upper limit for the lifetime of positron states where no o-Ps is involved ... 

The linewidth of annihilation from the free-positron state is Doppler-broadening measurements. In lifetime measurements the PsF component hides beneath the o-Ps component which has a similar lifetime. This is a case where the two-dimensional data analysis shows its great advantage As the Doppler broadening of each positron state is determined in its own time regime even positron states with similar features may be seperated from each other. Moreover, a tentative fitting procedure with only the three positron states as in pure water did not come to a satisfactory result with the AMOC histogram of the NaF solution. 

Introduction of negative energy (positron) states. The coupling between the ... 

For free particles this point of view has indeed some attractive features. There are, however, situations where the sign of the energy does not distinguish between electronic and positronic behavior. Consequently, transitions from electronic to positronic states cannot be excluded. A famous example is the Klein paradox, where a potential step divides space into two regions with a different interpretation of particles and antiparticles. If the step size is larger than twice... 

The energy is not conserved if the potentials depend on time. Thus an initial state with positive energy may turn into a superposition of electronic and positronic states. 

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