Electrons, with negative energy

Fig. 1.10 Apparatus used to study the photoelectric effect Electrons entitled from one electrode are cdected by tie second one. and a current can be measured. Applying a negative potential to the grid prevents electrons with Insufficient energy from being recorded.

All Cl plasmas contain electrons with low energies, issued either directly from the filament but deactivated through collisions, or mostly from primary ionization reactions, which produce two low-energy electrons through the ionization reaction. The interaction of electrons with molecules leads to negative ion production by three different mechanisms [5] ... 

Note the different behaviours that the electron can adopt towards the molecules. Electrons at thermal equilibrium, that is those whose kinetic energy is less than about 1 eV (1 eV = 98 k,l mol ), can be captured by molecules and yield negative radical anions. Those whose energy lies between 1 and a few hundred electronvolts behave as a wave and transfer energy to molecules, without any collisions . Finally, molecules will be transparent to the electrons with higher energies here we enter the field of electron microscopy. 

In spite of its successes, Dirac s hole theory of the positron is provisional rather than final. If a serious attempt is made to connect the theory with electrodynamics, while postulating that only free electrons and the holes, but not the great body of electrons of negative energy, shall act as generators of a field, the resulting formalism is extremely complicated, and seems far from satisfactory. Here theoretical physics is confronted with a serious problem. 

Even if problems with negative-energy states were eliminated by the construction of the energy functional, Eq. (3), they are re-introduced at the one-electron level, Eq. (10), due to the Kohn-Sham construction based on the Dirac form of the kinetic energy, Eq. (5) [41]. 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

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