Atomic momentum density ground state

The spherically symmetric atomic momentum densities, in contrast, exhibit monotonic as well as nonmonotonic behavior even in their ground states. Further, it was... 

We can conclude that the present method of correcting TF calculations provides adequate estimations of expectation values for ground state atoms taking into account the simplicity of the model and it self-consistent nature, where no empirical parameters are used. It provides information about the asymptotic behaviour of quantities such as p(0) and (r 2) that cannot be evaluated with the standard semi classical approach and allow us to estimate momentum expectation values which are not directly related to the density in an exact way. 

Figure 3.8. Molecular orbitals for the oxygen atom, with indication of their quantum numbers (main, orbital angular momentum and projection along axis of quantisation). Shown is the oxygen nucleus and the electron density (where it has fallen to 0.0004 it is identical for each pair of two spin projections), but with two different shades used for positive and negative parts of the wavefunction. The calculation uses density functional theory (B3LYP) and a Gaussian basis of 9 functions formed out of 19 primitive Gaussian functions (see text for further discussion). The first four orbitals (on the left) are filled in the ground state, while the remaining ones are imoccupied.

Another manifestation of the reciprocity of densities in r- and p-space is provided by Fig. 19.2. It shows the radial electron number density D r) = Aiir pir) and radial momentum density /(p) = Aitp nip) for the ground state of the beryllium atom calculated within the Hartree-Fock model in which the Be ground state has a ls 2s configuration. Both densities show a peak arising from the Is core electrons and another from the 2s valence electrons. However, the origin of the peaks is reversed. The sharp,... 

Numerical Hartree-Fock calculations, free from basis set artifacts, have been used to establish that the ground state momentum densities of all the atoms and their ions can be classihed into three types [84,85]. Type I and III momentum densities are found almost exclusively in metal atoms He, N, all atoms from groups 1-14 except Ge and Pd, and all the lanthanides and actinides. These momentum densities all have a global maximum at p = 0 and resemble the momentum density shown in Fig. 19.3 for the beryllium atom. The maximum atp = 0 comes mainly from the outermost s-subshell, 2s in this case. Type I and III densities dilfer in that the latter have a secondary maximum that is so small as to be invisible on a diagram such as Fig. 19.3. Type II densities are the norm for non-metallic atoms and are found in Ge, Pd and all atoms from groups 15-18 except He and... 

Commonly this equation and Eq. (35) are used to determine the normalized isotropic distribution. Consideration of Eq. (36) shows that various quantities of the collision processes and a few plasma parameters are involved in its coefficients and naturally have an immediate impact on its solution. With respect to the atomic data of the various collision processes, these are the momentum-transfer cross section Q (U), the total cross sections Qj U), the corresponding excitation or dissociation energies of the ground-state atoms or molecules, and the mass ratio m jM. With regard to the plasma parameters, the electric field strength E and the density N of the atoms or molecules occur, but only in the form of the reduced field strength E/N. All these quantities have to be known for a specific weakly ionized plasma in order to determine the isotropic distribution MU) by solving Eq. (36). 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

Next we proceed to develop the theory o resonance fluorescence experiments using the ensemble density matrix to describe the system of atoms. The important concepts of optical and radio-frequency coherence and of the interference of atomic states are discussed in detail. As an illustration of this theory general expressions describing the Hanle effect experiments are obtained. These are evaluated in detail for the frequently employed example of atoms whose angular momentum quantum numbers in the ground and excited levels are J =0 and Jg=l respectively. Finally resonance fluorescence experiments using pulsed or modulated excitation are described. 

---