Approximate Atomic Transition Amplitudes

In this section, a few examples are given of the approximate determination of matrix elements of electron field operators. Atomic units are used throughout. 

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We present exact calculations of the two-photon transition rates between discrete states of the non relativistic Hydrogen atom and we compare these results with those obtained with various approximation schemes. We plot in Fig. 1 the ls-3s resonant transition amplitude as a function of one of (he photon frequencies [3], where... 

Due to the muffin-tin approximation of the local multi-centre potential, the applicability of the scattered-wave (SW) method is limited to relatively small and high-symmetry clusters and molecules. On the other hand, it is the prerequisite for generalizing the in-out integration scheme of atomic one-electron wavefiinctions to the multi-centre case. This approach furnishes the multi-centre electronic structure in a particular atom sphere in its single-centre representation. Thus atomic transition-matrix elements can be readily rescaled in terms of near-nucleus electron amplitudes. 

The least ambiguous and most appropriate description of the atom after the collision is in terms of the density matrix (Blum, 1981), whose elements are bilinear combinations of scattering amplitudes for different magnetic substates. For the sake of simplicity we restrict ourselves to the most common case, in which the target is initially in an S state and the excitation involves the transfer of one electron from an s orbital to a p orbital in the independent-particle approximation. In atoms with one active electron the transition is — P. If there are two active electrons it is — P. We use the LS-coupling scheme. 

Larmor frequencies are typically of 500 MHz for a proton immersed in a static magnetic field of amplitude 5 = 12 T, a very strong, but nowadays current magnetic field. In the same magnetic field, the Larmor frequency of a fluorine nucleus F is a bit smaller, that of a carbon atom that represents more than 99% of all C-atoms in natural conditions has no spin and is consequently not directly seen in NMR) is four times smaller and that of a N about 0.07 that of the proton. This first approximation consequently allows us to define various ranges of radiofrequencies to be used to induce transitions between spin levels of a definite nucleus, H, F, N, N, etc. 

For the metal centre, the atomic orbitals (AO) describing the core electrons will not be considered for the construction of the complex s MO. This approximation can be justified by noting that the amplitude of these orbitals is significant only close to the nucleus, so they can therefore play only a negligible role in bond formation. One must, however, consider the valence AO that are occupied in the ground state of the isolated atom (nd and (n - - l)s), see Table 1.1), together with the (n + l )p orbitals, which, even though they are empty in the isolated atom, do contribute to bond formation in the complexes of transition metals. There are, therefore, nine atomic orbitals in all which participate on the metal, five d-type orbitals, one s-type, and three p-type orbitals. 

Intensity of Emission from the Vaience Band. In principle, the intensity of the electron emission from the valence state of the atom in the first-order process is determined by equations identical to those for the intensity of the emission from the core level [Eq. (23)]. The distinction lies in the matrix elements describing the atomic amplitude of this process. As mentioned above, the electron emission from the valence band may result from both the first- and the second-order processes. If the final state of the system formed as a result of these transitions is the same, these two processes must interfere. This interference is ignored in the present work. Such an approximation is justified by the fact that the final state of the system is determined by the secondary electron and the many-electron subsystem of the sample with a hole in the valence band. Neglect of the interference of the first- and second-order processes corresponds to the assumption that those processes give rise to different final states of the many-electron subsystem of the sample. Moreover, the contribution from the first-order processes of emission from the valence band is neglected in this work. The reason for that approximation is discussed in detail in Section 4. Thus, of all processes forming the spectrum of the secondary electron emission from the valence band of an atom, we shall consider only the second-order process. 

As discussed in Chapter 4, atoms are in a constant state of motion with the frequency and amplitude being related to the temperature. In a polymer, the motion is related to the amount of free volume and is small below the glass transition temperature and increases dramatically as the temperature is increased above the glass transition temperature. At the Tg the free volume in many polymers is approximately 1/40 or 2.5 % of the total volume. 

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