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Microanalysis in the STEM
Published in Robert J. Keyse, Anthony J. Garratt-Reed, Peter J. Goodhew, Gordon W. Lorimer, Introduction to Scanning Transmission Electron Microscopy, 2018
Robert J. Keyse, Anthony J. Garratt-Reed, Peter J. Goodhew, Gordon W. Lorimer
We use a model of the atomic electronic structure that will be familiar to all our readers; it is the ‘shell’ model, where the most tightly bound atomic electrons occupy the innermost energy levels (or ‘shells’). Quantum mechanics forbids an energy transfer which is less than the difference between the bound energy level and the next highest vacant energy level. One result of the interaction between an energetic (incident) electron and an atom can be the ionization of the atom or the creation of an inner-shell vacancy in the electronic structure. Hence, the energy spectrum of the energetic electrons leaving the specimen carries a signature of the electronic structure, and hence, amongst other things, the chemistry of the sample. Detection and study of this spectrum is termed ‘electron energy-loss spectroscopy’, and is usually referred to by the acronym ‘EELS’.
Elements, Isotopes, and Their Properties
Published in Robert E. Masterson, Nuclear Engineering Fundamentals, 2017
The shell model of the nucleus, which is an entirely different way to look at the same problem, is also described in the sections that follow. The shell model was developed in the 1940s, and it is based on the assumption that the nucleus has a number of distinct energy levels. The protons and neutrons within the nucleus attain these energy levels by orbiting in distinct orbital patterns or shells. Hence, the shell model is similar to the quantum mechanical model (see Sections 9.2 and 9.3) that describes the behavior of the electrons as they circle the nucleus in preferred orbital configurations that we previously called “energy levels,” “orbitals,” or “shells.” So, the liquid drop model and the shell model are two competing models of nuclear structure that are often used interchangeably in an attempt to explain the behavior of the nucleus as a whole. Both models have been revised and updated extensively over the years.
Nuclear Structure and Beta Decay
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
The shell model of the atomic nucleus (Mayer and Jensen 1955) is based on the assumption that the individual nucleons move in a potential well in discrete energy orbitals, just like the electrons in the atomic shell. The form of this potential well, which results from the sum of all nucleon–nucleon interactions, is (unlike the atomic shell) difficult to calculate because of our incomplete knowledge of the nuclear forces, and because of their complexity; it is usually specified phenomenologically. The simplest potential which to some degree approximates to reality is a harmonic oscillator potential which takes into account the spin–orbit interaction of the nucleons. This is very frequently used, since it provides simple analytic expressions for the majority of matrix elements. On the other hand, in addition, the wave functions of other more realistic potentials, such as for example the Woods–Saxon potential, or a potential calculated using the Hartree–Fock procedure, can be expanded using the wave functions of a harmonic oscillator potential as basis. All matrix elements may then be written as linear combinations of matrix elements derived using the oscillator potential. The orbitals of the nuclear potential are characterised by a principal quantum number n, orbital angular momentum ℓ (ℓ = 0,1,2,3,4,… denoted by the letters s,p,d,f,g …), total angular momentum j = ℓ ± 1/2, and a corresponding magnetic quantum number mj. Orbitals which differ only in mj form a j shell. Figure 3.10 shows the energetic sequence of the individual shells. The exact position ϵi of these shells (single-particle energies) is mostly adapted to experimental data from case to case.
Formation of ultralong-range fermionic Rydberg molecules in 87Sr: role of quantum statistics
Published in Molecular Physics, 2019
J. D. Whalen, R. Ding, S. K. Kanungo, T. C. Killian, S. Yoshida, J. Burgdörfer, F. B. Dunning
The Rydberg excitation spectrum for unpolarised 87Sr atoms measured in the vicinity of the 5s38s 3S atomic level is shown in Figure 5 and contains a number of well-defined individual lines. The largest feature corresponds to excitation of the 3S atomic state. The series of lines labelled , , and are associated with the formation of dimer Rydberg molecules having one bound ground-state atom that occupies, respectively, the ground (v=0), first excited (v=1), and second excited (v=2) vibrational levels. As seen in Figure 5, the positions of these levels are in good agreement with the binding energies predicted theoretically which are indicated by the positions of the axes of the corresponding molecular wavefunctions shown in Figure 1. The lines labelled , , and are associated with the formation of trimer molecules having two bound ground-state atoms. Since the two bound atoms do not interact strongly, the total molecular binding energies can be simply obtained by placing each atom in one of the available independent-particle bound states, the levels occupied being indicated by the subscripts. For example, corresponds to a molecule in which both ground-state atoms are in the v=0 vibrational state, and to the case where there is one atom in each of the v=0 and v=1 states. Careful inspection of the data in Figure 5 shows the binding energy of the state is twice that of the state, consistent with a simple ‘shell’ model as outlined above in which the bound ground-state atoms are distributed within the available vibrational levels and their individual binding energies summed. However, a more detailed theoretical analysis of tetramer Rydberg molecules containing three bound ground-state atoms shows that the binding energies could also be affected by the bending and stretching dynamics of the molecule [40].