Electron sub-levels

To emphasize the periodicity of 8, suggested by the number spiral, the periodic table is rearranged as shown in Figure 4.5. Closure of the eleven periods coincides with the completion of electronic sub-levels, except for atomic numbers 62 and 94 that split the /-levels into sub-sets of 6 and 8. Additional structure, in complete agreement with the experimentally known sub-level order of the elements, is revealed by the zig-zag profiles that define the field of nuclide stability in Figure 4.4. 

Two significant aspects of the symmetry observed in the analysis of periodicity are the inverted electronic energy levels and the approach of Z/N —> 1 for all nuclides. The inversion is explained by the computational observation that electronic sub-levels respond differently to compression of an atom. 

The different rows of elements are called periods. The period number of an element signifies the highest energy level an electron in that element occupies (in the unexcited state). The number of elements in a period increases as one traverses down the periodic table because as the energy level of the atom increases, the number of energy sub-levels per energy level increases. 

Figure 9.18 shows a typical energy level diagram of a dye molecule including the lowest electronic states Sq, and S2 in the singlet manifold and and T2 in the triplet manifold. Associated with each of these states are vibrational and rotational sub-levels broadened to such an extent in the liquid that they form a continuum. As a result the absorption spectrum, such as that in Figure 9.17, is typical of a liquid phase spectrum showing almost no structure within the band system. 

Strategy Since two electrons fill an orbital, multiply the number of orbitals in the sub-level by 2 to find its capacity. To find the total capacity of the principal level, add those of the individual sublevels. 

Given the rules referred to in Section 6.3, it is possible to assign quantum numbers to each electron in an atom. Beyond that, electrons can be assigned to specific principal levels, sub-levels, and orbitals. There are several ways to do this. Perhaps the simplest way to describe the arrangement of electrons in an atom is to give its electron configuration, which shows the number of electrons, indicated by a superscript, in each sublevel For example, a species with the electron configuration... 

The sublevels of a particular orbital half fill before electrons pair up in the sub-level. 

Whenever electrons are added to orbitals of the same energy sub-level, each orbital receives one electron before any pairing occurs. 

The orbital or azimuthal quantum number (/) defines form (i.e., eccentricity of elliptical orbit cf Pauling, 1948) and indicates which sub-level is occupied by the electron. It assumes integer values between 0 and n —. ... 

In quantum states with n > 1, quantum number / assumes different values for instance, for n = 3, / = 0, 1, 2. When / is equal to or greater than 1, several independent wave functions exist (21 + 1). An electron of the second level, sub-level p, can occupy three 2p orbitals of the same energy, described by three distinct wave functions. These orbitals, which, in the absence of external perturbation, rigorously have the same energy, are called atomic degenerate orbitals (ADs). 

UV absorption bands have fine structure due to the presence of vibrational sub-levels, but this is rarely observed in solution due to collisional broadening. As the transitions are associated with changes of electron orbitals, they are often described in terms of the orbitals involved, e.g. 

Just as the energy splitting within a band of levels will effect the physical properties of the cluster, so will the density of the sub-levels, the filling of the individual sub-levels, and the filling of the band of levels. For possible collective motion of the electrons within the cluster, the band must not be completely filled. In the case of AU55 this point is not completely clear. 

More complex Jablonski diagrams can include vibrational sub-levels of the electronic states. Rotational sub-levels are not shown because they are so closely spaced as to form a near continuum. One important feature of the Jablonski diagram is that spin-allowed transitions only are shown by vertical lines any transition which has a horizontal component is spin forbidden. 

Figure 3.8 Examples ofJablonski diagrams of pure electronic states of (a) formaldehyde and (b) 4-phenylbenzophenone, with (c) showing vibrational (V) and rotational (R) sub-levels...

The nucleus of an atom contains the heavier subatomic particles - the protons and the neutrons. The electrons, the lightest of the sub atomic particles, move around the nucleus at great distances from the nucleus relative to their size. They move very fast in electron energy levels very much as the planets orbit the Sun. 

The way in which the exclusion principle determines the order of hydrogenlike energy-level occupation in many-electron atoms, is by dictating a unique set of quantum numbers, n, /, mi and spin ms, for each electron in the atom. Application of this rule shows that the sub-levels with l = 0, 1, 2 can accommodate no more than 2, 6, 10 electrons respectively. In particular, no more than two electrons with ms = , can share the same value of mi. Each principal level accommodates 2n2 electrons. 

On compression of non-hydrogen atoms the energy levels, which in this case are occupied by electrons, respond in the same way. Apart from level crossings, interelectronic interactions now also lead to an internal transfer of energy and splitting of the magnetic sub-levels, such that a single electron eventually reaches the ionization limit on critical compression. The calculated ionization radii obey the same periodic law as the elements and determine the effective size of atoms in chemical interaction. 

The relative energies of sub-levels and hence the order in which these are occupied by electrons cannot be predicted in detail, but this pattern is revealed when the periodicity of the nuclides is examined as a function of atomic number. This distribution of stable nuclides as a function of neutron imbalance is shown in Figure 4.4. To ensure that the grouping into sets... 

The novel feature is that ns sub-levels become vacant towards the end of each (n — l)d sub-level to make room for 10 d electrons and to be filled again next. In this way each transition series consists of only 8 elements and not 10 as implied by the Schrodinger spectrum. 

The alternative derivation of atomic periodicity, based on the distribution of prime numbers and elementary number theory, makes firm statements on all of these unresolved issues. The number spiral predicts periodicities of 8 and 24 for all elements and nuclides respectively limits their maximum numbers, in terms of triangular numbers, to 100 and 300 respectively characterizes electronic angular-momentum sub-levels by the difference between successive square numbers (21 +1) and electron pairs per energy level by the square numbers themselves. In this way the transition series fit in naturally with the periodicity of 8. The multiplicity of 2, which is associated with electron spin, is implicit in these periodicity numbers. 

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