Subshells

The spherical shell model can only account for tire major shell closings. For open shell clusters, ellipsoidal distortions occur [47], leading to subshell closings which account for the fine stmctures in figure C1.1.2(a ). The electron shell model is one of tire most successful models emerging from cluster physics. The electron shell effects are observed in many physical properties of tire simple metal clusters, including tlieir ionization potentials, electron affinities, polarizabilities and collective excitations [34]. 

A transition element is an element whose atom has an incomplete d subshell, or which gives rise to a cation or cations with an incomplete d subshell. 

Filling up the 4/ orbital is a feature of the lanthanides. The 4/ and 5d orbitals are of similar energy so that occasionally, as in La, Ce and Gd, one electron goes into 5d rather than 4f. Similarly, in the actinides, Ac to No, the 5/ subshell is filled in competition with 6d. 

For atoms having an atomic number greater than 10, the electron filling the inner shell vacancy may come from one of several possible subshells, each at a different energy, resulting in families of characteristic X-ray energies, e.g., the Ka, P family, the La, P, y family, etc. 

The X-ray spectrum observed in PIXE depends on the occurrence of several processes in the specimen. An ion is slowed by small inelastic scatterings with the electrons of the material, and it s energy is continuously reduced as a frmction of depth (see also the articles on RBS and ERS, where this part of the process is identical). The probability of ionizii an atomic shell of an element at a given depth of the material is proportional to the product of the cross section for subshell ionization by the ion at the reduced energy, the fluorescence yield, and the concentration of the element at the depth. The probability for X-ray emission from the ionized subshell is given by the fluorescence yield. The escape of X rays from the specimen and their detection by the spectrometer are controlled by the photoelectric absorption processes in the material and the energy-dependent efficiency of the spectrometer. 

At the end of this section, let us return briefly to the spectra shown in Fig. 3. Notice the structure in the mass spectrum of QoCa, between the completion of the first metal layer at 32 and the second at 104. This structure is identical in the fragmentation mass spectra of fullerenes covered with Ca and with Sr. It is reminiscent of the subshell structure of pure Ca clusters. The subshells could be correlated with the formation of stable islands during the growth of the individual shells[10,l 1]. The sublayer structure we observe here may also give some clue to the building process of these layers. However, the data is presently insufficient to allow stable islands to be identified with certainty. 

Consider now the solutions of the spherical potential well with a barrier at the center. Figure 14 shows how the energies of the subshells vary as a function of the ratio between the radius of the C o barrier Rc and the outer radius of the metal layer R ui- The subshells are labeled with n and /, where n is the principal quantum number used in nuclear physics denoting the number of extrema in the radial wave function, and / is the angular momentum quantum number. 

There is no single best form of the periodic table since the choice depends on the purpose for which the table is used. Some forms emphasize chemical relations and valence, whereas others stress the electronic configuration of the elements or the dependence of the periods on the shells and subshells of the atomic structure. The most convenient form for our purpose is the so-called long form with separate panels for the lanthanide and actinide elements (see inside front cover). There has been a lively debate during the past decade as to the best numbering system to be used for the individual... 

These are shown in Fig. 2.3 and illustrate most convincingly the various quantum shells and subshells described in the preceding section. The energy required to remove the I electron from an atom of hydrogen is 13.606 eV (i.e. 1312 kJ per mole of H atoms). This rises to 2372 kJ mol for He (Is-) since the positive charge on the helium nucleus is twice that of the... 

No entirely satisfactory explanation of these observations has been devised, though they are paralleled by the similar reluctance of other elements following the completion of the 3d subshell to achieve their highest oxidation states — see particularly Se (p. 755) and As (p. 552) immediately preceding Br in the periodic table. The detailed kinetics of several oxidation reactions involving aqueous solutions of Br04 ... 

Untersatz, m. support, stand, base stay saucer assumption, -schale, /. supporting dish subshell (of electrons), -schali, m. (Phyei s) subsonics. 

This problem clearly did not worry Stoner, who just went ahead and assumed that three quantum numbers could be specified in many-electron atoms. In any case, Stoner s scheme solved certain problems present in Bohr s configurations. For example, Bohr had assigned phosphorus the configuration 2,4,4,41, but this failed to explain the fact that phosphorus shows valencies of three and five. Stoner s configuration for phosphorus was 2,2,2,4,2,2,1, which easily explains the valencies, since it becomes plausible that either the two or the three outermost subshells of electrons form bonds. 

Figure 7. In the conventional medium-long form of the periodic table, the elements axe shown with the 4-block (pink) between the s-block (blue) and the p-block (lavender), to reflect the order of subshell tilting shown in figure 10 and contrary to the order expected from figure 6.

Again it is a useful approximation to consider just the set of equivalent orbitals of the outermost subshell. Then... 

Let us now derive the Slater-Kirkwood38 formula in terms of our present quantities. A single subshell of equivalent electrons is assumed. Equation 29 may be rearranged to... 

Our earlier discussion of off-diagonal s, however, indicated that 2Q would substantially exceed in many-electron systems. Consequently, if Eq. 34 is used with N an empirical factor,31 we may expect the N values to exceed substantially the actual number of electrons of the outer subshell. 

The assumption that only the outermost subshell of electrons contributes to either a or EL. 

Slater-Kirkwood theory N = no. of electrons of outer subshell empirical evaluated from EL and correlate with actual number of electrons of both outer subshell and of whole system. 

The third quantum number required to specify an orbital is mh the magnetic quantum number, which distinguishes the individual orbitals within a subshell. This quantum number can take the values... 

There are 2/ + 1 different values of trij for a given value of / and therefore 2/ + 1 orbitals in a subshell of quantum number /. For example, when / = 1, mj= +1,0, — 1 so there are three p-orbitals in a given shell. Alternatively, we can say that a subshell with / = 1 consists of three orbitals. 

The hierarchy of shells, subshells, and orbitals is summarized in Fig. 1.30 and Table 1.3. Each possible combination of the three quantum numbers specifies an individual orbital. For example, an electron in the ground state of a hydrogen atom has the specification n = 1, / = 0, nij = 0. Because 1=0, the ground-state wavefunction is an example of an s-orbital and is denoted Is. Each... 

FIGURE 1.30 A summary of the arrangement of shells, subshells, and orbitals in an atom and the corresponding quantum numbers. Note that the quantum number m, is an alternative label for the individual orbitals in chemistry, it is more common to use x, y, and z instead, as shown in Figs. 1.36 through 1.38. 

There are three /7-orbitals in each subshell, corresponding to the quantum numbers / = +1, 0, —1. However, chemists commonly refer to the orbitals according to the axes along which the lobes lie hence, we refer to px, / v, and />. orbitals (Fig. 1.36). 

A subshell with 1 = 2 consists of five d-orbitals. Each d-orbital has four lobes, except for the orbital designated dz , which has a more complicated shape (Fig. 1.37). A subshell with 1=3 consists of seven -orbitals with even more compli cated shapes (Fig. 1.38). 

The location of an electron in an atom is described by a wavefunction known as an atomic orbital atomic orbitals are designated by the quantum numbers , l, and mi and fall into shells and subshells as summarized in Fig. 1.30. 

FIGURE 1.41 The relative energies of the shells, subshells, and orbitals in a many-electron atom. Each of the boxes can hold at most two electrons. Note the change in the order of energies of the 3d- and 4s-orbitals after Z = 20. 

Boron (Z = 5) has five electrons. Two enter the ls-orbital and complete the n = 1 shell. Two enter the 2s-orbital. The fifth electron occupies an orbital of the next available subshell, which Fig. 1.41 shows is a 2p-orbital. This arrangement of electrons is reported as the configuration 1 s22s22p1 or He 2s22/ 1 (5), showing that boron has three valence electrons. 

We need to make another decision at carbon (Z = 6) does the sixth electron join the one already in the 2p-orbital or does it enter a different 2p-orbital (Remember, there are three p-orbitals in the subshell, all of the same energy.) To answer this question, we note that electrons are farther from each other and repel each other less when they occupy different p-orbitals than when they occupy the same orbital. So... 

If more than one orbital in a subshell is available, add electrons with parallel spins to different orbitals of that subshell rather than pairing two electrons in one of the orbitals. 

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