L, quantum number

There are several ways of indicating the arrangement of the electrons in an atom. The most common way is the electron configuration. The electron configuration requires the use of the n and / quantum numbers along with the number of electrons. The principle quantum number, n, is represented by an integer (1,2,3. ..), and a letter represents the l quantum number (0 = s, 1 = p, 2 = d, and 3 = f). Any s-subshell can hold a maximum of two electrons, any p-subshell can hold up to six electrons, any d-subshell can hold a maximum of 10 electrons, and any f-subshell can hold up to 14 electrons. 

Fig. 3. Correlation diagram of the doublet states relating weak (left) and strong (right) tetragonal perturbations. In the strong field limit the splitting pattern is determined by pseudo-/l quantum numbers. Note the crossing point on the lowest energy curve...

The standard approach to solving the Schrodinger equation for hydrogenlike atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, 6, (p), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, fir) = 0, fid) = 0, and fiip) = 0. Solution of the fir) equation gives rise to the n quantum number, solution of the/(0) equation to the l quantum number, and solution of the fifi) equation to the mm (often simply called m) quantum number. For each specific n = n, 1 = 1 and mm = inm there is a mathematical function obtained by combining the appropriate fir), fi fJ) and /([Pg.101]

Fig. 1. (Left) The structure of the pHe+ atomcule, where the p with large-(n, l) quantum numbers circulates in a localized orbit around the He2+ nucleus, while the electron occupies the distributed Is state. (Right) The level scheme of large-(n, l) states of the pHe+ atomcule. The solid bars indicate radiation-dominated metastable states, while the broken lines are for Auger-dominated short-lived states. The ionized pHe++ states are also shown by dotted lines. From Ref. [2]...

This effect originates in the various f electron configurations of lanthanides and actinides. This effect is based on the correlation observed between the full pattern of the effect and the sequence of values of the L quantum number [52]. The correlation consists of the occurrence of the same double symmetry in (a) the series of L quantum number values of the ground terms of f element ions and (b) the sequence of relatively stabilized or destabilized f electron configurations (i.e.) the double-double effect. Accordingly... 

The drawback of this definition was, that it had been restricted to the atomic ground states and that it ignored the atomic shell structure, i.e. that large jumps in E(N) are expected, when atomic shells identified by their (n, l) quantum numbers are transgressed. 

To this point in the discussion of multielectron atoms, the spin and orbital angular momenta have been treated separately. In addition, the spin and orbital angular momenta couple with each other, a phenomenon known as spin-orbit coupling. In multielectron atoms, the S and L quantum numbers combine into the total angular momentum quantum number J. The quantum number J may have the following values ... 

Relatively more stable configurations have even values of L-quantum number (S = 0 and 1 = 6 terms) while relatively unstable configurations correspond to odd values of the L-quantum number (F = 3 and H = 5 terms). It should be noticed that relatively stabilized configurations display a relatively smaller ability to complex formation (less negative values of AG°) and vice versa. In Fig. 1 it is shown that the sequence of L-values has the same double symmetry as the double-double effect. However, no linear correlation has been experimentally observed between changes in the free energy of complex formation or lattice parameters of f-element compounds and the values of the L-quantum number of appropriate f-ions. The lack of such linear correlation is explained in Sect. 5. 

Fig. 9. Plot of the logarithm of the relative extraction coefficient vs L-quantum number for the extraction of trivalent lanthanides by HEH0P. Experimental data of static extraction at 25 "C69 ...

Fig. 11. Plot of lattice parameter ao vs L-quantum number for isostructural monoclinic actinide tetrafluorides. Data reported by Keenan and Asprey28 ...

Each of these arrangements corresponds to an electronic arrangement sometimes called a spec-troscopic state, which is describe by a full term symbol. The letter D indicates that the L quantum number has a value of two, P indicates that... 

The Inclined W Hypothesis, or the Linear Correlation of the Properties of the /-Transition Ions with their L-Quantum Number... 

Thus it seems that more than a passing reference is appropriate as to the relation of L-quantum number and the tetrad effect, and indeed Sin ha (58) has recently shown that the L-values exhibit the same periodicity as the tetrad effect in the lanthanide (actinide) series (Fig. 6), as well as that the properties of the/-transition metal ions vary linearly within each tetrad (58). It is a great pity that Klemm did not divide the lanthanide series based on the repeatation of the total angular momentum (L-values) values (see the above table), rather he showed and supported the classification of Endres (48). 

A being again the hyperangular or grandangular momentiun quantum number and fi is here a set of d-l quantum numbers. 

The XPS spectrum of 4f excitation often exhibits a series of peaks corresponding to the multiplet structure of the final 4f shell. The spin-orbit splitting in Yb has been discussed in section 3.3. The splitting of final states with different L quantum numbers in the Russell-Saunders scheme is evident in Sm and heavier members. The spin multiplet structure is seen in rare earths with more than half-filled shells (Tb-Tm). For example, the final state of Tb(4f ) may be in the state 4f (5 = 2) or 4ff4fi (5 = ). The splitting of these groups of states measures the exchange interaction between the 4f electrons. 

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