Spin-only formula

Now the total spin-angular momentum quantum number S is given by the number, n, of unpaired electrons times the spin angular momentum quantum number s for the electron, that is, S = nil. Substitution of this relationship into Eq. (5.11) yields an alternative form of the spin-only formula. 

The Chemical Relevance of Departures from the Spin-Only Formula... 

Equation (S6.1) is applicable to the salts of lanthanide ions. These have a partly filled 4f shell, and the 4f orbitals are well shielded from any interaction with the surrounding atoms by filled 5.9, 5p, and 6.9 orbitals, so that, with the notable exceptions, Eu3+ and Sm3+, they behave like isolated ions. For the transition metals, especially those of the 3d series, interaction with the surroundings is considerable. Because of this, the 3d transition-metal ions often have magnetic dipole moments corresponding only to the electron spin contribution. The orbital moment is said to be quenched. In such materials Eq. (S6.1) can then be replaced by a spin-only formula ... 

This is known as the spin-only formula. Many ions which in the free state have orbital angular momentum L 0), lose it, completely or partially, when incorporated into a complex. This phenomenon is called orbital angular momentum quenching. It can be shown that for A states the quenching is complete and that for T and E states it is incomplete. Because of this, the spin-only formula applies to more situations than might have been expected. 

For octahedral dl, d > d3, d , and d cases, the ground state is derived from the lowest term of the free ion for all values of the crystal field strength (defined by AJ. Hence the multiplicity, the number of unpaired spins and, if the spin-only formula applies, the effective magnetic moment must all be the same as that of the free ion, no matter how strong the interaction between the ion and the ligands. For octahedral d4, d , d , and d7 cases the ground state is derived from... 

In Table 12-9.1 calculated (spin-only formula) and experimental effective magnetic moments are listed for a number of ions, they are in accord with the previous discussion. 

A number of calculated and experimental magnetic moments for first-row transition metal complexes are given in Table 11.26, showing that the spin-only formula gives results that are in reasonably good agreement. 

The complexes [MntH-CO,]2. [Fe(HJOU>. [MnCIJ2 - and IFeCLJ" all have magnetic moments of nearly 5 92 BM. What does this tell you about the geometric and electronic structures of these complexes Why is the spin-only formula so precise in these cases ... 

The broadly successful application of these formulas to the paramagnetism of lanthanide complexes was due to the wide multiplet widths in the/block metals (large spin—orbit coupling coefficients X) and to the small effect of the ligand field on the deep-lying / orbitals. No comparably useful formula for the magnetic moments of d block complexes exists, except perhaps for the spin-only formula ... 

The determination of n from measurement of peff is the most familiar application of magnetic susceptibility measurements to inorganic chemists. To the extent that the spin-only formula is valid, it is possible to obtain the oxidation state of the central atom in a complex. Thus an iron complex with a peff of 5.9B.M. certainly contains Fe(III) (high-spin d5) and not Fe(II). The diamagnetism of AgO rules out its formulation as silver(II) oxide, because Ag2+ has an odd number of electrons (d9) and should be paramagnetic it contains Ag(I) and Ag(III), in equal amounts. There are, however, a number of pitfalls, especially if reliance is placed on a single measurement at room temperature. The Curie law is rarely obeyed within the limits of experimental error. This means that the measured peff is somewhat temperature-dependent. A number of factors can be responsible for deviations from ideal Curie (or even Curie-Weiss) behaviour, and/or from the spin-only formula. 

There may be a significant orbital contribution to the magnetic moment the spin-only formula, as the label implies, does not take this into account. The quantitative treatment of orbital contributions - which effectively alters the g-value from the free-electron value of 2.0023 - was one of the earliest and most important tasks of CF theory. The following generalisations can be made on the basis of the experimental results and the theoretical treatment for simplicity, we consider only octahedral and tetrahedral fields. 

Table 2 Theoretical Magnetic Moments (in BM) for Mn", Mnm and MnIV Compounds (Based on Spin Only Formula)...

Report your values of /jl in Bohr magneton and calculate the number of unpaired electrons using the spin-only formula, Eq. (15). 

This differs from the familiar /Ueff = -Jntn + 2) formula, where n is the number of unpaired electrons, [or. 45(5 + 1), spin-only formula], often applicable to the 3d" transition metal ions, as in the latter case the orbital contribution to the moment is quenched by the interaction of the metal s 3d orbitals with the ligands. 

This last equation is the spin-only formula where the Lande g factor has been introduced to take account of any spin-orbit coupling with excited states. The Lande g factor is given by... 

In practice, spin-orbit coupling is small for first-row d-block metal ions, so that equation 7.1 may be applied. Furthermore, the d orbitals of a transition metal ion usually show significant interactions with the ligand orbitals. In such cases, the orbital angular momentum contribution is usually quenched so that gS(S + 1) L(L +1). Where this happens, equation 7.1 can be simplified to the spin-only formula shown in equation 7.2 ... 

Selection rules 132 Laporte 132 spin 132 Self-assembly 90 Sepulchrate ligand 56 Sequestering agent 16 Shielding, electrons 20 Soft ligand 78 Soft metal 78 Solid angle factor 85 Solid angle sum 85 Spectrochemical series 112 Spin-only formula 146 Spin selection rule 132 Square planar complexes 108... 

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