Spin magnetic moments

The magnitude of the spin magnetic moment of an electron is (in SI units) 

The two possible orientations of an electron s spin and its associated spin magnetic moment with respect to an axis prodnce two energy levels in an externally applied magnetic field. In electron-spin-resonance (ESR) spectroscopy, one observes transitions between these two levels. ESR spectroscopy is applicable to species such as free radicals and transition-metal ions that have one or more nnpaired electron spins and hence have a nonzero total electron spin and spin magnetic moment. 

Many atomic nuclei have a nonzero spin angnlar momentum I. Similar to (10.4) and (10.5), the magnitude of I is [/(/ -l- where the nuclear-spin quantum number 

We convert this classical expression for the energy into a Hamiltonian operator by replacing the classical quantity f by the operator /. Thus, H = -yBI. Let Mj) denote the function that is simultaneously an eigenfunction of the operators P (for the square of the magnitude of the nuclear-spin angular momentum) and 1. We have 

Therefore, (10.59) gives the energy levels of the isolated nuclear spin in the applied magnetic field as 

Hie use of different charges and 2 for the Is and 2s orbitals destroys their orthogonality, so (10.56) is not normalized. The best values of the variational parameters are found by setting dW/dbi = 0 and dW/db = 0, where the variational integral W is given by the left side of Eq. (8.9). The results are [E. B. Wilson, Jr.,/. Chem. Phys., 1,210 (1933)] b = 2.686, bi = inib, and W = -201.2 eV. W is much closer to the true value -203.5 eV than the result -192,0 eV found in the last section. The value of shows substantial, but not complete, screening of the 2s electron by the Is electrons. 

We might try other forms for the orbitals besides (10.54) and (10.55) to improve the trial function. However, no matter what orbital functions we try, if we restrict ourselves to a trial function of the form of (10.56), we can never reach the true ground-state energy. To do this, we can introduce ri2, 23 and rjs into the trial function or use a linear combination of several Slater determinants corresponding to various configurations (configuration interaction). 

Recall that the orbital angular momentum L of an electron has a magnetic moment -(e/2mj)L associated with it [Eq. (6.128)]. It is natural to suppose that there is also a magnetic moment associated with the electronic spin angular momentum S. We might guess that would be times S. Spin is a relativistic phenomenon, how- 

Theoretical and experimental work subsequent to Dirac s treatment has shown that gf is slightly greater than 2 [see P. Kusch, Physics Today, Feb. 1966, p. 23) g, = 2(1 ajl it - - ) = 2.0023, where the dots indicate terms involving higher powers of a and where the fine-stmcture constant a is 

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Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

The spin magnetic moment Mg of an electron is proportional to the spin angular momentum S,... 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

When a magnetic field is applied to an electron or nuclear spin, the spin quantization axis is defined by the field direction. Spin magnetic moments... 

Hamiltonian with the energy from appropriate terms in the true Hamiltonian. The latter terms include the interaction between the external field and the magnetic moment produced by the orbiting electron, the interaction between the external field and the magnetic moment due to electron spin, and the interaction between the orbital magnetic moment and the spin magnetic moment. These interactions may be expressed as a perturbation to the total Hamiltonian for the system where... 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

The calculations discussed above considered spin magnetic moments but not the orbital magnetic moments. However, it is known that orbital correlation has a strong effect in low-dimensional systems, which leads to orbital polarized ground states.67-69 Based on this fact, Guirado-Lopez et al.69 and... 

The spin magnetic moments of Ni clusters calculated by Wan et al.48 are in reasonable agreement with density functional calculations,61,63,72 but both approaches, that is, TB and DFT, give values substantially smaller than the experimental magnetic moments. The results of Wan et al. improve by adding... 

Figure 11 Character of the arrangement of the spin magnetic moments, and average magnetic moment, in seven-atom clusters with a pentagonal bipyramid structure and interatomic distances ranging from <4ulk to 80% <4ulk- Reproduced with permission from Ref. 79.

Figure 13 Ground state structures and local spin magnetic moments (in pB) of Mn5 and Mn7 determined by DFT calculations. For Mng, the structure and local moments correspond to a relevant isomer 0.07 eV above the ground state. Some bond lengths are also given, in A. Reproduced with permission from Ref. 103.

Magnetogyric ratio y/107 rad T-1 s-1 Nucleide Spin Magnetic moment /x//xN Electric quadrupole moment Resonance frequency in a field of 2.35 T Percent natural abundance Receptivity at natural abundance... 

Interaction of particle spin magnetic moment with the external magnetic field (Zeeman term). 

Interaction of an electron s spin magnetic moment with the magnetic field it experiences by virtue of its motion relative to the external electric field (spin-orbit term). 

Dipole-dipole interaction between the spin magnetic moments of two electrons. 

The product INg(Ef) for the series is shown in Table 4. ThN and PaN should be nonmagnetic and, as expected, self-consistent pure spin magnetic moments are obtained for UN-AmN. The effect of a moment upon the f band pressure is to change the first term in (33) to )... 

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