Spin orientation

Experimentally, the spin orientations of a magnetic solid are determined by neutron diffraction measurements. From the viewpoint of theory, the spin orientation in 

An important consequence of SOC is that the spin gets a preferred orientation in space with respect to the crystal lattice. Before we examine how this comes about, it is necessary to recall that the orbital angular momentum states L, L ) follow the relationships 

To gain insight into how the S L term governs the spin orientation in space, it is necessary to employ two independent coordinate systems, that is, (x, y, z) for L and (x, /, z ) for S. Then, the preferred spin direction z is described by the two angles (0, / ), where 0 and 0 as the azimuthal and polar angles of the preferred spin direction with 

For a qualitative discussion of spin orientation, it is convenient to rewrite the SOC 

In determining the preferred spin orientation, the most important interaction between occupied and unoccupied spin levels is the one with the smallest energy gap Ae = (e,—Cj). The d-levels of same m values (e.g., between xz and yz, ancj between xy and x —y) interact through the operator z to give nonzero (/ Hsoli) ( Hso i). The d-levels of different m values with Am= i (e.g., between xz/yz and xy/(x — y ), and between z and xz/yz) interact through the ladder operators L+ and to give nonzero (/ IHjoIj) and (/ n soli). The z orbital cannot interact with the xy/ (x —y ) orbitals under SOC because their m values differ by Am = 2. 

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Fig. 12-5. The Magnetic Unit Cell of MnFa Showing the Spin Orientation of the Magnetic Ions. The fourfold axis is in the direction of the spin vectors. The nonmagnetic ions are not shown.

Figure 2.14. The molecular orbitals of gas phase carbon monoxide, (a) Energy diagram indicating how the molecular orbitals arise from the combination of atomic orbitals of carbon (C) and oxygen (O). Conventional arrows are used to indicate the spin orientations of electrons in the occupied orbitals. Asterisks denote antibonding molecular orbitals, (b) Spatial distributions of key orbitals involved in the chemisorption of carbon monoxide. Barring indicates empty orbitals.5 (c) Electronic configurations of CO and NO in vacuum as compared to the density of states of a Pt(lll) cluster.11 Reprinted from ref. 11 with permission from Elsevier Science.

Electron spin is the basis of the experimental technique called electron paramagnetic resonance (EPR), which is used to study the structures and motions of molecules and ions that have unpaired electrons. This technique is based on detecting the energy needed to flip an electron between its two spin orientations. Like Stern and Gerlach s experiment, it works only with ions or molecules that have an unpaired electron. 

The revision leads to a difference of 0.06 A. between the interatomic distance in the normal oxygen molecule and the sum of the double-bond radii. This may be attributed to the presence of an unusual structure, consisting of a single bond plus two three-electron bonds. We assign this structure both to the normal 2 state, with ro = 1.204 A., and to the excited 2 state, with ro = 1.223 A., the two differing in the relative spin orientations of the odd electrons in the two three-electron bonds. We expect for the double-bonded state the separation n 1.14 A. 

As mentioned in Section Wl, an electron has magnetism associated with a property called spin. Magnetism is directional, so the spin of an electron is directional, too. Like orbital orientation, spin orientation is quantized Electron spin has only two possible orientations, up or down. The spin orientation quantum number )... 

The last electron could be placed in any of the 3 p orbitals, because these three orbitals are equal in energy. The final electron also could be given either spin orientation. By convention, we place electrons in unfilled orbitals starting with the left-hand side, with spins pointed up. 

The electrons could occupy different 2 p orbitals with the same spin orientation (different Jitt/ values but... 

Arrangements 2 and 3 look spatially equivalent, but experiments show that a configuration that gives unpaired electrons the same spin orientation is always more stable than one that gives them opposite orientations. Hund s rule summarizes the way in which electrons occupy orbitals of equal energies. 

The most stable configuration involving orbitals of equal energies is the one with the maximum number of electrons with the same spin orientation. 

In, Hund s rule dictates that the five d electrons all have the same spin orientation. For these five electrons,... 

Ti A neutral titanium atom has 22 electrons. The ground-state configuration is (Ar] A 3 cf. The spins of the 4 electrons cancel, but the two electrons in 3 orbitals have the same spin orientation, so... 

One He atom has two electrons, so a He2 cation has three electrons. Following the aufbau process, two electrons fill the lower-energy cr 1 orbital, so the third must be placed in the antibonding crj orbital in either spin orientation. A shorthand form of the MO diagram appears at right. The bond order 1... 

Electron configurations of transition metal complexes are governed by the principles described in Chapters. The Pauli exclusion principle states that no two electrons can have identical descriptions, and Hund s rule requires that all unpaired electrons have the same spin orientation. These concepts are used in Chapter 8 for atomic configurations and in Chapters 9 and 10 to describe the electron configurations of molecules. They also determine the electron configurations of transition metal complexes. 

An alternative way to describe the phenomenon is to consider that the ground state of a chain is already divided into domains at any temperatures. In order for the system to follow a small variation of the magnetic field some domains have to reverse their spin orientation. This occurs through a random walk of the DWs, that is, equal probability for the DW to move backward or forward, which implies that the DW needs a time proportional to d2 to reach the other end of a domain of length d. Given that d scales as the two spins correlation length, ., which, for the Ising model, is proportional to exp(2///rB7 ), for unitary spins, the same exponential relaxation is found... 

Nylon blends, dyeing, 9 204 Nylon block copolymer, 19 762 Nylon carpet fibers, stain-resistant, 19 764 Nylon-clay nanocomposites, 11 313-314 Nylon extrusion, temperatures for, 19 789t Nylon feed yarns, spin-oriented, 19 752 Nylon fiber(s), 24 61 production of, 19 740 world production of, 19 7654 Nylon fiber surfaces, grafting of polymers on, 19 763-764... 

As with the unrestricted Hartree-Fock approximation, the LSDA allows for different orbitals for different spin orientations. The LSDA gives a simplified treatment of exchange but also includes Coulomb correlations. 

Stretching bending Spin orientation (In magnetic field) Electrons ESR Nuclei... 

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