Spin momentum

Photoelectron peaks are labelled according to the quantum numbers of the level from which the electron originates. An electron coming from an orbital with main quantum number n, orbital momentum / (0, 1, 2, 3,. .. indicated as s, p, d, f,. ..) and spin momentum s (+1/2 or -1/2) is indicated as For every orbital momentum / > 0 there are two values of the total momentum j = l+Ml and j = l-Ml, each state filled with 2j + 1 electrons. Flence, most XPS peaks come in doublets and the intensity ratio of the components is (/ + 1)//. When the doublet splitting is too small to be observed, tire subscript / + s is omitted. 

Spin-drehimpuls, m. spin angular momentum, -glied, n. (Math.) spin term, -momentdichte, /. spin momentum density. 

In addition, since the HPHF wavefunction exhibits a two-determinantal form, this model can be used to describe singlet excited states or triplet excited states in which the projection of the spin momentum Ms=0. The HPHF approximation appears thus as a simple method for the direct determination of excited states (with Afs=0)such as the usual Unrestricted Hartree Fock model does for determining triplet excited states with Ms = 1. 

An unpaired electron executes a spin about its own axis. The mechanical spin momentum is related to a spin vector which specifies the direction of the rotation axis and the magnitude of the momentum. The spin vector s of an electron has an exactly defined magnitude ... 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

For an electron, the magnetic moment resulting from the intrinsic spin momentum is customarily written... 

The unpaired d electrons in an isolated transition metal ion or atom possess orbital angular momentum in addition to the spin momentum, and these couple together to give a total angular momentum J. The Zeeman interaction for these ions can be expressed by Eqs. (1) and (2) if S is replaced by J and g by the Lande g factor ... 

Since the total angular momentum operator L and spin-momentum operator S commute with Eq. (18), the solution of the Hamiltonian containing the first two terms in Eq. (5) can be an eigenfunction of L2, Lz, 52, and Sz ... 

The spin-orbit operator LS given in Eq. (67) is expressed in terms of the individual electron-orbital and spin-momentum operators rather than the total momentum operators L and S. It can be shown (/, 5) that when evaluating integrals involving only LS functions of the same configuration, ls can be replaced by... 

Let us illustrate this method in the special case of two non-equivalent electrons ni/ir fe, described by four momenta two orbital li, I2 and two spin si, S2- Having in mind the commutativity of the addition as well as the fact that the interaction of the orbital momentum of a given electron with its own spin momentum is much stronger than that with... 

The first term is the Zeeman interaction depending upon the g(RS OW, q ) tensor, external magnetic field B0 and electron spin momentum operator S the second term is the hyperfine interaction of the th nucleus and the unpaired electron, defined in terms hyperfine tensor A (Rsklw, qj) and nuclear spin momentum operator n. The following terms do not affect directly the magnetic properties and account for probe-solvent [tfprobe—solvent (Rsiow, qJ)l ld solvent-solvent //solvent ( qj)] interactions. An explicit... 

In the more general case S 0 and the molecular angular momenta can be coupled in various ways. It is of primary importance to ascertain to what extent the interaction of the spin momentum S with the orbital momentum L is comparable to the rotation of the molecule, as well as to the interaction of each of the momenta L and S with the internuclear axis. An attempt to establish a hierarchy of interactions yields a number of possible, certainly idealized, coupling cases between angular momenta, first considered by Hund and known as Hund s coupling cases. Here we will discuss the three basic (out of five) cases of coupling of momenta in a linear molecule. 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

Through the interaction described above, the nuclear spin momentum is coupled to the rotation of the molecule with the result that the rotational levels of the molecule are split into a number of components, giving an associated hyperfine structure to the spectra. 

In this table, Ml is the total magnetic quantum number of the ion. Its maximum is the total orbital angular quantum number L. Ms is the total spin quantum number along the magnetic field direction. Its maximum is the total spin quantum number 5. / = L 5, is the total angular momentum quantum number of the ion and is the sum of the orbital and spin momentum. For the first seven ions (from La + to Eu +), J =L —S, for the last eight ions (from Gd + to Lu +), J = L + S. The spectral term consists of three quantum numbers, L, S, and J and may be expressed as. The value of L is indicated by S, P, D, F, G, H, and I for L = 0,... 

For atoms in S states, L = 0, / = 5, and therefore g = 2 but in all other states g < 2. Consequently observed g values indicate whether the paramagnetic moment arises from both orbital and spin momentum or from spin only. In the latter instance, observation of the maximum / arising from a given electron configuration indicates the number, iV, of unpaired spins, = N. 

The general expressions of the 12 wave functions of the " Tag (represented as the product of the orbital and spin parts are listed below (only the projection of the spin momentum is shown after the orbital part) ). > h ... 

Oxygen, with its 32g ground state, is one of the few stable molecules with a nonvanishing electronic spin momentum. The potential between 02 molecules is not only determined by the usual van der Waals interactions occurring between closed shell molecules, but it contains, moreover, the... 

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