Total angular moment

For a solid to interact with a magnetic field intensity, it must possess a net magnetic moment which, as discussed momentarily, is related to the angular momentum of the electrons, as a result of either their revolution around the nucleus and/or their revolution around themselves. The former gives rise to an orbital angular moment whereas the latter is the spin angular moment /ij. The sum of these two contributions is the total angular moment of an atom or ion, /ijon-... 

The total angular moment of an ion, with one unpaired electron, p, simply... 

An additional phenomenon is that electrons show a strong tendency to occupy as many states with the same spin as possible. Cr and Mn, for instance, both have five electrons with spin j in the 3d orbital. Electrons tend to occupy states with maximum spin angular moment, or at least with maximum total angular moment. In free atoms, the occupancy of states can be determined by Hand s rules (Hund, 1927 Condon Shortley, 1935) ... 

Within this multiplet, the total angular moment of rare earth ion can be expressed as... 

In an actual scattering calculation it is more convenient to work with states which are eigenfunctions of the total angular moment J [9,20,21,32,33]. These states, which correspond to Hund s case (e) coupling [13,19,33], can be obtained by vector coupling the iLSjmj>M2m 2> namely [30]... 

Here 7, is the total angular moment of the initial target state indices and are the incident and discrete state energies, respectively to the incident and captured muons. Further it is convenient to use the second quantization representation. In particular, the initial state of the system atom plus free muon can be written as a l final state is that of an atom with the discrete state electron, removed electron and captured muon in further 7 > represents one-particle (IQP) state, and F > represents the three-quasiparticle (3QP) state. The imaginary (scattering) part of the energy shift ImAE in the atomic PT second order (fourth order of the QED PT) is as follows [27,31] ... 

The spontaneous unimolecular reaction obeys two laws conservation of energy and conservation of total angular moment J. The rate of spontaneotis unimolecular reactions is described by the expression... 

Above we discussed various types of collisions. A reverse process corresponds to each of these processes. Cross sections and rate constants of forward and reverse elementary processes are mutually related. This relationship is determined by the laws of conservation of energy and total angular moment. 

When the three principal moment of inertia values are identical, the molecule is termed a spherical top. In this case, the total rotational energy can be expressed in terms of the total angular momentum operator J2... 

Angular Momentum (Moment of Momentum). Angular momentum is linear momentum (kg-m/s) times moment arm (m). Its SI unit is kg-m /s. For a rotating body the total angular momentum is equal to the moment of inertia I (kg-m ) times the angular velocity CO (rad/s or 1/s). 

The Dirac equation automatically includes effects due to electron spin, while this must be introduced in a more or less ad hoc fashion in the Schrodinger equation (the Pauli principle). Furthermore, once the spin-orbit interaction is included, the total electron spin is no longer a good quantum number, an orbital no longer contains an integer number of a and /) spin functions. The proper quantum number is now the total angular momentum obtained by vector addition of the orbital and spin moments. 

The dipole moment m depends on the total angular momentum, which may be written in terms of the orbital angular moment operator L and the total eleetron spin S. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

Finally, the magnetic moments resulting from the spin and the orbital motion interact. This spin-orbit coupling is taken into account by the total angular momentum quantum number J (Russel-Saunders coupling) ... 

Two kinds of environmental interactions are commonly important in the ESR spectrum of a free radical (i) To the extent that the unpaired electron has residual, or unquenched, orbital angular momentum, the total magnetic moment is different from the spin-only moment (either larger or smaller,... 

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