Magnetic dipole moment fields

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

A nucleus in a state with spin quantum number 7 > 0 will interact with a magnetic field by means of its magnetic dipole moment p. This magnetic dipole interaction or nuclear Zeeman effect may be described by the Hamiltonian... 

The magnetic field seen by the probe neutron is solely due to the magnetic dipole moment density of the unpaired electrons. In other words, the magnetisation density is simply related to the electron spin density by a multiplicative factor, and there is no ambiguity in its definition. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The magnetic hyperfine splitting, the Zeeman effect, arises from the interaction between the nuclear magnetic dipole moment and the magnetic field H at the nucleus. This interaction gives rise to six transitions the separation between the peaks in the spectrum is proportional to the magnetic field at the nucleus. 

Similarly, the interaction with a magnetic field can be written in terms of a magnetic dipole, quadrupole, and other moments. However, the first derivative in the Taylor series expansion, in the presence of magnetic field, is the permanent magnetic dipole moment the second derivative is the magnetizability. For a further study on the electrical and magnetic properties, one can refer to Dykstra [9]. 

If quark matter is in the ferromagnetic phase, it may produce the dipolar magnetic field by their magnetic moment. Since the total magnetic dipole moment Mq should be simply given as Mq = fjq (47t/3 rq)nq for the quark sphere with the quark core radius rq and the quark number density nq. Then the dipolar magnetic field at the star surface R takes the maximal strength at the poles,... 

Provided that a transition is forbidden by an electric dipole process, it is still possible to observe absorption or emission bands induced by a magnetic dipole transition. In this case, the transition proceeds because of the interaction of the center with the magnetic field of the incident radiation. The interaction Hamiltonian is now written as // = Um B, where is the magnetic dipole moment and B is the magnetic field of the radiation. 

The resulting property operators at the four-component relativistic level are listed in Table 1. From the property operators associated with uniform electric and magnetic fields one may directly read off the relativistic operators of electric dipole moment p,= —r, and magnetic dipole moment m = — c(r,o X a,), respectively [36]. 

A magnetic dipole moment, when placed in an external magnetic field, will have an energy of interaction E with the field which is the negative of the product of the magnetic field H and the component of the magnetic moment along the field direction fin... 

As was discussed qualitatively in Section II,A,2, the local magnetic fields produced at a nucleus in a solid by the magnetic dipole moments of nuclei around it are often responsible for the observed line widths. Van Vleck (73) has derived, in a rigorous manner, an expression for the second moment of the absorption curve of the nuclei in terms of the magnetic moments, spins, and internuclear distances of the nuclei. The second moment ((AH )) of the shape function g(H — Ho) normalized to unit area is... 

The ar 1 (Ml) term thus describes the interaction of the magnetic dipole moments of the electrons and nuclei with the magnetic field (of strength IHI = Ao k) of the light (which lies along the y axis) ... 

If the electric and magnetic dipole moments in the presence of frequency-dependent electric and static magnetic fields are expanded in a series, the leading terms give the following expression for (9)... 

In quantum mechanics this then becomes the vector product of the nuclear magnetic dipole moment and the distance vector between an electron i and the field-creating nucleus I (60)... 

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