Magnetic dipole matrix element

The magnetic dipole matrix element between the same two states. 

Magnetic dipole matrix element for a single spectral line in an oriented system 121... 

With the knowledge of the selection rule A/=0, 1, three different cases can be distinguished for the magnetic dipole matrix elements ... 

Special note After evaluation of the magnetic dipole matrix element and taking the absolute square, the calculated magnetic dipole strength is found, expressed in (Bohr magneton) or 0. The matrix element can also be expressed in esu cm, by applying the conversion... 

While the fine structure transitions are inherently magnetic dipole transitions, it is in fact easier to take advantage of the large A = 1 electric dipole matrix elements and drive the transitions by the electric resonance technique, commonly used to study transitions in polar molecules.37 In the presence of a small static field of 1 V/cm in the z direction the Na ndy fine structure states acquire a small amount of nf character, and it is possible to drive electric dipole transitions between them at a Rabi frequency of 1 MHz with an additional rf field of 1 V/cm. 

The magnetic dipole operator transforms as T g, while the direct square of eg irreps yields A g + A2g + Eg. Since the operator irrep is not contained in the product space, the selection rules will not allow a dipole matrix element between Cg orbitals. 

The importance of the generalized susceptibility function xi ) in the theory of magnetism in metals was discussed by many authors (see the review by Herring, 1966). Crudely speaking, xiq) is the approximate response function of the band electrons to a sinusoidally modulated magnetic field. The true response function contains the dipole matrix elements, but if they are approximated as constant or a simple function of q, the simple form of xi ) in eq. (3.69) follows. 

The term for magnetic interactions in Eq. (9.1) gives rise to a magnetic transition dipole matrix element (wi/,a) for a transition of a system between different states. A magnetic transition dipole is a vector analogous to an electric transition dipole but with the magnetic-dipole operator in replacing p ... 

Induced electric dipole transitions occur much more frequently than magnetic dipole transitions and therefore the largest part of this review is devoted to the former. However, induced electric dipole transitions have the disadvantage that the knowledge of wavefunctions is not sufficient for the calculation of electric dipole intensities and a parametrization is necessary. Judd and Ofelt developed independently the theoretical background for the calculation of the induced electric dipole matrix element. Their work is known under the common name Judd-Ofelt theory (sections 5 and 7). The papers of Judd... 

In terms of transition matrix elements of the electric and magnetic dipole moment operators, the transition dipole moments are... 

As stated in an earlier paragraph, the sharp emission and absorption lines observed in the trivalent rare earths correspond to/->/transitions, that is, between free ion states of the same parity. Since the electric-dipole operator has odd parity,/->/matrix elements of it are identically zero in the free ion. On the other hand, however, because the magnetic-dipole operator has even parity, its matrix elements may connect states of the same parity. It is also easily shown that electric quadrupole, and other higher multipole transitions are possible. 

One-electron submatrix elements of the spherical functions operator occur in the expressions of any matrix element of a two-electron energy operator and the electron transition operators (except the magnetic dipole radiation), that is why we present in Table 5.1 their numerical values for the most practically needed cases /, / < 6. 

The quantities A and B are usually called the constants of the magnetic dipole and electric quadrupole interactions. The nuclear matrix elements in (22.27) for k = 1 and 2 are proportional to the magnetic dipole p and electric quadrupole Q nuclear momenta, respectively ... 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

This result is an exact expression for the transition matrix element. Physically we have a dipole interaction with the vector potential and a dipole interaction with the magnetic field modulated by a phase factor. The problem is that this integral is difficult to compute. An approximation can be invoked. The wavevector has a magnitude equal to 1 /X. The position r is set to the position of an atom and is on the order of the radius of that atom. Thus K r a/X. So if the wavelength of the radiation is much larger than the radius of the atom, which is the case with optical radiation, we may then invoke the approximation e k r 1 + ik r. This is commonly known as the Bom approximation. This first-order term under this approximation is also seen to vanish in the first two terms as it multiplies the term p e. A further simplification occurs, since the term a (k x e) has only diagonal entries, and our transition matrix evaluates these over orthogonal states. Hence, the last term vanishes. We are then left with the simplified variant of the transition matrix ... 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

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