Matrix element magnetic transition

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

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. 

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. 

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 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. 

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

In equ. (8.51) the summation over the magnetic quantum numbers takes care of the unobserved substates, and the -function ensures energy conservation. The essential part which is of interest in the present context is the transition matrix element rfi(icams>, KbmSb ha) whose dependence on the wavevectors Ka and Kb of the emitted electrons with spin projections mSa and mSb and on the photon energy hot is indicated explicitly. Following the detailed discussion in [TAA87] this matrix... 

In-phase single-quantum coherence (SQC) is represented by nonzero values for the matrix elements that correspond to the single-quantum transitions. For example, I spin (1H) SQC corresponds to a superposition of the cqa s and /3 (xs states (row 1 and column 3), and the a Ps and A/ s states (row 2 and column 4). Real numbers are used for magnetization on the xf axis, and imaginary numbers are used for magnetization on the y axis. Notice that the downward transition A s — cqa s has a matrix element that is the complex conjugate of the upward transition oqas —> A s-... 

Table 1. The factors qlr cr for magnetically induced corrections to the matrix elements of the radiative n- and cr-transitions np — Is, 2s and ndm, ns — 2pm in hydrogen atom (fc) = 10fc...

Thus we may state that the matrix elements of the Lyman series transitions increase with the increase of the magnetic field. Similar increase is also characteristic of matrix elements for 7r-transitions to the state 2pl and n- and cr-transitions to the state 2s of the Balmer series. However this property does not hold in general. E.g. for 7r-transitions to the state 2p0 from many states of the upper-level diamagnetic manifold the coefficient q 0A 2po takes negative values (see table 2). So the magnetic field action on the radiative matrix elements is rather selective and depends on the structure of initial and final states and on the type of transition (7T or cr). 

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