Magnetic dipole transition probabilities

For the sake of simplicity and a more instructive description, we shah restrict ourselves to the case of unpolarized single line sources of 7 = 3/2v / = 1/2 magnetic dipole transitions (Ml) as for example in Fe, which has only a negligible electric quadrupole (E2) admixture. It will be easy to extend the relations to arbitrary nuclear spins and multipole transitions. A more rigorous treatment has been given in [76, 78] and [14] in Chap. 1. The probability P for a nuclear transihon of multipolarity Ml (L=l) from a state I, m ) to a state h, m2) is equal to... 

The dipole interaction depends on the distance between the ions (6.4). Therefore, the transition probability increases with increasing concentration of magnetic ions. Studies of the concentration dependence of the relaxation can be conveniently performed on samples of amorphous frozen solutions with a uniform distribution... 

Fig. 7.3 Effect of magnetic dipole interaction (7/m), electric quadmpole interaction (Hq), and combined interaction// = Hu + //q, Em> q on the Mossbauernuclear levels of Ni. The larger spacings between the sublevels of the ground state are due to the somewhat larger magnetic dipole moment of the nuclear ground state as compared to the excited state. The relative transition probabilities for a powder sample as well as the relative positions of the transition lines are indicated by the stick spectra below...

It is much more difficult to observe the Mossbauer effect with the 130 keV transition than with the 99 keV transition because of the relatively high transition energy and the low transition probability of 130 keV transition, and thus the small cross section for resonance absorption. Therefore, most of the Mossbauer work with Pt, published so far, has been performed using the 99 keV transition. Unfortunately, its line width is about five times larger than that of the 130 keV transition, and hyperfine interactions in most cases are poorly resolved. However, isomer shifts in the order of one-tenth of the line width and magnetic dipole interaction, which manifests itself only in line broadening, may be extracted reliably from Pt (99 keV) spectra. 

There are also electric quadrupole E2 terms of order —exiXjdFi/dxj Fpr/X and magnetic dipole Mi terms of order (r x v).Be/c Fpv/c Fpr/X since B0 = F0. These provide smaller transition probabilities by factors of the order of (r/X)2 10-8 in the optical region. However, when the dipole vanishes, they can give rise to forbidden lines indicated by square brackets, e.g. [O ill]. Still higher orders of transition are sometimes significant for nuclear y-rays. 

In a similar way, we can approximate the transition probability for a magnetic dipole allowed transition by... 

As shown in Example 5.2, it is easy to obtain that (A)(./(A) 10, where (A)m is the probability of spontaneous emission for a magnetic dipole transition. Thus, using the previous estimation of (A)e, we obtain that, for a magnetic dipole transition,... 

The induced magnetic dipole moment has transformation properties similar to rotations Rx, Rt, and Rz about the coordinate axes. These transformations are important in deducing the intensity of electronic transitions (selection rules) and the optical rotatory strength of electronic transitions respectively. If P and /fare the probabilities of electric and magnetic transitions respectively, then... 

Show that if there were no spin-orbit interaction in atoms, the probability of a magnetic-dipole transition between any two atomic states would be zero. 

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. 

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. 

Here is yet another bizarre result of quantum mechanics for you to ponder. The lx wavefunction for a hydrogen atom is unequal to zero at the origin. This means that there is a small, but nonzero probability that the electron is inside the proton. Calculation of this probability leads to the so-called hyperfine splitting —the magnetic dipoles on the proton and electron interact. This splitting is experimentally measurable. Transitions between the hyperfine levels in the lx state of hydrogen are induced by radiation at 1420.406 MHz. Since this frequency is determined by... 

In a multipole expansion of the interaction of a molecule with a radiation field, the contribution of the magnetic dipole is in general much smaller than that of the electric dipole. The prefactor for a magnetic dipole transition probability differs from the one for an electric dipole by a2/4 1.3 x 1 () 5. Magnetic dipoles may play an important role, however, when electric dipole transitions are symmetry-forbidden as, e.g., in homonuclear diatomics. 

Again expressing W in units of reciprocal seconds (s-1), AE in reciprocal centimeters (cm-1), and pmag in atomic units (h), the probability of a magnetic dipole transition from an initial state i to the final state f is given by... 

One other important difference between electric and magnetic dipole transition probabilities involves the inversion symmetry of all spatial coordinates (i.e. parity). A magnetic dipole moment is an axial vector that does not change sign under inversion, unlike an electric dipole moment. Consequently magnetic dipole transitions occur only between states of the same parity. 

Two further aspects need to be considered in order to understand the magnetic resonance spectrum, namely, the effects of an applied magnetic field, and the electric dipole transition probabilities. The effective Hamiltonian describing the interactions with an applied magnetic field, expressed in the molecule-fixed axis system q, is ... 

The quality of the SOC calculation in O2 can be checked by estimation of the fc Sj" — A3E transition probability. The transition is forbidden by selection rules for electric dipole radiation with account of SOC, and occurs as magnetic dipole spin-current borrowing intensity from microwave transitions between spin-sublevels of the ground state [41]. 

Magnetic dipole transitions play a role in the luminescence of some lanthanide ions, specially Eu +, when the local symmetry deviates little from inversion symmetry. They are parity-allowed between states ofthe3d or4f configurations but have a low probability. They are subject to selection rules AL = A5" = 0 and AJ = 0, 1 (0 0 forbidden). 

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