Electromagnetic Transition Rates

Determining the rate at which an excited state will decay by the emission of a photon is a very general quantum mechanical problem that is not limited to the world of nuclei. The detailed derivation of the transition rate is beyond the scope of this text, and we will only sketch out the results. The decay constant for the emission of a photon by a very well defined single state that has excess energy is shown in Appendix E to be given by the general expression  

The single-particle limit for magnetic multipole radiation obtained by assuming that the change in current is due to a single nucleon is 

One of the nagging features of these expressions is that the radial integral from the multipole expansion introduces a factor of r21, and thus the dimensions of B(E, l) and Bsp(E, l) depend on l. 

Either of the single-particle limits for the reduced electric or magnetic transition probability can be substituted into the expression for the transition rate to obtain 

Similar substitution into the expression for k (El) with 1 = 2 for electric quadrupole radiation will eventually yield 

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Conventional spherical shell model calculations have been undertaken to describe 90 88zr and 90 88y in these large scale calculations valence orbitals included If5/2 2P3/2 2Pl/2 and 199/2 The d5/2 orbital was included for 98Y and for high-spin calculations in 98Zr. Restrictions were placed on orbital occupancy so that the basis set amounted to less than 2b,000 Slater determinants. Calculations were done with a local, state independent, two-body interaction with single Yukawa form factor. Predicted excitation energies and electromagnetic transition rates are compared with recent experimental results. 

Selected electromagnetic moments are presented in Tables III. Agreement for the electric quadrupole moment and magnetic moment for the 8]+ state is not very good for 88Zr(8 +). There are several electromagnetic transitions rates that can be compared The calculated value B(E2 8]+... 

When the eigenvectors from these different fits for a given nucleus were then used to calculate electromagnetic properties, such as transition rates and quadrupole moments, the results were found to agree with one another within 10%. 

In this note, we discuss different approximation schemes for the evaluation of the two-photon transition rate between discrete states. Non relativistic atomic hydrogen is used as a test of the reliability of the methods. We consider a one particle system described by a Hamiltonian Ho, whose eigenstates and eigenvalues are denoted by n> and En, respectively. In the gauge with divA = 0, the interaction of the particle with the electromagnetic field has the usual form... 

The calculations give a reasonable description of the observed energy spectra, electromagnetic and P-decay transition rates, electromagnetic moments, and many other properties of nuclei with closed shells 2 nucleons. 

A simple method for predicting electronic state crossing transitions is Fermi s golden rule. It is based on the electromagnetic interaction between states and is derived from perturbation theory. Fermi s golden rule states that the reaction rate can be computed from the first-order transition matrix and the density of states at the transition frequency p as follows ... 

A chemical interconversion requiring an intermediate stationary Hamiltonian means that the direct passage from states of a Hamiltonian Hc(i) to quantum states related to Hc(j) has zero probability. The intermediate stationary Hamiltonian Hc(ij) has no ground electronic state. All its quantum states have a finite lifetime in presence of an electromagnetic field. These levels can be accessed from particular molecular species referred to as active precursor and successor complexes (APC and ASC). All these states are accessible since they all belong to the spectra of the total Hamiltonian, so that as soon as those quantum states in the active precursor (successor) complex that have a non zero electric transition moment matrix element with a quantum state of Hc(ij) these latter states will necessarily be populated. The rate at which they are populated is another problem (see below). 

Figures 2.13(a) and 2.13(b) illustrate the basis of a semiconductor diode laser. The laser action is produced by electronic transitions between the conduction and the valence bands at the p-n junction of a diode. When an electric current is sent in the forward direction through a p-n semiconductor diode, the electrons and holes can recombine within the p-n junction and may emit the recombination energy as electromagnetic radiation. Above a certain threshold current, the radiation field in the junction becomes sufficiently intense to make the stimulated emission rate exceed the spontaneous processes.

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

Example Problem Use the electromagnetic selection rules to identity the character of the isomeric transition from the hrst excited state at 0.439 MeV( + ) in 69Znm with the ground state ( ). Then calculate the Weisskopf single-particle rates for the allowed transitions. 

Abstract Spin-orbit coupling is a crucial parameter controlling the spin relaxation rate in solids. Here we review recent theoretical results on the randomness of spin-orbit coupling in two-dimensional structures and show that it exists in a form of random nanodomains. The spin relaxation rate arising due the randomness is analyzed. The random spin-orbit coupling leads to a measurable intensity of electric dipole spin resonance, that is to spin-flip transitions caused by the electric field of an electromagnetic wave. 

When a(f) = 1, the field E t) in Eq. (7.24) describes a continuous wave with amplitude Eq. The transition probability to the excited state is given by (X2(t) x2(t), and in this case a constant transition probability per unit time is found (after a few oscillations of the electromagnetic field). For a direct reaction, this is equal to the rate constant of Eq. (7.5), kn(hv). Using Eq. (7.28), it is found [3,4] that... 

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