Density of final states

Moreover, eq. (5.64) is nothing but the omnipresent golden rule. To see this just notice that the density of final states is identically equal to... 

The density of final states is obtained by noting that in the one-particle subspace the operator... 

The starting point for all calculations of transition probabilities is the well-known formula (22) sometimes called the Golden Rule. It expresses the transition probability per unit time A in terms of the density of final states... 

In principle, then, to calculate A, one simply must know the wave functions describing the states, the density of final states, and the nature of the perturbative Hamiltonian. In practice, however, this is an exceedingly difficult problem. 

He points out that the variation of lifetime with glass matrix is due to at least two causes, the first being the changes in refractive index. If the wave functions of the ion remain essentially the same from host to host, the spontaneous-transition probability will increase with increasing refractive index because of the increase in density of final states. The second cause is configuration mixing of 4/ and 5d states, which must reflect the size and symmetry of the crystal field produced at the ion by the surroundings. 

Here 0) is the frequency of the radiation, i denotes the initial state of the molecule, and f labels the final state of the photofragments, dv = Pj dE, where is the density of final states. Usually, the wavelength of the radiation considerably exceeds the size of the molecule, and one can use the dipole approximation (see, e.g., ref. 17). Then Hf d. fi (d is the component of the dipole moment along the external electric field) and the problem reduces to the analysis of the dipole matrix element... 

Here the wavefunctions i> and f> are eigenfunctions of the stationary atomic Hamiltonian, the -function ensures energy conservation, and the quantity p describes the density of final states in the photoprocess (see equ. (7.28g)). 

Therefore, in this particular case, Aj(u)) provides a direct measure of the density of occupied states at the adsorbate, lmGaa(co — coo), if there is no pronounced structure in the density of final states and if the variation of the matrix element (J t a) with the final state energy can be neglected over the width of the adsorbate density of states function. The angular distribution of Aj(m) is determined by the matrix element f r a and reflects the symmetry of the adsorbate orbital. 

In extreme cases a multiple-scattering, sharp resonant structure can result in which the electron is in a quasi-bound state (155). One example is the white line, which is among the most spectacular features in X-ray absorption and is seen in spectra of covalently bonded materials as sharp ( 2eV wide) peaks in absorption immediately above threshold (i.e., the near continuum). The cause of white lines has qualitatively been understood as being due to a high density of final states or due to exciton effects (56, 203). Their description depends upon the physical approach to the problem for example, the LiUii white lines of the transition metals are interpreted as a density-of-states effect in band-structure calculations but as a matrix-element effect in scattering language. 

The L( X-ray absorption edge of transition and rare earth metals does not usually have a white line, as opposed to the LiUii edges. This difference stands in connection to the fact that L initiates from the 2s state as opposed to the lPi/2 (Ejj) and 2p3/2 (Lm) states for LiUii. While the latter probe the s- and -symmetric portions, the L and K edges probe the p-symmetric portion of the density of final states. 

This expression is the exact form of Fermi s Golden Rule, familiar in time-dependent perturbation theory where F[, 0)) is approximated by o) (Merzbacher, 1970). p( ,) is the density of final states. 

We must first find the density of final states, which we characterise in terms of the relative momentum /c . The permitted values of k in the normalisation box are given by (4.7). 

For a large number of accessible acceptor levels, the summation over all the terms of the Franck-Condon factor reduces to the unweighted density of final states [32, 33],... 

---