Nuclear time scale

The CT/ET free energy surface is the central concept in the theory of CT/ ET reactions. The surface s main purpose is to reduce the many-body problem of a localized electron in a condensed-phase environment to a few collective reaction coordinates affecting the electronic energy levels. This idea is based on the Born-Oppenheimer (BO) separation " of the electronic and nuclear time scales, which in turn makes the nuclear dynamics responsible for fluctuations of electronic energy levels (Eigure 1). The choice of a particular collective mode is dictated by the problem considered. One reaction coordinate stands out above all others, however, and is the energy gap between the two CT states as probed by optical spectroscopy (i.e., an experimental observable). 

Obviously, the details in the time-profile, 7, and the frequency spectrum, Fp, of the incident X-pulse, depend on the experimental setup. However, if the duration of the pulse is either sufficiently short or sufficiently long compared to the time scale of the nuclear dynamics, 7 may be replaced by either a delta function or a constant on the nuclear time scale. Likewise, if the width of Fp can be neglected (known as the static approximation ), we can obtain simplified expressions for the differential scattering signal. However, as pointed out earlier, the frequency widths of X-ray pulses obtained from, e.g., synchrotron radiation are typically on the order of percent of the carrier frequency. Hence, in order to simulate the finer details of the experimental signal, the actual frequency distribution of the incident X-ray pulse must be taken into account [29],... 

A concept of great relevance in considering the interaction of neutrons of low or moderate energy with nuclei is that of the compound nucleus. When the neutron interacts with the nucleus, it is first captured by the nucleus (Z, N) to form the heavier nucleus (Z, N 4- 1). The lifetime of this compound nucleus (typically 10 s) is long on the nuclear time scale, i.e., it is much longer than the time which the neutron would have taken to travel through a distance equal to the nuclear diameter, which is of the order of 10 s for a 1-MeV neutron (of velocity approximately 10 ms" ) incident on a nucleus of diameter 10" m. 

Radiation Damage. It has been known for many years that bombardment of a crystal with energetic (keV to MeV) heavy ions produces regions of lattice disorder. An implanted ion entering a soHd with an initial kinetic energy of 100 keV comes to rest in the time scale of about 10 due to both electronic and nuclear coUisions. As an ion slows down and comes to rest in a crystal, it makes a number of coUisions with the lattice atoms. In these coUisions, sufficient energy may be transferred from the ion to displace an atom from its lattice site. Lattice atoms which are displaced by an incident ion are caUed primary knock-on atoms (PKA). A PKA can in turn displace other atoms, secondary knock-ons, etc. This process creates a cascade of atomic coUisions and is coUectively referred to as the coUision, or displacement, cascade. The disorder can be directiy observed by techniques sensitive to lattice stmcture, such as electron-transmission microscopy, MeV-particle channeling, and electron diffraction. 

The products of nuclear fission reactions are radioactive and disintegrate according to their own time scales. Often disintegration leads to other radioactive products. A few of these secondary products emit neutrons that add to the pool of neutrons produced by nuclear fission. Very importantly, neutrons from nuclear fission occur before those from radioactive decay. The neutrons from nuclear fission are termed prompt. Those from radioacth e decay arc termed delayed. A nuclear bomb must function on only prompt neutrons and in so doing requires nearly 100 percent pure (or Pu) fuel. Although reactor... 

Some preliminary laboratory work is in order, if the information is not otherwise known. First, we ask what the time scale of the reaction is surely our approach will be different if the reaction reaches completion in 10 ms, 10 s, 10 min, or 10 h. Then, one must consider what quantitative analytical techniques can be used to monitor it progress. Sometimes individual samples, either withdrawn aliquots or individual ampoules, are taken. More often a nondestructive analysis is performed, the progress of the reaction being monitored continuously or intermittently by a technique such as ultraviolet-visible spectrophotometry or nuclear magnetic resonance. The fact that both reactants and products might contribute to the instrument reading will not prove to be a problem, as explained in the next chapter. 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

Following a description of femtosecond lasers, the remainder of this chapter concentrates on the nuclear dynamics of molecules exposed to ultrafast laser radiation rather than electronic effects, in order to try to understand how molecules fragment and collide on a femtosecond time scale. Of special interest in molecular physics are the critical, intermediate stages of the overall time evolution, where the rapidly changing forces within ephemeral molecular configurations govern the flow of energy and matter. 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

The binding of calcium ion to calmodulin, a major biochemical regulator of ion pumps and receptors, occurs on a time scale about a thousand times shorter than that observed for RNA conformational change. This Ca2+-calmodulin binding, which can be followed successfully by nuclear magnetic resonance (NMR), occurs in about ten milliseconds. 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

There is no distinction between electrons and nuclei and no mechanism whereby relative atomic positions can be localized. However, since the lighter electrons may reasonably be assumed to move much faster than the nuclei, the nuclear coordinates may to first approximation be assumed to remain fixed at Q on the time scale of electronic motion. The electronic wave equation then reduces to... 

In this expression p is a mass parameter associated to the electronic fields, i.e. it is a parameter that fixes the time scale of the response of the classical electronic fields to a perturbation. The factor 2 in front of the classical kinetic energy term is for spin degeneracy. The functional f [ i , ] plays the role of potential energy in the extended parameter space of nuclear and electronic degrees of freedom. It is given by. 

To more fully appreciate the equilibrium models, like SCRF theories, and their usefulness and limitations for dynamics calculations we must consider three relevant times, the solvent relaxation time, the characteristic time for solute nuclear motion in the absence of coupling to the solvent, and the characteristic time scale of electronic motion. We treat each of these in turn. 

The SCRF models assume that solvent response to the solute is dominated by motions that are slow on the solute electronic motion time scales, i.e., Xp Telec. Thus, as explained in Section 2.1, the solvent sees the solute electrons only in an averaged way. If, in addition to the SCRF approximation, we make the usual Bom-Oppenheimer approximation for the solute, then we have xs Xelect-In this case the solute electronic motion is treated as adjusting adiabatically both to the solvent motion and to the solute nuclear motion. 

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