Energy-Resolved Quantities

We now focus attention on energy-resolved quantities, such as the cross section. Measurements of these quantities can be performed using cw excitation sources that excite only a single energy or by using pulsed sources and extracting energy-resolved information after the pulse is over. 

The photodissociation probability into the state characterized by n at energy E, Pn(E i), is given by the square of Aa(E i), the photodissociation amplitude for observing the free state exp(—iEt/1i) E, n 0) in the long-time limit. That is, 

Because the bound and continuum wave functions usually belong to different electronic manifolds, they are orthogonal to one another, and the only term that contributes to 4n(E i) derives fromVP/, the excited part of the wave packet. It follows from the boundary conditions on E, n ) [Eq. (2.57)], that the t — oo limit of Eq. -(2.73) can be written as  

This formula forms the basis for many of the computations of detailed photodissociation cross sections and of angular distribution of photoifagments reported in the literature [14-27]. 

by comparing Eqs. (2.77) and (2.73), that the coefficient of the state E, n ) at time t — 0 in Eq. (2.76) is exactly An(E t), the long-time photodissociation amplitude. Thus, we obtain the cmcial insight that the probability of obtaining product in the state E, n 0 is given solely by the probability of preparing the state E, n ) at the time of preparation. 

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Note also that the form of the pulse does not appear in the expression [Eq. (2.78)] for the cross section. This is because resolving the energy, embodied in the orthogonality expression [Eq. (2.47)], extracts a single frequency component of e(to), whose contribution is canceled in the division by the incident light intensity. Therefore, as shown in Appendix 2A, we can use any convenient pulse shape to compute energy-resolved quantities. This is not the case if we want to follow the real-time dynamics of the system, where the pulse shape is intimately linked with the observables. Indeed this link prevents a pulse-free definition of concepts such as the lifetime of a state. This issue is addressed in Appendix 2A. 

Time-independent picture. The opposite extreme from short-pulse excitation involves the use of nearly monochromatic radiation. Practically, this means that the interaction between molecule and radiation field is of longer duration than Tnr. In this limit, the quantity measured is the absorption lineshape. It will be shown below that the linewidth observed in an energy-resolved experiment is related in a very simple way to the predissociation lifetime in the time-resolved experiment. 

The energy and state resolved tiansition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves [F)... 

Prompt instrumentation is usually intended to measure quantities while uniaxial strain conditions still prevail, i.e., before the arrival of any lateral edge effects. The quantities of interest are nearly always the shock velocity or stress wave velocity, the material (particle) velocity behind the shock or throughout the wave, and the pressure behind the shock or throughout the wave. Knowledge of any two of these quantities allows one to calculate the pressure-volume-energy path followed by the specimen material during the experimental event, i.e., it provides basic information about the material s equation of state (EOS). Time-resolved temperature measurements can further define the equation-of-state characteristics. 

If we now calculate Cm from Eq. (7), the results of the foregoing analysis yield numerical values for the entropy of dilution parameters ypi in the various solvents. From the 0 s obtained simultaneously, the heat of dilution parameter Ki — 0 pi/T may be computed. To recapitulate, the value of in conjunction with gives at once Cm i(1--0/T). Acceptance of the value of Cm given by Eq. (7) as numerically correct makes possible the evaluation of the total thermodynamic interaction i(l —0/7"), which is equal to ( i—/ci). If the temperature coefficient is known, this quantity may be resolved into its entropy and energy components. 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

If excitation is weak or partner concentration is small, then the free ions are produced in low concentration and their bimolecular recombination is too slow to be seen in the timescale of the geminate reaction. Therefore the kinetics of the latter is often studied separately with a fast time-resolved technique. Alternatively, the free-ion quantum yield found from the initial concentration of ions participating in the slow bimolecular recombination can be used to calculate

charge separation quantum yield tp is the usual subject of numerous investigations. Here we will concentrate only on two of them, where this quantity was studied as a function of not only the recombination free energy but of the solvent viscosity as well. These investigations were carried out on the following systems ... 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

Time-resolved CO laser resonance absorption used to 518 study vibrational energy disposal pattern in CO following photofragmentation of 3-cyclopentenone 220—270 nm photolysis of succinic anhydride shown to 519 produce COj, CO, and CjH but not in quantities... 

For valence structure this range is up to about 50 eV. The momentum distribution is observed for each resolved cross-section peak corresponding to an ion eigenvalue —cf. In order to characterise the observation of the target—ion structure we choose a quantity that is as independent as possible of the probe characteristics such as total energy. In conditions where the plane-wave impulse approximation is valid we consider the reaction as a perfect probe for the energy—momentum spectral function... 

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