Probability per unit time

The absorption spectmm, 0(01), is the ratio of transition probability per unit time/incident photon flux. The incident photon flux is the number of photons per unit area per unit time passing a particular location, and is... 

Actually, we are always interested only in the transition probability per unit time to a group of final states with density pf = dnfldEf. This transition rate is given by... 

The expression for the transition probability per unit time W taking into account the process of activation has the form... 

Here, the W j represent the probability per unit time that the system performs a transition from state i to state j. In the present case of relaxation between the two states denoted as LS and HS, we may associate the matrix elements W 2 and 1 21 with the rate constants /clh and Ichl fc)r the conversion process LS HS and HS LS, respectively. The diagonal elements of the relaxation matrix W... 

The problem of the physical meaning of the quantity Hx and of the reorganization energy of the medium Es has been analyzed in Ref. 11. Following Ref. 11, we write the expression for the transition probability per unit time in the form3... 

To formulate the basic model, we consider the transfer of a proton from a donor AHZ,+1 to an acceptor B 2 in the bulk of the solution. For reactions in the condensed phase, at any fixed distance R between the reactants, the transition probability per unit time W(R) may be introduced. Therefore, we will consider first the transition of the proton at a fixed distance R and then we will discuss the dependence of the transition probability on the distance between the reactants. 

Let us assume that all the nuclear subsystems may be separated into several subsystems (R, q9 Q, s,...) characterized by different times of motion, for example, low-frequency vibrations of the polarization or the density of the medium (q), intramolecular vibrations, etc. Let (r) be the fastest classical subsystem, for which the concept of the transition probability per unit time Wlf(q, Q,s) at fixed values of the coordinates of slower subsystems q, Q, s) may be introduced. 

We may also introduce the transition probability per unit time at fixed values of the coordinates of slower subsystems, Wlf(q9 Q) and WfXq, < ), and consider the master equations for the corresponding probability densities RXq, Q) and Rf(q, Q), etc. 

Thus, in this limit, P(t) increases linearly with t and the concept of the transition probability per unit time w may be introduced. The calculation for long times leads to w decreasing with an increase... 

Thus unlike the previous case where the transition probability per unit time exists at some small time and is determined by the frequency characteristics of the reactive oscillator, here the concept of the transition probability per unit time exists only at some sufficiently long time. Note two more differences between the formulas (161)-(162) and (171)-(172). In the first case the frequency factor transition probability (i.e., preexponential factor) is determined mainly by the frequency of the reactive oscillator co. In the second case it depends on the inverse relaxation time r l = 2T determined by the interaction of the reactive oscillator with the thermal bath. 

Say we do the experiment first with just one door open call this door f (which stands for fluorescence). We sit outside of the f-door, and after we start the experiment (at time T0 — 0), we record whether the monkey is still in the room at some time T, and record this. This is the same as a photon counting experiment. We divide up our observational/recording times into At equal increments. The probability that the monkey will still be in the room at time T() + At, if he had started there at time T(h is P(T0 + At) = (1 - k/At), where kf is the time-independent rate (probability per unit time) for finding the f-door. From now on we define Tq = 0. We repeat this experiment a large number of times, each time we record the time T + At when the monkey emerges through the f-door (so he was still in there at time T). For each experiment,... 

The probability per unit time for photodestruction of the donor ( pb,z>) is always the same, in the presence and absence of the acceptor. However, in the presence of the competing process of energy transfer the overall rate of photobleaching is less. Therefore, we can use the rate of photobleaching to measure the rate of energy transfer. This method uses measurements recorded in the second to minute range in order to measure rates in the nanosecond range. 

Here fi2 = 1/Lv(t + r), L is the mean free path of radicals at thermal velocity v, and the initial spur radius r0 and the fictitious time T are related by r2 = Lvr. On random scattering, the probability per unit time of any two radicals colliding in volume dv will be ov/dv, where <7 is the collision cross section. The probability of finding these radicals in dv at the same time t is N(N - 1 )p2 dv2, giving the rate of reaction in that volume as crvN(n - 1 )p2 dv. Thus,... 

Fig. 8.24. Schematic behaviour of gas mass, total mass and metallicity in the Simple model (left), the extreme inflow model of Larson (1972) (middle) and a model with time-decaying inflow (right). The abscissa is u = /( co(t )dt where o> is the (constant or otherwise) transition probability per unit time for gas to change into stars. The initial mass has been taken as unity in each case.

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

If the system is kept at fixed temperature, its thermal fluctuations will explore the configurations around X and may eventually reach the activated configuration, from where it descends to Y. The system thus has a definite probability per unit time, R, of escaping from its metastable state near X. The rate of transfer increases with temperature. 

We consider an ensemble of reactants in the reduced state situated at the interface. Their concentration is kept constant by an efficient means of transport. We denote the perturbation describing the interaction between one reactant and the electrode by M(r,R). According to time-dependent first-order perturbation theory, the probability per unit time that a reactant will pass from the initial to the final state is ... 

Electron transfer processes induce variations in the occupancy and/or the nature of orbitals which are essentially localized at the redox centers. However, these centers are embedded in a complex dielectric medium whose geometry and polarization depend on the redox state of the system. In addition, a finite delocalization of the centers orbitals through the medium is essential to-promote long-range electron transfers. The electron transfer process must therefore be viewed as a transition between two states of the whole system. The expression of the probability per unit time of this transition may be calculated by the general formahsm of Quantum Mechanics. 

The coefficient Cb ,(t) is then obtained by a standard first-order perturbation calculation which takes into account the initial conditions defined by Eq. (7). This gives the transition probability per unit time from the initial state Xav l a to the isoenergetic continuum of states Xbw l b in the form ... 

In the example discussed above, the transition X- X sta s simply for a single spin flip at a randomly chosen lattice site, and W(X - ) = 1 if 5< 0 while W(X- X ) = Q — 3 lkgT) for >0 should be interpreted as transition probability per unit time. Note that other choices for W would also be possible provided they satisfy the principle of detailed balance ... 

Pj is the probability of finding a mobile ion at site i, Wij is the probability per unit time that an ion will hop from site j to site i and is equal to zero... 

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