Transition per unit time

First, consider the case in which the spin system and the lattice are in equilibrium at temperature T in field Hq. This means that the number of transitions upward must equal the number downward. If Wu is the probability of an upward transition per unit time and is the probability of a downward transition, then =... 

The quantity bm 2 represents the probability of the transition m - n. Clearly, the number of transitions per unit time depends on the intensity of the incident radiation, which is proportional to < J 2, and the square of the matrix element (m px n). The latter determines the selection rules for spectroscopic transitions (see the following section). 

As a consequence, expression (A2.21) turns into the sum of two quantities, namely, the probabilities for transitions per unit time accompanied by phonon absorption,... 

Since our system is in equilibrium, the number of absorption transitions i f per unit time must be equal to the number of emission transitions / / per unit time. Considering that the light-matter interaction processes described in Chapter 2 (Figure 2.5) are taking place, in equilibrium the rate of absorption must be equal to the rate of (stimulated and spontaneous) emission. That is ... 

Finally we mention detailed balance, which will be studied in section 6. Equation (4.1) merely states the obvious fact that in equilibrium the sum of all transitions per unit time into any state n must be balanced by the sum of all transitions from n into other states n. Detailed balance is the stronger assertion that for each pair n, n separately the transitions must balance ... 

Vc = % leading to the following expression for the probability of transition per unit time,... 

Suppose that the probability of finding a system in state a at time t is p(a,t) and that the rate of transitions per unit time from a... 

Equation (6.302) tells us that the probability of an electric dipole transition per unit time from state a to state b under the influence of electromagnetic radiation polarised in the X direction is... 

The time-sequence of any kinetic process is evidently determined if the particle population of certain valleys, i, in the crystal is known for die initial state, and if the elementary transition probabilities wy. (transitions per unit time) into a valley k, neighboring i> can be given. Retaining the assumption that defect interactions are to be excluded, we shall limit ourselves here to i — k transitions of elementary particles located at a sufficient distance from all other defects. In formulating the theoretical expression for w we can make use of die treatment by Jost (10) and that by Schottky (ref. 10, p. 155). It is found that Wjj. can be represented as the product of a frequency... 

In equilibrium, a detailed balance must be satisfied, meaning that the number of transitions per unit time from any state i to any state j is on average equal to the number of transitions per unit time from / to i. Then, the number of transitions from any state i to any state j is proportional to the product of the probability P, of being in state i, the probability gi j of making an attempt to move from state i to state j and the probability Pi j of accepting this attempted transition. Therefore, detailed balance can be written as a simple equation ... 

If there is no interference of transition probabilities, i.e., for times sufficiently short so as not to deplete the initial state appreciably, we may define a probability of transition per unit time from the initial state yi,- as follows ... 

Excited states are prone to decay because of their high energy. This can take place by collisions with molecules, atoms, electrons, or ions, or by emission of electromagnetic radiation. In the latter case, the wavelength is given by Planck s law. For levels q and p, the number of spontaneous transitions per unit time is given by ... 

In the simplest case for the interaetion of a two-level system with cw (continuous wave) laser radiation (steady-state mode), the number of transitions per unit time from level 1 to level 2 (absorption) is equal to the number of transitions from level 2 to level 1 (stimulated and spontaneous emission) ... 

We now rephrase Eq. 9.3 in terms of the microscopic dynamics of the two-level system. If W i-2 and W 2-i are the probabilities for upward and downward transitions per unit time in a two-level system, the change in beam intensity will be... 

In the expressions (8.2.12), (8.2.13) <7 is the area of a cell, Pc is the peirtial pressure of c-th component, Ud is the frequency of molecule vibration in normal direction, and ai e the correction factors taking into eiccount the difference between the probability of reaction transition per unit time and the probability of unit-act reactive transition from initial to final configuration. 

According to quantum mechanics, the probability of an induced transition per unit time per unit radiation density between states i) and f) is given by Fermi s Golden Rule ... 

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