Einstein’s expression

Ln-L distance, energy transfer occurs as long as the higher vibrational levels of the triplet state are populated, that is the transfer stops when the lowest vibrational level is reached and triplet state phosphorescence takes over. On the other hand, if the Ln-L expansion is small, transfer is feasible as long as the triplet state is populated. If the rate constant of the transfer is large with respect to both radiative and nonradiative deactivation of T, the transfer then becomes very efficient ( jsens 1, eqs. (11)). In order to compare the efficiency of chromophores to sensitize Ln - luminescence, both the overall and intrinsic quantum yields have to be determined experimentally. If general procedures are well known for both solutions (Chauvin et al., 2004) and solid state samples (de Mello et al., 1997), measurement of Q is not always easy in view of the very small absorption coefficients of the f-f transitions. This quantity can in principle be estimated differently, from eq. (7), if the radiative lifetime is known. The latter is related to Einstein s expression for the rate of spontaneous emission A from an initial state I J) characterized by a / quantum number to a final state J ) ... 

Consider now the observed values of the equivalent conductivity for the various species of ions given in Table 2 [disregarding the ions (OH)-and H+, which need special consideration]. If we ask, from this point of view, why such a wide variety of values is found, this must be ascribed to the wide variety in the character of the random motion executed by different species of ions in the absence of an electric field. We shall not go into the details of Einstein s theory of the Brownian motion but the liveliness of the motion for any species of particle may be expressed by assigning a value to a certain parameter for a charged particle in an... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

An important process has not been included in the analysis. It is the possibility of spontaneous emission. Were it not for such a process, in the absence of electromagnetic radiation a molecule in the excited state ro would be forced to remain there forever. Thus, in Einstein s analysis of this problem three competing processes were considered to be in equilibrium, leading to tbf expression... 

However, one of the consequences of Einstein s special theory of relativity (in 1905) is that a photon has an energy that can be expressed as... 

Gamma rays, which are like x-rays, are assumed not to change the chemical identity of the emitting atom. It was recognized that the change in mass from the Einstein s relativistic expression (E = me1 2 3) is so small that it could be ignored... 

Note that, in Equation 12.3, is a negative quantity since it is opposed to the direction x (see Fignre 12.5) and this explains the negative sign in the second member of Equation 12.3 where the absolnte qnantity It/ l is introduced. Equation 12.4 relates (4 to P- Useful alternative expressions for F are obtained by combining Equation 12.4 with Einstein s equation... 

The delta function corresponds to Einstein s equation, which says that the kinetic energy of the emitted electron Ef equals the difference of the photon energy h(a and the energy level of the initial state of the sample, The final state is a plane wave with wave vector k, which represents the electrons emitted in the direction of k. Apparently, the dependence of the matrix element 1 j) on the direction of the exit electron, k, contains information about the angular distribution of the initial state on the sample. For semiconductors and d band metals, the surface states are linear combinations of atomic orbitals. By expressing the atomic orbital in terms of spherical harmonics (Appendix A),... 

Wu L, Einstein M, Geissler WM, Chan HK, Elliston , Andersson S. Expression cloning and characterization of human 17 P-hydroxysteroid dehydrogenase type 2,... 

Figure 3.3a). The total area under the curve gives the integrated absorption intensity, J ej [Pg.63]

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

Even without solving this equation one can draw an important conclusion. It has the same form as the diffusion equation (IV.2.8) and in fact it is the diffusion equation for the Brownian particles in the fluid. Consequently a2 is identical with the phenomenological diffusion constant D. On the other hand, a2 is expressed in microscopic terms by (2.4) or by (1.6). This establishes Einstein s relation... 

For solids the matter is not quite so simple, and the more exacting theories of Einstein, Debye, and others show that the atomic heal should be expected to vary with the temperature. According lo Debye, there is a certain characteristic temperature lor each crystalline solid at which its atomic heal should equal 5.67 calories per degree. Einstein s theory expresses this temperature as hv /k. in which h is Planck s constant, k is Bolizmanns constant, and r, is a frequency characteristic of ihe atom in question vibrating in the crystal lattice. 

As stated above, expression (9) for the rate constant of transition in Einstein s crystal was first calculated analytically by the method of the straight search in the pioneer works of Pekar [1] and Kun and Rhys [2]. Their analytic expression remains till now the unique exact expression for multi-phonon transition probability in the time unit. Then, there appeared different methods that permit to derive the integral expressions for the rate constant in the general case of the phonon frequencies dispersion the operator calculation method [5], the method of generating polynomial [6], and the method of density matrix [7]. The detailed consideration of these methods was made in the Perlin s review [9],... 

The integral in the expression (20) can be calculated exactly only in the absence of the frequency dispersion of the phonons, i.e. for Einstein s model of the crystal cos — a>. Then, the expression for the rate constant of multiphonon transition results from the formula (20) ... 

The concept of a mass point remains valid, but a time interval dt can no longer be treated as a nondynamical parameter. Einstein s basic postulate [323, 393] is that the interval ds between two space-time events is characterized by the invariant expression... 

In this book the value of A is given in energy units rather than units of mass, which is possible because of the equivalence of mass and energy expressed in Einstein s... 

The period between these two dates saw the publication of two papers by Einstein (1905) 07 and of two by Smoluchowski (1906)208 on the Brownian motion. In the present context the reasoning of Einstein s second paper is of special interest. Let us assume that the N molecules discussed in connection with the expression given Eq. (78) compose a microscopically small particle suspended in a liquid which is in thermal equilibrium. In order to determine the instantaneous state of motion of... 

Einstein s hypothesis, then, led to two definite predictions. In the first place, there should be a photoelectric threshold frequencies less than a certain limit, equal to 4>/h, should be incapable of ejecting photoelectrons from a metal. This prediction proved to be verified experimentally, and with more and more accurate determinations of work function it continues to hold true. It is interesting to see where this threshold comes in the spectrum. For this purpose, it is more convenient to find the wave length X = c/v corresponding to the frequency /h. If we express in electron volts, as is commonly done, (see Eq. (1.1), Chap. IX), we have the relation... 

---