Amplitude factor

It is a property of this family of differential equations that the sum or difference of two solutions is a solution and that a constant (including the constant i = / ) times a solution is also a solution. This accounts for the acceptability of forms like A (t) = Acoscot, where the constant A is an amplitude factor governing the maximum excursion of the mass away from its equilibrium position. The exponential form comes from Euler s equation... 

Since p is a complex number, it may be expressed in terms of the amplitude factor tan P, and the phase factor exp jA or, more commonly, in terms of just P and A. Thus measurements of P and A are related to the properties of matter via Fresnel coefficients derived from the boundary conditions of electromagnetic theory. ... 

For an unstable electrode system, the asymmetrical fluctuations first become unstable, then cascadelike transitions to the unstable state of the symmetrical fluctuations occur, if possible. As shown in Eqs. (42a) and (42b), when the amplitude factor becomes positive for certain wave numbers, the fluctuations become unstable, and the pits start to grow. When the amplification factor is negative for all wave numbers without exception, the growth of pits is depressed. From Eq. (43), the amplitude... 

Figure 35. Amplitude factor of the symmetrical fluctuation for anodic dissolution through a metal oxide layer with complex formation. Dm = 1.0 x 10-9 m2 s-1, Jt = 1.0 x 10"5 nr s-1 mol-1, m = 2, m = 2 1.Curves 1,2, and 3 correspond to the surface concentrations of the anion, (C (jr, yt 0)) = 10, 50, and 100 mol m-J, respectively.

Here the critical wavelength ACT = 2n/k is thought to represent the minimum value of pit diameter. Some values of ka are shown in Table 1. From Eq. (80), the maximum amplitude factor is written by... 

Then the reciprocal of the maximum amplitude factor, r l/Pmax is thought to express the induction time for pit generation. Some calculated values are shown in Table 2. 

Here I have divided the expression into two terms which correspond to an amplitude factor (a) and an oscillatory component (b). I will now consider each of these terms in some detail. 

It is known that the cohesion of a metal is ensured by the electrons partially filling a conduction (or valence) band. The wave functions of these conduction electrons are Bloch functions, i.e. amplitude modulated plane waves. Even though these wave functions are linear combinations of the electronic wave functions in the isolated atoms, reminiscence of the atomic orbitals is lost (or is eventually contained in the amplitude factor). The conduction electrons are, of course, originally, the outer or valence electrons of the atoms but in a metal, to describe them as s, p, d or f, i.e. with the quantum number proper to the atomic case, has little meaning. They may be considered to many purposes to be free electrons . 

The cross-sections for itinerant electrons, as, e.g., electrons in broad bands, are evaluated by taking into account that the electrons in the initial as well as in the final state may be represented by Bloch-wavefunctions P = u,t(/ ) exp(i R) (see Chap. A). In these wavefunctions atomic information is contained in the amplitude factor Uj (i ), whereas the wave part exp (i R) is characterized by the wavenumber k of the propagating wave (proportional to the momentum of the electron). 

To form a photon-like particle, the elementary normal EMS modes now have to be superimposed to create a wavepacket of finite axial extensions and of finite linewidth in wavelength space. Here we are free to choose an amplitude factor Go of the generating function (90) having the form... 

This expression has the same relaxation time as equation 4.20, but a different amplitude factor. 

The second point to be noted is that kf and kr cannot be assigned without a knowledge of the amplitude factor. This basic symmetry in the relaxation times occurs in many cases, and, in general, the rate constants for unimolecular reactions cannot be assigned unless the concentrations of A and B at equilibrium may... 

Two points should be noted (1) Because the rate constants are pseudo-unimolecular, there is a concentration dependence, so ka and koff may be resolved without the amplitude factor. (2) There is a lower limit to 1/r that is, 1/t cannot be less than koS. This sets a limit on the measurement of these rate constants. A good stopped-flow spectrophotometer can cope only with rate constants of 1000 s 1 or less, and many enzyme-substrate dissociation constants are faster than this. 

The amplitude factor of 1/16 was derived by Stockmayer82) by solving Eq. (B.36) and (B.45) in the limit of large q, analytically. As already mentioned, the amplitude cannot be derived from scaling arguments. 

Here is the pre-exponential (amplitude) factor corresponding to lifetime tv The fractional intensity contribution (fj from component i is related to the pre-exponential factors by ... 

The mass function tends to be large for hydrides for example for N2 and HI it is respectively 0.07 and 1. This is because the large amplitude of vibration of a light atom favours efficient coupling with translation, which is also obvious on classical considerations. The amplitude factor gives rise to the special behaviour of hydrides in the Lambert-Salter30 correlation (see Section 4.2). 

N, is the number of surface atoms, A, is the amplitude of vibration of the sth atom of mass m,. The expression inside the brackets is called an amplitude factor, written by Stretton33 as (A2). In general the orientation factor may be taken as N,/6, since for each atom we need to consider orientation over a complete sphere. 

From the above equations, it is seen that the amplitude factor for a homonuclear diatomic molecule is (2m)-1, which for H2 in atomic mass units would be 0.5. For H20, the amplitude of vibration in the bending mode would be very nearly the same as that of a hypothetical H2 molecule with the same frequency. The amplitude factor is therefore again 0.5. Considering v3 in CH4, in which opposite pairs of H atoms are pinched together, the potential energy is shared between the two pairs. Considering Hooke s law, clearly the squared amplitude is lower by a factor of 2 compared to the hypothetical H2 of the same frequency, and therefore the amplitude factor of CH4(v3) is 0.25. (Note that for CH4, H20 and H2, the product of the amplitude factor and orientation factor is constant.) Stretton33 has compiled a useful list of amplitude factors for substituted methanes. 

Hence, since q2rk = Dkl, the amplitude factors in the multiexponential t,ket(t) are strictly proportional to the concentrations c0>k in the case of short exposures, where the memory function g(t) is measured directly. In the long exposure limit, the amplitude factors are proportional to co k/Dk. 

From Eq. (39), the amplitude factors in Eq. (33) can be expressed as a function of molar mass and concentration, and the normalized decay function for a dilute polymer solution, which is et(t) in case of TDFRS andgj (t) in case of PCS, can be written as... 

Fig. 23. Top Characteristic frequency fc versus surfactant (AOT) concentration in cyclohexane, 22.0 °C. Curve through data points calculated according to43). Bottom Amplitude factors of the field effect measurements normalized with respect to the applied dc field of AOT/CgHij solutions Upper curve (positive amplitude, solid circles) Chemical excess losses. Lower curve (negative amplitudes, open circles) orientational field effect [Ber. Bunsenges. Phys. Chem. 79, 667 (1975)]...

The amplitude factor will vary on a timescale comparable to the transit time of a scattering element as it convects through the scattering volume. If this volume is characterized by a length scale, L, that timescale is... 

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