Long wavelength limit

If 5 aj(r) is slowly varying in space, tire long-wavelength limit x-j(k —> 0) reduces to a set of static susceptibilities or thennodynamic derivatives. Now, since for t > 0 the external fields are zero, it is useful to evaluate the one-sided transfomi... 

These are both long ranged in the long-wavelength limit > 0 S q) due to broken translational synnnetry and S q) due to broken time reversal synnnetry. S q) vanishes for flucUiations around equilibrium, and S q) is the same for both NESS and equilibrium. The results above are valid only for iy where the two... 

The long-wavelength limit which depends on the cathode material is ca. 2 = 650 nm m the red region of the spectrum. If this does not suffice for the determination an antimony-alkaline metal alloy is employed as the cathode material [56—58]. 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

Fig. 3.6. (a) Imaginary and (b) real parts of the long-wavelength limit of the... 

The dynamical behaviour of the atoms in a crystal is described by the phonon (sound) spectrum which can be measured by inelastic neutron spectroscopy, though in practice this is only possible for relatively simple materials. Infrared and Raman spectra provide images of the phonon spectrum in the long wavelength limit but, because they contain relatively few lines, these spectra can only be used to fit a force model that is too simple to reproduce the full phonon spectrum of the crystal. Nevertheless a useful description of the bond dynamics can be obtained from such force constants using the methods described by Turrell (1972). 

Thus the static dielectric constant diverges in the long wavelength limit as q -> 0. 

No data are available for an exact determination of AH/°(CH2), but an upper limit can be obtained87 by considering the long wavelength limit for the photolysis of CH2CO, which is about 3700 A.96 105 117 A wavelength of 3700 A. corresponds to an energy of about 77 kcal., which must therefore be sufficient to effect the process... 

Thus, the nucleus is not an effective antenna due to its small size compared to the wavelength of the radiation, y Rays are in the long wavelength limit and are not very sensitive to the detailed internal structure of the emitting nucleus. 

The electric dipole approximation - or long-wavelength limit is obtained by taking k—> 0. For afinite path, C, Lebesgue s dominated convergence theorem justifies file interchange ofthe order ofintegration and the limit k h>0. so that... 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

M. Thachuk, M. Yu. Ivanov, and D. M. Wardlaw, /. Chem. Phys., 105, 4094 (1996). A Semiclassical Approach to Intense-Field Above-Threshold Dissociation in the Long Wavelength Limit. 

The Rydberg constant for Li24" is 109,729 cm-1. (a) What is the long-wavelength limit in the absorption spectrum... 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

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