Dimensionless coupling

Fig. 27. Logarithm of normalized rate constant ln(fc/nto) versus dimensionless coupling strength C /Q for PES (4.28) with Q = 0.1, n = 1, F /a>o = 3. Separate points and dashed line correspond to instanton result and numerical data [Hontscha et al. 1990].

The basic thermodynamic inequality contained in Eq. (7) can conveniently be expressed in terms of the dimensionless coupling strength. 

In order to obtain a formula for the energy difference between the complete ground state energy E and the noninteracting energy Eks, and thus for E c, the interaction Hamiltonian H = H- is supplemented by a dimensionless coupling strength parameter g in such a way,... 

For 7=1 the Hamiltonian reduces to the Ising model and for y = 0 to the XY model. For the pure homogeneous case, /,, +i = / and = B, the system exhibits a quantum phase transition at a dimensionless coupling constant... 

The above equation is an integrodifferential equation that has an unusual structure. Here (co2)1 2 is the frequency of the free oscillator and y is the damping constant. The fourth term on the left-hand side of Eq. (226) has the form of the memory kernel, and its strength is controlled by the dimensionless coupling constant X which contains the contribution from the vertex function. 

It will be useful to express the self-energy of the polaron in the two models in units of a dimensionless coupling constant which is considerably important in polaron theory ... 

TABLE 17-2 Empirically determined values of the dimensionless coupling constant. 

Fig. 14 Vibronic energy levels in the linear E e case (in units of hco) versus dimensionless coupling constant k (after Muramatsu and Sakamoto [60]). Encircled is the domain of coupling constants where the vibronic ground state is well energy separated from excited vibronic states and the concept of vibronic pseudo spin applies...

Here the r = 3 case corresponds to the so-called FPU a model and the r = 4 case to the FPU (1 model. The degree of nonlinearity is adjusted either by the coupling constant 8 or by the total energy E of the system, but a convenient way of parameterization is to take the dimensionless coupling constant,... 

In general, the dimensionless coupling coefficient k must depend on time, due to the decomposition (191). However, since this coefficient is small, its variations of the order of ek can be neglected in comparison with the relative variation of the eigenfrequency 800/00 . So k is assumed to be constant. 

Suppose that the lowest cavity mode is resonant. Then one can evaluate the dimensionless coupling constant asK (e2/In mc2L) (here we return to the dimensional variables). The maximum value of parameter is (see the discussion in Section X) max dmaxvs/2nc, where 5max 0.01 is the maximal possible relative deformation in the material of the wall, and vs 5 103 m/s is the sound velocity inside the wall. Then the ratio e/k cannot exceed the value 5max (mirL/Hne2) 0.05 for L 1 cm and m the mass of electron (for these parameters k 2 10 7). Consequently, one may believe that in the real conditions e/k [Pg.369]

These scaling relations can be written in a differential form and we get for the dimensionless couplings... 

For example, Q1Q2, the dimensionless coupling term for the ubiquitous 2 1 anharmonic resonance (that arises from the = Q1Q2 term in the Taylor series expansion of V(Qi, Q2)), becomes... 

Dimensionless coupling constant, 397fr, 405f pr Dimerization reconstruction, Schlier and Farnsworth, 240... 

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