Gap equation

After having fixed the input parameters, the temperature dependence of Q(r), A(r) as well as the GL parameter k = Q(t)/A(t) are established by solving the gap equations (22) and (24) numerically for the Gaussian and the post Gaussian approximations, respectively.The results are presented in Fig.1,where solid curve corresponds to the post... 

Clearly, the solutions of nonlinear gap equations are not unique. In numerical calculations we separated the physical solutions by observing the sign of 4>q and that of the effective potential at the stationary point W//( o)- The temperature dependence of these two quantities are presented in Fig. 2. It is seen that 4>q (solid line) is positive in the large range of r and goes to zero when r is close to r = 1. Similarly, the depth of the effective potential at the stationary point, Veff(4>o), becomes shallow when r —> 1 and vanishes at T = Tc. 

The non-perturbative gap equation requires the renormalization of the bare coupling go. The resummation of the set of chain graphs yields for the coupling at the scale of Mandelstam s... 

The actual values for A, A and M follow from the condition that the stable solutions correspond to the absolute minimum of Q with respect to these quantities. Imposing cXl/dA = 0 we obtain the following gap equation for... 

The values we obtain for A strongly depend on p, the value of Ht and, as a consequence of the factor (1 — p] /s) in the gap equation (11), on value and form of the cutoff. Using values of the parameters leading to reasonable vacuum properties and a sharp 3-momentum cutoff, we find A of the order of 1 MeV with the instanton value for Ht and A 10 MeV with twice this value for Ht. Both results are in agreement with earlier expectations [8] that A is small compared with A. 

In spite of these uncertainties we can derive some more general results from the above gap equations. With increasing temperature both condensates, 5 and 5, are reduced and eventually vanish in second-order phase transitions at... 

The partial densities and the gap equation for the up and down paired quarks can be found from the fixed points of the thermodynamic potential density Q... 

Figure 2. The graphical representation of the solution to the charge neutrality conditions (thick dash-dotted line) and the solution to the gap equation for three different values of the diquark coupling constant (thick solid and dashed lines). The intersection points represent the solutions to both. The thin solid line divides two qualitatively different regions, A < S/i and A > S/i. The results are plotted for fi = 400 MeV and three values of diquark coupling constant Go = r/Gs with i] = 0.5, i] = 0.75, and i] = 1.0.

Gathering all these stuffs to put them in the self-consistent equations, we have the coupled gap equations for As,... 

At the present stage, we include only contributions of the first family of leptons in the thermodynamic potential. The conditions for the local extremum of ilq correspond to coupled gap equations for the two order parameters [Pg.387]

Figure 6. Solutions of the gap equations and the charge neutrality condition (solid black line) in the /// vs //, plane. Two branches are shown states with diquark condensation on the upper right (A > 0) and normal quark matter states (A = 0) on the lower left. The plateau in between corresponds to a mixed phase. The lines for the /3-equilibium condition are also shown (solid and dashed straight lines) for different values of the (anti-)neutrino chemical potential. Matter under stellar conditions should fulfill both conditions and therefore for //,( = 0 a 2SC-normal quark matter mixed phase is preferable.

Poly(pyrido[3,4- ]pyrazine vinylene) 693 has been synthesized via condensation of 3,4-diamino-2,5-dibtomopyridine 691 with l,2-bis[3-(2 -ethylhexyloxy)phenyl]-ethane-l,2-dione 692 followed by coupling with l,2-bis(tri- -butyl-stannyl)ethylene in DMF at 110°C in the presence of tetrakis(triphenylphosphine)palladium. The vinylene polymer 693 showed improved stability toward photooxidation compared with similar polymers with purely aliphatic side chains and also had smaller band gaps (Equation 58) <2002SM(131)53>. 

The emergence of superconductivity in electronic systems close to a ferromagnetic instability has recently been studied by solving a linearized gap equation within the Eliashberg formalism [3-5], In both two- (2D) and three-dimensional (3D) systems, it was found that the superconducting transition temperature, Tlc (T stands for linearized) substantially decreases as the sys-... 

A mean-field treatment gives the following gap equation [71] for singlet superconductivity ... 

It is important to test the validity of the slab model to reproduce the surface properties of isolated surfaces. The two main parameters are clearly the vacuum gap introduced and the thickness of the slab employed. Both of these are system dependent but Figure 8.12 shows the effect of the vacuum gap on the calculated surface energy for the unrelaxed surface of a-AbOs taken from localized basis set GGA-DFT calculations using the DSOLID code. At small gap distances the surface energy is underestimated, since in the limit of a zero vacuum gap Equation 8.26 would give = 0. The surface energy in this case is clearly converged above a... 

Vbias is the bias voltage and p (0 < p< 1) a parametric representation of the bias voltage distribution in the tuimeling gap. Equation (6-10) gives a maximum when t = = j, i.e. close to the equilibrium... 

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