Threshold energy exponents

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

Table 2. Exponent of the temperature dependence of the Maxwellian averaged nuclear cross section (middle column) for four different nuclear reactions, evaluated at a temperature of 1.5 107 K. The last column gives the Coulomb threshold energy.

In contrast to the steplike behavior of the cross section for direct photoionization, the cross section for direct ionization by electron impact to form a single quantum state of the ion is zero at threshold and varies approximately linearly with energy above threshold. Thus if we represent these cross sections a by the equation threshold energy, and a is a constant, then the exponent n is approximately unity or zero for ionization by... 

The term a is a symmetry factor for the energy threshold for the passage of electrons and is approximately equal to 0.5. In Fig. 2-4, the value of a was chosen as Vs for better distinction integer exponents are chosen for Tq, G and Gq for clarity,... 

An important quantity whieh has been frequently studied is the mean ehain length, (L), and the variation of (L) with the energy J, following Eq. (12), has been neatly eonfirmed [58,65] for dense solutions (melts), whereas at small density the deviations from Eq. (12) are signifieant. This is demonstrated in Fig. 6, where the slopes and nieely eonfirm the expeeted behavior from Eq. (17) in the dilute and semi-dilute regimes. The predieted exponents 0.46 0.01 and 0.50 0.005 ean be reeovered with high preeision. Also, the variation of (L) at the threshold (p, denoted by L, shows a slope equal to... 

Ashley, Moxom and Laricchia (1996) measured the positron impact-ionization cross section in helium and found that its energy dependence up to 10 eV beyond the threshold was quite accurately represented by a power law, as in equation (5.8), but with the exponent having the value 2.27 rather than Klar s value of 2.651. This discrepancy prompted Ihra et al. (1997) to extend the Wannier theory to energies slightly above the ionization threshold using hidden crossing theory. They derived a modified threshold law of the form... 

With these definitions we can go a step beyond Eq. (7) and define the critical exponent a for the near-threshold behavior of the bound-state energy... 

By studying the ground-state energy as a function of b at a fixed value of c, he showed that there exists a critical point, the wave function is normalized at the threshold, and the critical exponent for the energy is a = 1. A set of long-range potentials exactly solvable only at E 0 (a 1) is discussed in Ref. 58. 

Creep tests were conducted on Sn-Ag and Sn-Zn eutectic solders at 25 °C and 80 °C over the stress range from 10 to 22 MPa and the creep data of log steady state strain rate vs. log stress were fit to straight hnes as shown in Fig. 5. The resulting activation energies were 82 and 68 kJ/mol, respectively [7]. These values are lower than the activation energy for creep of Sn. The exponents n were determined to be 11 and 6, respectively. The value 11 is high and may indicate threshold-stress-type behavior. The creep rates in the alloys are lower than that observed in pure Sn. This is mainly attributed to changes in the preexponential term. A, which is a function of microstructure. The theory for this is not well in hand. 

Mavoori et al. [7] analyzed their stress relaxation at constant strain data for Sn-3.5Ag solder using Eq. (12) and determined the activation energy, Q, to be 34 kJ/mol and a stress exponent of 6. The threshold stress was found to be 10 MPa. How stress relaxation occurs in this solder is insufficiently known to attempt to interpret these numbers. 

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