Gamow function

A generalized version of Eq. (19), suitable for complex matrices and producing both energies and widths of resonances, has also been introduced and applied in the framework of the present theory [37b, 92,93]. As it was pointed out in Refs. [37b, p. 487], "This variational principle was first given with the Gamow functions in mind [92]. However, given the equivalence demonstrated (in Ref. [37b]), it is obvious that a completely analogous principle holds for H(0)." H(0) is the complex Hamiltonian with scaled coordinates, r see Section 5.1). 

Gamow functions, si p ), which are normally chosen as (more flexible forms are possible). 

The importance of this fact for statistical mechanics was stressed by A.J. Khinchin, Mathematical Foundations of Statistical Mechanics (G. Gamow, transl., Dover Publications, New York 1949) p. 63. But he called A a sum function only if n = 1. 

Gamow factor, which increases with increasing energy. The product of these two terms produces a peak in the overlap region of these two functions called the Gamow peak (Fig. 12.9). This peak occurs at an energy E0 = (bkT/2)2 3. 

Examples of large-basis shell-model calculations of Gamow-Teller 6-decay properties of specific interest in the astrophysical s-and r- processes are presented. Numerical results are given for i) the GT-matrix elements for the excited state decays of the unstable s-process nucleus "Tc and ii) the GT-strength function for the neutron-rich nucleus 130Cd, which lies on the r-process path. The results are discussed in conjunction with the astrophysics problems. 

We have again utilized the Lanczos algorithm to compute the B strength function for the l3°Cd(0+) l30In(1 + ) Gamow-Teller transitions. Low-... 

Random Phase Approximation Calculations of Gamow-Teller /3-Strength Functions in the A = 80-100 Region with Woods-Saxon Wave Functions... 

A consequence of the steepness of both functions defining the Gamow-peak is an extreme sensitivity of the Maxwellian-averaged cross section to temperature ... 

The final state nucleus (deuterium in its ground state) in the reaction p + p —> d + e+ + z/e, has JJ = 1 +, with a predominant relative orbital angular momentum If = 0 and Sf = 1 (triplet S-state). For a maximally probable super-allowed transition, there is no change in the orbital angular momentum between the initial and final states of the nuclei. Hence for such transitions, the initial state two protons in the p + p reaction must have /j = 0. Since the protons are identical particles, Pauli principle requires S) = 0, so that the total wave-function will be antisymmetric in space and spin coordinates. Thus, we have a process Si = 0, Z = 0 >—>, S y- 1,/y- 0 >. This is a pure Gamow-... 

Equation (31) shows a characteristic feature of the resonance wavefunc-tions, namely that the integral over radial space (not over the spherical harmonics), involves the square of the function itself and not of its absolute value. However, due to the asymptotic behavior (25a), the integral (31) (as well as the one of Eq. (30)), is infinite, since the integrant goes like where a is the imaginary part of the complex momentum of the free Gamow orbital. 

In the CESE approach, the coordinates of fhe Hamiltonian are real. Special attention is given to the accurate, state-specific, single- or multi-state calculation of ho. Complex coordinates are used only in certain orbitals of fhe function space, which is chosen appropriately so as to follow the two-part form of fhe resonance eigenfunction discussed above. We have called these orbitals "Gamow orbitals." They may be chosen to have particular forms in harmony wifh fhe corresponding open channels, or fhey can be expanded in standard forms, such as fhe one of Slater orbitals, albeit with a complex coordinate. Specifically, in fhe case of one-electron decay, the simplest version of fhe SSA square-infegrable resonance wavefuncfion for an N-electron atom is (r symbolizes collectively the real coordinates of fhe bound orbitals). 

Instead, Nicolaides and Beck [37b, p. 506] considered the exactly solvable problems of the harmonic oscillator and of hydrogen and found that in fact, a single function can diagonalize H(rd ). This is the rotated function of the unrotated solution, i.e., [H(rd ) — e ] (rd ) = 0, for each state n > and real e . (The proof is straight-forward). In fact, we stated two "theorems" that are basic to the development of the practical implementation of the CESE-SSA [37b, p. 505], since, in practice, they allow the difficult electron correlation calculations to be done only once on the real axis, and then continue the computation in the complex energy plane where the Gamow orbitals are optimized until the complex energy is stabilized. 

Figure 7.11 provides a plot of T(x,f) as a function of the distance in units of the length L at a fixed time t = lOr, for the double-barrier resonant system described above (solid line). Notice that the probability density exhibits a propagating wavefront. For comparison. Figure 7.11 shows also the solution for the probability density using the Gamow s... 

Figure 7.11 Plot of Ln I (x,t)p as a function of the distance in units of the potential radius x/L (solid line) for the double-barrier resonant potential with the same parameters as in the previous figure, at time t = lOr. Also shown is the purely growing exponential Gamow s solution (dashed line). See text.

N. Fernandez-Garcia, O. Rosas-Ortiz, Gamow-Siegert functions and Darboux-deformed short range potentials, Ann. Phys. 323 (2008) 1397. 

If the barrier is robust, as it is for HI fusion, Wentzel-Kramers-Brillouin (WKB) logic can be used to generate the transmission coefficient (Gamow penetration factor) as a function of the energy of relative motion s (as it is for spontaneous alpha-decay). 

The theoretical Gamow-Condon-Gurney formula (O Eq. (2.65), Ghap. 2) describes the alpha decay constant as a function of the alpha-energy (or rather the Q-value of the alpha decay), as well as the nuclear radius and the atomic number of the daughter. 

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