Energy loss straggling of fast charged particles

Abstract

Energy loss straggling of fast charged particles colliding with atoms have been considered in the eikonal approximation. The result is represented in the form of the Fano formula with a nonperturbative correction. The known nonperturbative Titeica formula (which is transformed to the Fano formula when perturbation theory is applicable) is obtained only under certain approximations in eikonal calculations. It has been shown that straggling calculated with allowance for nonperturbative effects at large charges of the projectile can be significantly different (by an order of magnitude) from the results obtained by Titeica, Fano, and Bohr. Energy loss straggling of fast highly charged ions on hydrogen and copper atoms have been calculated. The latter results are compared to experimental data.

Josephson current through interacting double quantum dots with spin–orbit coupling

“…The respective ground state for the 0-phase is a non-degenerate singlet with {s = 0, s z = 0} (Nambu particle number 2). For the π-phase, the ground state is associated with a free spin [10,22,39,[66][67][68][69]; at B = 0, this is given by the degenerate doublet with {s = 1/2, s z = ±1/2} (namely Nambu particle number 1 or 3). The triplet state (Nambu particle number 0 or 4) has {s = 1, s z = ±1}, respectively.…”

“…through the Kondo effect [3][4][5][6][7][8][9][27][28][29][30][31][32][33][34][35][36][37][38]. Theoretical studies dealt with the single-level Anderson impurity coupled to BCS leads as a minimal model for the analysis of phase boundaries and the related transition in the Josephson current [22][23][24][31][32][33][34][35][36][37][38][39][40][41][42].…”

Magnetic interference patterns in superconducting junctions: Effects of anharmonic current-phase relations

“…Then the spectrum of spin-degenerate Andreev states takes the form E i,± (χ) = ±|∆| 1 − D i sin 2 (χ/2), which formally coincides with the spectrum of superconductor-insulatorsuperconductor point contacts [26]. The transparency D i is described here by the Breit-Wigner resonance function, taken at the energy of the i-th localized state [22][23][24]28]. The coefficient D i can take any value between 0 and 1, depending on the energy of the state and its position x i,0 across the interlayer.…”

Fluctuations of energy losses of charged particles colliding with an oscillator

“…The velocity of a projectile should always be much higher than the char acteristic velocities of electrons of a target. Since the average ionization potential I, average Fano ionization potential I F , and quantity (4/3)K are close to each other, the Titeica formula is often written in the form (4) It is noteworthy that the cited approaches involve various approximations based on an inexact nonper turbative consideration, which results in some errors for fluctuations [9,10] and energy losses [11,12]. Energy losses are often calculated within approaches in which an atom is considered as a quantum oscillator with the frequency ω corresponding to the Bethe ion ization potential I [13,14]; therefore, ω ≈ I ≈ I F ≈ (4/3)K. Considering the atom as an oscillator and using perturbation theory, one can obtain, though not in an analytical form, corrections to the Bethe theory [13][14][15], in particular, the shell correction and Barkas correction for energy losses.…”