The stopping and range of ions in matter is physically very complex, and there are few simple approximations which are accurate. However, if modern calculations are performed, the ion distributions can be calculated with good accuracy, typically better than 10%. This review will be in several sections: 

a)

A brief exposition of what can be determined by modern calculations.




b)

A review of existing widely-cited tables of ion stopping and ranges.




c)

A review of the calculation of accurate ion stopping powers.

The stopping and range of ions in solids

Wave-particle duality is an inherent peculiarity of the quantum world. The double-slit experiment has been frequently used for understanding different aspects of this fundamental concept. The occurrence of interference rests on the lack of which-way information and on the absence of decoherence mechanisms, which could scramble the wave fronts. Here, we report on the observation of two-center interference in the molecular-frame photoelectron momentum distribution upon ionization of the neon dimer by a strong laser field. Postselection of ions, which are measured in coincidence with electrons, allows choosing the symmetry of the residual ion, leading to observation of both, gerade and ungerade, types of interference.

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Double-slit photoelectron interference in strong-field ionization of the neon dimer.

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

Magnetic and electric dipole transitions between levels of the 4fx configuration perturbed by a static crystalline field are treated. The expression obtained for the pure‐electronic electric‐dipole transition probability involves matrix elements of an even‐order unit tensor between the two 4fx states involved in the transition. The contributions to the transition probability from interactions, via the crystalline field, with the nd 94fx−1, 4fx−1 nd, 4fx−1 ng configurations are shown to add linearly, in such a manner as to multiply each odd k crystal‐field parameter Ak q by a constant. If ``J mixing'' in the 4fx configuration is neglected ΔJ between the upper and lower 4fx levels is restricted to six units or less. If ``L mixing'' is neglected then ΔL is also restricted to six units or less. Application is made to the fluorescence spectra of PrCl3 and EuCl3. Many of the missing and weak transitions are explained.

Intensities of Crystal Spectra of Rare‐Earth Ions

Hartree-Slater subshell photoionization cross-sections at 1254 and 1487 eV.

A method of analysis of all level positions in multiple strongly perturbed series of levels is presented and illustrated by examples from the rare gas and Ba spectra. The method is suggested by Seaton's multichannel quantum defect theory but is presented here as an empirical approach. It emphasizes the dependence of a perturbation on the periodicity of the perturbing series and aims at extracting significant information from experimental data in compact form; this goal will be pursued in later works.

Graphic Analysis of Perturbed Rydberg Series

The quantum-defect-theory (QDT) treatment of an electron in the Coulomb field surrounding an ionic core is recast in a form largely independent of field characteristics and thus applicable, e.g., to square wells or to the Morse fields of diatomic molecules. The reformulation parallels Seaton's classification of alternative Coulomb-field wave functions, and makes it applicable to other fields. Wronskians of alternative pairs of base functions have an important role in the theory. For electron energies $\ensuremath{\epsilon}l0$ in a Coulomb field these Wronskians reduce to trigonometric functions of $\ensuremath{\nu}={(\ensuremath{-}2\ensuremath{\epsilon})}^{\ensuremath{-}\frac{1}{2}}$, familiar in the QDT; other fields lead to trigonometric functions of different arguments. Quantum defects are generalized to eigenvalues of a reaction matrix, as in Seaton's work, but this matrix can now be calculated even below threshold energies with the introduction of a "smooth" Green's function appropriate to QDT applications.

General form of the quantum-defect theory

Correlations between the orientations of two particles with spin \textonehalf{} are formulated in terms of a complete set of operators appropriate to display the spectrum of oscillations under the influence of the spin interactions. The operators are classified for this purpose according to space and permutation symmetries. The treatment is intended as a model for any pair of two-level systems. Its extension to multilevel systems and the eventual emergence of dissipative behavior are outlined. The correlation of two-photon polarizations in the Einstein-Rosen-Podolsky paradox is described as an illustration.

Pairs of two-level systems

Broad peaks observed in the photoabsorption spectra of rare earths \ensuremath{\sim}10-20 eV above the $4d$ edge are attributed to transitions $4{d}^{10}4{f}^{N}\ensuremath{\rightarrow}4{d}^{9}4{f}^{N+1}$. Exchange interaction splits the $4{d}^{9}4{f}^{N+1}$ configuration and raises some multiplets by \ensuremath{\sim}20 eV. Autoionization to $4{d}^{9}\ensuremath{-}4{f}^{N}\ensuremath{\epsilon}f$ broadens the high levels. The interpretation extends to $p\ensuremath{\rightarrow}d$ processes observed in the absorption of synchrotron light by transition elements.

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Raising of Discrete Levels into the Far Continuum

A technique is presented which adapts the Stokes method to quantum mechanics and serves to calculate polarization effects by means of Pauli matrices. Illustrative examples indicate that this technique may be not only convenient mathematically but also physically more transparent than previous methods.

Ugo Fano

Papers

Graphic Analysis of Perturbed Rydberg Series

General form of the quantum-defect theory

Pairs of two-level systems

Raising of Discrete Levels into the Far Continuum

A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics