Hydrogen-like Wave Functions Research Articles

Ionisation induced by neutrons

The ionisation of an atom by neutron bombardment is studied in a quantum-mechanical formulation. The total K-shell ionisation probability is calculated for hydrogen-like electronic wavefunctions. The semiclassical results of Migdal (1941, 1977) can be derived under certain assumptions; time delay effects are clearly exhibited and the experimental possibilities are briefly discussed.

Hydrogenlike orbitals of bound states referred to another coordinate

A hydrogenlike wave function is expanded in terms of a series of hydrogenlike wave functions referred to an arbitrary center in space. The expansion coefficient is given as a function of the distance between two centers. The coefficient for the radial part contains an integral of a spherical Bessel function with two Gegenbauer polynomials of different arguments. This integral is evaluated analytically to be a primitive polynomial multiplied by an exponential function of the distance. The coefficient for the angular part is represented in terms of the spherical harmonics.

Ultraviolet-infrared double-resonance laser spectroscopy ofnd(n=12−17) Rydberg states inHe3

The hyperfine structure of the singlet and triplet $\mathrm{nd}$ Rydberg states ($n=12\ensuremath{-}17$) in $^{3}\mathrm{He}$ have been studied by Doppler-free ultraviolet-infrared double-resonance laser spectroscopy. Using the frequency-doubled output of a cw dye ring laser we have populated the $1s5p^{3}P$ state of $^{3}\mathrm{He}$ by excitation from the metastable $1s2s^{3}S$ state at 294.5 nm. In a second step, an infrared single-mode cw color center laser is used to excite $d$ Rydberg states. The experimental results for the $n^{1,3}D$ hyperfine splittings ($n=12\ensuremath{-}17$) are in excellent agreement with semiempirical calculations using hydrogenlike wave functions. Anomalous isotopic shifts in the electrostatic energy intervals of 5 to 20 MHz were found for the states $n=12\ensuremath{-}17$. These shifts are smaller than the shifts of about 35 MHz found for the states $n=5\ensuremath{-}8$ with the help of anticrossing spectroscopy. No satisfactory explanation of these shifts has yet been found.

Studies of differential cross sections for electron impact ionisation of helium

Triple differential cross sections are obtained using different wavefunctions of the normal helium atom and with different expressions for the effective charges of the outgoing electrons. The results obtained are compared with experimental findings at intermediate and high energies. The exchange effect is considered at low energy. Double differential cross sections are calculated with a hydrogen-like wavefunction for helium and results are compared with experimental observations.

EPR Study of the Relaxed Excited State of F‐Centres in Alkali Halides

AbstractEPR spectra in the relaxed excited state (RES) of F‐centres in isotope‐enriched crystals 41KCl, 39KCl, 85RbCl, and 87RbCl are studied by means of an optical detection technique. The EPR linewidth is established to be due to the hyperfine interactions of the F‐centre electron with the ligand nuclei. It is shown that the EPR data for the RES of F‐centres in alkali halides can be accounted for with a rather localised RES wavefunction. A hydrogenlike 2s wavefunction with the extremum in the region of the first coordination sphere is used as an envelope function. Additional lines in the optically detected spectra of F‐centres are studied.

Coulomb deflection factors for K-shell ionization by protons and deuterons: The role of the dipole amplitudes

The semi-classical approximation (SCA) method has been used to calculate Coulomb deflection factors. The monopole and dipole parts of the perturbing potential are included, together with the target nucleus recoil. Proton and deuteron projectiles, travelling on hyperbolic and straight-line trajectories are investigated. Relativistic and non-relativistic scaled hydrogen-like electronic wavefunctions are used. It is shown that for low projectile energies the inclusion of the dipole amplitudes including recoil, and relativistic electronic effects, becomes very important.

On the relation between quantum-mechanical and impact parameter amplitudes in charge exchange

The relation between the impact parameter capture amplitude a( rho ,v) and the quantum-mechanical one f(q,v) in charge transfer is obtained using the Brinkman-Kramers (1930) approximation. Analytical expressions for a( rho ,v) and calculated cross sections are given for some capture reactions using hydrogen-like wavefunctions.

The anomalous scattering of X-ray photons at the K absorption edge of free atoms and molecules, calculated by means of the modified Kramers-Heisenberg dispersion formula

The energy dependence of the anomalous atomic scattering factor in the vicinity of the critical energy for excitation of the 1s electrons is calculated. The effect of the finite lifetime of the intermediate states, as well as the influence of strong exciton states, is included. The evaluation of the dipole matrix elements in the generalised and modified Kramers-Heisenberg formula is based upon hydrogen-like wavefunctions. Numerical results are given for the atoms Ne, Ar, Kr and the molecule HCl.

Compton scattering from electrons in bound states: Binding and "off-mass"-shell effects

The Compton scattering of a photon from an unpolarized electron bound in a central field is treated relativistically. The distribution of the emitted electrons for a photon scattering in a given direction is obtained as a function of the radial components of the initial Dirac wave function. This expression, in the lowest power of the coupling constant, leads to the Klein-Nishina formula. The off-mass shell and binding corrections to the free Compton scattering formula, which are important near the ionization threshold, are found. Using exact hydrogenlike wave functions, various approximate formulas as functions of the atomic number $Z$ are discussed, and some possible limitations of the usual description of Compton profiles are considered.

Comparison of methods for calculating atomic shielding factors

A new method for determining atomic shielding factors to use in hydrogenlike wave functions which gives "exact" Compton profiles in close agreement with experiment was recently developed by the present authors. These shielding factors ${Z}^{*}(0)$ are determined by imposing the requirement for each atomic orbital that the "impulse" hydrogenic Compton-profile value at its center, $q=0$, match the Hartree-Fock Compton-profile value. We now compare the use of ${Z}^{*}(0)$ hydrogenic values in the calculation of the moments $〈r〉$, $〈{r}^{2}〉$, and $〈{r}^{4}〉$ with values of these moments obtained using Slater, Clementi, Burns, and binding-energy-matched ${Z}^{*}'\mathrm{s}$ used in hydrogenic wave functions for $K$-, $L$-, and $M$-shell orbitals for rare-gas atoms up through xenon. Relative accuracy is determined by comparison to Hartree-Fock results for these moments. It is found that for $1s$, $2p$, and $3d$ states, our ${Z}^{*}(0)$ values give the most consistently accurate results. A table of recommended ${Z}^{*}$ values is given for all atoms and states considered.

Asymptotic form of the charge-exchange cross section in three-body rearrangement collisions

A three-body general-type rearrangement collision is considered where the initial and final bound states are described by hydrogenlike wave functions. The solution obtained is for the first Born approximation where the full interaction potential is taken into account. When the initial state is the ground state, it is shown that for n/$mu$Z very-much-greater-than 1, where n,$mu$, and Z are, respectively, the principal quantum number, the reduced mass, and the nuclear charge of the formed atom, the capture cross section at all incident energies and for capture into the s, p, and d angular momentum states behaves as C/n$sup 3$ + O (1/n$sup 5$), where C depends on masses and charges of the particles, the final angular momentum, and the incident energy. An analytic expression for C is given. It is shown that for the low-lying levels the 1/n$sup 3$ scaling law at all incident energies is only approximately satisfied. The only exception is for capture into the s states according to the Oppenheimer- Brinkman-Kramers approximation. The case for the symmetric collisions is considered and it is shown that for high n and high incident energy E, the cross section behaves as 1/E$sup 3$. Zeros and minima in the differential crossmore » sections are given in the limit of high n for electron capture by protons from atomic hydrogen, and for positronium formation by proton--atomic-hydrogen collisions.« less

M shell ionization in α decay

The results of a simple calculation of the M shell ionization probability in α decay are presented. The sudden approximation has been used with non-relativistic hydrogen-like wave functions and an attempt is made to estimate the likely result of such a calculation using relativistic self-consistent field wave functions. The predicted M X-ray yield is about half of the experimental value and is close to the recent theoretical result of Hansen.

Theoretical determination of the imaginary parts of the atomic scattering factors of Si

The imaginary parts of the atomic scattering factors of Si have been calculated for 12 characteristic wavelengths by using both Hartree-Fock and hydrogen-like wave-functions for the bound Si electrons. Since the values resulting from the two calculations do not differ significantly and both agree very well with existing experimental values, the use of the simpler formulae obtained by Wagenfeld using hydrogen-like wavefunctions for bound as well as excited Si electrons is suggested for the calculation of the imaginary parts of the atomic scattering factors without substantial loss of accuracy.

Relativistic Hyperfine Structure Correction Factors for d and f Electrons Calculated with Hydrogen Wave Functions

By use of hydrogen-like wavefunctions n-dependent relativistic correction factors (RCF) to hyperfine parameters have been calculated for d and f electrons. The RCF have been evaluated for the mean values of r-3 in the effective magnetic dipole and electric quadrupole Hamiltonians. These new RCF are compared with earlier commonly used ones and with factors determined from self-consistent-field (SCF) wavefunctions. It has been found that for d and f electrons no significant improvement is obtained using these new factors compared with the earlier ones given by Kopfermann. A rather large deviation has been shown to exist between the SCF factors and those calculated in this work. It seems that the only reliable way to take into account relativistic effects for d and f electrons is to perform relativistic SCF calculations.

Shallow Donor Levels of Germanium in an Intermediate Magnetic Field

The magnetic field dependence of the shallow donor levels 1 s , 2 p 0 , 2 p -1 and 3 p -1 in germanium was calculated by using two types of variational wave functions each of which is appropriate either for weak magnetic fields or strong magnetic fields. In the range up to γ=0.7 ( H =43 kOe), it became evident that for the 1 s and 2 p 0 states the hydrogen-like wave functions give lower eigenvalues, whereas for the 2 p -1 and 3 p -1 states the harmonic oscillator-type wave functions become more appropriate above about 10 kOe. The effect of the magnetic field on the triplet and singlet ground states was evaluated with inclusion of the central-cell correction for both arsenic and antimony impurities. The optical transitions in the case of H //[111] and E ⊥[111] were considered.

Hole states in helium-like ions

An analysis is made of the poles of the single-particle Green's function of helium-like ions in which electron correlation is taken into account on the basis of the model of singly filled orbitals. The energy of Hartree-Fock pole calculated through the use of the hydrogen-like wave functions for the orbitals turns out to be approximately equal to the exact value. The second pole, which corresponds to a different hole state, has a residue which vanishes as z increases and must be considered fortuitous.

Screening numbers and the scaled hydrogen-like wave functions for some systems with large atomic numbers

Values of the screening numbers σnl for obtaining the analytic scaled hydrogen-like wave functions are calculated for a large number of atoms and ions from Z = 87 to Z = 103.

Effective Spin–Orbit Hamiltonian

Deviations from the Landé interval rule governing separations between the J multiplets of atomic terms are known to stem from term mixing by the spin–orbit interaction. A phenomenological interaction of the form HE = Σnλn(L·S)n with 1 ≤ n ≤ p can account for any such separations if p = 2S, one less than the multiplicity of the term. The general theory of effective operators is applied to the spin–orbit interaction for electron configurations of the form lN, and it is shown that, through pth order in the perturbation expansion of the spin–orbit Hamiltonian h = ηΣi1i·si, the above form for HE obtains where λn = ηΣm(η / F2)m−1λnm, n ≤ m ≤ p. Assuming hydrogenlike wavefunctions for f electrons, the λnm are shown to be constants depending only on the term being considered and the number of equivalent electrons N. Using η and F2 values given in the literature, the λn are calculated for the ground terms of the triply ionized rare earths. Values for the λn have been reported for Nd3+, Tb3+, Ho3+, and Er3+ in CaWO4 where the Stark-split energy levels of the ground terms were analyzed by treating the λn and the crystal-field parameters in HX = ΣkmBkm†Ckm as adjustable. The theoretical and experimental values for λ1, λ2 and λ3 agree to within 2.6%, 7.0%, and 23%, respectively. The calculation is then inverted and η and F2 values for the four ions are determined by requiring the theoretical λn parameters to equal those obtained empirically. The η and F2 values obtained in this manner differ from those reported by 2.3% and 5.1%, respectively. It is therefore feasible, through the use of the effective spin–orbit Hamiltonian HE, to determine η and F2 values for the ground configuration of an ion by considering the ground term alone.

Effective Z values for wave functions in multi-electron atoms and their application to the calculation of shake-off probability

In the multi-electron atom, as shown previously by Mande and Damle (1966), the interelectronic interaction can be accounted for in the orbital energies by introducing a single screening parameter. It has been shown in this paper that, using the effective Z values for the energies, the effective Z values for wave functions of hydrogen-like orbitals can be determined. The hydrogen-like wave functions thus obtained are found to be close to the wave functions given by Jerman and Skillman (1963), specially for the 1s, 2s and 2p orbitals. The present wave functions can be employed to calculate the shake-off probability in the K shell following the beta decay process. The calculated value of the shake-off probability for yttrium is found to agree fairly well with the experimental and theoretical results obtained earlier by other workers.

The graphical representation of hydrogen-like wave functions

Clarifies concepts and graphs related to the representation of hydrogen-like wave functions.