Recent Articles

Variational Multistate-Ket Derivation of the Heisenberg Spin Hamiltonian

D. J. Klein and David C. Foyt

Phys. Rev. A 8, 2280 (1973) - Published 1 November, 1973

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A new variational derivation scheme for the Heisenberg exchange Hamiltonian is presented. A many-body variational Ansatz for the multistate ket appearing in the derivation is described and is applied to a simple model problem containing the qualitatively important different types of exchange contributions. Advantages of the present derivation over others are briefly indicated.

$L$ X-Ray Intensity Ratios of U, Th, Pb, Ta, and Lu by $α$ -Particle-Induced Emission

Roger K. Wyrick and Thomas A. Cahill

Phys. Rev. A 8, 2288 (1973) - Published 1 November, 1973

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The $L$ X-ray intensity ratios $\frac{L α_{1, 2}}{L_{l}}$ , $\frac{L α_{1, 2}}{L β_{2, 15}}$ , $\frac{L α_{1, 2}}{L β_{6}}$ , $\frac{L β_{1}}{L η}$ , and $\frac{L β_{1}}{L γ_{1}}$ were measured for the elements U, Th, Pb, Ta, and Lu by ionization of $L$ -subshell electrons from 26-MeV $α$ -particle bombardment of thin elemental foils. Comparison of the measured ratios with theoretical transition-rate ratios obtained from a relativistic Hartree-Fock-Slater model showed agreement within experimental error for those cases in which photopeaks were separable.

Atomic Aftereffects and the Line Shape of Muonic X Rays

P. Vogel

Phys. Rev. A 8, 2292 (1973) - Published 1 November, 1973

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The muonic x-ray energy, by virtue of the electron screening, depends on the depletion of the electron states by Auger transitions and on the time needed for the refilling of the electron vacancies. The effect of these processes on the line shape of the muonic x rays is studied here. Equations for the electron $1 s$ -level population and for the muonic-x-ray center of gravity are derived and applied to representative examples of medium- and high- $Z$ muonic atoms. It is shown that the atomic aftereffects (i.e., refilling of the $1 s$ state) result in almost full screening in wide range of atomic numbers. The calculated energy shift with respect to the full screening is only 1-3 eV. The uncertainty, related to the uncertain width of the electron $1 s$ vacancy is discussed and estimated.

Semiempirical Approach to Density Distributions of Heavy Atoms

S. C. Roy and A. M. Ghose

Phys. Rev. A 8, 2298 (1973) - Published 1 November, 1973

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A method to determine radial electronic charge-density distributions of heavy elements has been indicated using the experimental coherent-scattering cross-section values of relativistic photons. In the absence of a generally valid relativistic theory of coherent scattering of $γ$ rays, relativistic effects have been taken into account semiempirically. In the present paper the radial electronic charge-density distributions have been determined for lead ( $Z = 82$ ) using this method. The results are then compared with the theoretical values obtained from nonrelativistic Hartree-Fock and relativistic Dirac-Fock self-consistent-field-model calculations.

Absolute Cross Sections and Polarization for Electron-Impact Excitation of the $K$ and $H$ Resonance Lines of the ${Ca}^{+}$ Ion

Paul O. Taylor and Gordon H. Dunn

Phys. Rev. A 8, 2304 (1973) - Published 1 November, 1973

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Crossed beams of electrons and ${Ca}^{+}$ ions have been used to measure absolute cross sections for electron-impact excitation of the resonance $K$ and $H$ lines of Ca II at 3934 and 3968 Å, respectively. Polarization fractions of the light were also measured. The cross sections presented are absolute in the sense that all measurables including photon flux, have been compared to relevant standards. The cross section for excitation of the $K$ line is observed to have a value of about $(18 \pm 2) π a_{0}^{2}$ at the 3.15-eV threshold, and to decrease to a magnitude of $(1.5 \pm 13) π a_{0}^{2}$ at 700-eV electron energy. Experimental uncertainties have been presented at the 98% confidence level, typically three standard deviations of random fluctuations combined in quadrature with the systematic uncertainties. The experimental results for both the $K$ and $H$ emissions are in agreement at 350 eV with the Coulomb distorted-wave calculation of Burgess and Sheorey but lie about 35% below the low-energy three-state close-coupling calculations of Burke and Moores. The ratio of the cross sections for the $K$ and $H$ emission is found to be 2.0 at all energies. Detailed study of the cross section at low energies demonstrates the expected finite value at threshold (within the accuracy allowed by the electron energy spread of 0.3 eV), and does not indicate the presence of a large (greater than 5%) contribution from cascade. Structure in the $K$ cross section about 2 eV above threshold suggests interactions with autoionizing levels of Ca I, presumably belonging to the $5 s n l$ or $4 d n l$ series. The polarization fraction at low energy is about 25% higher than the calculation of Saraph which is based on the Burke and Moores close-coupling calculation.

Production of Copper $L$ X Rays in Heavy-Ion—Atom Collisions

T. M. Kavanagh, R. C. Der, R. J. Fortner, and M. E. Cunningham

Phys. Rev. A 8, 2322 (1973) - Published 1 November, 1973

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Copper $L$ x-ray production in heavy-ion-atom collisions is discussed. Thick-target x-ray yields and cross sections for x-ray production are presented for copper in the role of both target and projectile. Collision energies were in the range 40 keV to 1.1 MeV and a wide range of collision partners was used. X-ray spectra obtained with a Bragg spectrometer are presented for several of the collision systems. The data, which show strong "level-matching" effects, are interpreted in terms of electron promotion via molecular orbitals, as discussed for asymmetric collisions by Barat and Lichten. Fluorescence yields for the multivacancy states produced in these collisions are discussed, and several solid-target effects are noted.

X-Ray Production by Protons of 2.5-12-MeV Energy

R. B. Liebert, T. Zabel, D. Miljanić, H. Larson, V. Valković, and G. C. Phillips

Phys. Rev. A 8, 2336 (1973) - Published 1 November, 1973

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Characteristic $K$ -shell x -rays produced by proton bombardment of Mn, Fe, Ni, Cu, Zn, Y, Mo, Ag, and Cd targets have been observed for incident energies between 2.5 and 12 MeV. Absolute cross sections for $K$ -shell ionization have been determined and compared to the predictions of the plane-wave Born approximation (PWBA) and binary-encounter approximation (BEA) theories. The (BEA) theory is found to give better agreement at lower bombarding energies and higher $Z$ , while the PWBA fits the data better at lower $Z$ and higher bombarding energies. $L α$ , $L β$ , and $L γ$ cross sections for Pb are also presented as a function of energy between 2.5 and 12 MeV.

Measurements of Electron-Pair-Production Cross Sections

Munaim Mashkour

Phys. Rev. A 8, 2342 (1973) - Published 1 November, 1973

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A number of cross sections associated with electron pair production have been measured in the photon energy range 10-300 MeV in Ilford $G - 5$ emulsion ( $Z_{eff} ≃ 21$ ). Measurements include the distribution of the recoil momentum and scattering angle of the target nucleus, the divergence angle between the pair members, the energy partition, the effect of photon polarization, and the absolute pair production cross section. Agreements and disagreements with the theory and previous measurements are discussed.

Elastic and Inelastic Cross Sections for Electron-Hg Scattering from Hg Transport Data

Stephen D. Rockwood

Phys. Rev. A 8, 2348 (1973) - Published 1 November, 1973

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Elastic- and inelastic-scattering cross sections for low-energy electrons in Hg have been obtained through comparison of experimental and calculated transport data. Electron-electron interactions and superelastic collisions are incorporated into the numerical solution for the electron-energy distribution. The electron-electron interactions are shown to have a pronounced effect upon computed drift velocity and characteristic energies at low $\frac{E}{N}$ , while superelastic collisions principally alter the cascade ionization rate. Details of the present method for solving the dc Boltzmann equation, including the above process, are discussed.

Investigation of the Wigner Spin Rule in Collisions of $N^{+}$ with He, Ne, Ar, $N_{2}$ , and $O_{2}$

John H. Moore, Jr.

Phys. Rev. A 8, 2359 (1973) - Published 1 November, 1973

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The Wigner electron-spin-conservation rule has been investigated in collisions of 1.5-3.5-keV $N^{+}$ with He, Ne, Ar, $N_{2}$ , and $O_{2}$ by the technique of ion-impact energy-loss spectrometry. Inelastic and superelastic collisions involving both ground-state and metastable $N^{+}$ have been observed in the energy-loss spectrum between -2.1 and 4.1 eV. The intensities of many of these transitions have been measured and the beam composition and transition cross sections have been determined. In collisions of $N^{+}$ with the singlet-state targets, spin-conservative transitions are nearly three orders of magnitude more probable than spin-nonconservative transitions. In the $N^{+}$ - $O_{2}$ collisions there is some evidence that the Wigner rule does not hold as well as in less complex collisions.

$K$ -Shell Ionization of $N_{2}$ and C $H_{4}$ by 50- to 600-keV $H^{+}$ , $D^{+}$ , $H_{2}^{+}$ , and ${He}^{+}$ Impact

N. Stolterfoht, D. Schneider, and K. G. Harrison

Phys. Rev. A 8, 2363 (1973) - Published 1 November, 1973

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A crossed-beam scattering chamber was used to measure absolute cross sections for Auger-electron ejection by 50- to 600-keV $H^{+}$ , $D^{+}$ , $H_{2}^{+}$ , and ${He}^{+}$ impact on $N_{2}$ and C $H_{4}$ . Auger electrons emitted at 150° with respect to the incident ions were analyzed by an electrostatic parallel-plate spectrometer with a resolution of 1.1% full width at half-maximum. The intensities of Auger-satellite lines were determined to study the simultaneous inner- and outer-shell ionization of $N_{2}$ by $H_{2}^{+}$ and ${He}^{+}$ at different energies. The Auger spectra were integrated with respect to energy and angle of ejection of the electrons to obtain total cross sections for $K$ -shell ionization. The experimental results are compared with existing theories. Good agreement is obtained for $H^{+}$ , $D^{+}$ , and $H_{2}^{+}$ , but discrepancies appear for ${He}^{+}$ impact at low energies. Ratios of Auger-electron and x-ray production cross sections are determined to study the effect of multiple ionization on the fluorescence yield.

Differential Scattering of Metastable $He (2 {}^{3}S)$ on $He (1 {}^{1}S)$ at Energies between 5 and 10 eV

R. Morgenstern, D. C. Lorents, J. R. Peterson, and R. E. Olson

Phys. Rev. A 8, 2372 (1973) - Published 1 November, 1973

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Differential cross sections of the scattering of $He (2 {}^{3}S)$ on $He (1 {}^{1}S)$ at seven energies between 5 and 10 eV have been measured in a new apparatus. The cross sections display several types of structure. A large rainbow peak is observed at a reduced angle $τ = E θ$ of approximately 161 eV deg for all energies. Fine structure on the rainbow peak and at smaller angles is also resolved. From the rainbow scattering we can deduce that the depth of the ${He}_{2} (a {}^{3}Σ_{u}^{+})$ potential curve is $D_{e} = 2.00$ eV. This value translates to a well depth for the $A {}^{1}Σ_{u}^{+}$ state [ $He (2 {}^{1}S) + He (1 {}^{1}S)$ ] of 2.50 eV, which is in agreement with the latest spectroscopic analysis. At reduced scattering angles less than 120 eV deg the observed structure is not reproduced in the two-state elastic-scattering theoretical calculations based on the adiabatic states $a {}^{3}Σ_{u}^{+}$ and $c {}^{3}Σ_{g}^{+}$ . It is suggested that this disagreement is due to a strong interaction between the $c {}^{3}Σ_{g}^{+}$ state of $He (2 {}^{3}S) + He (1 {}^{1}S)$ and the $b {}^{3}Π_{g}$ state of $He (2 {}^{3}P) + He (1 {}^{1}S)$ . This interaction severely modifies the differential elastic-scattering cross sections at angles less than 120 eV deg.

Theoretical Low-Energy Inelastic-Scattering Cross Sections for $He (2 {}^{3}S) + He (1 {}^{1}S) \to He (2 {}^{3}P) + He (1 {}^{1}S)$ : Curve Crossing between the $c {}^{3}Σ_{g}^{+}$ and $b {}^{3}Π_{g}$ States of ${He}_{2}$

L. Lenamon, J. C. Browne, and R. E. Olson

Phys. Rev. A 8, 2380 (1973) - Published 1 November, 1973

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Low-energy (5-100 eV) cross sections have been computed for the inelastic scattering of $He (2 {}^{3}S) + He (1 {}^{1}S) \to He (2 {}^{3}P) + He (1 {}^{1}S)$ . Two-state quantum-mechanical close-coupling calculations have been performed between the $c {}^{3}Σ_{g}^{+}$ and $b {}^{3}{II}_{g}$ states of the ${He}_{2}$ molecule. The necessary molecular wave functions, potential curves, and rotational-coupling matrix elements were obtained by ab initio computation. The inelastic transition probabilities are found to be quite large, ${| S_{12} |}^{2} \approx 0.8 - 1.0$ for impact parameters close to the curve crossing of the $c {}^{3}Σ_{g}^{+}$ and $b {}^{3}{II}_{g}$ potential curves at $R_{x} \approx 3.3 a_{0}$ . The inelastic total cross sections range from 1.0 × $10^{- 16}$ ${cm}^{2}$ at 5 eV to 4.0 × $10^{- 16}$ ${cm}^{2}$ at 100 eV. Reduced plots of the inelastic differential cross sections show a sharp maximum at a $τ = E θ$ value of approximately 100 eV deg. The strong inelastic process is also found to significantly alter the elastic differential cross sections in the $τ \approx 25 - 125$ -eV deg region. The rotational coupling between the $c {}^{3}Σ_{g}^{+}$ and $b {}^{3}{II}_{g}$ states is found to be the dominant inelastic process at energies up to 500 eV.

Low-Energy Inelastic Scattering of $He (2 {}^{3}S)$ on $He (1 {}^{1}S)$

R. E. Olson, R. Morgenstern, D. C. Lorents, J. C. Browne, and L. Lenamon

Phys. Rev. A 8, 2387 (1973) - Published 1 November, 1973

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Time-of-flight scattering measurements at 8.65 eV (c.m.) display strong inelastic scattering from $He (2 {}^{3}S) + He (1 {}^{1}S)$ to $He (2 {}^{3}P) + He (1 {}^{1}S)$ . Close-coupled quantum-mechanical calculations employing ab initio potential curves and matrix elements show the inelastic process is due to strong rotational coupling between the $c {}^{3}Σ_{g}^{+}$ state of $He (2 {}^{3}S) + He (1 {}^{1}S)$ and the $b {}^{3}Π_{g}$ state of $He (2 {}^{3}P) + He (1 {}^{1}S)$ . Good agreement is found between the calculated and the experimentally measured inelastic-scattering cross sections.

Analytic Expressions for Transient Signals in the Optical Pumping of Alkali-Metal Vapors

F. A. Franz and C. E. Sooriamoorthi

Phys. Rev. A 8, 2390 (1973) - Published 1 November, 1973

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Rate equations for the optical pumping of alkali-metal vapors simultaneously subject to pumping, relaxation in the excited state, and relaxation in the ground state are shown to yield analytic solutions for the alkali-metal electronic and nuclear-spin polarizations in the limit of weak pumping. Nuclear-spin and hyperfine-interaction effects are properly included. Analysis of the double-exponential form of the pumping transient of the ground-state electronic-spin polarization is shown to yield new methods for the measurement of cross sections for collisional relaxation in both the ground and excited states. Formulas are derived that relate relaxation cross sections to experimentally measurable parameters. Formulas describing the influence of nuclear spin on signals obtained from standard depolarization of resonance-radiation experiments also are provided. Ambiguities in earlier optical-pumping experiments and calculations arising from oversimplified descriptions of relaxation processes are discussed.

$Z_{1}^{3}$ -Dependent Stopping Power and Range Contributions

J. C. Ashley, R. H. Ritchie, and Werner Brandt

Phys. Rev. A 8, 2402 (1973) - Published 1 November, 1973

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The theory of the $Z_{1}^{3}$ -dependent contribution to the stopping of charged particles in matter is cast into simple formulas for stopping powers and range-energy relations which apply in a comprehensive manner to all targets. Formulas are given for compound targets and applied to the standard nuclear emulsion.

Determination of the Interatomic Potential of Krypton

U. Buck, M. G. Dondi, U. Valbusa, Michael L. Klein, and G. Scoles

Phys. Rev. A 8, 2409 (1973) - Published 1 November, 1973

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New measurements of the differential collision cross section for Kr + Kr are presented and, together with previously published data, are analyzed in terms of a Morse—Spline—van der Waals interatomic potential function. By means of a least-squares fitting procedure a potential is obtained which best describes the measurements. This potential also predicts second virial coefficients and vibrational-level spacings that agree well with experiment. Some of the difficulties that arise in attempting the inversion of scattering data in this and similar systems are discussed. Finally, we present a comparison with some low-temperature solid-state data.

Experimental Electron-Transfer Cross Sections for Fluorine Ions in Argon at Energies from 8 to 54 MeV

S. M. Ferguson, J. R. Macdonald, T. Chiao, L. D. Ellsworth, and S. A. Savoy

Phys. Rev. A 8, 2417 (1973) - Published 1 November, 1973

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Electron-capture and -loss cross sections have been measured for fluorine ions passing through argon gas in the energy range from 8 to 54 MeV. Cross sections for single- and multiple-electron capture and loss in a single collision were obtained using a computer analysis in real time for fitting the data with an iterative procedure. A detailed description of the computer analysis is given. The capture of as many as four electrons in a single collision has been observed in several cases. The velocity dependence of the single-capture cross section is described by a power of velocity with the power generally increasing for lower-incident-charge states. For multiple-capture processes the cross section decreases more rapidly than a power of velocity. The capture cross section for the fully stripped fluorine nucleus amounts to less than 50% of the theoretical predictions of Nikolaev scaled from a Brinkman-Kramers calculation for protons in argon. The ratio of double- to single-electron-capture cross sections has been discussed in terms of a geometry-independent component of the capture process. The maxima observed in this ratio were rather sharp and close to the same energy for all incident-charge states, making it impossible to identify from the excitation functions the specific shells involved in the capture process. Electron-loss cross sections generally exhibit a broad maximum at velocities approximately twice the orbital velocity of the electron that is lost in the collision. The velocity dependence of multiple-loss cross sections is generally not similar to those for single loss.

$K$ -Shell Photoelectric Cross Sections for 145-keV Gamma Rays

Ramakrishna Gowda and B. Sanjeevaiah

Phys. Rev. A 8, 2425 (1973) - Published 1 November, 1973

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The spectra of electrons resulting from the interaction of 145-keV $γ$ rays with Zr, Ag, Sn, Ta, and Pb have been obtained using a well-type plastic-scintillation spectrometer. The number of $K$ -shell photoelectrons in each case is estimated by fitting a Gaussian to the respective resolved photoelectron spectrum, and hence the $K$ -shell photoelectric cross sections are estimated. They are found to be 32 ± 2, 68 ± 4, 83 ± 5, 348 ± 21, and 537 ± 32 b/atom, respectively. These values are in good agreement with the interpolated values of Schmickley and Pratt.

Differential Charge-Transfer Studies in $H^{+} + Ne \to H (1 s, 2 p, 2 s) + {Ne}^{+}$

P. J. Martin and D. H. Jaecks

Phys. Rev. A 8, 2429 (1973) - Published 1 November, 1973

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Single collisions of protons with neon atoms have been studied at a laboratory scattering angle of 3° for proton energies between 2 and 20 keV. The measured differential-scattering cross section for the sum of elastic, inelastic, and charge-transfer collisions compares in order of magnitude with classical potential-scattering calculations based on the work of Dose and Bingham, with the best agreement at higher energies. The measured total cross section for charge transfer to $H (2 p)$ shows good agreement in shape and magnitude with previous measurements of Pretzer, VanZyl, and Geballe. The measured probability of total charge transfer and the measured probabilities of change transfer to $H (2 s)$ and $H (2 p)$ are discussed in terms of a two-state theory.

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Variational Multistate-Ket Derivation of the Heisenberg Spin Hamiltonian

$L$ X-Ray Intensity Ratios of U, Th, Pb, Ta, and Lu by $α$ -Particle-Induced Emission

Atomic Aftereffects and the Line Shape of Muonic X Rays

Semiempirical Approach to Density Distributions of Heavy Atoms

Absolute Cross Sections and Polarization for Electron-Impact Excitation of the $K$ and $H$ Resonance Lines of the ${Ca}^{+}$ Ion

Production of Copper $L$ X Rays in Heavy-Ion—Atom Collisions

X-Ray Production by Protons of 2.5-12-MeV Energy

Measurements of Electron-Pair-Production Cross Sections

Elastic and Inelastic Cross Sections for Electron-Hg Scattering from Hg Transport Data

Investigation of the Wigner Spin Rule in Collisions of $N^{+}$ with He, Ne, Ar, $N_{2}$ , and $O_{2}$

$K$ -Shell Ionization of $N_{2}$ and C $H_{4}$ by 50- to 600-keV $H^{+}$ , $D^{+}$ , $H_{2}^{+}$ , and ${He}^{+}$ Impact

Differential Scattering of Metastable $He (2 {}^{3}S)$ on $He (1 {}^{1}S)$ at Energies between 5 and 10 eV

Theoretical Low-Energy Inelastic-Scattering Cross Sections for $He (2 {}^{3}S) + He (1 {}^{1}S) \to He (2 {}^{3}P) + He (1 {}^{1}S)$ : Curve Crossing between the $c {}^{3}Σ_{g}^{+}$ and $b {}^{3}Π_{g}$ States of ${He}_{2}$

Low-Energy Inelastic Scattering of $He (2 {}^{3}S)$ on $He (1 {}^{1}S)$

Analytic Expressions for Transient Signals in the Optical Pumping of Alkali-Metal Vapors

$Z_{1}^{3}$ -Dependent Stopping Power and Range Contributions

Determination of the Interatomic Potential of Krypton

Experimental Electron-Transfer Cross Sections for Fluorine Ions in Argon at Energies from 8 to 54 MeV

$K$ -Shell Photoelectric Cross Sections for 145-keV Gamma Rays

Differential Charge-Transfer Studies in $H^{+} + Ne \to H (1 s, 2 p, 2 s) + {Ne}^{+}$

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