Bohr Formula Research Articles

Decay and fission of oriented nuclei

The angular distributions of fragments originating from the binary decay of oriented spherical and deformed nuclei are investigated with allowance for correct transformation properties of wave functions under time inversion. It is shown that, as in the case of protonic decay, the adiabatic approximation for collective rotational degrees of freedom of the systems under investigation is inapplicable in describing the angular distributions of fragments of the deep-subbarrier alpha and cluster decays of nuclei. It is demonstrated that this approximation is justified in describing spontaneous and induced low-energy nuclear fission. The dependence of partial fission widths on the orientation of intrinsic axes, spins, and projections of spins and relative orbital angular momenta of fission fragments is analyzed by using the formalism of the unified theory of nuclear reactions and the theory of open Fermi systems. It is shown that the adiabatic approximation leads to the coherent interference between the wave functions for the relative motion of fragments, whereby the universal angular distributions of fission fragments of oriented nuclei is formed. Deviations from the A. Bohr formula are investigated for these distributions.

Universal ionic valence velocity scaling

This work combines the usual empirical formula for the effective charge of a beam penetrating a target with the Bohr charge equilibrium condition. The result is a logarithmic scaling law, which provides the rms valence velocity and virialized ionization potential for any atomic species Z at any charge state $0<~Q<Z.$ Comparisons with Thomas-Fermi modeling and the Bohr formula for hydrogenic valence velocities are performed.

Comparison of the clearance of 99mTechnetium-DTPA from the lung and lung function indices derived from the single breath diagram for carbon dioxide in horses with chronic obstructive pulmonary disease.

The elimination of carbon dioxide from the lung can be used to monitor ventilation and perfusion, and 99mtechnetium-diethylenetriaminepentaacetate (99mTc-DTPA) clearance is a highly sensitive index of alveolar epithelial integrity, which is a presupposition for effective gas exchange. The aim of this study was to determine whether a correlation exists between indices of lung function derived from the single breath diagram for CO2, and 99mTc-DTPA clearance in horses with chronic obstructive pulmonary disease. The rate of 99mTc-DTPA elimination was negatively and substantially correlated with the dead space according to Bohr's formula, and the physiological and alveolar dead space. Strong and significant correlations existed between the three dead space indices and the alveolar fraction of CO2, and the ratio A1/A2 as an index of alveolar efficiency. The associations found were dependent on the clinical status of health and the mode of calculation of 99mTc-DTPA alveolar clearance.

Single electron loss of Kr+ ions in gaseous media

Differential and total cross sections for single electron losswere measured for Kr+ ions on gaseous media in the energyrange of 1.0 to 5.0 keV. The single electron loss crosssections for all the targets studied are found to be in theorder of magnitude between 10-22 and 10-20 cm2,and show a rapid monotonically increasing behaviour as afunction of the incident energy; no remarkable differences inthe energy dependence of σ12 were found for any atomstudied. The single electron loss cross sections show a weakerZt dependence in the large Zt region than that given bythe Bohr formula. A striking feature of the results is theexperimental evidence of saturation in the single electron losscross section as the nuclear charge of the target increases.

Slowing down of relativistic heavy ions and new applications

New precision experiments using powerful accelerator facilities and high-resolution spectrometers have contributed to a better understanding of the atomic and nuclear interactions of relativistic heavy ions with matter. Experimental results on stopping power and energy-loss straggling of bare heavy projectiles demonstrate large systematic deviations from theories based on first order perturbation. The energy-loss straggling is more than a factor of two enhanced for the heaviest projectiles compared to the relativistic Bohr formula. The interaction of cooled relativistic heavy ions with crystals opens up new fields for basic research and applications, i.e., for the first time resonant coherent excitations of both atomic and nuclear levels can be measured at the first harmonic. The spatial monoisotopic separation of exotic nuclei with in-flight separators and the tumor therapy with heavy ions are new applications based on a precise knowledge of slowing down.

Energy-Loss-Straggling Experiments with Relativistic Heavy Ions in Solids.

Energy-loss straggling measurements performed at the high-momentum-resolution magnetic spectrometer fragment separator with bare and highly charged {sub 8}O, {sub 54}Xe, {sub 79}Au, and {sub 92}U ions with specific kinetic energies (700{endash}1000)MeV/u are reported. The results are in good agreement with rigorous calculations recently reported by Lindhard and So/rensen and reveal systematic deviations from the well-known relativistic Bohr formula, which was obtained within the framework of the first-order Born approximation. {copyright} {ital 1996 The American Physical Society.}

Influence of energy straggling on quantitative PIXE analysis

Examples of numerical calculations aimed to estimate the effect of proton energy straggling on quantitative PIXE are presented. Special attention has been taken to low energy PIXE and external beam PIXE. X-ray yield calculations including beam energy straggling for beam energies ranging from 0.3 to 3 MeV were carried out and the X-ray yield variations due to the energy straggling were determined. Proton energy straggling was calculated using the uncorrected Bohr's formula. K-and L-shell ionization cross sections variations were considered for the X-ray yields calculations. Results indicate that in some cases the error induced on the produced X-ray yield may be up to a few percent on low energy PIXE. For external beam PIXE and high energy PIXE the X-ray yield variations are very small (less than 0.5%).

Space distribution and energy straggling of charged particles via Fokker-Planck equation

The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-known Bohr or Bethe formula is found as a first-order approximation of the Fokker-Planck equation.

Energy loss of heavy ions at high velocity

The slowing down of heavy ions by electronic stopping at high velocity is discussed. The ions are nearly fully stripped and have a well defined charge with relatively small fluctuations. Owing to the large charge of the ions, the classical Bohr formula applies instead of the Bethe formula, which is based on a quantum perturbation calculation. It is essential to include the Barkas effect in the description since it becomes quite large for heavy ions, especially in high-Z materials. In Lindhard's treatment [Nucl. Instr. and Meth. 132 (1976) l], the Barkas correction is viewed as an effect of dynamic screening of the ion potential in the initial phase of a collision with an electron, which reduces the relative velocity and therefore enhances the cross section. With inclusion of this enhancement factor for all impact parameters, as evaluated by Jackson and McCarthy for distant collisions [Phys. Rev. B 6 (1972) 4131], the description reproduces within a few percent measurements for 15 MeV/u Br on Si, Ni, and Au and for 10 MeV/u Kr on Al, Ni, and Au. The procedure is shown also to apply at lower velocities near the stopping maximum, albeit with less accuracy. The straggling in energy loss has been analyzed for a measurement on Si and it is well described by a combination of about equal contributions from fluctuations in the number of violent collisions with single electrons (Bohr straggling) and from fluctuations in ion charge state.

Cardiac output determined by the CO2 rebreathing method during arm exercise

Since arm exercise affects the respiratory muscles the CO2 rebreathing method for determining cardiac output (Q) has to be evaluated during arm exercise. The purpose of this study was (1) to compare three different methods of determining arterial CO2 tension (PaCO2) during arm exercise, (2) to verify the linearity of the relationship between Q and oxygen uptake (VO2) during arm exercise, and (3) to investigate whether the CO2 rebreathing method according to Collier can determine accurately Q during arm exercise. Sixty male subjects performed arm-cranking exercise at 20%, 40% and 60% of their individual maximal load. Carbon dioxide output (VCO2) was measured by gas exchange measurement, and mixed venous CO2 tension (PvCO2) was determined from the CO2 rebreathing plateau at each exercise level. PaCO2 was estimated in three different ways: (A) by the modified Bohr formula for dead space, (B) by an arterialized blood sample from the hyperaemic ear-lobe, and (C) by the end-expiratory CO2 tension. A, B, and C were used to calculate Qa, Qb and Qc, respectively. The Pearson's correlation coefficient was high (P < 0.01) among the three different ways of estimating PaCO2. The Q-VO2 relationship proved to be linear (P < 0.01). The Q-values showed a good agreement with the direct Fick measurements, and were in the same range compared to other results obtained by dye dilution, electrical impedance cardiography and the exponential CO2 rebreathing method during arm exercise. In conclusion, the CO2 rebreathing method appeared to be accurate to determine Q during submaximal arm exercise.

Geometry of the three-electron ionization ladder and its corresponding spectrum.

We have computed correlated wave functions for a special class of $^{4}$${\mathit{S}}^{0}$ triply excited states of Li that exhibit strong localization at the corners of an equilateral triangle as the three-electron ionization threshold is approached. Starting from this finding, we have derived the corresponding spectral formula -${\mathit{E}}_{\mathit{n}}$=Cn(n-1)/${\mathit{r}}_{\mathit{n}}^{2}$, with ${\mathit{r}}_{\mathit{n}}$\ensuremath{\sim}${\mathit{n}}^{2}$ for n\ensuremath{\ge}3. C is dependent on the solid angle of the pyramid only for low n, acquiring a ${\mathrm{\ensuremath{\theta}}}_{\mathit{n}}$-independent value of 3.36 for n\ensuremath{\ge}12, where ${\mathrm{\ensuremath{\theta}}}_{\mathit{n}}$ is the angle formed by the position vectors of any pair of electrons. The above simple formula characterizes other many-electron ionization ladders as well and, for large n, scales like the Bohr formula for the hydrogenic spectra.

Squeezed coherent states of the hydrogen atom

The O(4) algebra of the hydrogen atom is made use of to define its squeezed coherent states. It it shown that the effect of squeezing is to increase the energy of the atom in its levels. The corrections to the Bohr formula and the spectral transitions due to squeezing are calculated.

Electron-loss and excitation cross sections for a He+ ion colliding with various atoms.

A unitarized impact-parameter method is applied to calculate the electron-loss and excitation cross sections for ${\mathrm{He}}^{+}$ ions colliding with atoms. The projectile ionization and excitation are dominantly caused by the average potential field of the target atom (atomic number ${Z}_{2}$). The inelastic process of exciting the target atom contributes negligibly except for light target elements. We adopt the Moli\`ere potential to describe this average potential field. The energy dependences of the electron-loss cross sections in He, ${\mathrm{N}}_{2}$, and Ar targets are in good agreement with the reported data. In the case of the Kr target, the present theory yields larger cross sections than the data, especially below 1 MeV impact energy of a ${\mathrm{He}}^{+}$ projectile. The calculated loss cross sections at impact velocity ranging from 2${v}_{0}$ to 6${v}_{0}$ (${v}_{0}$=2.18\ifmmode\times\else\texttimes\fi{}${10}^{8}$ cm/s) show a weaker ${Z}_{2}$ dependence in the large ${Z}_{2}$ region than that given by the Bohr formula. As for the cross section for exciting the ground state of a projectile to the first excited state, a similar weak ${Z}_{2}$ dependence can be found. The recent experimental results using 40-MeV ${\mathrm{F}}^{8+}$ ions colliding with He, Ne, Ar, and Kr targets have supported this tendency.

Z13and Z12corrections to the electronic stopping power formula

Abstract Higher-order corrections have been estimated for the Bethe stopping power formula. The Z1 3 correction formula of Jackson and McCarthy has been modified considering the screening owing to electrons around the projectile. Instead of Z1 4 corrections predicted by Lindhard, Z1 2 corrections have been considered as the difference between a modified Bohr formula and a modified Bethe formula. The mean excitation energy has been estimated by using the improved Bloch relation. The calculated results of stopping power are in good agreement with experimental results for various ions and targets to within a few percent for projectile energy E ≥ 1 MeV/amu and several to 10% for E < 1 MeV/amu.

Single- and multiple-electron-loss-cross-section measurements from 20-MeVFe4+on thin gaseous targets

Single- and multiple-electron-loss cross sections have been measured with 20-MeV ${\mathrm{Fe}}^{4+}$ on thin He, Ne, Ar, Kr, Xe, ${\mathrm{H}}_{2}$, ${\mathrm{N}}_{2}$, ${\mathrm{O}}_{2}$, C${\mathrm{H}}_{4}$, C${\mathrm{O}}_{2}$, CH${\mathrm{F}}_{3}$, C${\mathrm{F}}_{4}$, and S${\mathrm{F}}_{6}$ gaseous targets; cross-section-per-atom data for each system are presented. Cross-section-per-atom data from targets with approximately the same average atomic number indicate that both nonadditivity and density effects, depending on the final charge state of the ion, play roles in electronloss processes during projectile interactions with molecular targets. A modified Bohr formula is developed which reproduces single-electron-loss-per-atom data for atomic targets.

Determination of the alveolar dead space

The author suggests a method of finding the alveolar dead space (AIDS), i.e. the part of the functional dead space (FDS) which presents the greatest difficulties for determination. A1DS determination is based on the difference in the CO2 concentration in the venous blood flowing to the lungs for oxygenation under conditions of respiration at rest, and of CO2 concentration in the alveolar portions of expriation following 10-sex breath withholding at the level of the usual inspiration. The mean A1DS volume (VDA1) in % to the alveolar part of the tidal volume for 20 healthy persons in sitting position and in 12 of them in lying position was found: VDA1 (in sitting position)= = 8.0 +/- 1.5% (P = 0.95), sigma = 3%; VDA1 (in lying position) = = 5.0 +/- 0.7% (P = 0.95), sigma = 2.6%. It is suggested that FDS determined by this method be used to judge the efficacy of using the inspired air in the lung ventilation, to determine AIDS in the diagnosis of lung embolism, to judge the pulmonary circulation shunt by the difference between the value found according to Bohr's formula for FDS and the FDS value established according to the suggested formula.

A differential equation for the energy eigenvalues of relativistic hydrogenic atoms, and its solution

By using various general theorems, differential equations for the energy are written down for Klein–Gordon and Dirac hydrogenic atoms. When coupled with the fact that in the nonrelativistic limit one must get the Bohr formula, these equations are shown to determine the exact energy eigenvalues. Also, in the case of the Klein–Gordon equation, a recursion relation is derived from which the average values of powers of the radius can be found.

Thermal-neutron fission of 226Ra

Thermal-neutron fission cross-section of 226Ra was measured by fission track detectors (polycarbonate) and it was found that the cross-section was 4·7 mbarn which is more than ten times larger than the previously reported values. Fission-product yields were also investigated as a function of complementary fragment mass ratio by measuring the track lengths of fragment pairs. The range-energy conversion was carried out by using a simplified Bohr's formula. It was found that symmetric fission mode is dominant in the 226Ra thermal-neutron fission in contradiction to the generally accepted view that asymmetric mass division is favored at low excitation energy. The accuracy of the track length measurement was checked by the well-known fission systems of 235U thermal-neutron fission and 252Cf spontaneous fission.

Relativistic Bohr Formula

SYNGE1 has discussed a relativistic extension of the Bohr formula, hv′=E′−E, for the frequency of the emitted light quantum between two levels of a quantum system: illustration where m′ and m are the total masses of the two states. Equation (1) is a simple consequence of the covariant energy-momentum conservation at the vertex m′→m+y, evaluated in the rest frame of the state with mass m′. In the rest frame of m, we have instead of (1), hv=(m′2−m2)/2mc. It is also clear that equation (1) can be written formally as hv′ = E′ − E, in the rest frame of m′2, because E′ = m′ in the rest frame of m′ and because the correct equation is hv′ = m′−E, all quantities in the rest frame of m′. If the emitted quantum is not the light quantum but any system (or system of particles) with the total invariant mass μ we obtain, again by the energy-momentum conservation at the vertex m′−m+μ, instead of (1), illustration where hv′ is the total relativistic energy of the emitted quantum. Open image in new window Open image in new window

Phase-Integral Approximation in Momentum Space and the Bound States of an Atom

The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrödinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman-type functional integral which was recently proposed by Garrod for the energy-momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound-state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.