Total Transition Probability Research Articles

Coherent-state representation of a charged relativistic boson field

An explicit construction of a complete coherent-state representation of a charged relativistic boson field is given. We apply the results to the emission of soft, charged, massless bosons in a scattering process. We obtain a closed expression for the total transition probability for the source to create radiation in a given energy interval and in the case of no self-interaction among the bosons.

Collisional radiative destruction of some atomic excited states in an argon glow discharge

Destruction mechanisms of 3p2, 3p6, 2p2 and 2p3 (Paschen notation) Ar I excited states are investigated in an argon glow discharge by means of a laser perturbation method. Total radiative transition probabilities (in good agreement with accepted values) and thermally averaged cross sections for de-excitation by collisions with ground-state atoms are deduced from measured quenching rates for different discharge conditions.

Charged-particles scattering in the presence of a homogeneous magnetic field

We study the scattering of charged particles in the presence of a homogeneous magnetic field. Using the Green's function formalism, an appropriate transition amplitude for the scattering process is defined, and application is made on scattering by a Coulomb potential in the high-energy approximation. For this case, the transition amplitude is obtained in a closed form; its behavior with the magnetic field intensity and initial translational energy is qualitatively discussed for transitions from the first Landau level. In the ultrastrong-field limit, the total transition probability presents periodic threshold structures with increasing values of the initial translational energy.

Calculation of dynamically induced electronic transitions of matrix-isolated atomic nitrogen

The parity-forbidden transitions 2P→2D and 2D→4S within the ground configuration (2p)3 of atomic nitrogen have been observed as phonon- and vibron-induced emission bands in luminescence spectra of solid nitrogen and have been attributed to dipole radiation induced by dynamic crystal fields of odd parity with respect to the substitutionally trapped atom in the N2 lattice. We confirm this interpretation by an a priori calculation of the absolute transition probabilities and band shapes, using known atomic, molecular, and crystal properties of nitrogen and no adjustable parameters. The induced dipole moments are calculated from the vibrationally modulated quadrupole fields of the host molecules at the site of the polarizable guest atom. The dynamics of the doped lattice are treated by a normal-coordinate analysis of the ’’supermolecule’’ consisting of the atom and its 18 nearest radially relaxed N2 neighbors suspended in the unperturbed surrounding N2 lattice. The calculated dipole-weighted densities of phonon states and the resulting total transition probabilities agree well with observations. The corresponding emissions in rare-gas matrices are represented via superpositions of empirical pair overlap dipole moments.

Theoretical radiative lifetime of the metastable c3Πu state of H2

Magnetic dipole and electric quadrupole interactions are considered here as the mechanism for the forbidden radiative transition from the metastable c3Πu state of H2 to the dissociative b3Σu+ state. The discrete vibrational wavefuncion for the c3Πu state is generated from Browne’s potential. The continuum vibrational wavefunction for the b3Σu+ state is generated from an exponential type pontential function, which fits closely with Kolos and Wolniewicz’s potential. By virtue of this exponential type potential function, the phase integral and the exponential integral of the continuum vibrational wavefunction become integrable and an analytical expression is given for the vibrational wavefunction of b3Σu+. The electronic wavefunctions, used here to calculate the transition moments, are two-center wavefunctions previously derived by Chiu (for c3Πu) and by Hirschfelder and Linnett (for b3Σu+). It is found that the translational expansion of the magnetic dipole operator (originally at the center of mass) at the atomic centers generates an additional gradient operator, which contributes significantly towards the magnetic dipole transition probability. From the total transition probability which is a sum of that due to magnetic dipole transition and that due to electric quadrupole transition, a lifetime of τ=1.76 msec is calculated for the lowest vibrational level of the c3Πu state. For a comparison, the δ-function approximation is used for the continuum vibrational wavefunction. This approximation gives rather good agreement with the accurate calculation.

Eigenstates and radiative transition probabilities for Tm 3+ (4f 12) in phosphate and tellurite glasses

Electronic wavefunctions of Tm 3+ in intermediate coupling were obtained and used to calculate the U (λ) matrix elements between all possible states of the 4f 12 configuration. The Judd-Ofelt intensity parameters Ω λ obtained for Tm 3+ in phosphate and tellurite glasses were used in conjunction with the U (λ)'s to calculate the forced electric dipole line strengths. The total electric and magnetic transition probabilities are calculated. The entire theoretical spectrum involving the ground and excited levels (from 129 nm to 16447 nm) is given.

Exact long-term total transition probability rates and Fermi's 'golden rules'

Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.

New measurements of SVL lifetimes and quantum yields in benzene vapor: Herzberg-Teller comparisons of radiative transition probabilities

The forbidden 1 B 2u − 1 A 1g (S 1 −S 0 ) radiative transition probabilities for seven single vibronic levels (SVL) in 1 B 2u benzene vapor have been determined from new measurements of SVL fluorescence yields and lifetimes. New yields are also given for 17 other levels in C 6 H 6 and 8 levels in C 6 D 6 . The radiative probabilities are each sums of many single channel probabilities representing the various pathways possible by first- and higher-order Herzberg-Teller terms. From these sums, the value for the dominant first-order pathway (via the e 2g mode ν 6 ) has been isolated by SVL fluorescence band intensity measurements which reveal the fraction of the total SVL transition probability contributed by this radiative channel. The fraction is variable between 70 and 24 percent, but decreases nearly monotonically as the SVL energy increases. The single channel radiative transition probabilities are compared on a relative basis with the predictions of Herzberg-Teller theory. Substantial deviations from prediction are found. These deviations become most severe for high levels where the observed probabilities fall well below the theoretical values. A number of possible causes are considered; none can be shown definitely to account for the deviations.

Lower and upper bounds for time-smoothed total transition probabilities and their rates

A formalism of time-smoothed total transition probabilities and their rates is developed which employs Laplace averages of these quantities. Under conditions pertinent to scattering processes, the Laplace-average formalism is shown to yield results equivalent to those obtained from a stationary-state formalism. Rigorous lower and upper bounds are obtained for the Laplace-averaged quantities which reduce to equalities for two-level systems. The lower bounds appear to be potentially useful estimating lower bounds for total cross sections of various processes.

Quantum-mechanical impulse approximation for vibro-rotational excitation in atom-diatom collisions

The quantum-mechanical impulse approximation (IA) has been successfully applied to vibrational and rotational excitation in atom-diatomic-molecule collisions. An additional closure approximation used previously to calculate total vibrational transition probabilities is discussed. A comparison with exact IA calculations shows that the closure approximation overestimates large vibrational transitions. This discrepancy is due to the fact that the vibrational transitions are accompanied by large rotational excitations.

Semiclassical close-coupling calculations for rotational transitions in polar diatom-atom collisions

An approximate semiclassical treatment of rotational transitions in the close-coupling domain has been applied to the system Ar-CO. The distortion of the curved planar trajectory is accounted for by the use of the weighted mean energy and angular momentum in the equations of motion. The close-coupling between several rotational levels is considered in the higher order matrix terms and the total transition probability from a given state never exceeds unity. Cross sections were computed at several energies and different values of the repulsive anisotropy parameter. The results bring out some interesting features of rotational transitions in atom-diatom collisions and in particular the important role of the repulsive anisotropy in the interaction potential.

The kinetics of donor-acceptor pair transitions with strong phonon coupling in GaP

Donor-acceptor pair luminescence in GaP was studied by time-resolved spectroscopy, by measuring and analysing the integral band decay over many decades of intensity and time and by measuring and analysing the temperature dependence of the luminescence intensity. We report here the results obtained with some or all of these techniques for pairs involving deep as well as shallow centres. Those with a deep centre are: S P-Si P and Si Ga-Si P, in which the acceptor is the deep centre, and O P-C P and O P-Zn Ga, in which the donor is the deep centre. These pairs all have a broad, phonon-dominated luminescence band. The pairs involving shallow centres include Si Ga-C P, S P-C P and Te P-C P. Due to the different way of momentum conservation in these pairs, those involving a donor on a Ga-site have strong phonon co-operation and weak zero-phonon (ZP) transitions, whereas those with a donor on a P-site have strong ZP transitions. With time-resolved spectroscopy, well-resolved structure, due to a ZP pair band and its phonon replicas, is obtained for nearly all pairs. A comparison is made with the structure obtained by measuring in the stationary state at very low excitation densities. Some trends in the strength of phonon co-operation are noted. In the case of Si Ga-C P pairs, sharp replicas of ZP pair lines with a momentum-conserving phonon are also resolved. Using the integral band decay technique and results from time-resolved spectroscopy we have determined the transition probabilities for the total luminescence band as well as for the ZP band for nearly all pairs mentioned. The influence of incomplete saturation of the pairs during excitation with a focussed beam of an argon-ion laser is discussed and approximately corrected for. Extrapolated to zero pair separation, the total transition probability ranges from ∽ 10 5s -1 for Si Ga-C P pairs to ∽ 15 × 10 5s -1 for O P-C P and O P-Zn Ga pairs. The ZP probability ranges from ∽ 4 × 10 3s -1 for Si Ga-C P and Si Ga-Si P pairs to ∽ 4 × 10 5s -1 for S P-C P pairs. The transition probability for pairs in GaP is discussed in relation to the site occupied by the donor and the depth of the centres involved. The relatively high dope concentrations used in some cases, ∽ 1 to 2 × 10 18cm -3, are discussed briefly in relation to “concentration quenching”. The temperature dependence of the luminescence intensity of S P-Si P pairs is satisfactorily analysed with a simple linear model. The same model is applied to similar literature data on the S P-C P pair luminescence. Using in addition data on the average pair transition probability, trapping rate constants for hole capture by the Si P and C P acceptors were calculated to be ∽ 10 -9 cm 3s -1. Evidence is presented that the trapping rate constant for holes by the Zn Ga-O P complex is about 2 × 10 -9 cm 3s -1.

Spectral shape and nuclear matrix elements in the β-decay of 206Tl

The spectral shape, endpoint energy and half-life of the β-decay of 206Tl were measured. The β-particles were analysed in a 4π Si(Li) sandwich spectrometer. The sources used were isotopically separated and therefore extremely thin. The measured spectrum was corrected for the escape of bremsstrahlung. The extracted shape factor shows only small deviations from the statistical form, the endpoint energy of the decay is 1527 ± 4 keV, its half-life 4.27 ± 0.05 min. The nuclear matrix elements of this 0 − → 0 + decay have been calculated from single-particle wave functions. The resulting theoretical spectral form describes the experimental data very well, whereas the total transition probability is underestimated theoretically by a factor of 7.

Excitation of the CO Fourth Positive Band System by Electron Impact on Carbon Monoxide and Carbon Dioxide

Absolute excitation cross sections were measured for five vibrational bands of the CO(A 1Π–X 1Σ+) Fourth Positive band system. The bands were produced by electron impact on CO and CO2 and the cross sections were measured from threshold to 350 eV. Relative intensity measurements on 28 bands were used to determine the dependence of Re on the r centroid, which is Re = 1.9(1.0 − 0.6r̄υ′υ″). Absolute transition probabilities were computed using this functional form for Re and normalizing the total transition probability of the υ′ = 2 level to published experimental results. The total f-value for this transition is then 0.15. Absolute excitation cross sections for the first five vibrational levels (υ′ = 0 → 4) of the A 1Π state were determined for each parent gas using the measured cross sections of specific bands and the absolute transition probabilities. When CO was the parent gas, the relative populations of these vibrational levels were proportional to the Franck–Condon factors, qυ′0. The sum of the cross sections for these levels is 1.87 × 10−17 cm2 ± 15% at 300 eV. This represents 87.4% of the total excitation cross section for the A 1Π state. The relative populations of the first five vibrational levels were also determined for dissociative excitation of CO2 and they differed from the relative populations which resulted from direct excitation of CO. The sum of the cross sections for these five levels is 1.4 × 10−18 cm2 ± 17% at 300 eV.

Lifetimes and transition probabilities for some Fe II levels by the beam-foil method

The lifetimes of five levels in Fe II have been determined by the beam-foil method. Measurements were made at several energies between 200 and 400 keV using both 5 and 10 μg/cm 2 carbon foils. The various results for a given level agree to within ±8% and show no significant variation with energy. There is good agreement with the results of Andersen. The experimental total transition probability, or reciprocal of the lifetime, was compared to the transition probability calculated from the oscillator strengths of Warner, corrected by using the normalization function of Warner and Cowly. There is disagreement by roughly a factor of three which would be significant in solar iron abundance calculations based on Fe II. It will be necessary to determine accurate branching ratios for all the transitions which depopulate the upper-levels studied in order to resolve this discrepancy.

Theory of non-radiative transitions in molecules

Reports a study of the effect on photon absorption cross sections of non-radiative transitions (damping) in molecules. The analysis is carried out in the Heisenberg representation in which the time dependence of the dynamical variables used is that given by Senitzky's theory of the damped quantum oscillator and the damped two-level system. The formulation of the non-radiative problem presented is used to study the effect of damping of molecular vibrations, damping of vibronic states, and internal conversion or intersystem crossing followed by radiative re-emission-fluorescence or phosphorescence. In all cases examined the effect of damping appears as a Lorentzian line shape. Integration over the envelope of Lorentzian energies yields the total transition probability for a photon absorption.

Possible Characterization of Short-Range Interactions

In the preceding paper [J. Math. Phys. 10, 2047 (1969)], we investigated possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2 which were valid only for certain types of sequences of wavepackets. In this paper we generalize the assumption on the transition probability so that it will apply to an arbitrary wavepacket. We do this by utilizing a measure G[R, ψ] of how much of a wavepacket ψ is within a sphere of radius R about the scattering center at all times. We then characterize a ``short-range'' interaction by the existence of a positive constant A and positive functions B(p) and C(p) such that for all R ≥ 0, ‖Tψ‖2≤AG[R,ψ]+〈C(p)e−RB(p)〉ψ.We then show that for such interactions the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse the size of which depends upon B(p). We compare our results with the similar results obtained by the method in the preceding paper and show that these new results much more closely approximate the behavior of the Yukawa interaction.

Short-Range Interactions and Analyticity in Momentum Transfer

We investigate possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2. We desire to require that the total transition probability decrease exponentially as the displacement of the wavepacket ψ increases. It is shown that this can only be done for certain types of sequences of wavepackets, and a criterion is developed for selecting these ``proper'' wavepackets. We then show for all such interactions that what is essentially the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse which always includes the physical region. We then compare these results with the Schrödinger potential theory and thereby relate the type of the exponential decrease of a potential to the type of the exponential decrease of the corresponding transition probability. Finally, our results are compared with similar results previously obtained by others working on this problem.

Radiative Lifetimes and Total Transition Probabilities for Some Polyatomic Species

Radiative lifetimes have been measured in CS2+, SO2, C2N2, and N2O+ systems using the phase-shift technique and electron-beam excitation. The measured lifetimes ranged from 12.2 ± 2.5 nsec for the SO2 emission in the 2300–2700-Å range to 294 ± 30 nsec for the CS2+ B 2Σu+−X 2Πg−(i) transition. The transitions ascribed to SO2 and C2N2 show complications due to radiative cascading, while the CS2+ and N2O+ transitions observed were cascade free. These new data are discussed in light of prior spectroscopic results.

Lifetimes and Total Transition Probabilities for NH, SiH, and SiD

Radiative lifetimes have been measured for the NH (d 1Σ–c 1Π), (c 1Π–a 1Δ), (c 1Π–b 1Σ), and (A 3Π–X 3Σ) systems, and for the SiD and SiH (A 2Δ–X 2Π) transitions. The phase-shift technique yielded lifetimes ranging from 10 nsec for the NH (d 1Σ–c 1Π) transition to 700 nsec for the SiH (A 2Δ–X 2Π) band. All the lifetimes of upper states measured showed the effects of radiative cascading. In particular, the c 1Π state of NH shows radiative cascading originating at least in part from the observed d 1Σ state. The measured d 1Σ state lifetime is near the nominal lifetime computed for the state cascading into the c 1Π state and is considerably shorter than the c 1Π state lifetime. The NH (A 3Π–X 3Σ) and the SiD and SiH (A 2Δ–X 2Π) transitions showed cascading that arises from states which have not been spectroscopically identified. All the systems studied, with the exception of the NH (d 1Σ–c 1Π) transition, showed cascading in which the cascading state lifetime was shorter than the measured state lifetime. This situation has not been completely discussed previously and is given consideration as to the effect on themeasured phase shifts and calculated radiative lifetimes.