An expansion method for calculating atomic properties. VIII. Transitions in the Hartree-Fock approximation

1966 ◽  
Vol 293 (1434) ◽  
pp. 359-364 ◽  
Keyword(s):  
Quantum Number ◽  
Principal Quantum Number ◽  
Expansion Method ◽  
Matrix Elements ◽  
Order Error ◽  
First Order ◽  
Hartree Fock ◽  
Transition Matrix Elements ◽  
Active Electron

According to Brillouin’s theorem, diagonal matrix elements of one-electron operators are given correct to first order by the Hartree-Fock approximation. Transition matrix elements of one-electron operators are not given correct to first order because of the contributions from virtual excitations involving a change in the azimuthal quantum number of the active electron but no change in the principal quantum number. The electric dipole matrix elements connecting the 1 s 2 s 1,3 S states of helium to the 1 s 2 p 1,3 P states are calculated exactly to first order in inverse powers of the nuclear charge and the differences from the Hartree-Fock approximation are shown explicitly to correspond to virtual transitions of the inner shell 1 s electron of the type 1 s — np . A generalization of the Hartree-Fock approximation is discussed which eliminates most of the first order error.

An expansion method for calculating atomic properties X. 1 s 2 1 S -1 snp 1 P transitions of the helium sequence

1967 ◽  
Vol 301 (1466) ◽  
pp. 253-260 ◽  
Keyword(s):  
Cross Sections ◽  
Nuclear Charge ◽  
Expansion Method ◽  
Probable Error ◽  
Oscillator Strengths ◽  
Matrix Elements ◽  
Isoelectronic Sequence ◽  
First Order ◽  
Hartree Fock ◽  
Atomic Properties

The electric dipole matrix elements connecting the 1 s 2 1 S and 1 snp 1 P states of the helium isoelectronic sequence are calculated exactly to first order in inverse powers of the nuclear charge Z and the differences from the Hartree-Fock approximation are shown to correspond to virtual transitions of the 1 s electrons. Comparison of the oscillator strengths predicted by a screening approximation with more accurate values reveals a regular variation in the error contained in the screening approximation, the correction of which allows the prediction of oscillator strengths and probabilities of 1 s 2 1 S – 1 snp 1 P transitions for all values of n and all values of Z within a probable error of 2% (table 5). Values of the photoionization cross-sections at the spectral heads are also presented.

scholarly journals Nuclear transition matrix elements for neutrinoless double-β decay within mechanisms involving light Majorana neutrino mass and right-handed current

2019 ◽  
Vol 28 (01n02) ◽  
pp. 1950001
Author(s):  
Yash Kaur Singh ◽  
R. Chandra ◽  
K. Chaturvedi ◽  
Tripti Avasthi ◽  
P. K. Rath ◽  
...  
Keyword(s):  
Neutrino Mass ◽  
Transition Matrix ◽  
Weak Coupling ◽  
Majorana Neutrino ◽  
Nuclear Transition ◽  
Matrix Elements ◽  
Β Decay ◽  
Hartree Fock ◽  
Transition Matrix Elements ◽  
Majorana Neutrino Mass

Employing the projected-Hartree-Fock-Bogoliubov (PHFB) model in conjunction with four different parametrizations of pairing plus multipolar effective two-body interaction and three different parametrizations of Jastrow short-range correlations, nuclear transition matrix elements for the neutrinoless double-[Formula: see text] decay of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] isotopes are calculated within mechanisms involving light Majorana neutrino mass and right-handed current. Statistically, model specific uncertainties in sets of twelve nuclear transition matrix elements are estimated by calculating the averages along with the standard deviations. For the considered nuclei, the most stringent extracted on-axis limits on the effective light Majorana neutrino mass [Formula: see text], the effective weak coupling of right-handed leptonic current with right-handed hadronic current [Formula: see text], and the effective weak coupling of right-handed leptonic current with left-handed hadronic current [Formula: see text] from the observed limit on half-life [Formula: see text] of [Formula: see text] isotope are [Formula: see text][Formula: see text]eV, [Formula: see text] and [Formula: see text], respectively.

An expansion method for calculating atomic properties IV. Transition probabilities

1964 ◽  
Vol 280 (1381) ◽  
pp. 258-270 ◽  
Keyword(s):  
Wave Function ◽  
Matrix Element ◽  
Transition Probabilities ◽  
Expansion Method ◽  
The State ◽  
Diagonal Matrix Element ◽  
Expectation Value ◽  
First Order ◽  
Hartree Fock ◽  
Atomic Properties

An earlier expression for the expectation value of a single-electron operator which isstationary with respect to first-order variations of the state wave function has been generalized to the case of an off-diagonal matrix element connecting two different states. Explicit calculations are carried out of the probabilities of dipole transitions between configurations 1 s a 2 s b 2 p c and 1 s a 2 s b–1 2 p c+1 for all members of the isoelectronic sequences from helium to neon and the importance of taking into account the mixing of degenerate configurations is demonstrated. The accuracy is at least comparable to that of the Hartree-Fock approximation and in cases where degeneracy is important it is much superior.

Spin–spin interaction in f5 electron configurations

1969 ◽  
Vol 47 (24) ◽  
pp. 2779-2783 ◽  
Author(s):  
Gulzari Malli ◽  
K. M. S. Saxena
Keyword(s):  
Order Perturbation ◽  
Spin Splitting ◽  
Spin Interaction ◽  
Electron Configuration ◽  
Matrix Elements ◽  
First Order ◽  
Hartree Fock ◽  
Splitting Factor ◽  
Complete Matrix ◽  
First Order Perturbation

The complete matrix of the spin–spin interaction for the f5 electron configuration is evaluated for general usage. The results of the diagonal matrix elements are tabulated here in terms of the corresponding spin–spin splitting factor for usage in the first-order perturbation calculations. These are used to evaluate the Hartree–Fock values of the splitting factors for Pm(4f5), Sm3+ (4f5), Tb(4f9), and Dy3+ (4f9).

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: III. DIPOLE TRANSITIONS IN THREE-ELECTRON SYSTEMS

1967 ◽  
Vol 45 (5) ◽  
pp. 1661-1673 ◽  
Author(s):  
Maurice Cohen ◽  
Paul S. Kelly
Keyword(s):  
Excited States ◽  
Wave Functions ◽  
Electron Systems ◽  
Matrix Elements ◽  
Isoelectronic Sequence ◽  
Hartree Fock ◽  
Transition Matrix Elements ◽  
Simplified Procedure ◽  
Good Agreement ◽  
Dipole Length

Hartree–Fock wave functions for a number of S, P, and D states of the lithium isoelectronic sequence have been calculated, using a simplified procedure described in an earlier paper. Transition matrix elements for all permitted dipole transitions between these states have been computed using both the dipole length and the dipole velocity formulations. The results are in good agreement with earlier calculations.

Nuclear transition matrix elements for neutrinoless double-β decay of 76Ge isotope within PHFB approach

2019 ◽  
Vol 28 (11) ◽  
pp. 1950096
Author(s):  
P. K. Rath ◽  
A. Kumar ◽  
R. Chandra ◽  
R. Gautam ◽  
P. K. Raina ◽  
...  
Keyword(s):  
Transition Matrix ◽  
Tensor Decomposition ◽  
Nuclear Transition ◽  
Matrix Elements ◽  
Spin Tensor ◽  
Β Decay ◽  
Hartree Fock ◽  
Transition Matrix Elements ◽  
Majorana Neutrinos ◽  
Double Β Decay

Employing projected–Hartree–Fock–Bogoliubov (PHFB) model, nuclear transition matrix elements (NTMEs) [Formula: see text] for the neutrinoless double-[Formula: see text] decay of [Formula: see text]Ge isotope are calculated within mechanisms involving light as well as heavy Majorana neutrinos, and classical Majorons. By considering the spin-tensor decomposition of realistic KUO and empirical JUN45 effective two-body interactions, it is noticed that the effect due to SRC on NTMEs [Formula: see text] and [Formula: see text] involving the exchange of light and heavy Majorana neutrinos, respectively, is maximally incorporated by the central part of the effective two-body interaction, which varies by a small amount with the inclusion of spin-orbit and tensor components. Presently, the model-dependent uncertainties in the average NTMEs [Formula: see text] and [Formula: see text] turn out to be about 10% and 37%, respectively.

Comparison of multiple scattering approximations in a three body problem

1978 ◽  
Vol 56 (10) ◽  
pp. 1365-1371
Author(s):  
V. S. Kulhar ◽  
A. K. Jain ◽  
C. S. Shastry
Keyword(s):  
Multiple Scattering ◽  
Delta Function ◽  
Equal Mass ◽  
One Dimension ◽  
Matrix Elements ◽  
Three Body Problem ◽  
Mechanical Problem ◽  
First Order ◽  
Transition Matrix Elements ◽  
Three Body

An analysis of the nonrelativistic quantum mechanical problem of three particles of equal mass interacting via attractive delta function potentials in one dimension is carried out. Following the methods of McGuire and Dodd with two body potentials of equal strength, exact matrix elements for the process of elastic and rearrangement scattering are rederived and some mistakes occurring in their papers have been corrected. Then expressions are obtained for transition matrix elements for these processes in the first order using Newton, Born, Newton–Faddeev, and Lovelace–Faddeev formulation. Numerical results obtained are compared with the exact results. It is found that above breakup threshold the Newton–Faddeev scheme gives better results in comparison to other schemes for elastic as well as rearrangement scattering. Thus, this paper provides us with a comparative analysis of several multiple scattering expansions available in the literature.

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