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Journal of Experimental and Theoretical Physics

Radiative properties of diamagnetic levels in atoms: Dependence of transition probability on magnetic field strength

V. D. Ovsyannikov, V. V. Chernushin

Published Oct 1, 1999• 2 citations
Publisher

Abstract

We study the effect of diamagnetic interaction on the probability of radiative transitions of an atom from states split by a field. We write the analytic expressions for the diamagnetic corrections to the matrix elements of transitions belonging to the Lyman and Balmer series and of transitions between arbitrary nondegenerate states in hydrogen. We also discuss the perturbation theory for transitions from degenerate diamagnetic states. The theory is based on expanding in powers of the field strength the eigenfunctions and eigenvalues of the matrix of diamagnetic interaction in the subspace of states with given principal and magnetic quantum numbers. The field changes the coefficients in both the superposition and the degenerate basis. To derive the analytic expressions for the higher-order matrix elements, we use the Sturm expansion of the reduced Coulomb Green’s function. We also elaborate on the features of the frequency dependence of the corrections to the radiative matrix elements, which correlate with the structure of the diamagnetic spectrum of excited levels. Finally, we establish that the magnetic field acts selectively on the diamagnetic components of emission lines: as the field strength increases, an increase in the intensity of certain lines is accompanied by a decrease in the intensity of the other lines.

DOI
10.1134/1.559022

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We study the effect of diamagnetic interaction on the probability of radiative transitions of an atom from states split by a field. We write the analytic expressions for the diamagnetic corrections to the matrix elements of transitions belonging to the Lyman and Balmer series and of transitions between arbitrary nondegenerate states in hydrogen. We also discuss the perturbation theory for transitions from degenerate diamagnetic states. The theory is based on expanding in powers of the field strength the eigenfunctions and eigenvalues of the matrix of diamagnetic interaction in the subspace of states with given principal and magnetic quantum numbers. The field changes the coefficients in both the superposition and the degenerate basis. To derive the analytic expressions for the higher-order matrix elements, we use the Sturm expansion of the reduced Coulomb Green’s function. We also elaborate on the features of the frequency dependence of the corrections to the radiative matrix elements, which correlate with the structure of the diamagnetic spectrum of excited levels. Finally, we establish that the magnetic field acts selectively on the diamagnetic components of emission lines: as the field strength increases, an increase in the intensity of certain lines is accompanied by a decrease in the intensity of the other lines.
Under the conditions of the total Paschen-Back effect the diamagnetic interaction determines the dependence of the intensity of the Zeeman components of atomic radiation lines on a magnetic field. The change in the matrix elements of the radiative transitions is due to the magnetically induced corrections to the wavefunctions of the initial and final states, whose contributions are of the same order of magnitude for the head lines of the optical series. For the high-frequency lines the positive corrections to the matrix element from the wavefunction of the upper level dominate. A magnetic field also induces dipole radiative transitions with selection rules for the orbital angular momentum | Δ l | ≥ 3. The matrix elements of such transitions increase rapidly with the energy of the upper level, making possible efficient single-photon population of the dipole-inaccessible Rydberg states in moderate magnetic fields.
Second-order contributions of diamagnetic interaction to the energy shift and splitting are studied in the basis of the angular momentum eigenfunctions for two kinds of Rydberg states in atoms: (i) hydrogen-like degenerate states and (ii) nondegenerate Rydberg states with small magnetic quantum number, m ? 3, in many-electron atoms. General formulae are presented for calculations of higher order energy corrections in degenerate states with the use of reduced Green's function. The analytical expressions are derived for the second-order radial matrix elements of operator r2 and for the second-order diamagnetic susceptibilities of hydrogen-like states with m ? n-5. The quasiclassical quantum defect method is used to calculate the irreducible components ?nl(p) of the diamagnetic susceptibility ?nlm(2) for Rydberg states of alkali atoms. The numerical results are presented for susceptibilities of degenerate hydrogen substates, and for susceptibilities of alkali atoms in Rydberg s-, p-, d-states. The asymptotic dependence of susceptibilities in alkali atoms on the effective principal quantum number ? is determined numerically and the deviation from that of the hydrogen-like states is discovered. The data is also presented for the scaling parameters cl(p), determining the irreducible parts of the diamagnetic susceptibilities in states with high ?, according to the asymptotic formula ?nl(p) = cl(p)?11.
Analytical expressions for the dependence of the intensity of Zeeman components of doublet lines on the magnetic field are obtained. Sharp changes of these function on passing from the anomalous Zeeman effect to the Paschen-Back effect lead to the disappearance of marginal lines and the equalization of intensities of remaining lines. In the region of the complete Paschen-Back effect, a strong influence on these dependences is produced by the dynamic atom-field interaction, which weakens the paramagnetic effect in the states with a positive magnetic quantum number m and enhances the effect in the states with a negative m . Simple analytical expressions are obtained that take into account the effect of the diamagnetic interaction on line intensities. The role of the diamagnetic interaction increases in Rydberg atomic states with a large spin-orbit splitting. For the states with m > 0, it can lead to the “diamagnetic reversal” of the Paschen-Back effect, i.e., the recovery of the anomalous Zeeman effect.
The wavefunctions, matrix elements, and probabilities of radiative transitions between Stark sub-levels of atomic multiplets are calculated as a function of the strength of a dc electric field. The general expressions for the wavefunction of a multiplet state in a field obtained by perturbation theory for close-lying levels with the use of completely reduced Green’s function allow one to determine the field dependence of both the dipole-allowed and dipole-forbidden radiation amplitudes. A decomposition of the second-order amplitude for the transition between fine-structure sublevels of two levels of equal parity into irreducible components is obtained. Numerical calculations of the probabilities of radiative transitions between triplet states of helium show the possibility of experimental observation of the emergence and vanishing of Stark lines of radiative transitions in the vicinity of anticrossing fields.
General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.
Analytical dependences on a static electric field F 0 are derived for the wave functions, matrix elements and probabilities of radiation transitions between multiplet substates interacting in field in pairs, ranging from ordinary doublet states with spin S =1/2, pair-wise interacting sublevels of triplet and quintet states with the magnetic quantum number M =0, to the most general case of states with arbitrary angular L and spin S momenta and maximal magnitude of the magnetic quantum number | M |= L + S -1. Equalization of doublet line intensities in the anticrossing field region and vanishing of one of the two doublet lines in the high-field regime are demonstrated. A general relation is determined between the anticrossing field F A , multiplet splitting and tensor polarizability.