Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Resonance scattering of Lyman- α radiation by hydrogen in the ground state

H. E. Moses

Published Nov 1, 1980• 1 citation
Publisher

Abstract

We calculate the cross section for the resonance scattering of Lyman-$\ensuremath{\alpha}$ radiation by spinless nonrelativistic hydrogen atoms in the ground state using a two-level model. A generalization of Dirac's resonance scattering theory is used together with the exact matrix elements for the electromagnetic interaction. In contrast to the usual treatments in which only the dipole approximation for the matrix elements are taken, the shift in position of the resonance is finite and has a value of 75% of the Lamb shift of the $n=1$ state. Whether this latter fact is significant in renormalization calculations is left open.

DOI
10.1103/physreva.22.2069

Related papers

Quantum Field Theory Description of the Hydrogen Lyman α Transition: From QED to the Bohr–Maxwell Correspondence

Daniel Scretching

This paper presents a quantum-field-theoretic formulation of the hydrogen 2p → 1s Lyman-α transition, replacing the semiclassical Bohr–Maxwell description with the modern framework of quantum electrodynamics. The initial and final bound-electron states are described by the Dirac equation in the proton’s Coulomb field, with the Schrödinger equation recovered as the leading nonrelativistic limit. The electromagnetic radiation field is quantized, and photon emission is generated by the QED interaction Hamiltonian. The transition amplitude is evaluated from the atomic current and quantized electromagnetic field, leading through Fermi’s golden rule to the spontaneous-emission rate and Einstein coefficient A₂₁. Energy and momentum conservation require the emitted photon to carry the atomic transition energy, including the small correction associated with proton recoil. At leading order, the Lyman-α photon has ΔE ≈ 10.2 eV, ν ≈ 2.47 × 10¹⁵ Hz, and λ ≈ 121.6 nm. In the electric-dipole approximation, the field-theoretic calculation reduces to the standard matrix element of the interaction Hamiltonian H_int = −μ·E, thereby connecting QED directly to the transition dipole moment, oscillator strength, Einstein A coefficient, and radiative lifetime. The calculation reproduces the accepted numerical results for hydrogen while identifying their origin in the coupling between the quantized electron field and the quantized electromagnetic field. The paper also explains how classical electrodynamics is recovered from QED. Maxwell’s equations govern the expectation values of the electromagnetic-field operators in the appropriate classical-field limit, particularly for coherent states containing many photons. A single spontaneously emitted Lyman-α photon, however, remains a genuinely quantum excitation rather than a classical electromagnetic wave. Leading corrections beyond the nonrelativistic Coulomb approximation are then considered, including relativistic fine structure, the Lamb shift, and atomic recoil. The resulting treatment provides a rigorous bridge from the historical Bohr model and Schrödinger wave mechanics to Dirac theory and full QED, verifying the Lyman-α energy, frequency, wavelength, transition probability, and radiative decay structure within a unified quantum-field framework.

•Jul 12, 2026•0 citations

Angular Distribution of Lyman- α Radiation Emitted by H ( 2S ) Atoms in Weak Electric Fields

William L. Lichten

S>The angular distribution of the Lyman- alpha radiation emitted by hydrogen atoms in the 2S state upon the application of an electric field is studied by considering the 2P/sub 1/2/ levels; the radiation is found to be completely unpolarized and its angular distribution isotropic. The consequences of this result for the value of the absolute cross section for excitation of the 2S state of hydrogen by electron impact are discussed. (D.L.C.)

Physical Review Letters•Jan 1, 1961•24 citations

Shift and width of the Balmer-α and Lyman-β lines of neutral hydrogen due to electron collisions

J. Callaway, K. Unnikrishnan

The contribution from electron scattering to the width and shift of the ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ and ${\mathrm{Ly}}_{\mathrm{\ensuremath{\beta}}}$ lines of neutral hydrogen is evaluated. Elements of the S matrix for electron-atom scattering calculations are obtained from close-coupling calculations including pseudostates, either of the variational or the optical potential type. The convergence of sums over angular-momentum variables is discussed. Results are compared with some other calculations and with experiment where possible.

Physical Review A•Sep 1, 1991•6 citations

Resonance Scattering of Light from Atomic Hydrogen

Lewis C. Himmell, Peter R. Fontana

The angular distribution of scattered Lyman-$\ensuremath{\alpha}$ photons from atomic hydrogen is calculated as a function of an applied magnetic field. The intensity of the scattered radiation is a function of the energy difference of the excited-$2P$ levels and shows an extremum at the point where two levels cross. Neglecting hyperfine structure, there are two level crossings of interest: one at 3484 G ($2{P}_{\frac{3}{2,\ensuremath{-}\frac{3}{2}}}\ensuremath{-}2{P}_{\frac{1}{2},\frac{1}{2}}$), and the other 5223G ($2{P}_{\frac{3}{2,\ensuremath{-}\frac{3}{2}}}\ensuremath{-}2{P}_{\frac{1}{2},\ensuremath{-}\frac{1}{2}}$), respectively. The coher$\stackrel{\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}{\mathrm{e}}$nt and incoherent resonance scattering is determined near these crossing points by using a Hamiltonian that includes hyperfine structure and higher-order radiative effects. In the low-field region (3484 G), the hyperfine crossings that contribute to the coherent signal are at 3476.06 and 3491.93 G, respectively. The radiative width of the levels is considerably larger than the hyperfine splittings, and the total scattering intensity has a single peak at 3483.97 G. In the high-field region (5223 G), two hyperfine levels with ${m}_{F}=\ensuremath{-}1$ repel each other (anticross) with a minimum energy separation of 15.3 Mc/sec at 5200.1 G. Resonances occur at 5220.6 and 5223.9 G, and the maximum of the total signal is at 5222.6 G.

Physical Review•Oct 5, 1967•9 citations

Two Photon Processes within the Hyperfine Structure of Hydrogen

Spencer Dylan Percy

The hyperfine levels for the ground state of hydrogen are normally connected by magnetic dipole transitions. However, for astrophysical applications, it may be that two-photon electric dipole processes are also important over cosmological distances. This thesis sets up a general mathematical formalism for the calculation of two-photon processes within the hyperfine structure of an atom with spin $1/2$, and applies it to the ground state of hydrogen. Rates are calculated for both spontaneous emission and absorption of radiation. Due to an extraordinary degree of cancellation in the matrix elements, the rates turn out to be too small to have a significant impact on astrophysical processes involving the absorption and emission of radiation. The results are nevertheless important in reducing the uncertainty due to possible contributions from these processes. Other two-photon processes involving Raman scattering are also considered and the relevant cross sections calculated. The calculations of the decay rate, absorption coefficient and Raman scattering cross section are extended to heavier hydrogenic ions as a function of the nuclear charge $Z$, using an approximate interpolation formula for the hyperfine splitting in the ground state. Results are presented for the hydrogenlike ions from He$^+$ to Pb$^{81+}$. Modifications for the case of isotopes with a nuclear spin greater than $1/2$ are briefly considered

•Jan 1, 2018•0 citations

Simple method for obtaining exact nonrelativistic formulas for two-photon (n, l, m) continuum dipole transitions for hydrogenic atoms

Adrian Costescu, I. F. Schneider, Ion Nicolae Mihailescu +1 more

The Kramers-Heisenberg matrix element for two-photon transitions between a bound (n, l, m) state and a continuum is obtained in terms of the Coulomb Green's function matrix element for the ground-state ? continuum transition. The number of the F(1) Appell functions involved is minimized. The analytical results for the ns ? continuum transitions are given, and both the resonant behavior and the transparency region of the two-photon cross section are predicted.

Optics Letters•Jul 1, 1986•11 citations
We calculate the cross section for the resonance scattering of Lyman-\ensuremathα\ensuremath{\alpha}\ensuremathα radiation by spinless nonrelativistic hydrogen atoms in the ground state using a two-level model. A generalization of Dirac's resonance scattering theory is used together with the exact matrix elements for the electromagnetic interaction. In contrast to the usual treatments in which only the dipole approximation for the matrix elements are taken, the shift in position of the resonance is finite and has a value of 75% of the Lamb shift of the n=1n=1n=1 state. Whether this latter fact is significant in renormalization calculations is left open.
This paper presents a quantum-field-theoretic formulation of the hydrogen 2p → 1s Lyman-α transition, replacing the semiclassical Bohr–Maxwell description with the modern framework of quantum electrodynamics. The initial and final bound-electron states are described by the Dirac equation in the proton’s Coulomb field, with the Schrödinger equation recovered as the leading nonrelativistic limit. The electromagnetic radiation field is quantized, and photon emission is generated by the QED interaction Hamiltonian. The transition amplitude is evaluated from the atomic current and quantized electromagnetic field, leading through Fermi’s golden rule to the spontaneous-emission rate and Einstein coefficient A₂₁. Energy and momentum conservation require the emitted photon to carry the atomic transition energy, including the small correction associated with proton recoil. At leading order, the Lyman-α photon has ΔE ≈ 10.2 eV, ν ≈ 2.47 × 10¹⁵ Hz, and λ ≈ 121.6 nm. In the electric-dipole approximation, the field-theoretic calculation reduces to the standard matrix element of the interaction Hamiltonian H_int = −μ·E, thereby connecting QED directly to the transition dipole moment, oscillator strength, Einstein A coefficient, and radiative lifetime. The calculation reproduces the accepted numerical results for hydrogen while identifying their origin in the coupling between the quantized electron field and the quantized electromagnetic field. The paper also explains how classical electrodynamics is recovered from QED. Maxwell’s equations govern the expectation values of the electromagnetic-field operators in the appropriate classical-field limit, particularly for coherent states containing many photons. A single spontaneously emitted Lyman-α photon, however, remains a genuinely quantum excitation rather than a classical electromagnetic wave. Leading corrections beyond the nonrelativistic Coulomb approximation are then considered, including relativistic fine structure, the Lamb shift, and atomic recoil. The resulting treatment provides a rigorous bridge from the historical Bohr model and Schrödinger wave mechanics to Dirac theory and full QED, verifying the Lyman-α energy, frequency, wavelength, transition probability, and radiative decay structure within a unified quantum-field framework.
The contribution from electron scattering to the width and shift of the ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ and ${\mathrm{Ly}}_{\mathrm{\ensuremath{\beta}}}$ lines of neutral hydrogen is evaluated. Elements of the S matrix for electron-atom scattering calculations are obtained from close-coupling calculations including pseudostates, either of the variational or the optical potential type. The convergence of sums over angular-momentum variables is discussed. Results are compared with some other calculations and with experiment where possible.
The angular distribution of scattered Lyman-$\ensuremath{\alpha}$ photons from atomic hydrogen is calculated as a function of an applied magnetic field. The intensity of the scattered radiation is a function of the energy difference of the excited-2P2P2P levels and shows an extremum at the point where two levels cross. Neglecting hyperfine structure, there are two level crossings of interest: one at 3484 G ($2{P}_{\frac{3}{2,\ensuremath{-}\frac{3}{2}}}\ensuremath{-}2{P}_{\frac{1}{2},\frac{1}{2}}$), and the other 5223G ($2{P}_{\frac{3}{2,\ensuremath{-}\frac{3}{2}}}\ensuremath{-}2{P}_{\frac{1}{2},\ensuremath{-}\frac{1}{2}}$), respectively. The coher$\stackrel{\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}{\mathrm{e}}$nt and incoherent resonance scattering is determined near these crossing points by using a Hamiltonian that includes hyperfine structure and higher-order radiative effects. In the low-field region (3484 G), the hyperfine crossings that contribute to the coherent signal are at 3476.06 and 3491.93 G, respectively. The radiative width of the levels is considerably larger than the hyperfine splittings, and the total scattering intensity has a single peak at 3483.97 G. In the high-field region (5223 G), two hyperfine levels with ${m}_{F}=\ensuremath{-}1$ repel each other (anticross) with a minimum energy separation of 15.3 Mc/sec at 5200.1 G. Resonances occur at 5220.6 and 5223.9 G, and the maximum of the total signal is at 5222.6 G.
The hyperfine levels for the ground state of hydrogen are normally connected by magnetic dipole transitions. However, for astrophysical applications, it may be that two-photon electric dipole processes are also important over cosmological distances. This thesis sets up a general mathematical formalism for the calculation of two-photon processes within the hyperfine structure of an atom with spin 1/21/21/2, and applies it to the ground state of hydrogen. Rates are calculated for both spontaneous emission and absorption of radiation. Due to an extraordinary degree of cancellation in the matrix elements, the rates turn out to be too small to have a significant impact on astrophysical processes involving the absorption and emission of radiation. The results are nevertheless important in reducing the uncertainty due to possible contributions from these processes. Other two-photon processes involving Raman scattering are also considered and the relevant cross sections calculated. The calculations of the decay rate, absorption coefficient and Raman scattering cross section are extended to heavier hydrogenic ions as a function of the nuclear charge ZZZ, using an approximate interpolation formula for the hyperfine splitting in the ground state. Results are presented for the hydrogenlike ions from He+^++ to Pb81+^{81+}81+. Modifications for the case of isotopes with a nuclear spin greater than 1/21/21/2 are briefly considered