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Physical Review

Resonance Scattering of Light from Atomic Hydrogen

Lewis C. Himmell, Peter R. Fontana

Published Oct 5, 1967• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.162.23

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Observation of Level Crossing in H, n=2

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We have observed the crossing of the atomic hydrogen levels ${\ensuremath{\beta}}_{B}(2{S}_{\frac{1}{2}}, {m}_{J}=\ensuremath{-}\frac{1}{2}, {m}_{I}=\ensuremath{-}\frac{1}{2})$ and ${e}_{B}(2{P}_{\frac{1}{2}}, {m}_{J}=+\frac{1}{2}, {m}_{I}=\ensuremath{-}\frac{1}{2})$ near 605 G. A new atomic-beam method is used. H atoms are excited to the metastable $2S$ state by electron bombardment. The $2S({m}_{J}=\ensuremath{-}\frac{1}{2})$ state is quenched in the bombardment region. The $2S({m}_{J}=+\frac{1}{2})$ beam passes through a zero-field region where $\frac{1}{3}$ of it is converted to the single hyperfine level ${\ensuremath{\beta}}_{B}$. The ${\ensuremath{\beta}}_{B}$ atoms enter a uniform magnetic field parallel to the beam. With no externally applied electric field, 92% of them reach the detector. When we apply an electrostatic field of about 0.7 V/cm perpendicular to the beam, the ${\ensuremath{\beta}}_{B}$ atoms are strongly quenched near the ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ crossing point. Observed quenching agrees with the Bethe-Lamb theory of the $2S$ lifetime in external fields. We measure magnetic field by the nuclear magnetic resonance frequency of protons in water. Natural asymmetry and minor experimental corrections reduce the observed center of the ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ quenching resonance by 1 part in ${10}^{4}$. The corrected ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ crossing occurs at 2577.57\ifmmode\pm\else\textpm\fi{}0.25 kc/sec, proton nmr in water. From this, we calculate the Lamb shift $\mathcal{S}$ in H, $n=2$. With a diamagnetic shielding correction, and the accurately known $2S$ hyperfine interval, we get $\mathcal{S}=1058.07\ifmmode\pm\else\textpm\fi{}0.10$ Mc/sec. This disagrees with $\mathcal{S}=1057.77\ifmmode\pm\else\textpm\fi{}0.10$ Mc/sec from the Lamb experiments.

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The crossed-level method of atomic spectroscopy is discussed and the angular distribution formula for both the incoherent and coherent resonance scattering is derived. The form of this distribution function, as given here, explicity displays the geometric factors depending on radiation propagation vectors. With the application to hydrogen in mind, the distribution function is expressed explicitly for single electron transitions with external fields possessing axial symmetry. The properties of the distribution function are discussed with emphasis on the case of unpolarized radiation. For the case of hydrogen there are two possible applications of major interest. The first concerns the possibility of a precision measurement of the $2p$ fine structure splitting and, hence, a determination of the fine structure constant. Explicit results for the shape and other properties of the resonance line with a uniform magnetic field obtained. The other application is concerned with the possibility of measuring the $2s\ensuremath{-}2p$ Lamb splitting. This requires an electric field parallel to the magnetic field. Unfortunately, the level crossings which are sensitive to the Lamb splitting cannot radiate sufficiently rapidly while those which do radiate appreciably occur at field strengths which are extremely insensitive to the Lamb splitting.

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Ulrich D. Jentschura, Vincent Debierre, Chandra Mani Adhikari +2 more

The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasidegenerate $2{P}_{1/2}$ levels that are removed from the $2S$ states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift, and the hyperfine effects. The van der Waals shift becomes commensurate with the $2S\ensuremath{-}2{P}_{3/2}$ fine-structure splitting only for close approach ($R<100{a}_{0}$, where ${a}_{0}$ is the Bohr radius) and one may thus restrict the discussion to the levels with $n=2$ and $J=1/2$ to a good approximation. Because each $S$ or $P$ state splits into an $F=1$ triplet and an $F=0$ hyperfine singlet (eight states for each atom), the Hamiltonian matrix a priori is of dimension 64. A careful analysis of the symmetries of the the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states that are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to $1/{R}^{3}$ and $1/{R}^{6}$ and are evaluated within the hyperfine manifolds. When both atoms are metastable $2S$ states, we find an interaction energy of order ${E}_{h}\ensuremath{\chi}{({a}_{0}/R)}^{6}$, where ${E}_{h}$ and $\mathcal{L}$ are the Hartree and Lamb shift energies, respectively, and $\ensuremath{\chi}={E}_{h}/\mathcal{L}\ensuremath{\approx}6.22\ifmmode\times\else\texttimes\fi{}{10}^{6}$ is their ratio.

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Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.

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Coherent Scattering of Photons by Atomic Hydrogen

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The angular distribution of scattered Lyman-\ensuremathα\ensuremath{\alpha}\ensuremathα 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 (2P32,\ensuremath−32\ensuremath−2P12,122{P}_{\frac{3}{2,\ensuremath{-}\frac{3}{2}}}\ensuremath{-}2{P}_{\frac{1}{2},\frac{1}{2}}2P2,\ensuremath−23​3​​\ensuremath−2P21​,21​​), and the other 5223G (2P32,\ensuremath−32\ensuremath−2P12,\ensuremath−122{P}_{\frac{3}{2,\ensuremath{-}\frac{3}{2}}}\ensuremath{-}2{P}_{\frac{1}{2},\ensuremath{-}\frac{1}{2}}2P2,\ensuremath−23​3​​\ensuremath−2P21​,\ensuremath−21​​), respectively. The cohere\ifmmode˘\else˘\fi\stackrel{\mathrm{\ifmmode \breve{}\else \u{}\fi{}}}{\mathrm{e}}e\ifmmode˘\else˘\fint 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 mF=\ensuremath−1{m}_{F}=\ensuremath{-}1mF​=\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.
We have observed the crossing of the atomic hydrogen levels ${\ensuremath{\beta}}_{B}(2{S}_{\frac{1}{2}}, {m}_{J}=\ensuremath{-}\frac{1}{2}, {m}_{I}=\ensuremath{-}\frac{1}{2})$ and ${e}_{B}(2{P}_{\frac{1}{2}}, {m}_{J}=+\frac{1}{2}, {m}_{I}=\ensuremath{-}\frac{1}{2})$ near 605 G. A new atomic-beam method is used. H atoms are excited to the metastable 2S2S2S state by electron bombardment. The $2S({m}_{J}=\ensuremath{-}\frac{1}{2})$ state is quenched in the bombardment region. The 2S(mJ=+12)2S({m}_{J}=+\frac{1}{2})2S(mJ​=+21​) beam passes through a zero-field region where 13\frac{1}{3}31​ of it is converted to the single hyperfine level ${\ensuremath{\beta}}_{B}$. The ${\ensuremath{\beta}}_{B}$ atoms enter a uniform magnetic field parallel to the beam. With no externally applied electric field, 92% of them reach the detector. When we apply an electrostatic field of about 0.7 V/cm perpendicular to the beam, the ${\ensuremath{\beta}}_{B}$ atoms are strongly quenched near the ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ crossing point. Observed quenching agrees with the Bethe-Lamb theory of the 2S2S2S lifetime in external fields. We measure magnetic field by the nuclear magnetic resonance frequency of protons in water. Natural asymmetry and minor experimental corrections reduce the observed center of the ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ quenching resonance by 1 part in 104{10}^{4}104. The corrected ${\ensuremath{\beta}}_{B}\ensuremath{-}{e}_{B}$ crossing occurs at 2577.57\ifmmode\pm\else\textpm\fi{}0.25 kc/sec, proton nmr in water. From this, we calculate the Lamb shift S\mathcal{S}S in H, n=2n=2n=2. With a diamagnetic shielding correction, and the accurately known 2S2S2S hyperfine interval, we get $\mathcal{S}=1058.07\ifmmode\pm\else\textpm\fi{}0.10$ Mc/sec. This disagrees with $\mathcal{S}=1057.77\ifmmode\pm\else\textpm\fi{}0.10$ Mc/sec from the Lamb experiments.
The crossed-level method of atomic spectroscopy is discussed and the angular distribution formula for both the incoherent and coherent resonance scattering is derived. The form of this distribution function, as given here, explicity displays the geometric factors depending on radiation propagation vectors. With the application to hydrogen in mind, the distribution function is expressed explicitly for single electron transitions with external fields possessing axial symmetry. The properties of the distribution function are discussed with emphasis on the case of unpolarized radiation. For the case of hydrogen there are two possible applications of major interest. The first concerns the possibility of a precision measurement of the 2p2p2p fine structure splitting and, hence, a determination of the fine structure constant. Explicit results for the shape and other properties of the resonance line with a uniform magnetic field obtained. The other application is concerned with the possibility of measuring the $2s\ensuremath{-}2p$ Lamb splitting. This requires an electric field parallel to the magnetic field. Unfortunately, the level crossings which are sensitive to the Lamb splitting cannot radiate sufficiently rapidly while those which do radiate appreciably occur at field strengths which are extremely insensitive to the Lamb splitting.
Breit has derived formulae giving the polarization P(H)P(H)P(H) and the angle of maximum polarization $\ensuremath{\theta}$ of a resonance line showing hyperfine structure, as a function of a weak magnetic field applied in the direction of observation of the resonance radiation. The formulae give a means of estimating the effect of hyperfine structure on the calculation of the mean life $\ensuremath{\tau}$ of an excited atom from experiments on the magnetic depolarization and rotation of the plane of polarization of resonance radiation. The calculation has been carried through for the case of the resonance line of $\mathrm{Cd}(\ensuremath{\lambda}3261)$ and that of $\mathrm{Hg}(\ensuremath{\lambda}2537)$, and it is found that the differences between the value of $\ensuremath{\tau}$ calculated from hyperfine structure data and that calculated from $tan2\ensuremath{\theta}$ by the usual non-hyperfine structure method lie well within the experimental error. P(H)P0\frac{P(H)}{{P}_{0}}P0​P(H)​ is found to be practically the same whether hyperfine structure is taken into account or neglected. This is due to the fact that the greatest contribution of the polarization in these cases comes from the isotopes having no nuclear spin and the ggg-value for the upper state of these isotopes is larger than any other upper hyperfine structure state involved. The mean lives of the 73S17^{3}S_{1}73S1​ state of mercury have been recalculated from Richter's data on the polarization of stepwise radiation. The values of $\ensuremath{\tau}$ obtained by this method of calculation are 7.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}9}$ for $\ensuremath{\lambda}4047$, 1.69\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ for $\ensuremath{\lambda}4358$ and 1.53\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}8}$ for $\ensuremath{\lambda}5461$. A discussion of Richter's results is given.
The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasidegenerate 2P1/22{P}_{1/2}2P1/2​ levels that are removed from the 2S2S2S states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift, and the hyperfine effects. The van der Waals shift becomes commensurate with the $2S\ensuremath{-}2{P}_{3/2}$ fine-structure splitting only for close approach (R<100a0R<100{a}_{0}R<100a0​, where a0{a}_{0}a0​ is the Bohr radius) and one may thus restrict the discussion to the levels with n=2n=2n=2 and J=1/2J=1/2J=1/2 to a good approximation. Because each SSS or PPP state splits into an F=1F=1F=1 triplet and an F=0F=0F=0 hyperfine singlet (eight states for each atom), the Hamiltonian matrix a priori is of dimension 64. A careful analysis of the symmetries of the the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states that are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to 1/R31/{R}^{3}1/R3 and 1/R61/{R}^{6}1/R6 and are evaluated within the hyperfine manifolds. When both atoms are metastable 2S2S2S states, we find an interaction energy of order ${E}_{h}\ensuremath{\chi}{({a}_{0}/R)}^{6}$, where Eh{E}_{h}Eh​ and L\mathcal{L}L are the Hartree and Lamb shift energies, respectively, and $\ensuremath{\chi}={E}_{h}/\mathcal{L}\ensuremath{\approx}6.22\ifmmode\times\else\texttimes\fi{}{10}^{6}$ is their ratio.
Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.