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Physical Review

Metastability of 2s States of Hydrogenic Atoms

J. Shapiro, GREGORY A. BREIT

Published Jan 1, 1959• 181 citations
Publisher

Abstract

It has been found by Breit and Teller that double photon emission is the principal cause of decay of interstellar hydrogen atoms from their metastable $2s$ state and the mean life corresponding to this decay was bracketed by them between upper and lower limits. In the present note an account is given of a more accurate evaluation of the transition probability ${A}_{\ensuremath{\tau}}=(8.226\ifmmode\pm\else\textpm\fi{}0.001){Z}^{6}$ ${\mathrm{sec}}^{\ensuremath{-}1}$, where $Z$ is the atomic number. Relativistic effects on the atomic wave functions have been neglected in the calculations which are therefore accurate only for small $Z$.

DOI
10.1103/physrev.113.179

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It has been found by Breit and Teller that double photon emission is the principal cause of decay of interstellar hydrogen atoms from their metastable 2s2s2s state and the mean life corresponding to this decay was bracketed by them between upper and lower limits. In the present note an account is given of a more accurate evaluation of the transition probability A\ensuremathτ=(8.226\ifmmode±\else\textpm\fi0.001)Z6{A}_{\ensuremath{\tau}}=(8.226\ifmmode\pm\else\textpm\fi{}0.001){Z}^{6}A\ensuremathτ​=(8.226\ifmmode±\else\textpm\fi0.001)Z6 sec\ensuremath−1{\mathrm{sec}}^{\ensuremath{-}1}sec\ensuremath−1, where ZZZ is the atomic number. Relativistic effects on the atomic wave functions have been neglected in the calculations which are therefore accurate only for small ZZZ.
The mean life of the 2S state of hydrogen in the absence of collisions is primarily determined by the simultaneous emission of two photons. Owing to this cause the mean life is `-*4 sec. It is expected that the mean life of is 2S `S of He i is of the same order of magnitude and is much shorter than that of is 2S 3S of He i. The non-adiabatic collisions with electrons are found to be more important than the static effects at small electron densities, leading to a transition probability `-*-`~h sec ` for an electron temper- ature `-.--`io,ooo° C and an electron density 30 cm3. The effect of static electric fields on the 2S state of hydrogen is worked out in more detail than previously by taking into account the hyperfine structure. The mean life in static electric fields which are strong enough to make double emission of little importance and yet weak enough to give a mean life > io7 sec is found to be the sum of two lives: (i) the mean life calculated neglecting hyperfine structure and (2) the mean life calculated neglecting radiation coupling. In both (i) and (2) the mixing of 2~ into 2S by the Stark effect is taken into account. Radiations due to quadrupole and magnetic dipole effects are considered qualita- tively for is 2S 3S of helium and quantitatively for 2S of hydrogen. For the latter there exists a small radiative magnetic dipole effect on Dirac's theory of the electron which is absent in the ordinary spin model
The two-photon decay mode of hydrogenic atoms from an arbitrary state (n1{n}_{1}n1​, l1{l}_{1}l1​, m1{m}_{1}m1​) to an arbitrary state (n2{n}_{2}n2​, l2{l}_{2}l2​, m2{m}_{2}m2​) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the 2s2s2s metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.
The two-photon transition rate for hydrogen is evaluated with the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}$ and the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}$ interactions including the continuum. We find that the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}$ interaction approaches the final result with a small number of intermediate states and consequently is the one to be used in any approximate calculation. Numerical values of the two-photon absorption coefficient as a function of frequency are given and the resonance enhancement is exhibited.
Laser resonance transitions between normally metastable states of antiprotonic helium atoms were observed making use of state dependent quenching effects caused by small admixtures of H2{\mathrm{H}}_{2}H2​ molecules. By selectively shortening the lifetimes of states with higher principal quantum number nnn as compared to those of lower nnn, this method for the first time provides access to all initially populated metastable states of p‾He+\overline{p}{\mathrm{He}}^{+}p​He+ atoms. This was demonstrated by observing the transitions $(n,l)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(38,l)\ensuremath{\rightarrow}(39,l+1)$, l=35,36,37l\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}35,36,37l=35,36,37 and $(n,l)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(37,l)\ensuremath{\rightarrow}(38,l+1)$, l=34,35,36l\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}34,35,36l=34,35,36.
We report the first detailed decay studies of trapped metastable (2S)(2S)(2S) hydrogen. By two-photon excitation of ultracold H samples, we have produced clouds of at least $5\ifmmode\times\else\texttimes\fi{}{10}^{7}$ magnetically trapped 2S2S2S atoms at densities greater than $4\ifmmode\times\else\texttimes\fi{}{10}^{10}{\mathrm{cm}}^{\ensuremath{-}3}$ and temperatures below $100\ensuremath{\mu}\mathrm{K}.$ At these densities and temperatures, two-body inelastic collisions of metastable atoms are evident. Experimental values for the total two-body loss rate constant are ${K}_{2}{=1.8}_{\ensuremath{-}0.7}^{+1.8}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}{\mathrm{cm}}^{3}{\mathrm{s}}^{\ensuremath{-}1}$ at $87\ensuremath{\mu}\mathrm{K}$ and ${K}_{2}{=1.0}_{\ensuremath{-}0.5}^{+0.9}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}{\mathrm{cm}}^{3}{\mathrm{s}}^{\ensuremath{-}1}$ at $230\ensuremath{\mu}\mathrm{K}.$ These results are in the range of recent theoretical calculations for the total $2S\ensuremath{-}2S$ inelastic rate constant. The metastable clouds were excited in a gas of ground-state (1S)(1S)(1S) hydrogen with peak densities reaching $7\ifmmode\times\else\texttimes\fi{}{10}^{13}{\mathrm{cm}}^{\ensuremath{-}3}.$ From the one-body component of the metastable decay, we derive experimental upper limits for K12,{K}_{12},K12​, the rate constant for loss due to inelastic $1S\ensuremath{-}2S$ collisions.
The relativistic and nonrelativistic approaches for the calculations of the two-photon decay rates of highly excited states in hydrogen are compared. The dependence on the principal quantum number ( n ) of the n s, n d and n p initial states is investigated up to n = 100 for the nonresonant emissions. For the n s states together with the main E1E1 channel the contributions of higher multipoles (M1M1, E2E2 and E1M2) are considered. For the n p states the E1M1 and E1E2 channels are evaluated. Moreover, the simple analytical formula for the E1M1 decay is derived in the nonrelativistic limit.