3.3.2 High Precision Determination of Helium Ionization Energy

K.J.Coakley
Statistical Engineering Division, ITL C.Sansonetti and J.D.Gillaspy
Atomic Physics Division, PL

By measuring the binding energies of the low-lying levels of helium, one can stringently test the accuracy of theoretical calculations of two-electron quantum electrodynamic (QED) effects in atoms. In 1992, NIST reported a fully empirical determination of the 2¹S ionization energy based on the representation of the long series of transitions 2¹S - n¹P by a Ritz formula. The NIST estimate disagreed with an estimate from a Yale experiment. In order to search for a possible resolution of the disagreement, we have improved our analysis of the NIST data. First, we correct the data for systematic error due to phase dispersion effects in our Fabry-Perot interferometer. Second, we use a improved physical model. A model for the systematic error is determined from a calibration experiment. In a two-stage bootstrap resampling scheme, we estimate the total uncertainty of our estimate of the 2¹S ionization energy. In the first stage, we simulate a realization of calibration data and get a realization of the wave number correction model parameters. In the second stage, conditioned on the particular realization of the wave number correction factors, we simulate a bootstrap replication of the observed helium wave number data. Our new estimate of the ionization energy is $\hat{E}_{\infty} = 32033.2288455(50)$ cm^-1. Based on the old analysis, the discrepancy between the Yale and NIST estimates was 2.8 sigma. Based on our new analysis, the discrepancy is reduced to 1.8 sigma.

We model the nth quantum state as

$$\hat{E}(n) ~=~ E _ { \infty } ~-~ \frac {R} { (~ n~-~\delta (n) ~ ) ^ 2} ~+~ S~n^7 ~+~ \Delta E (n_i).$$

Above, $E_{\infty}$ is the ionization energy, R is the finite mass Rydberg constant and $\delta (n)$ is the quantum defect which we model as

$${\delta (n)} ~=~~ \frac {B}{ ( n -{\delta (n)} ) ^2 } ~ +~ \frac {C}{(n- {\delta (n)})^4}$$

where B and C are adjustable parameters and

$$\Delta E(n) ~=~ -2R \left[ ~ ({\mu}/{M} ~ + ({\mu}/{M} ) ^ 2 ... ...6} \alpha^2 ( {\mu}/{M} ) ^ 2 Z^2 (Z-1)^2 / (2n^2) ~ \right]$$

$$~+~3R{\alpha}^2 (Z-1)^4 / ( 4n^4) .$$

For helium, Z=2. The finite mass Rydberg constant is $R~=~ (1 - \mu / M ) R_{\infty}$where $\mu = \frac{ M} { m + M}$where the mass of the electron and the mass of the alpha particle (helium nucleus) are m and M. The constants are $\alpha~=~$1/137.0359895(61), $\mu/M~=~1.37074562(3) \times ~ 10 ^ {-4}$, $R_{\infty} ~=~ 109737.315682(9)$ cm^-1 .

Figure 15: Upper left: residuals computed from observed data. Upper right: a bootstrap replication of residuals. Lower left: variability of ionization energy estimate (about mean value) due to calibration experiment errors. Lower right: variability of ionization energy estimate (about mean value) due to errors in both calibration and primary experiment.

       

Date created: 7/20/2001
Last updated: 7/20/2001
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