# Ionization Energy of Atomic Hydrogen

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## Introduction

In previous articles relating to various transition energies in Hydrogen, Helium and Deuterium we have employed the following formula for electron energy given a particular primary quantum number n:

$$E_{n}=\mu c^2\sqrt{1-\frac{Z^2\alpha^2}{n^2}}$$

where ## \alpha ## is the fine structure constant and ## \mu ## the reduced electron mass for a single electron bound to whichever nucleus. Z=1 for Hydrogen and Deuterium, Z=2 for Helium. Reduced mass is calculated from electron mass and nuclear mass as follows: $$\mu = \frac{m_e\times m_{n}}{m_e+m_{n}}$$.

## Calculation of Ionization Energy: Atomic Hydrogen

Perhaps one of the simplest applications of this formula is the determination of the ionization energy of atomic hydrogen – ie the energy required to move the single Hydrogen electron from its ground state to the point where in essence it becomes a ‘free electron’. At this point, primary quantum number n=∞ and electron energy is given by the straightforward formula:

$$E_{∞}=\mu c^2$$

Hence the transition energy from ground state to the above is simply:

$$E_i=E_∞ – E_1=\mu c^2-\mu c^2 \sqrt{1-\alpha^2}=\mu c^2(1-\sqrt{1-\alpha^2})$$

Carrying out the above calculation on Wolfram Alpha we obtain a first approximation for this value in electron volts or equivalent SI units. This result needs to be corrected or perhaps ‘fine tuned’  by subtracting the most accurately available value of “Ground State Lamb shift”: an energy ‘offset’ described in our previous “Insights” article: The Deuterium Lyman Alpha line. The value for this offset (8172.874(60) MHz) is given in the abstract for the following paper:

## Correcting for Ground State Lamb Shift

We convert the offset value from MHz to electron volts and then subtract from our original calculation above obtaining the following result:

13.5984345 eV .

Wolfram Alpha calculation details can be viewed here.

The  calculated value is in very close concordance with the corresponding value given on the NIST spectral database (just enter “H I” on the entry form).

13.59843449 eV.

## Conclusion

As in all previous articles employing the simplified Dirac Energy Equation, we obtain a precision result agreeing with the NIST value to as many significant figures as Wolfram Alpha is prepared to calculate!

## References

 [1] A. Kramida, Yu. Ralchenko, J. Reader, and and NIST ASD Team. NIST Atomic Spectra Database (ver. 5.7.1), [Online]. Available: https://physics.nist.gov/asd [2019, December 6]. National Institute of Standards and Technology, Gaithersburg, MD., 2019. [ bib ] [2] M. Weitz, A. Huber, F. Schmidtkaler, D. Leibfried, W. Vassen, C. Zimmermann, K. Pachucki, T.W. Hansch, L. Julien, and F. Biraben. Precision-measurement of the 1s ground-state lamb shift in atomic-hydrogen and deuterium by frequency comparison. Physical Review A, 52(4):2664-2681, 1995. [ bib | DOI ] We have measured the hydrogen and deuterium 1S Lamb shift by direct optical frequency comparison of the 1S-2S and 2S-4S/4D two-photon transitions. Our result of 8172.874(60) MHz for the 1S Lamb shift in hydrogen is in agreement with the theoretical value of 8172.802(40) MHz. For the 1S Lamb shift in deuterium, we obtain a value of 8183.807(78) MHz, from which we derive a deuteron matter radium of 1.945(28) fm. The precision of our value for the 1S Lamb shift has surpassed that of radio frequency measurements of the 2S-2P Lamb shift. By comparison with a recent absolute measurement of the hydrogen 1S-2S transition frequency,we deduce a value for the Rydberg constant R(i)nfinity = 109 737.315 684 9(30) cm(-1).

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2 replies
1. mfb says:
as many significant figures as Wolfram Alpha is prepared to calculate!

Unless you click on "more digits".

The WolframAlpha image has a low resolution in the article and you could have added a link to WA with the calculation filled in.

2. neilparker62 says:
Many thanks – the article wasn’t actually finished and I accidentally pressed submit. Anyway I’ll chat to Greg and either temporarily ‘withdraw’ it or just do some online editing. It’s very ‘junky’ as it stands – not even any references.

Re "more digits": I’m not sure if this option is available when you are using scientific constants as in this calculation. WA seems to put some kind of limit on the number of significant figures it will display.