What would be the atomic number of the atom whose 1s electron moves at 0.99c (the speed of light)?

analianopolisca 2022-08-07 Answered
What would be the atomic number of the atom (may be hypothetical) whose 1s electron moves at 0.99 c (the speed of light)?
Quantum mechanics might have an answer, but I do not know the necessary maths to calculate.
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Answers (1)

Isaias Archer
Answered 2022-08-08 Author has 11 answers
You can get a back of the envelope notion of the energy of a inner-most orbital by just treating the problem as a hydrogen-like atom (not entirely fair and almost certainly a slight over-estimate but at least it is easy). You get
E 1 s R y Z 2 = ( 13.7 e V ) Z 2 .
Where Z is the atomic number of the atom in question and R y = 13.6 e V is the Rydberg constant.
If you are asking for a speed of β = 0.99 ( γ = 7.1) then you are suggesting an kinetic energy of about T = ( γ 1 ) m e c 2 = 6.1 ( 5.11 × 10 5 e V ) = 3.1 × 10 6 e V . Which suggests:
Z 2 = ( γ 1 ) m e c 2 R y 6.1 ( 5.2 × 10 5 e V ) 13.7 e V = 2.3 × 10 5 Z 480 ,
give or take a small factor.
Even for β = 0.9 ( γ = 2.3) you get Z 220.
- For β = 0.75 ( γ = 1.5) I find Z 140.
- For β = 0.65 ( γ = 1.3) I find Z 110.
- For β = 0.55 ( γ = 1.2) I find Z 86.
All of these values are thoroughly relativistic, but as you can see the ultra relativistic regime requires unreasonable heavy nuclei.

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