# What are the consequences of having an element with an atomic number above 137?

In other words, is an element such as Feynmanium the last chemical element that can physically exist on a planet such as Earth?

• 137 is in no way special. Commented Mar 19, 2018 at 5:00
• I think he is referring to a predicted limit for Bohr atoms. When you look at the shell model of atoms 137 is the limit, because 138 is supposed to have electrons that are moving faster than the speed of light. I think when you add modern quantum mechanics you can show that the number should be higher but I can't find that reference anymore. Commented Mar 19, 2018 at 7:33
• Almost duplicate: chemistry.stackexchange.com/questions/8073/… Commented Mar 19, 2018 at 7:50

The critical value "137" occurs in calculations based upon relativistic theories that treat the nucleus as a point, such and Dirac theory or Sommerfeld theory.

However, when the nucleus is correctly treated as having a non-zero size, a significantly higher value is calculated.

The most recent calculations I see are in New method for solving the $$\bf Z > 137$$ problem and determining hydrogen-like energy levels (2014) Phys.-Usp. 57, 189. (alternative link to pre-print):

Following the development of the Dirac theory in 1928, Dirac [reference 2], Darwin [reference 3] and Gordon [reference 4] obtained expression (1) as a result of exact solution of the Dirac equation in the Coulomb field of a point charge (-Ze). ... Formula (1) [results in a complex number] if Z [is greater than 137]...the complexity of energy levels in (1) is often called the “Z>137 catastrophe”.

In 1945, Pomeranchuk and Smorodinsky [6] considered an atomic system with [the nucleus being a sphere of radius $$0.8 \times 10^{-12}$$cm and having uniform charge within the nucleus and found 175 to be the critical value].

The article goes on to explain that since the 1945 calculation, various calculations, with various nuclear charge distribution models, have calculated critical values in the 170-175 range. The article's own calculations find 178.

• I believe you are overthinking it, and that by a wide margin. The "critical" value 137 occurs in calculations based upon non-relativistic (and not even completely quantum) Bohr theory, when you equate the electron velocity and the speed of light. Refuting it does not require anything nearly as sophisticated. Its reappearance on a different level of theory is hardly relevant, though of course not accidental. Commented Mar 19, 2018 at 12:07
• @IvanNeretin I think I'm biased because of a previous question that asked whether the 137 still occurs in relativistic theories. chemistry.stackexchange.com/questions/41645/… Commented Mar 19, 2018 at 12:20
• @IvanNeretin chemistry.stackexchange.com/questions/41645/… Commented Mar 19, 2018 at 15:41