Atomic radius by dimensional analysis

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SUMMARY

The discussion centers on deriving the atomic radius formula R=Z^(-1/3)*(hbar^2)/(m*e^2) using dimensional analysis, based on the Thomas-Fermi theory. The user seeks clarification on the Z^(-1/3) dependence, specifically how it relates to the constants h (reduced Planck's constant), m (mass), and e (elementary charge). The user concludes that Z, representing nuclear charge and electron count, must be combined with these constants in a specific manner, leading to the need for justification of the differing powers assigned to each constant.

PREREQUISITES
  • Understanding of Thomas-Fermi theory
  • Familiarity with dimensional analysis
  • Knowledge of fundamental constants: h (reduced Planck's constant), m (mass), and e (elementary charge)
  • Concept of nuclear charge (Z) in atomic physics
NEXT STEPS
  • Study the derivation of the Thomas-Fermi model in quantum mechanics
  • Explore dimensional analysis techniques in physics
  • Investigate the role of nuclear charge (Z) in atomic structure
  • Learn about the implications of the atomic radius on chemical properties
USEFUL FOR

Students and researchers in physics, particularly those focusing on atomic theory, quantum mechanics, and dimensional analysis. This discussion is also beneficial for anyone studying the relationship between atomic structure and fundamental physical constants.

areyouserious
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Hi,
So according to the Thomas-Fermi theory, the formula for atomic radius is
R=Z^(-1/3)*(hbar ^2)/m*e^2. (in other words Z^(-1/3) times the Bohr radius)

While the total energy has a Z^(7/3) dependence.

I need to get this from dimensional analysis. I can get the dependence on h, m, and e easily, but the Z^(-1/3) is a mystery to me. I understand that the Z, being unitless, must be tacked on to another unit, and working backwards, I can see that it works out if I attach a Z term to h and to e, and attaching a Z^(1/3) term to m. But how can I justify this? Why only 1/3 power for m, but first power for e and h? Please help, I am completely stuck.
Thanks
 
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OK, Z means what? It's the nuclear charge, as well as the number of electrons. So where might you see Z's? The combination Ze makes sense in a way Zh does not, no? You can also see Z's appear if you are counting something - like electrons.

I hope this points you in the right direction.
 

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