For what Z values does an atom begin to differ from a nonrelativistic model?

In summary, my professor stated that relativistic effects are rather small in the hydrogen atom, but more significant in higher-Z atoms. He also mentioned that for a crude guess, it is acceptable to combine quantum mechanical results with classical relationships. He also said that you can consider the electrons orbiting in a circular orbit according to the Bohr model. For a more realistic calculation, he said you need to use relativistic momentum and velocity. He also mentioned that you are in the same class as me at UC Santa Cruz.
  • #1
hb1547
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Homework Statement


"Relativistic effects are rather small in the hydrogen atom, but not so in higher-Z atoms. Estimate at what value of Z relativistic effects might alter energies by about a percent and whether it applies equally to all orbiting electrons or to some more than others. For this crude guess, it is acceptable to combine quantum mechanical results you have learned, related to energy, angular momentum, and/or probable radii, with some classical relationships."

My professor also added the note:
"You can consider the electrons to be orbiting in*a circular*orbit according to the Bohr model. Ignore screening in your calculation, but mention what effect it would have on a more realistic calculation."

Homework Equations


[tex]E_{n} = -E_{0}\frac{Z^{2}}{n^{2}}[/tex]

The Attempt at a Solution


Just generally not sure where to begin with this. Do I look at it from a Bohr model standpoint, and try to find a point where Gamma becomes 1.01? Do I look at it from a quantum mechanical view, and try to use quantum numbers to find an energy level high enough that implies an electron is going a certain speed?
 
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  • #2
I would do the latter, the equation you posted is a good one, you have to be able to relate the Energy of the orbiting electron with its velocity; you can plug in gamma = 1.01 the relativistic momentum equation and use:
image012.gif

to find the corresponding energy, then plug that into E in the equation you posted and Z=1 for hydrogen and solve for n!

Also, do you go to UC Santa Cruz? It sounds like you're in the same class as me... ;)
 

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  • #3
Gotta love Professor Smith
 
  • #4
Yeah, he's pretty awesome.

(Also, that equation I gave is for total E, you need KE and it's actually way easier to use KE = (gamma)mc2-mc2, you don't need to calculate momentum or velocity)
 
  • #5
Awesome thanks! That helped a lot -- that was a much simpler problem than I was making it in my head. Thanks again, good luck on tomorrows quiz, haha
 

1) What does it mean for an atom to differ from a nonrelativistic model?

An atom that differs from a nonrelativistic model means that the behavior of its electrons cannot be accurately described using classical, nonrelativistic theories. Instead, relativistic effects must be taken into account.

2) How does the Z value of an atom affect its behavior in a nonrelativistic model?

The Z value, or atomic number, of an atom determines the number of protons in its nucleus. In a nonrelativistic model, the behavior of atoms with high Z values will start to deviate from predictions due to the increased strength of the electric field experienced by the electrons.

3) What is the significance of Z values in determining the accuracy of a nonrelativistic model?

Z values play a crucial role in determining the accuracy of a nonrelativistic model because they indicate the strength of the electric field experienced by the electrons. As Z values increase, so does the strength of the field, leading to deviations from nonrelativistic predictions.

4) At what point do Z values begin to significantly impact the behavior of an atom in a nonrelativistic model?

Z values start to significantly impact the behavior of an atom in a nonrelativistic model when they reach values of around 30 or higher. At this point, the electric field experienced by the electrons becomes strong enough to cause noticeable deviations from classical predictions.

5) Why is it important for scientists to take relativistic effects into account when studying atoms with high Z values?

It is important for scientists to consider relativistic effects when studying high Z atoms because these effects can significantly impact the behavior and properties of the atom. Neglecting relativistic effects can lead to inaccurate predictions and understanding of the atom's behavior, which can hinder advancements in fields such as quantum mechanics and atomic physics.

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