Inside radius of atomic electron cloud vs Z

[NLB]

I am interested in knowing if the inside radius of the inner most part of the electron cloud is a constant versus Z. For example is it always the Bohr radius, or does this inner radius change as a function of Z? What is the experimental evidence?

 

[Simon Bridge]

How are you defining/thinking of the radius?
By combining the concept of a "radius" and a "cloud" in the context of electrons, you appear to be mixing up two different models.

All atoms have at least one s1 electron - the radial wavefunction for that state is continuous from 0 to infinity - where would you put the "minimum radius"? Here's how the radial wavefunctions vary with Z. You can google for the shapes.

...anyway, the atomic diameter is roughly the same order of magnitude regardless of Z, which means that the electrons get more tightly packed as Z increases. Which I suspect is what you are asking about.

 

[NLB]

Thanks for your reply. I appreciate the information. Very useful.

I am asking about the radius at which the maximum of the radial probability of the electron occurs. I am not mixing up two different concepts. This is how the Bohr radius is defined now--as the radius of the maximum of radial probability of the s1 electron cloud, for a Hydrogen atom.

I'm not interested in knowing the "minimum radius" of that s1 electron cloud. Rather, I am interested in the most likely radial position of that s1 electron, as a function of Z. From your information, I see that it varies, quite a bit. Thank you.

 

[Simon Bridge]

Cool - glad to be able to help.

I know, it can be vexing when people don't just "get" what you are asking, but I have to get a clearer picture if I am to avoid unwittingly side-tracking the thread. Thanks for your understanding on this. :)

 

[NLB]

Thank you again for your help. Another question, though. In the link you gave me, both functions rR(r) and (rR(r))^2 are plotted, but with no explanation or definition.

So, I assumed that rR(r) was the radial probability, and I took the formula of rR(r) and differentiated it with respect to r. Then I set the resulting derivative equal to zero, and solved for r, which would be the r for the maximum of the function rR(r). This should give me what I want--the maximum radial probability as a function of Z. What I got was rmax = a0/Z. (I'm assuming a0 is the Bohr radius of H1 atom, 5.29E-11 meters.) If all this is right, then for U238, where Z=92, then this maximum radial probability of the s1 inner radius is:

(5.29E-11)/(92)=(2.62E-13) meters.

Which, of course, is a quite a bit smaller than the Bohr radius for the H1 atom.

Does this make sense? Am I doing this right, or do I need to work with the function (rR(r))^2 instead?
Thanks again for your help.

 

[Simon Bridge]

$R_{nl}(r)$ is the radial component of the single-electron wavefunction.
The probability density function is the square modulus.

i.e. The probability of finding an electron in state |n,l> between r and r+dr is $p(r)dr=|R_{nl}(r)|^2dr$ so $\int_0^\infty p(r)dr = 1$ right?

The mean radius is $$\langle r \rangle=\int_0^\infty R_{nl}^\star r R_{nl} dr$$
... since $R_{nl}$ is real, that is where they get the $r|R_{nl}(r)|^2$ from.
It's just the expectation value of r: E[r].

Similarly $$\langle r^2 \rangle=\int_0^\infty r^2 R_{nl}^2 dr$$ ... would be E[r^2].