# Size of atoms based on atomic number

I'm interested in the size of atoms (extent of the outermost electron cloud), in particular its cross-sectional area and how it scales by increasing Z. (I trust that this won't be affected much by the number of neutrons?)

I recognize that for a given atom/ion one can simply look up the data, but I'm interested in a more 'first-principles' kind of view. The distances of electrons in given orbitals is not constant across different atoms, since as the nucleus gains protons the orbitals tend to get crushed somewhat closer I believe.

(In fact my final goal will be to determine scattering for charged particles, which is based on much more than the geometry, but I became interested in this particular subproblem apart from that ultimate question.)

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Atoms have no sharp outer boundaries. All these things and clouds are probabilistic. The electrons always have a nonzero chance to be arbitrarily far from the nucleus, and so on.

Moreover, the visual size of an atom will depend on the frequency of light one uses to "see" the atom, and so on.

Assuming that you understand all these disclaimers and you only want some expectation value, and only an estimate of it because no analytic formula exists beyond Hydrogen, here is why all atomic radii are comparable to the Bohr radius $a_0$, independently of $Z$:

Let's neglect the electron-electron interactions. In that case, we have $Z$ electrons in the radial electrostatic potential. The Bohr radius itself is $$a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_e e^2} .$$ If the nucleus has charge $Ze$, there will be an extra $Z$ in the denominator i.e. $Ze^2$ instead of $e^2$. However, we also need to fill $Z$ electrons in the Hydrogen-like potential. That will fill the shells up to $n=Z^{1/2}$ or so because there are $2n_{\rm max}^2$ of electrons with $n\leq n_{\rm max}$.

However, just like in the Bohr model of the atom that gets the scaling right, the radius of the orbit for the quantum number $n$ - that is somewhat misinterpreted as $l$ in Bohr's model - goes like $n^2$. So if $n\sim Z^{1/2}$, then $n^2\sim Z$, and this enhancement of the radius by $Z$ cancels the reduction of the size caused by the stronger potential induced by the stronger nucleus, as discussed at the beginning of the calculation.

So the radius will be close to $a_0$ regardless of the $Z$. The electrons' self-interactions will only change the typical outer shell's radius by "dozens of percent", to put it verbally. As Vladimir says, negative ions will include more electrons so they will be substantially larger than $a_0$ while the positive ions will remove some electrons, so the positive ions will be smaller than $a_0$.

Because various elements differ in electronegativity, the atoms such as Li, Na, K, Rb etc. will be larger - with the extra electron in the new shell - while the atoms such as He, Ne, Ar, Kr, Xe, Rn will be smaller, just having filled the last shell - much like positive ions. See a visual table of the size of atoms here:

http://intro.chem.okstate.edu/1314F00/Lecture/Chapter7/Lec111300.html

One can see that when things are evaluated, the "covalent radii" still increase with $Z$ a little bit. It's because the central potential, which I assumed to be $Ze$, is actually growing less quickly with $Z$ once the electron-electron interactions are taken into account (because from the outer electron's viewpoint, the inner electrons screen much of the electric charge of the protons in the nucleus).

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Atomic sizes are surprisingly close to $a_0$ when in the ground states. It is the negative/positive ions that are significantly larger/smaller.

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Wikipedia has two periodic tables with atomic radii, one empirically measured covalent radii and the other theoretically modelled for single atoms. They give different numbers and patterns.

In the modelled table, atomic radius has a fairly smooth pattern, growing as Period increases and as Group falls (i.e. moving down and left across the table). The reason given for this is that extra shells are larger, but more electrons (and protons) bind more tightly.

In the empirical table there is no data for elements which do not have easily measurable covalent compounds, so no figures for noble gases. For the data it does have, there are peaks near Caesium and Tellurium; non-metals tend to be noticeably smaller. The patterns are not monotonic: for example Molybdenum is larger than either Chromium or Tungsten while Osmium is smaller that either Rhenium or Iridium.

The cross-sectional area will be proportional to the square of the radius.

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there is a non conformity between the plot at right 'showing the calculated atomic..' and the table 'Calculated atomic radii' at least in the hydrogen data. One display 25 and the other 53. A typo or something else? –  Helder Velez Mar 29 '11 at 11:09