Unravelling the Mystery of Rare Gases and Binding Energy

[PhysicsKin]

Hi!

I saw something on my lecture notes that I don't really understand. It reads "Rare gases have filled s and p-sub-shells, which leads to a spherically symmetric charge distribution. Since electrons are indistinguishable they take on a common wavefunction. The point is that this results in a higher binding energy for each one of the electrons."

I don't really understand the last bit, especially why having a common wavefunction leads to a higher binding energy. Does anyone care to explain?

Thanks!

 

[Simon Bridge]

Its quantum. I don't think there is a classicalesque analogy for it.

Basically, you can pack electrons in tighter when there is a lot of symmetry.
Look up the wavefunctions of Helium for a "simple" example. Compare with Hydrogen and Lithium.
But it is not really cause and effect - the "common wavefunction" and the higher binding energy go hand-in-hand.

 

[PhysicsKin]

Thanks for the reply.

I am guessing the first point is due to exchange symmetry? I am not sure how to visualise this... is there a mathematical proof to this?

 

[Simon Bridge]

You cannot prove an empirical truth by mathematics alone ... but there is a demonstration that the regular Schrödinger formulation for fermions results in these symmetries.
It's part of the usual mathy description.

 

[Astronuc]

Hi!

I saw something on my lecture notes that I don't really understand. It reads "Rare gases have filled s and p-sub-shells, which leads to a spherically symmetric charge distribution. Since electrons are indistinguishable they take on a common wavefunction. The point is that this results in a higher binding energy for each one of the electrons."

I don't really understand the last bit, especially why having a common wavefunction leads to a higher binding energy. Does anyone care to explain?

I believe the correct statement should be that none of the 6 electrons has a different or distinguishable binding energy compared to the others. The high binding energy has to do with paired electrons with opposite spin. The binding energy of a given electron has to with the Z and probable 'distance' from the nucleus. Note that the group I elements S¹ are readily ionized or give up one electron, while group IV readily attract one electron to fill the outer shell.

Note that He has a filled S shell, 1s². Ne has 1s² 2s² 2p⁶.

The theory for multi-electron atoms is found in Hartree-Fock theory.
http://en.wikipedia.org/wiki/Hartree–Fock_method
http://www.eng.fsu.edu/~dommelen/quantum/style_a/hf.html

http://en.wikipedia.org/wiki/Ionization_energy

 

[DrDu]

Its quantum. I don't think there is a classicalesque analogy for it.

I am convinced this to be an effect which can be understood classically.
Electrons in the same shell (or orbit, classically) are not very efficient in shielding each other from the nuclear charge. Hence the effective nuclear charge seen by an electron increases steadily when a shell is filled an reaches a maximum when the shell is full. This trend is paralleled by the binding energy. Approximate values for the effective nuclear charge and also binding energy can be obtained from Slater's rules:
http://en.wikipedia.org/wiki/Slater's_rules
Specifically, the maximum of binding energy for filled or half filled shells is not a mysterious effect. The stabilization increases continuously, however a shell can take only a finite amount of electrons (or of electrons with the same spin, in the case of half filled shells), hence the maximum.