Total angular momentum of atom in its ground state

[daleklama]

Homework Statement

The question is to find the total angular momentum of the following atoms in their ground state - Na (11 electrons), and Rb (37 electrons). That's all the info given.

Homework Equations

I have no idea - that's what I can't find!

The Attempt at a Solution

I've pored through my notes and the textbook and I can't find what equation I use/method I use to solve this! :( I'm happy to do the work myself of course, but could anyone tell me the equation or where I could find it? I really have no idea how to approach this! Thank you :)

 

[TSny]

Think about the electron configuration of Na. If an atom has a full subshell of electrons, what is the total angular momentum of all the electrons in that subshell?

Is there a subshell in Na that is only partially filled? If so, what is the total angular momentum of the partially filled subshell? (Total angular momentum includes both orbital and spin angular momentum.)

 

[zmth]

In so far as relativistic effects such as magnetic couplings like spin - orbit coupling which is a good approximation esp for light atoms, no spin dependant terms in the energy, can be neglected then the total orbital angular momentum of the atom is ALWAYS zero ! and that applies to an atom or ion with 5 electrons or however many - odd or even as long as the spin dependence is negligible which is generally true for not too high atomic number nucleons - for the ground state. It is also true for a molecule in its ground state.

 

[zmth]

you can solve it in principal by brute force methods. Using hermite-gaussian type basis functions for each electron in relation to the nucleus. Then the coulomb potential between all pairs of particles matrix elements integrals can be done in exact closed form and also the kinetic energy matrix elements. Of course to save some work one knows that in so far as can neglect relativistic effects such as spin-orbit and other magnetic couplings then the total angular momentum commutes with the Hamiltonian and can be diagonalized simultaneously. Just solve for basis functions that have a given constant total angular momentum which will be discrete quantum numbers like integers and not half an odd integer times perhaps some function of Plancks constant depending upon which units one is using, we are neglecting spin dependent forces, and they will not mix - there will be no matrix elements between those basis functions with different total angular momentum quantum numbers will be such that the Hamiltonian will have zero matrix element between them. then you can verify that the system with total angular momentum of 0 will be the lowest energy - the ground state. When you get up to higher atomic numbers for sure such as mercury say then one does have to consider the spin angular momentums. In that case the total orbital angular momentum is not a constant of the motion - does not commute with the Hamiltonian because we need to include spin dependant terms such as spin orbit couplings at least. Only total J - that is orbital plus spin is a constant of the motion and in that case am not sure that the total J will always be 0 for even number of electrons - likely not. And not sure whether it will always be the minimum possible of 1/2 for an odd number of electrons - it probably will not be in general.

 

[zmth]

Think about the electron configuration of Na. If an atom has a full subshell of electrons, what is the total angular momentum of all the electrons in that subshell?

Is there a subshell in Na that is only partially filled? If so, what is the total angular momentum of the partially filled subshell? (Total angular momentum includes both orbital and spin angular momentum.)

IT is important to note that the shell model is only a rough approximation and for a so called " partially filled shell" which is not entirely ever true in reality BUT regardless one can be SURE even in this 'misnomer or misleading' statement it will always be true that the total orbital angular momentum will always be zero in so far as relativistic effects such as spin orbit couplings can be neglected..! This is one thing that people often get confused - the shell model is based upon the premise of excited state of the single electron of a one nucleon system such as hydrogen and it is no longer the case when one has more than one electron - it is a different thing all together - no longer a single reduced mass system for one.

 

[zmth]

In so far as relativistic effects such as magnetic couplings like spin - orbit coupling which is a good approximation esp for light atoms, no spin dependant terms in the energy, can be neglected then the total orbital angular momentum of the atom is ALWAYS zero ! and that applies to an atom or ion with 5 electrons or however many - odd or even as long as the spin dependence is negligible which is generally true for not too high atomic number nucleons - for the ground state. It is also true for a molecule in its ground state.

correction it is not always zero but often is the case due to the electron repulsions

 

[zmth]

IT is important to note that the shell model is only a rough approximation and for a so called " partially filled shell" which is not entirely ever true in reality BUT regardless one can be SURE even in this 'misnomer or misleading' statement it will always be true that the total orbital angular momentum will always be zero in so far as relativistic effects such as spin orbit couplings can be neglected..! This is one thing that people often get confused - the shell model is based upon the premise of excited state of the single electron of a one nucleon system such as hydrogen and it is no longer the case when one has more than one electron - it is a different thing all together - no longer a single reduced mass system for one.

correction total angular momentum it is not always zero but often is the case due to the electron repulsions