I have a question concerning the valence electrons in nickel and palladium crystals. There are 10 valence electrons for each atom and the face centered cubic crystal form indicates that each interior atom bonds with 12 adjacent atoms. Using palladium as an example, is there a shared "sea" of the 4d10 electrons so that at any instant in time each atom shares 10/12 of each atoms valence total or does each of the adjacent atoms' 4d10, plus 2 of the 4p6 or 4s2 electrons fill in 1 of 12 of each adjacent atoms' 4f14 sub shell? In either case there would be a 2 electron vacancy in the 10/12 percentage or the 4f14 subtle it would seem?

 

Since each palladium atom has a full shell of 18 why would there need to be any sharing at all? What is the presumed separation in the lattice from each neighbor? What is the probability of a 4d10 electron being far enough away from it's nucleus?

 

That is part of my question, the N or 4 shell is not full only the d subshell. The 4f subshell is empty. Palladium forms a fcc crystal, so the sharing or covalent bonding is an empirically determined condition. The lattice separation from nucleus to nucleus is twice the covalent radius of 139 pm. I don't know about probabilities, but the they are presumably defined by the 4d10 orbitals. (or vice versa.)

 

I suppose your question revolves around which nucleus an electron belongs to in some sense. We know that energy is quantized and that moving an electron to a higher state requires the input of energy. Similarly when an electron "falls" into a lower energy state they will emit energy as photons. In the absence of this known process occurring we conclude that a mechanism exists for keeping all of these outer electrons in some form of energy equilibrium where they can simultaneously belong to one and all, neither absorbing energy, nor emitting photons. We also know that we can't pin down the location of an electron or it's momentum with any degree of precision. An electron, no matter what it's quantum numbers, has some small probability of being practically anywhere. Orbitals are not ORBITS. I know this does not answer your question, but I think your question reveals a naive way of looking at lattice dynamics.

These articles may be helpful
http://en.wikipedia.org/wiki/Covalent_bond
http://en.wikipedia.org/wiki/Metallic_bonding
http://www.chemguide.co.uk/atoms/properties/orbitsorbitals.html

Also keep in mind that the 5s shell has a lower energy than the 4f shell. Silver (Ag) the next element in the periodic table after Palladium has a 5s1 electron.

 

Perhaps. I understand the probablistic aspect, of Heisenberg uncertainty. Still crystal lattices are real and I'm not sure you have to think in terms of Feynman diagrams to understand whats going on. My understanding is that it takes very little energy to move electrons between sub shell orbitals, but it does take energy involving photon emission, usually, to move between shells. (I know they are not orbits. in fact I don't believe electrons are point particles either. It is just convenient to think of them that way for mathematical reasons. I actually believe they are spinning waves, the nodes of which are registered experimentally as a point. But that's another story.) My question is do the electrons in the 4d shell (I also understand that the shell defines an energy space, perhaps a potential space.) which are shared by adjacent atoms, move between sub shells but not shells, with respect to energy and move physically outside the covalent radius, which is an empirically defined parameter, to create what has been called an electron sea.

 

Of course crystal lattices are real, I never meant to imply anything else. An isolated atom has a set of orbitals that define the energy level and the probability of an electron occupying a particular region of space. Atoms in a crystal lattice have a different set of orbitals that overlap, and that define the energy levels and the probability of an electron(s) occupying a particular region of space. What happens within those orbitals is not known and in fact cannot be known. Whatever the shape and location of the covalent bonding orbitals they apparently allow electrons to be shared. I'll leave it at that.

 

Based on what you just said, I looks like it could be interpreted either way with equal validity.
Thanks, Papbravo.
I like your avatar.

 

Thanks. You're the first in over nine years.

 

I used to look like one of those guys, but now I have to wear a hat to keep my head warm.

 

I am merely puzzled because they look like Russians to me, but the location is listed as Michigan.
Probably just showing my ignorance.

 

Papabravo
An isolated atom has a set of orbitals that define the energy level and the probability of an electron occupying a particular region of space. Atoms in a crystal lattice have a different set of orbitals that overlap, and that define the energy levels and the probability of an electron(s) occupying a particular region of space. What happens within those orbitals is not known and in fact cannot be known. Whatever the shape and location of the covalent bonding orbitals they apparently allow electrons to be shared.

The first part of this is a pretty fair summary but more is actually known than this these days.

There are basically two approaches from opposite directions to the question of what happens to outer electrons in what amounts to super- giant molecules that are formed in the metallic bond.

In the ionic bond the electron(s) are definitely reattached from one nucleus to another froming the ions which then transport the electrons with them so the electrons have the same mobility as the ions.

In the covalent bond the electrons are shared between nuclei and small, medium and large molecules are built up this way.
The shared electrons are delocalised from individual nucleii, but belong to the group in the molecule as a whole.

When two atoms join covalently the number of orbitals does not suddenly change, but the overlap or shared ones split into a new set.
This new set contains half the new orbitals in a slightly higher energy state than the unbonded atoms. These are called antibonding orbitals.
The other half of the orbitals occupy a slightly lower state than the unbonded atom, and are called bonding orbitals so the net energy of the two, bonding and antibonding is the same as the original. Thus preserving the conservation of energy requirement. This happens because of the quantum requirement, called the Pauli exclusion principle, that no two levels can have exactly the same set of quantum numbers.

The electrons fill the minimum energy levels first (ie the bonding orbitals) leaving the antibonding orbitals empty.

Now metals and semiconductors that are 'covalently' bonded comprise superlarge arrays of enormous numbers of atoms.
So the numbers of electrons wanting to enter the bonding orbitals is correspondingly huge.
There are nowhere near enough quantum numbers to satisfy Pauli so the bonding (and antibonding) orbitals split again into a myriad of closely spaced (in energy) sublevels called bands.

Once again the bonding levels are filled preferentially, but this time depending upon the substance, the bottom levels of the antibonding band may be close to (semiconductors) or even lower than (conductors) the top of the bonding band. The difference is called the bandgap.

In these circumstances it does not take much thermal or photonic energy to promote an electron from the bonding to the antibonding band.

We also call the antibonding band the conduction band.

I said at the beginning that there are two approaches.
The second approach is to say the electron has left any particular nucleus and is now subject to a continuous array of potential wells in one, two or three dimensions, due to the array of positively charged nuclei.
Then apply the Schrodinger equation to that electron and solve it.
An analytical solution, known as the Kronig-Penny model can be found for one dimension.
This produces the same results as the synthetic model of building up more and more covalent bonding and antibonding orbitals.
Other dimensions have only been solved numerically.

I am sorry I am not able to reproduce diagrams at the moment, these do make the concepts easier, but feel free to ask questions.

 

I am merely puzzled because they look like Russians to me, but the location is listed as Michigan.
Probably just showing my ignorance.

They are Elbonians, occasional characters in a well known comic strip.
http://en.wikipedia.org/wiki/Dilbert
Definitely Eastern European though.

 

The first part of this is a pretty fair summary but more is actually known than this these days.

There are basically two approaches from opposite directions to the question of what happens to outer electrons in what amounts to super- giant molecules that are formed in the metallic bond.

In the ionic bond the electron(s) are definitely reattached from one nucleus to another froming the ions which then transport the electrons with them so the electrons have the same mobility as the ions.

In the covalent bond the electrons are shared between nuclei and small, medium and large molecules are built up this way.
The shared electrons are delocalised from individual nucleii, but belong to the group in the molecule as a whole.

When two atoms join covalently the number of orbitals does not suddenly change, but the overlap or shared ones split into a new set.
This new set contains half the new orbitals in a slightly higher energy state than the unbonded atoms. These are called antibonding orbitals.
The other half of the orbitals occupy a slightly lower state than the unbonded atom, and are called bonding orbitals so the net energy of the two, bonding and antibonding is the same as the original. Thus preserving the conservation of energy requirement. This happens because of the quantum requirement, called the Pauli exclusion principle, that no two levels can have exactly the same set of quantum numbers.

The electrons fill the minimum energy levels first (ie the bonding orbitals) leaving the antibonding orbitals empty.

Now metals and semiconductors that are 'covalently' bonded comprise superlarge arrays of enormous numbers of atoms.
So the numbers of electrons wanting to enter the bonding orbitals is correspondingly huge.
There are nowhere near enough quantum numbers to satisfy Pauli so the bonding (and antibonding) orbitals split again into a myriad of closely spaced (in energy) sublevels called bands.

Once again the bonding levels are filled preferentially, but this time depending upon the substance, the bottom levels of the antibonding band may be close to (semiconductors) or even lower than (conductors) the top of the bonding band. The difference is called the bandgap.

In these circumstances it does not take much thermal or photonic energy to promote an electron from the bonding to the antibonding band.

We also call the antibonding band the conduction band.

I said at the beginning that there are two approaches.
The second approach is to say the electron has left any particular nucleus and is now subject to a continuous array of potential wells in one, two or three dimensions, due to the array of positively charged nuclei.
Then apply the Schrodinger equation to that electron and solve it.
An analytical solution, known as the Kronig-Penny model can be found for one dimension.
This produces the same results as the synthetic model of building up more and more covalent bonding and antibonding orbitals.
Other dimensions have only been solved numerically.

I am sorry I am not able to reproduce diagrams at the moment, these do make the concepts easier, but feel free to ask questions.

This was helpful. Now another question. If a free hydrogen atom, protium, is absorbed into the lattice, does it tend to form a covalent bond with the lattice atoms, ie, contributing its electron to the lattice while sharing a lattice electron to fill its 1s2 shell? Or does it stay unbonded in the lattice interstices? Or fluctuate between bonded and unbonded? I know the shell concept is antiquated, but so am I. It is helpful to me in terms of keeping track of the potential states.

 

You are getting into some serious chemistry with that question
(Which is just me avoiding the answer)
Setting aside the question of where would single hydrogen atoms come from, hydrogen is far and away the lightest substance on Earth and combined with the small size of even the molecule tends to diffuse through solids, including metals.
However there seem to be papers on the formation of (metal) hydrides by direct insertion of monatomic H into lattices.

https://www.google.co.uk/#q=formation+of+hydrides