Physics 249 Lecture 24, Oct 31st 2012
Reading: Chapter 9
HW available on web site, due Nov 2nd
1) Filling ml and ms states, Considerations:
i) For the first electron in a given orbital.
An electron orbiting counterclockwise around a proton will have angular momentum oriented up and magnetic field oriented down, due to the negative charge. In the electrons rest frame it sees the proton rotating counterclockwise around it such that the proton has angular momentum up and magnetic field up. Orbital mechanics requires the magnetic fields of the electron and proton to be oriented in opposite directions. This will be true regardless of the direction of the orbital angular momentum or the orbital angular momentum projection on an axis. Note that this means the orientation of the orbit of the first electron is arbitrary.
The electron will orient with spin angular momentum down and thus magnetic field momentum up relative to the orbital angular momentum, since this minimizes the potential energy. U=-μ∙B
Since the orientation of the initial electron orbital angular momentum is arbitrary the conventional choice to invert the z axis so that the orbital angular momentum points down with maximum negative ml in order to define a the atomic magnetic field as pointing up and has maximum projection on the z axis. For l=1 this will define the first electron as having ml=-1, ms=1/2 and total angular momentum j=1/2 and projection mj = ml + ms = -1/2. The magnetic field relative to the z axis would be μ=12μB. Note that we are defining the axis of the system so that the maximum angular momentum projection is along the z axis and |mj|=j.
ii) Filling of subsequent electron orbitals with the same spin projection.
For subsequent electrons we want to continue to fill the orbitals in a way that minimizes the magnetic potential. The choices are to fill the ms -1/2 state or another ml state. The next ml state has smaller spacial overlap with the first ml state and will minimize the electrostatic potential energy compared to filling the same ml state. For that ms state same ms state will be filled because it again lowers the magnetic potential energy both with the nucleus and with the other electrons. Note lower ml values will be filled first since they allow the electron spin to orient opposite proton magnetic field and minimize the potential.
This process will continue filling all ml levels with electrons oriented in the same direction. Note the final electrons will have ms and ml oriented in the same direction, which increases the potential energy. However, the alignment of the all the electron spins will give a larger negative contribution to the potential energy and this is favored over filling spins in the opposite direction.
iii) Filling electron orbitals with opposite spin.
Once all ml states are filled with electrons with the same spin orientation we consider how to fill the electrons with opposite spin orientation. The overall orbital angular momentum is zero so there is no preference in filling ml by aligning with ml. By filling the highest ml=l state first the ms=-1/2 can be orient the spin in the opposite direction and minimize the magnetic potential energy. Similarly to above you will fill the states progressively now in reverse order.
This process will happen similarly for any l>0 orbital.
iv) d orbital
Now consider the d orbital. The first d orbital gives the ferromagnetic elements. Note these are the only atoms with large magnetic moments low enough in the periodic table to be common ferromagnets. The defining characteristic of ferromagnetic atoms is that they have a strong enough magnetic field that they can align other nearby atoms.
For the d orbital all the inner orbitals have been completely filled and so have no net angular momentum. The magnetic field will be caused by the total mj value of the d orbital, which can be calculated by adding the ml and ms values of all the electrons in the orbital. Note again that we have chosen the z axis such that |mj|=j so mj measures the maximum magnetic field and the direction.
#e, Atom, value of the total angular momentum projection, configuration of ml and ms
d1 : -32 -2 ↑- -1 -- 0 -- +1 -- +2 --
d2 : -2 -2 ↑- -1 ↑- 0 -- +1 -- +2 --
d3 : -32 -2 ↑- -1 ↑- 0 ↑- +1 -- +2 --
d4 : 0,3 -2 ↑- -1 ↑- 0 ↑- +1 ↑- +2 --trans
d5 : +52 -2 ↑- -1 ↑- 0 ↑- +1 ↑- +2 ↑-
d6 Fe: 4 -2 ↑- -1 ↑- 0 ↑- +1 ↑- +2 ↑↓
d7 Co: 92 -2 ↑- -1 ↑_ 0 ↑- +1 ↑↓ +2 ↑↓
d8 Ni: 4 -2 ↑- -1 ↑- 0 ↑↓ + 1 ↑↓ +2 ↑↓
d9 :52,12 -2 ↑- -1 ↑↓ 0 ↑↓ + 1 ↑↓ +2 ↑↓ trans
d10 : 0 -2 ↑↓ -1 ↑↓ 0 ↑↓ +1 ↑↓ +2 ↑↓
The transition atoms are a case where potential energy can be minimized by moving an s orbital electron into the d orbital. In these cases by moving one electron you will fill an entire sequence of ml values. This creates an approximately spherically symmetric charge distriubtion where the electric charge at lower radius will feel no electrostatic force from charge at higher radius lowering the potential energy. Also the spins can be maximally aligned lowering the magnetic potential energy between electrons. The rest of the atoms follow the pattern discussed above.
I have indicated the value of total angular momentum projection since the total magnetic field will be proportional to that.
The largest values are for Fe, Co and Ni, the three common ferromagnetic atoms.
Start chapter 9:
2) Ionic bonding (in more detail)
Ionic bonding occurs between atoms with low ionization potential, typically atoms with few electrons in an outer s shells, and atoms which can lower their potential energy by absorbing an extra electron. For an atom to absorb an electron it must have an empty slot in an orbital with large probability at low radius. An electron orbiting at high radius around an atom would see an electrically neutral system. At lower radius the nucleus will not be totally shielded and the electron can be bound to the system lower the potential energy.
The energy to disassociate an electron is called the ionization energy.
The energy released by binding an extra electron is called the electron affinity.
The magnitude of the electron affinity energies are lower than the ionization potentials so the dissociation and binding of the electron is not enough to lower the potential energy of the system. However the coulomb attraction of the negative and positive atoms contributes a negative term to the potential energy.
Also the binding must be stable so there must be a repulsion that keeps the two atoms from falling into each other. Naively you might say that the electron clouds of the two atoms repulse each other when they in close proximity. However, aside from the outer ionized electron the distribution of the negative charge is spherically symmetric around the point-like nucleus so whole system acts like a point like charge concentrated at the nuclear and is attractive since one ion is positive and the other negative. Once the atoms start to penetrate the spherical symmetric distribution the outer charge will cause no net force and eventually there would be repulsion as the unshielded nuclei repulse each other. However, we see a repulsive force at larger radii. The explanation for this is that as the full outer (or same n,l) orbitals of the atoms overlap then electrons with the same quantum numbers are starting to occupy the same space. The Pauli exclusion principle would force some of the electrons into higher energy states raising the potential energy. Therefore, the lowest potential energy occurs when the ions maintain a separation that prevents the overlap.
The overall potential energy is:
Ur=EEM+Eex+Eion-Eaff=-ke2r+Arn+Eion-Eaff
A and n depend on the ionic bond being considered.
The equilibrium position occurs when the forces due to the electrostatic attraction and the exclusion principle repulsion balance. Moving to lower or higher radius will cause a restoring force toward the equilibrium position. As noted previously the system acts approximately like a harmonic oscillator.
The dissociation energy is the energy to go from the sum of electrostatic and exclusion principle energies at equilibrium to the difference between the ionization minus electron affinity energies.
A note on exclusion principle repulsion. Compare the equilibrium separations of:
NaCl 0.236nm
NaBr 0.250nm
Bromine is a atom with an additional complete shell with 4s, 3d and 4p orbital subshells compared to Cl. You would expect the electron probability distribution to be substantially larger. However, the increase in size is not that large since Bromine has more protons which bind the inner electron more tightly. In fact Br- is 15% larger than Cl-. This is not consistent with the 6% difference in the equilibrium separations. However, even though Bromine has the extra electrons they do not overlap Na+ until the corresponding outer 1p orbital of Na+ overlaps the 1p orbitals of Cl and Br. The observed separation is used to measure the size of Na+, 0.116nm which is consistent with the size of 1p orbitals for atoms in the B – F series.
On the size of 1p orbitals. The Bohr radius, a, goes as the inverse of the charge so we can estimate the adjusted Borh radius using the Zeff*q for the 1p orbitals. Zeff ~ 3. Hydrogen p orbitals have substantial probability to order 6a so with a Zeff of 3 we would expect orbital repulsion to start around 6a/3 = 2a = 0.106 times 2.