Physics 249 Lecture 25, Nov 2nd 2012
Reading: Chapter 9
HW 7 due today. HW 8 available on web site, due Nov 9th
0) Spin orbit splitting and excited states of atoms.
The simplest type of excited state would be to raise the n of an electron in an atom, with
no other changes in quantum numbers. However this type of transition does not occur.
The answer why has to do with conservation of total angular momentum. Like the
electron the photon has spin angular momentum. For the electron the spin is ½ and for
the photon it is 1. This is actually another point of separation between particles that are
symmetric and anti-symmetric under exchange symmetry. The anti-symmetric type
particles have half integer spin and obey the Pauli exclusion principle. The symmetric
particles have integer spin. In this case when a photon is absorbed by the electron raising
it’s energy the electron also must absorbed the angular momentum of the particle and
therefore must be excited to a state with larger or smaller angular momentum. Thus only
raising the n state is forbidden.
Because the form of the radial wave functions leads to different energies for different l
states raising l also raises the energy. Since the photon has spin angular momentum 1
transition that change l by +-1 are allowed.
The spin orbit effect can also cause new energy levels to be available. For a l=>1 orbit
j = |l-s| or |l+s|. For l=1 j = 1/2 or 3/2. The 1/2 state has lower energy since the spin
angular momentum is oriented opposite the orbital angular momentum of the electron (or
proton in the electron rest frame) and the magnetic dipole momentum is aligned in the
same direction lowering the energy. Thus you can raise energy differently by going to
different j states. We label these states by their j. nlj where n is the energy quantum
number, l the orbital angular momentum state s p d …, and j the total orbital angular
momentum. Note that the energies are now depending on n and j.
Since again transitions are caused by spin 1 photons you can have transitions that
conserve angular momentum if l changes by +-1 and j changes by 0,+-1 though not delta
l = 0 (this violates exchange symmetry for identical electrons) or j=0  j=0 (not typically
applicable to spin ½ particles).
For instance from
2s1/2 -> 2p1/2: +1 to l
2s1/2 -> 2p3/2: +1 to l and +1 to j
3p1/2 -> 2d3/2: +1 to l and +1 to j
3p3/2 -> 2d3/2: +1 to l
3p3/2 -> 2d5/2: +1 to l and +1 to j
not 3p1/2 -> 2d5/2: +2 to j
not 3p1/2 -> 2p3/2: +0 to l
1) 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.
Ionic bonding is most common between alkali metals, which of an s orbital with only one
electron, and the halogen group, which has a p orbital with one open postion.
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:
𝑈(𝑟) = 𝐸𝐸𝑀 + 𝐸𝑒𝑥 + 𝐸𝑖𝑜𝑛 − 𝐸𝑎𝑓𝑓 = −
𝑘𝑒 2 𝐴
+ 𝑛 + 𝐸𝑖𝑜𝑛 − 𝐸𝑎𝑓𝑓
𝑟
𝑟
A and n depend on the ionic bond being considered.
You should include unshielded proton repulsion at low separation distance, but this is
typically small in ionic bonding.
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.