Angular momentum and atomic structure

area enclosed by current loop Orbital magnetism Orbiting electrons form a current loop which give rise to a magnetic field. Current flow defined as opposite to direction of electron flow. Magnetic moment of a current loop: Since the current is defined as the direction of flow of positive charge, the orientation of the magnetic moment will be antiparallel to the angular momentume of the electron and can be found using the right hand rule. area enclosed by current loop current

The current loop of the orbiting electron sets up a magnetic dipole which behaves like a bar magnet with the north-south axis directed along m.

Calculating the magentic dipole moment of an orbiting electron Kepler’s Law of areas: A line that connects a planet to the sun sweeps out equal areas in equal times. v A

Remember that the z component of angular momentum is quantized in units of so the magnetic dipole moment is quantized as well: e where The Bohr magneton

Dipole in a Magnetic Field Magnetic dipole tends to want to align itself with the magnetic field but it can never align due to the uncertainty principle! Torque exerted:

Analogy: Precession of a spinning top Here, the gravitational force provides the torque in place of a magnetic field and the angular momentum comes from the spinning of the top.

Precession of electron: Larmor frequency

Zeeman effect No magnetic field Magnetic field applied

How many lines would we expect if the atom were placed in a magnetic field? allowed transitions forbidden transition angular momentum must be conserved …photons carry angular momentum. Remember that not all transitions are allowed. “Satellite” lines appear at the plus or minus the Larmor frequency only and not at multiples of that frequency. You expect a number of equally spaced “satellite lines” displaced from the emission lines by multiples of the Larmor frequency.

Electron orbits as a dipole moment So we have seen that the current loop created by an electron orbiting in an atomic creates a dipole moment that interacts like a bar magnet with a magnetic field. but wait, there's more For many atoms, the number and spacing of the satellite lines are not what we would expect just from the orbital magnetic moment….there must be some other contribution to the magnetic moment.

Electron spin The electron has its own magnetic moment, and acts as a little bar magnet as well. dq Classically: you could imagine a scenario where the electron had some volume and the charge were distributed uniformly throughout that volume such that if the electron spun on its axis, it would give rise to current loops.

The spin magnetic moment dq In analogy to the orbital magnetic moment: the magnetic moments contributed by a differential elements of charge can be summed to be: the “spin” magnetic moment the “spin” angular momentum More generally, if the charge is not uniform: the “g” factor

Magnetic force on a spinning electron

The Stern Gerlach experiment Beam split into two discrete parts! Outer electron in silver is in an s state (l=0), magnetic moment comes from the spin of the outer electron.

Spin contribution to the magnetic moment In addition to the orbital magnetic moment, we must take into account the spin. The spin orientation: Electrons come in “spin up” and “spin down” states. The magnitude of the spin angular momentum is:

Indistinguishability of particles The first two pictures give the same outcome. Even though a and b are identical, you can tell them apart by following them along their unique paths. a b q q a a b a ?? b b p - q b a Quantum mechanically, each particle has some probability of being somewhere at a particular time, which overlaps greatly at the collision point. Which particle emerges where? In wave terms, they interfered.

Probability Y ( ) Y( ) Y ( ) Y( ) x2 x1 x1 x2 Consider two particles in a box, one in the n=1 state, the other in the n=2 state. Y ( ) Y( ) Y ( ) Y( ) x2 x1 x1 x2 A B A B valley hill valley hill Wavefunction not generally symmetric under exchange of identical particles!!

Symmetric and Antisymmetric Wavefunctions Antisymmetric: probability generally highest when particles are furthest apart. “avoiding one another”. Symmetric: probability generally highest when particles are closest together. “Huddling”.

The Pauli Exclusion Principle

4 quantum numbers needed to specify an electron's state The allowed energy levels are quantized much like or particle in a box. Since the energy level decreases a the square of n, these levels get closer together as n gets larger. There are a total of n subshells, each specified by an angular momentum quantum number , and having an angular momentum of

There are a total of orbitals within each subshell, these can be thought of as projections of the angular momentum on the z axis. The electron has an intrinsic magnetic moment called “spin”. The orientation of the angular momentum vector of this apparent rotation motion can only have a manitude of ½.

The Pauli Exclusion Principle The total wavefunction: must be antisymmetric for electrons!

Chemical properties of an atom are determined by the least tightly bound electrons. Factors: Occupancy of subshell Energy separation between the subshell and the next higher subshell.

Electron’s Constraints Hydrogen-Like with Zeff 2) s=1/2 always. l=0,1,2,3,n-1 3) m_j = -j ... +j m_l = -l ... +l m_s = -s ... +s 4) Pauli Exclusion Principle for fermions (e.g. electrons/protons) 5) Charge Screening (i.e. Zeff) affects energy levels 6) Electrons lower energy: Stay away to avoid Pauli and electric repulsion

A pattern starts to emerge s shell l=0 Helium and Neon and Argon are inert…their outer subshell is closed. p shell l=1 Alright, what do we add next???

Uh-oh…3d doesn’t come next…why???

Influence of the radial wavefunction Note the “pulling down” of the energy of the low angular momentum states with respect to hydrogen due to the penetration of the electric shielding. 2s state more tightly bound since it has a nonzero probability density inside the 1s shell and “sees” more of the nuclear charge.

Sodium...not what we expect? Excited state 4s lies lower than 3d!! The energy separating shells becomes smaller with increasing n. Electrons in lower angular momentum states penetrate shielding more, and thus are more tightly bound. As the energy levels become closer together, some lower angular momentum states of higher n may actually have a lower energy.

Now we know where to put the transition metals 4s comes BEFORE 3d… …and apparently 5s comes before 4d, and 6s comes before 5d…

General order of filling shells Note the filling of aligned spins before “doubling up”.

Alright, so where on the periodic table do we make room for all of those f orbitals?? Lanthanide series (or “rare earths” due to low abundance) Actinide series Stick them at the bottom to keep things from getting too awkward.

Viola...the periodic table demystified