Hi everyone. Today we'll resolve the paradox posed by
the so called anomalous Zeeman effect in atomic spectroscopy, by looking at the
consequences of having both orbital and spin angular momentum at play in the
atomic electron Hamiltonian. But first I want to tell you a little bit
more about spin and its interaction with magnetic fields.
First, let's review the elements of our theory of electron spin.
The spin vector components are represented as operators by the two by
two Pauli matrices. In direct analogy to orbital angular
momentum, the square of the total spin is h bar squared times the half integer spin
quantum number, little s, times s plus 1. Eigenvalues of the individual components
are simply the spin magnetic quantum number, m sub s.
Times the quantum of action h bar. In a magnetic field along z, we have a
trivial 2 by 2 Hamiltonian whose eigenvalues are given by the diagonal
elements. With corresponding eigenvectors which are
purely one spin component or the other. We see that the first moves up in energy
which we expect from a magnetic moment pointed anti-parallel to the magnetic
field. Because of spin angular momentum and its
magnetic moment or antiparallel, however, this corresponds to spin parallel to the
field. And so we call it spin up.
The other moves down in energy, consistent with a magnetic moment
parallel to the field, which has angular momentum opposite to the field.
It can be called spin down. Note that we can compactly represent the
eigen energies by this simple function of the spin magnetic quantum number, m sub
s. We can use the stationary states of the
simple spin Hamiltonian to calculate the state evolution of a spin in a magnetic
field in which it is not in eigenstate. For example, let's say we prepare a spin
as an eigenstate of Sx. For example, by placing it in a magnetic
field long x and waiting for slow relaxation processes that we ignored in
our Hamiltonian, to take the system to its lowest energy state.
For a magnetic field along the negative x direction, that's the state with a
positive eigen value of S X. The ket labeled plus.
Now, let's suddenly rotate the magnetic field along Z.
The state is not a eigen state of the new Hamiltonian, but we can decompose it into
a super position of eigen states that diagonalize the Hamiltonian.
And which each evolve in time with a different complex exponential given by
its energy. We can now use this to determine how the
expectation value of spin along x changes with time.
We simply sandwich the operator by the state.
Remembering to not only transpose but also complex conjugate the bra on the
left converting it from a column vector to a row vector.
The result is a quantity which oscillates in time.
But how is spin conserved then? If s x is changing, where does the spin
go? By calculating the expectation value of
spin along the y direction, s y we see that it too is oscillating in time, 90°
out of phase with spin along x. This is simply the quantum mechanical
version of a process we are all familiar with, the Larmor precession of an object
with angular momentum subject to a torque, such as a gyroscope.
In this classical example, it's a mechanical torque due to gravity.
Whereas the spin undergoes procession due to a mechanic, magnetic torque, caused by
a magnetic moment in a magnetic field. Note that the procession frequency is
controlled by the Zeeman splitting from magnetic field magnitude.
This field control over spin procession Larmor frequency is precisely how we can
spacially image using magnetic resonance imaging.
Powerful electromagnets are used to create strong magnetic fields and field
gradients. So that only a small volume will the
Zeeman splitting that corresponds to a resonant exotation from narrow band radio
frequency electromagnetic waves. This selective auscultation is then
sensed by inductive detectors. Pickup coils.
By scanning the resonance slice around in space by changing the electromagnetic
coil currents. Images of deeply buried structures are
created. MRI works by acting on the spin of the
nuclei, not electrons. Which is enabled by the much smaller
nuclear magneton due to much larger mass of the nuclei.
And correspondingly small zamon energies and radio frequencies for excitation.
Although impractical for imaging human biology, electrons spin resonance at
around 20 gigahertz per tesla for the free electron.
Is, however, very useful for study of inanimate samples, such as organic
molecules and solid state materials. Now we can get back to the issue of
anomalous Zeeman effect of atomic spectra in a magnetic field.
Both the orbital and spin angular momentum of the electron have magnetic
moments which couple to the magnetic field.
But as we saw, the relativistic nature of the spin makes inclusion of a non-unity
Thomas G factor necessary. We can of course, diagonalize the
Hamiltonian, and determine the Zeeman splitting for each term separately.
But how do we go about diagonalizing this expression with vectors l and s, that
are, in general, not parallel? The key is to notice that the total
angular momentum, l plus s, is conserved. We expect that the energetic splitting
will be on the same form as before, but with an effective g factor.
Let's see how this works out. First we project both vectors onto the
conserved angular momentum axes by taking the inner product with the unit vector
along j l plus s. For a magnetic field along z, this then
picks out the z component of j, which has eigenvalues h bar times total magnetic
quantum number, m sub j. Carrying out the dot product yields an
expression that is mostly known quantities in terms of the individual
quantum numbers. But what's the dot product between the
spin and the orbital angular momentum? This can be seen by once again looking at
the conserve quantity, j squared. Substituting this quantity into our
Hamiltonian entirely converts it into a scalar energy quantity.
Which is proportional to the magnaton times the magnetic field times the
magnetic quantum number, m sub j. With the coefficient out front, it
depends on little l, s and j. This is the so called Lande G-factor.
Here's a few useful examples. When the orbital quantum number is zero.
We just have the magnetic field interacting with the spin.
Which we know has a g factor of two. Lande correctly predicts this.
For l equals one, the total angular momentum j can be one plus one half,
three halves. Or minus one half, yielding one half.
In the first case, the Lande G-factor is four thirds.
And in the second, two thirds. So from the six l equals one states,
that's three orbital projections times two spin projections.
Four of the j equal to three halves states, where m sub j is minus three
halves minus one half plus one half and plus three halves, have one G factor and
the two remaining, j is equal one half states with m sub j equal to plus and
minus one half have another G factor of half its value.
In a magnetic field, the six split like this.
Now, let's look at transitions to the l = 0 derived states.
The selection rules with Landau Spin Orbit Coupling are analogous to the
orbital selection rules. Changes in quantum number j, are plus or
minus 1, and changes in the total magnetic quantum number, m sub j are 0
and plus 1 or minus 1. This means that only the four j equal
three halves states can radiatively transition to the j equals one half
states. And only the m sub j equal to plus or
minus one half states can transition to both l equals zero states.
This leaves six distinct transitions. The energy of the spectroscopic shifts
that we observe by, by dispersing the emitted photos through the spectrometer
are proportional to the difference in the products of lambda g factor times the
total magnetic quantum number m sub j for initial and final states.
We can now see how multiplates more than the Lorenz triplet are created.
In this example, spectral, spectroscopic line splits into six in a magnetic field.
Each separated by two thirds of the [UNKNOWN] splitting, an all line shift
with magnetic field. It is this spin orbit coupling, that
explains the Anomalous Zeeman effect.