In order to understand the theory behind Molecular
Beam Spectroscopy, one must examine quantum physics and molecules.
Each molecule is a combination of atoms, each with their own
set of quantum numbers, such as spin (s) or angular momentum
(l, m). These numbers are used to describe the atoms at the quantum
level. When atoms bond to form a molecule, the quantum numbers
of these atoms combine, not by strict adding but through quantum
mechanical procedures. S is a vector representing the
total spin (intrinsic angular momentum), and the vector L
represents the orbital angular momentum. It is often more convenient
to use J=L+S in the case of the molecule where J
is the total angular momentum. S and L are directly
related to their respective quantum numbers. These new vectors
determine the "state" of the molecule. Each set of
quantum numbers dictates the expected behavior of the molecule
when measurements are taken. Nearly all molecules, when bonded,
align in such a way that all electron spin pairs are anti-aligned
or the net electron-spin component is zero. This means that the
only contribution to spin comes from the nucleus itself. In the
case of a molecule with two atoms, the first atom's nuclear spin
would be represented by I1, and the second atom's nuclear spin would be represented
by I2. Each
molecule has a vibrational quantum number (v) as well. The quantum
combination of the I values and the rotational J
value for a specific v value yields the states of the system.
The intermediate angular momentum, labeled F1, is found by combining I1 and J. By combining
F1 and I2, the
total F angular momentum is obtained. Their relationship
to each other is pictured below. Through quantum mechanics, one
knows that each molecule only has a finite number of spin states
for specific J and v values due to the values of the nuclear
spins. Since J and v have an infinite number of values,
there are an infinite number of total states in the molecule.
Diagram 1: This diagram shows
how splitting can occur. It does not, however, contain all states
and the levels are relative to the frequency range.
Molecular beam spectroscopy focuses on the
hyperfine structure of the molecule. The hyperfine structure
arises when examining the Schrödinger equation, a key element
in quantum mechanics. The hyperfine structure correction is necessary
because the Schrödinger equation lacks this physical attribute
of the molecule. The hyperfine structure characterizes the orientation
and spin of the moment. Any nuclear interaction involving the
nuclear spin is considered a hyperfine interaction. Any energy
associated with changing the orientation of the nucleus affects
the hyperfine structure. In molecular beam spectroscopy, the
final goal is to find the constants of the hyperfine interactions
that describe mathematically the orientation and shape of the
nucleus.
A molecule does not need to stay in one particular
state. Transitions occur when a molecule changes from one state
to another. Not all transitions between states are as likely
as other transitions. Some of the transitions occur only in the
rarest of rare cases. By examining the molecule through quantum
mechanics, one can determine what these "forbidden"
transitions are. Each transition is unique by the fact that certain
physical properties are glued to specific transitions. The most
common property is the dependence on light frequency and emissions.
Each transition has an associated frequency to it, corresponding
to the energy required to either induce a transition or energy
emitted in a spontaneous transition. Molecular beam spectroscopy
induces transitions by applying the correct frequency and looking
to see if a transition did indeed occur as theory states. For
hyperfine transitions, the frequency is in the radio frequency
(RF) range or 100-60000 kHz.
The molecular beam spectrometer (MolBeam) is a complex
apparatus that must do everything that will allow us (as researchers)
to examine the hyperfine structure of molecules. The apparatus
currently being used at St. Olaf is pictured at the right. The
picture was taken when the MolBeam was still in use at Harvard.
The explanation of the apparatus is quite involved and complex.
To help the reader follow the explanation, a link to the schematic of the MolBeam
is provided. This will open in a new window to help explain the
how the MolBeam works. It will be used as a reference several
times.
The MolBeam has five distinct regions, each
of which does something integral for the entire apparatus. The
regions are named source, A, B, C and detector. These are clearly
marked on the schematic. The beam travels from the source through
A, C, then B, and finally into the detector region. Most regions
can be sealed from each other with slide valves to allow the
researchers to do mechanical work on a particular region without
disturbing the others. The explanation will follow this path.
First, the source region.
The source region is where it all begins.
The two main components of the source region are the oven source
itself and the large 10" diffusion pump. The source region
needs to be pumped down to 10^-7 torr in order to function properly.
This is done through the operation of the diffusion pump. To
operate the diffusion pump must be connected to a foreline pump
and kept at a pressure of around 10^-2 torr. The MolBeam at St.
Olaf uses a mechanical pump for the foreline. Then the pump can
do its magic. The diffusion pump heats some very expensive
silicon oil in the base of the pump which starts to vaporize
the oil. As the oil rises, the water cooling system cools the
oil, and it settles back into the base. The oil carries any gas
molecules that were around it when it cooled. In the base, these
air molecules are pushed out through the foreline, hence lowering
the pressure. Once the pressure is low enough, the source oven
can be activated.
The source oven is where the sample is placed.
With St.Olaf's MolBeam, we tend to examine alkali-halide salts,
but gases and hydroxides have also been examined. The source
sits in the oven, connected to a current supply. To heat the
oven, currents of up to 80 amps are passed through it. Typically
the oven temperature is near 550 degrees C, which vaporizes the
salt. The source then exits the tube and oven through a small
opening. This creates the molecular beam.
The beam travels out of the source region
into the A region. The A region contains the quadruple lenses
and a 6" 
diffusion
pump. The region needs to be at 10^-7 torr as well. The most
important part of the region is the lenses. These lenses select
the specific J state to be examined. The lenses are four
metal rods arranged in a diamond pattern. This configuration
is shown in the associated diagram (diagram 2). The diagram represents
a cross-section of the quadrupole lens. The lens arrangement
utilizes the dipole nature of molecules. Each J state
is associated with a rotational "orientation" of the
molecule. The lenses utilize this with the configuration of the
quadrupole rods. The voltage equipotential lines for this configuration
is 
. The electric field is then the
negative gradient of this voltage which is 
where i and j are the unit vectors in the x and
y directions respectively. The energy associated with the interaction
of the electric field and the molecule is proportional to the
square of the electric field. This yields 
,
which means the energy and focusing varies radially. By specifying
a quadrupole voltage, V0,
the beam will focus a specific J value (center of diagram
2); all other J values should not focus (outside portion
of diagram 2). Theory can predict the voltage to a certain degree.
Once a starting place is given by theory, experimentation is
needed to find the exact V0 for the specific J states. The lenses do not
focus only the correct J state for a quadrupole voltage,
but it does focus lower J states as well. The lower J
states are significantly weaker, however, they are largely
insignificant; but they can corrupt data sets. The lenses, with
J states, also select a certain orientation of the other states
as well. This insures that the main bulk of the beam will be
the same J value and having the same spin orientation.
The beam is now a parallel beam due to the
lenses in the A region. The beam now travels into the C region
where the beam passes through two meter long RF plates. The RF
plates are connected to a frequency generator which is controlled
by a computer. The generator can output frequencies in the RF
range with steps of 10 Hz. The plates act as a capacitor where
the RF signal and DC power can be applied to create an oscillating
electric field. These plates with the electric field can induce
transitions in the molecular beam, near the predicted quantities
found through quantum theory. The C region needs to be at 10^-7
torr and in pumped down through the A and B regions.
After the beam is subjected to the RF and
DC voltages, it travels into the B region. The B region contains
another set of lenses and a 6" diffusion pump. This region
must be on the order of 10^-7 torr. This lens works in the same
manner as the one in the A region. If a transition does occur
in the C region, the previously focused beam will now become
unfocused in the B region and fly out of the beam as the unselected
states did in the A region.
The beam ends its journey in the detector
region. The detector region contains the detector and two ion
pumps. Pressures must near 10^-8 torr in the region. This could
have been done with diffusion pumps, but the oil splash from
the pump would have interfered with the detector. Ion pumps work
by running a high voltage through plates inside the pump. These
plates ionize the air molecules which attracts them to grounded
titanium plates. These ions are embedded in the plates, permanently.
The ion pumps do not have an exhaust system or oils to pump with.
The detector is a tantalum ribbon covered
with tungsten. The detector is created by vaporizing tungstenhexacarbonyl
which decomposes, leaving the tungsten on the heated tantalum.
It is a delicate process that requires some patience. Once the
tungsten detector is made, it is placed within an apparatus that
sits in the detector region. Current is passed through the ribbon
in order to heat it. When the beam hits the ribbon, it will ionize
the molecule. The ionization is correlated with the work function
of tungsten. If the detector cannot ionize the beam, oxygen is
added to change the work function. The molecule will then take
an electron from the detector system, changing the current. This
change in current is directly related to the amount of beam hitting
the detector.
The MolBeam does not take data with voltage
applied to the RF plates the entire time. Actually, it measures
the difference in the beam when the RF plates are on and when
the RF plates are off. The computer can take this data and subtract
the difference to see if the beam did indeed change when the
RF plates were on. The difference will be noticeable if a transition
occurred. We subtract RF "on" data from RF "off"
data in order to have peaks instead of dips in the data curve.
|
What is the Useful
Information |
The data is collected through computer automation
and saved. What can people get from the data? First the data
needs to be fitted. The data is fitted using the Rabi function.
The Rabi function can correctly determine the frequencies at
which the transitions occur, but it is not merely that simple.
Other physical corrections need to be accounted for in order
to get the best fit possible. When DC and RF signals are applied
to the beam, strange things can occur. Stark effect may alter
the data. Stark effects are another level of splitting which
would split from the "F state" splits (see Diagram
1). The stark interactions not only split the F states
but in can also shift the peaks. The quantum numbers of the transition
predict the behavior of the stark components. Once the stark
components are taken into effect and the Rabi function adjusted
accordingly, the data can be fitted. The fitting can be done
in a math manipulation program such as MathCad or with a stand-alone
program such as St. Olaf's nSimp program, which utilizes the
Simplex method of fitting. The following diagram shows a fit
of KF data using MathCad2000. Notice the stark components with
the stark stick spectrum. The base of the stark stick spectrum
is the frequency that is of most interest.
Diagram 3: The KF data taken
in Summer 2001. The data has been fitted with MathCad2000 using
the Rabi function.
The fits yield the frequencies based on the
transitions. These frequencies, with the uncertainties, can be
used to adjust the molecular constants. The molecular constants
are related to the shape of the nucleus, quadrupole shape, and
other interaction constants. The new constants can be used to
generate a new set of theoretical predictions. The new predictions
can yield better starting places for future scans. The new scans
can find more frequencies to adjust the molecular constants.
The molecular constants are pinpointed with this bootstrapping
technique. Most of the bootstrapping is done with the aid of
a computer program. By finding the molecular constants, the hyperfine
structure is better known. This concludes the in-depth explanation
of Molecular Beam Spectroscopy.
Contacts
The MolBeam Group: molbeam@stolaf.edu
Advisors
Dr. Jim Cederberg: ceder@stolaf.edu
Dr. Duane Olson: olsondn@stolaf.edu
Dr. David Nitz: nitz@stolaf.edu
Explanation written by Evan Frodermann, MolBeam
summer 2001, St.Olaf Class of 2002. With help from Mike Bongard(Summer
2001), Katie Huber(Summer 2001), Heather Tollerud(Summer 2001),
and Dr. Jim Cederberg (faculty).
References
Modern Physics by
Rex and Thornton
Intro to Quantum Mechanics by David J. Griffiths
Physics of Atoms and Molecules by Bransden and Joachain