Micro wave spectroscopy
Theory
In the experiment entitled Molecular Spectroscopy we
learn that to a gross approximation the total energy of a gaseous
molecule can be expressed as a sum of electronic vibrational,
rotational and translational parts. If the molecule possesses
magnetic or electric moments and if there are external fields,
further energy terms arise. And of course, interactions among
the various motions, for example vibrational-electronic
(vibronic),
spin-rotation,
and
rotational-vibrational
(rovibrational) must be added as (usually small) correction terms
to the total energy of the molecule.
The rotational energy of a linear molecule, approximated as a
rigid rotor is
BJ (J+1), where J = 0,1,2,3… is the rotation
quantum number, and where B=h/8π2cI is the rotational
constant in reciprocal centimeters (cm-1). For molecules other
than hydrides, the moment of inertia I is such that B ≤ 1 cm-1,
and the rotational levels are so close together that only “optical”
spectrographs or spectrometers of the highest resolving power
are capable of detecting the rotational structure on vibrational or
vibronic transitions; and even these instruments fail when B <<
1 cm-1 as is the case for most molecules containing four or more
heavy atoms.
Multiplication of B(cm-1) by the velocity of light c shows
that in frequency units B is of the order or less than 30 GHz (1
GHz = 1 gigahertz = 109 cycles/sec). This places transitions
between the pure rotational energy states of most molecules in
the microwave region of the spectrum, where resolution of
rotational structure is much better than that in the optical region.
Just as in the classical problem where an oscillatory electrical
field can cause an electric dipole to rotate with the frequency of
the impressed field, the oscillatory electric vector of
electromagnetic radiation can similarly “drive” a quantum
mechanical rotor possessing a permanent electric dipole
moment. Similarly, the magnetic vector of the electromagnetic
wave can interact with a rotor possessing a permanent magnetic
moment. Thus molecules such as OCS and BrCN, having
electric moments, and O2, having a magnetic moment, have
rotational spectra in the microwave region, while a molecule
such as carbon dioxide does not.
For a quantum mechanical rotor, the energy of the
electromagnetic field is absorbed only when the frequency of
the field is near that corresponding to the energy difference
between two discrete energy states of the rotor. The resulting
absorption lines for the quantum oscillator constitute the
microwave spectrum that one observes. Working backward from
the spectrum then, one can ascertain the rotational energy states
of the molecule, the moments of inertia of the molecule, and
therefore some information about the dimensions of the
molecule. In certain cases, one can obtain sufficient information
to obtain all the bond distances in the molecule, or for a
nonlinear molecule, all the bond distances, bond angles, mass of
isotopes(according to reduced mass) and moment of inertia.
This is one of the primary goals of microwave spectroscopy.
Intensities of Transitions and Selection Rules
As mentioned in the Introduction, charges or magnetic
moments can interact with the E and H vectors associated with
electromagnetic radiation. The selection rules for pure rotational
transitions in a linear rotor. These are
ΔJ=±1,
ΔM=±1
x,y-polarized
ΔJ=±1,
ΔM=0
z-polarized
The rotational energy levels have constant distance between
each two rotation levels that equal to 2B, this come out by
differences of two levels through energy value as following:
∆𝜺 = 𝜺J+1 - 𝜺J = B J+1(J+1+1) - B J(J+1)
∆𝜺=2B(J+1)
In this stage there are constant in rotational levels 2B,4B,6B,
and so on….
Can be make a table of energy measurements for several values
of rotational quantum numbers:J
𝜺J
J-…..J
0
1
2
3
4
0
2B
6B
12B
20B
------0---1
1-----2
2------3
3------4
Absorption spectral line
Cm-1
--------First line at 2B
Second line at 4B
Third line at 6B
Fourth line at 8B
Stark Effect
An atom or molecule possessing electrical charges can
interact with a static external electrical field. The result of such
interaction is called the Stark Effect.
The energy of a stationary dipole in a field is given by,
W=−μo.E =−μo.Ecosθ, where θ is the angle between the dipole
and the field E. It is convenient to define the field direction in
space as the z-axis, since then θ, as before, is just the angle
between the space-fixed z and the molecule-fixed z' axes. A
rotating dipole, with angular momentum vector perpendicular to
μo represents the linear rotor in microwave spectroscopy. If the
dipole is rotating very rapidly, the effect of the field largely
averages out. However, there is a tendency for the field to
change the angular momentum, and therefore the energy of the
rotating dipole, a tendency that naturally becomes more
pronounced as the angular velocity of the rotor decreases. The
quantum mechanical description for this change of angular
momentum is one based upon the “mixing” of J-states for the
linear rotor. The angular momentum states of the molecule in
the field are therefore no longer pure J-states of the free
molecule, i.e. J is no longer a “good quantum number”. Because
of the form of the Hamiltonian, H'=−μo.Ecosθ, the mixing of
states in this case follows the same “selection rules” as do dipole
transitions; namely, only states differing in J by ± can mix.
The energy level diagram given in Fig. 1 is useful in
visualizing the effects of the Stark field. The vertical arrows
indicate the transition J=1←0,ΔM=0with and without the Start
field. Note that this transition is shifted to higher energy by an
amount proportional to the square of the molecular dipole
moment, μ0. The Stark effect is thus useful in determining
dipole moments.
Figure 1.Energy level diagram for ridged rotator, illustrating the
stark effect.
Hyperfine Structure
In addition to angular momentum due to molecular rotation,
some molecules possess nuclear angular momentum because of
the nuclear spin of one or more of their nuclei. The proper
description of the rotational states must account for both the
molecular and nuclear-spin angular momentum contributions to
the total angular momentum. If no interaction exists between the
orientation of the nucleus and that of the molecule, the energy of
a given molecular rotation state designated by J is unaffected
by the nuclear spin, I. Such interactions exist however, and
lead to hyperfine structure in molecular spectra.
In most atoms, hyperfine structure mainly arises because
of interaction between magnetic moments of the nuclei and of
the electrons. In most molecules the electronic magnetic
moment is zero, and angular-dependent electrostatic terms are
often the largest in the hyperfine interaction. These terms arise
because of the interaction of electric charges in the nuclei with
electrons in the molecule. The angular dependence of the
interaction naturally vanishes if either charge distribution is
spherically symmetric, or this is the case for S-states of atoms or
when the nuclei themselves have spin I<1.
Expanding the interaction energy in a multipole expansion,
the lowest-order non-zero angular-dependent term is that where
the electric quadrupole moment of the nucleus interacts with the
gradient of the electrostatic potential due to the electrons. The
dependence of this interaction upon the orientation of the
quadrupolar nuclei gives rise to small energy corrections to the
rotational states of the molecule. For a linear molecule, the
nuclear quadrupole hyperfine energy terms are given by,
F is the total angular momentum quantum number inclusive of
nuclear spin I. F takes on all values I+J,I+J−1……I−J. WQ
vanishes for I=0 or I=1/2 since in both these cases the nuclear
quadrupole moment Q is zero. The quantity ∂2V/∂z'2, usually
denoted q, is the second derivative of the electrostatic potential
at the quadrupolar nucleus along the molecular axis due to all
charges outside a small sphere containing the nucleus. The
correction term to take into account electrons that penetrate into
the nucleus can be shown to be very small.
The selection rules for transitions among the hyperfine levels
are as follows :- ΔJ=0,±1 ΔF=0,±1 ΔI=0
A transition J=1←0 with I=32 will thus be split into three
lines corresponding to as shown in Figure 2. Note that the
ΔJ=+1 and ΔF=0,±1 sign of eqQ is important in determining the
appearance of a hyperfine multiplet.
A detailed examination of electric quadrupole interactions
reveals considerable information relating to the electronic
charge distribution in molecules. One of the principal
difficulties in interpretation of the data is in the separate
determination of the quantities in the product eqQ, known as the
quadrupole coupling constant. However, by measuring this
product for a series of molecules incorporating a given nucleus,
Q thus remaining constant, the relative variation of q with
molecular structure can be investigated.
Figure 2. Energy level diagram for linear rigid rotator.
Rotational Motion
The rotational motion of a molecule is determined by the
moments of inertia and the angular momenta.
1. Classically, any object has three orthogonal principal
moments of inertia (diagonals of inertia tensor) with
corresponding simple expressions for the rotational
energy and angular momentum.
2. This carries over to quantum mechanics, and it is
customary to classify the rotational properties of
molecules according to the values of the principle
moments of inertia.
The principle moments of inertia are designated
Ia, Ib, and Ic in order of increasing magnitude
Linear Molecules & Symmetric Tops
A molecule which is linear or has an axis of
rotational symmetry is called a symmetric top
Either Ic = Ib > Ia or Ic > Ib = Ia
At this time the same equation of two linear molecule
has been using for the distance between the spectral
lines 2B but with small value of bond distortion constant
(D) to give up the real estimation of bond according(non
rigid rotate).example// O=C=S,O=N-H,H≡C-Cl
• Linear molecules (methyl halids)have a small I about the
axis of the
molecule so they are of the first type and are called
Prolate symmetric tops
• Other molecules, e.g., benzene, have the largest
moment of inertia about the symmetry axis & are called
oblate symmetric tops
• Molecules which are spherically symmetric, e.g, methane
have three equal moments of inertia and are called
spherical tops
– Molecules with Ic ≠ Ib ≠ Ia are asymmetric tops
Prelaboratory Questions
1) Calculate the exact abundances and masses of the isotopic
variants of OCS.
2) Describe the normal mode vibrations of OCS. Look up the
vibrational frequencies and indicate any degeneracies, where
appropriate.
3) With the available experimental apparatus, only rotational
transitions of OCS with are observed. How could transitions
with ΔM'J=O ; ΔM'J=±1 be observed?
4) Calculate the stark field (volts cm-1) required to produce a
stark splitting of 100 MHz for the most abundant isotropic
species (use the literature parameters given in the notes).
5) Major contaminants in the vacuum system are N2, O2, Ar,
CO2, and H2O (assuming it is not a smoggy day). Would you
expect any of these species to give lines in the 8-12 GHz portion
of the microwave spectrum?
Laboratory Experiments
Procedure
The purpose of the experiment is to study a portion of the
microwave spectrum of the linear molecule OCS. Various
modifications of the molecule are possible because of the
presence of isotopes of O, C, and S in natural abundance.
Prominent in the microwave spectrum are 16O12C32S, 16O12C34S,
16 13 32
O C S, 16O12C33S. In addition, rotational transitions in
thermally populated excited vibrational levels of the most
abundant species 16O12C34S can also be detected. The spectra
allow the study of:
1) The rotational states of OCS and determination of the bond
distances in the molecule;
2) The Stark splitting, the dependence of this splitting on E, and
a determination of the dipole moment of OCS;
3) The nuclear quadrupole splitting due to the 33S nucleus in the
molecule 16O12C33S, and a determination of the quadrupole
coupling constant.
Assignment
1) Measure the line positions for the 16O12C32S, 16O12C34S,
16 13 32
O C S, 16O12C33S (in this case refer to the diagram to
calculate the line position in the absence of quadrupole
splitting), and (if possible), 18O12C32S species and calculate the
bond distances.
2) Measure the stark splitting for the
calculate the dipole moment.
16
O12C32S species and
3) Measure the line positions for the 16O12C33S multiplet and
calculate. You should think about a method to determine the line
spacing with reasonable accuracy.
4) Measure the position of the line for the 16O13C32S species and
that of a nearby less intense line. Verify that the latter line arises
from a vibrationally excited species by cooling the stark cell
with dry ice.
Notes on the Microwave Spectroscopy
Experiment
Calculating moments of inertia, I, from the known bond
distances is straightforward given one of the formulas discussed
below. You should do this to determine where to look for the
lines for each isotopic species: 16O12C32S, 16O12C34S, 18O12C32S,
16 13 32
O C S, 16O13C33S. What are the relative abundances and do
they correctly correspond to the observed relative intensities?
The predicted line positions should help to identify the line due
to a vibrationally excited species. This line can be positively
identified by cooling the cell to reduce the population of this
species and thus its relative intensity. Use dry ice, but be careful
not to freeze the o-ring seals at the ends of the cell. What
information can you get from this line position?
When you have collected your data and converted it to moments
of inertia, you should use it to calculate the bond distances. Line
positions for any two isotopic species can be used to obtain two
bond distances. What assumption is made in doing this
calculation? The bond distances can be calculated to all pairwise combinations of the isotopic species for which you have
obtained line positions. Which pairs should give the most
accurate bond distance values? Why?
The moments-of-inertia to bond-distance calculation are not
straightforward. These notes outline one way that you might
precede. Two isotopic species differing only in their sulfur
masses, m3, are considered as an example. A change of
subscripts allows these calculations to be applied to isotopic
species differing at oxygen. The case of isotopic species
differing at carbon is different and is left as an exercise.
Moments of Inertia from Bond Distances We'll start with the
relation defining the COM
m1r1+m2r2+m3r3=0
(1)
And the definition of the moment of inertia about the COM
I=m1r12+m2r22+m3r32.
The bond distances are
l23=r3−r2orr3=r2−l23
(2)
l12=r2−r1orr1=r2−l12
(4) and let
l13=r3−r1=l12+l23
(3)
(5)
Substituting (3) and (4) into (1) to eliminate r1 and r3
m1r2−m1l12+m2r2+m3r2+m3l23=0
Mr2=m1l12−m3l23
(6) Where M=m1+m2+m3
(7)
Substituting (3) and (4) into (2) to eliminate r1 and r3
I=(m1r2−l12)2+(m2r22+m3r2+l23)2
I=Mr22+2(r2m3l23−m1l12)+m1l122+m3l232
(8)
Now, we may substitute (6) into (8) to obtain
I=M(m1l12−m3l23)2/M2+2[(m1l12−m3l23)(m3l23−m1l12)]/M
+m1l122+m3l232
I=m1l122+m3l232−(m1l12+m3l23)2/M
(9)
This is an expression for the moment of inertia in terms of the
bond distances. It may be rewritten:
I=1/M.[m12l122+m1m2l122+m1m3l122+m1m3l232+m2m3l232+m32l232−
m12l122+2m1(m3l12l23−m32l232)]
=1/M.[m1.m2l122+m2m3l232+m1m3.(l122+2l12l23+l232)]
I=1/M.[m1m2l122+m2m3l232+m1m3l132]
(10)
Where (5) has been used in the last step. So far, we have not
considered isotopic substitution so the results are perfectly
general. Bond Distances from Moments of Inertia Starting with
the moments of inertia I and I' for two species differing in mass
m3, we wish to obtain the bond distances. One approach starts
with equation (10):
I=1/M.[m1m2l122+m2m3l232+m1m3l132]
(10)
For the isotopic species
I'=1/M.[m1m2l122+m2m'3l232+m1m'3l132]
(11)
Multiplying (10) and (11) by m'3M and M3M' respectively
m'3MI=m1m2m'3l122+m2m3m'3l232=m1m3m'3l132
(12)
2
2
2
m3M'I'=m1m2m3l12 +m2m3m'3l23 +m1m3m'3l13 (13) Subtracting
(13) from (12)
m'3MI−m3M'I'−m1m2(m'3−m3).l122
(14)
or
l122=m'3MI−m3M'I'/(m1m2m'3−m3)
(15)
This may be substituted into (10) using (5) and the resulting
quadratic equation solved for l23.
Diagrammatic figure of molecular bonds.
Example//
Rotation spectrum of 76Br19F has a serious lines in equal
distance from each other (0.71433cm-1).calculate the rotation
constant, moment of inertia, and length bond of molecule.
Answer //
1-Equal distance= 2B =0.71433cm-1 ⇒
The rotation constant=B=0.35717 cm-1
2- Moment of inertia=h/8π2BC ⇒ I=7.83x10-96 kg.m2
3- length bond of molecule= r= (I/𝛍)0.5=0.1755nm