CHEM 2070 Structure And Spectroscopy Michael K. Denk

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Time, PlaceMWF, 13:00 - 14:20,MacKinnon 224

Instructor

TextbookLecture notes as posted on the WWW:

- Introduction
- Vibrational Spectroscopy
- Electronic Spectroscopy
- Nuclear Magnetic Resonance Spectroscopy |pdf-file|
- ESR Spectroscopy
- Microwave Spectroscopy
- Atomic Spectra

Grading

- 2 Term Tests (30 %)
- 1 Final Exam (40 %)
- Midterm 1 Fri Feb. 12th in class
- Midterm 2 Fri Mar 11th in class
- Final April 22nd 8:30-10:30 am

Hours:After class, by appointment, by e-mail

e-mail:

BackgroundReading Material:

P. Atkins, Julio de Paula- Physical Chemistry, 7th edition, 2002
- Freeman

- Not as clearly written as earlier editions. My personal favorite is the 3rd. ed.
Ira Levine- Molecular Spectroscopy
- Wiley
- The classic text on the physical foundations of spectroscopy. Good use of mathematical methos.
D. H. Williams, I. Fleming- Spectroscopic Methods in Organic Chemistry
- McGraw Hill
- Covers UV-Vis, NMR, IR and MS, good for the bench chemist, somewhat outdated
A. D. Baker, D. Betteridge- Photoelectron Spectroscopy - Chemical and Analytical Aspects
- Pergamon, 1972
- covers PES and ESCA, very readable
H. Günther- NMR Spectroscopy
- 602 pages ; John Wiley & Sons; 2 edition (July 25, 1995); ISBN: 047195201X. One of the earliest textbooks on the topic. Well written and well structured. Discusses many issues that are "avoided" by other textbooks. Can be highly recommended for all bench chemists
Basic One- and Two-Dimensional NMR Spectroscopy, 3rd Revised Edition- by Horst Friebolin (Author)
- Paperback - 410 pages 3rd Revised Edition edition (November 24, 1998)
- John Wiley & Sons Canada, Ltd. ; ISBN: 3527295135
- An excellent book covering nearly all aspects of NMR including 2D techniques. Easy to read, well organized, a good balance between fundamental principles and applications.
J. A. Wertz, J. R. Bolton- Electron Spin Resonance
- Chapman and Hall, 1986
- A classic text, full coverage of transition metals and mathematical derivations
Valery F. Traven- Frontier Orbitals and Properties of Organic Molecules
- Ellis Horwood, 1992
- Nice presentation of MO diagrams and PES spectra of organic molecules, very good introduction covering theoretical chemistry

Tools

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Terminology

Spectroscopic methods use artificial terms (often in the forms of acronyms like "NMR" "IR" ) but also terms that seem to be familiar but are used in a very narrow specific sense.

Fortunately, there are resources that compile and explain the terminology:

- V. Gold
- Glossary of terms used in Physical Organic Chemistry
Pure. Appl.Chem.1983,55, 1281-1371R. Jones, J. F. Bunnett

NOMENCLATURE FOR ORGANIC-CHEMICAL TRANSFORMATIONS

Pure. Appl.Chem.1989,61, 725-768R. D. Guthrie

SYSTEM FOR SYMBOLIC REPRESENTATION OF REACTION-MECHANISMS

Pure. Appl.Chem.1989,61, 23-56.S. E. Braselevsky

SELECTED TERMS AND SYMBOLS IN PHOTOCHEMISTRY

J. OF PHOTOCHEMISTRY AND PHOTOBIOLOGY B-BIOLOGY 1 (2): 261-270 DEC 1987

The Historical Development of Spectroscopy

The historical roots of spectroscopy go back to the very beginnings of chemical studies.

Around 1860, Robert Bunsen who had previously left his mark on chemistry by studying "cacodyloxide", and hence inventing organometallic chemistry, revolutionized science by demonstrating that each element has the unique capability of changing the color of a flame. To observe the correct color, Bunsen needed a clean burning flame which in turn led him to invent a special gas burner that bears his name and is in use to this day. Flame tests are VERY sensitive and this led Bunsen to discover two low abundance elements that had previously remained undetected : Rubidium and Cesium.

The modern version of flame photometry is Atomic Absorption Spectroscopy (AAS). It uses a plasma instead of a flame, is quantitative and allows the routine determination of elements down to the ppm level. The quantity of the analyzed material is obtained through comparison with commercially available AAS standards.

Robert Wilhelm Bunsen

Bunsen's discovery shows the crucial elements of other, more modern spectroscopic techniques. You need a good idea, the right equipment and some understanding of the underlying physical principles. Because this is a lecture course, we will focus on the third component, the underlying physical principles. Unfortunately, the subject is nontrivial and comprises most of the advanced physics, in particular quantum mechanics and occasionally even relativistic effects.

A Classification of Spectroscopic Methods (I): Particles and Waves

Spectroscopy in the broad sense of the word comprises all phenomena where electromagnetic radiation interacts with matter.

Electrons and other particles behave as waves under certain circumstances and are indeed useful for spectroscopic investigations. Examples are Low Energy Electron diffraction (LEED) a method used to determine the structures of surfaces (e.g. catalysts) and Rutherford Back Scattering (RBS), a method useful to determine the relative amount of light elements in materials (e.g. TiN - Ti(C,N) - TiC).

It should be noted that the distinction between particles and waves is somewhat arbitrary. Many light particles, in particular electrons, can show diffraction which is a typical property of electromagnetic waves. A characteristic property of particles that waves do not share is rest mass.

The Basic Elements of a Spectrometer

A simplistic block diagram of a
spectrometer consists of a source of radiation (light...), a
wavelength selector, the sample cell and detectors. If only the
attenuation of a source radiation is required, the detector is placed
at the opposite side of the radiation source, other arrangements,
particular the 90^{o} position of detector B below are useful
fro the measurement of emitted light. The emission of light can
occour virtually instantaneous (so called Fluorescence) or deleayed
(so called Phsophorescence.

Light can be emitted as a resultt of a photochemical excitation but also as a result of a chemical reaction ("Chemoluminescence). Examples for chemoluminescence are abundant i nature: fireflies, many marine organisms. The classical laboratory experiment in chemoluminescence is the Hydrogenperoxide - Luminol reaction.

Note that the Frequency selection device becomes unnecessary if a Laser is used as light source (laser light is monochromatic).

The block diagram above describes a so called single beam instrument. It is often desirable to compare the sample to a reference sample and this leads to the so called "double beam" design. The double beam design is a duplication of the above block diagram except that some parts like the detector or the frequency selector can be shared by toggling between sample and reference.

Diffraction gratings provide an inexpensive way to select frequencies.

Individual diffracted beams of radiation cancel (destructive interference) unless they obey the selection rule

nl= 2d (sinq-sinf)

where n= 1, 2, 3, ... is the diffraction order, d = distance between grooves and l is the wavelength. The angle q is that of the incident radiation, the angle f is that of the scattered radiation.

For a given n, q and Df (slit mask), a narrow range of frequencies Dl is selected.

A disadvantage of the diffraction grating (and indeed most monochromators e.g. prisms) is that most of the light (ca. 99 % ) is lost in the process. The individual frequencies are generate by adjustment of q which is usually a slow process.

Spectral Lines - Signal or Noise ?

All spectrometers and samples produce "noise" These are randomly occurring spikes that are caused by

- The detector
- The amplifier
- The sample
- The light source

To identify a signal as signal (rather than a random spike that happens to be unusually strong) it is required that its intensity is at least 3 times as strong as that of the noise.

The "electronic noise" caused by the detector and by the amplifier is due to the thermal excitation of electrons in the microcircuits and can be greatly reduced by cooling the equipment to very low temperatures (liquid nitrogen temperature, -196 C or lower). This is rarely practical or necessary in chemical investigations (exception: reflective IR of powders) but common for astrophysical spectroscopy.

Noise caused by the sample is typically due to the thermal motions of the molecules and can sometimes be reduced by cooling of the sample.

A Classification of Spectroscopic Methods (II): Types of Electromagnetic Radiation

Spectroscopically useful radiation covers the whole electromagnetic spectrum from high energy X-rays to low energy radio frequencies and a vast number of different radiation sources has accordingly been developed, in many cases for the specific purpose of generating radiation for spectroscopic applications.

Spectroscopic Methods - Overview

The smooth transition from one part of the electromagnetic spectrum to the next creates the false impression that there is in fact only one type of spectroscopy with different light sources and different detectors.

The following table sums up the differences between the methods by listing the type of information gained from the individual methods together with experimental details like sensitivity, details of the interaction between light and matter, and restrictions.

X-ray diffraction Scattering, mainly by electrons,
followed by interference (l
= 0.01-1 nm) Electron density map of
crystal 10 crystal ~10 Difficult to locate light atoms in
presence of heavy atoms Difficult to distinguish atoms with
similar number of electrons Neutron diffraction Scattering, mainly by nuclei, followed
by interference (l
= 0.1 nm) oriented internuclear
distances 10 crystal ~1 cm Best method to locate hydrogen
atoms Allows determination of magnetic
ordering Requires large crystals (1mm and
up) Electron diffraction Diffraction mainly by nuclei, but also
by electrons (l
= 0.01-0.1 nm) Scalar distances due to random
orientation 10 100 Pa (1 Torr) Thermal motions cause blurring of
distances. Molecular model required Gas phase and surface only Absorption of radiation due to dipole
change during rotation (l=
0.1-30 cm; 300-1 GHz in frequency) Mean value of
10 10 (10 Mean value of
Absorption of radiation due to dipole
change during vibration (l=
10 Symmetry of molecule Functional groups 10 100 Pa (1 Torr) Useful of characterization. Some
structural information from number of bands, position and
possibly isotope effects. All states of matter Scattering of radiation with changed
frequency due to polarizability change during a
vibration (l
= visible usually) Symmetry of molecule Functional groups 10 10 (100 Torr) Useful for characterization. Some
structural information from number of bands, position,
depolarization ratios, and possibly isotope effects. All
states of matter Absorption of radiation due to dipole
change during an electronic transition (l
= 10-10 Qualitative for large
molecules 10 1 Pa (10 All states of matter Interaction of radiation with a nuclear
transition in a magnetic field (l =
10 3 KHz to 300 MHz Symmetry of molecule through number of
magnetically equivalent nuclei Many many others 10 10 Applicable to solutions and gases. In
conjunction with molecular weight measurements may be
possible to choose one from several possible
models Mass Spectroscopy Detection of fragments by
charge/mass Mass number,fragmentation
patterns 10 (10 Gas phase only Fragmentation pattern changes with
energy of excitation Extended X-ray absorption fine
structure (EXAFS) Back scattering of photoelectrons off
ligands Radial distances, number, and types of
bonded atoms 10 Any state Widely used for metallo enzymes and
heterogeneously supported catalysts

After I.R. Beattie,
**Technique**
**Nature of the
effect**
**Information**
**Interaction
time**
**Sensitivity**
**Potential
Problems**
^{-18}
s but averaged over vibrational motion
^{-3}
cm^{3}
^{-18}
s but averaged over vibrational motion
^{3}
^{-20}
s but averaged over vibrational motion

**Microwave****Spectroscopy**
*r*-^{2}
terms; potential function
^{-10}
s
^{-2}
Pa^{-4}
Torr)
*r*-2
does not occur at *r*e
even for harmonic motion. Dipole moment necessary. Only one
component may be detected. Analysis difficult for large
molecules of low symmetry
^{-1}-10^{-4}
cm)
^{-13}
s
^{-14}
s
^{4}
Pa
^{2}
nm)
^{-15}
s
^{-2}
Torr)
^{2}-10^{7}
cm;
^{-1}-10^{-9}
s
^{3}
Pa
^{-9}
Pa^{-11}Torr)
^{-18}
s, but averaged over vibrational motion
*Chem. Soc.
Rev.*, **1975**, *4*, 107.

Electromagnetic Radiation (I) - Generation

In the early days of spectroscopy, the luminosity, long term stability and the price of the radiation source dominated all other issues.

Today, the emphasis is often on generating short and ultra-short pulses of the required radiation to do time resolved experiments (NMR, study of ultrafast processes through femtosecond laser spectroscopy).

SourceRadiationOscillating Crystal

Radiowaves

Klystron

Microwaves

Nernst filament, Laser

IR

Tungsten lamp, laser

UV-VIS

X-ray electrode, synchrotron

X-rays, hard UV

g-rays

g -emitters like

^{60}Co

The prieces of light sources ranges from a few dollars for a Nernst filament or a small semiconductor laser to about 100 Mio $ for a synchrotron facility.

A synchrotron is essentially a storage ring for very fast electrons. Acceleration of the electrons produces a very intense light that can be tuned from the X-ray region all the way to UV.

Synchrotrons are essential tools for many problems in materials science biochemistry as well as fundamental studies. The very intense synchrotron radiation allows the study of molecules in low concentrations (enzymes, catalytists etc).

Electromagnetic Radiation (II) - Physical Description

The description of light as "electromagnetic" waves was first given by James Maxwell (1831-1879). He achieved one of the first great unifications of previously unrelated phenomena by showing that electric and magnetic phenomena are related and can be described by only four fundamental equations, the so called Maxwell equations.

These equations are not only important in their own right but were also the starting point for much of Einstein's work. It is also noteworthy that Maxwell's fundamental contributions have continuously been available in print since they were first published. Biography

In this course, we will not use Maxwell's equations but restrict ourselves to the underlying basic concepts of electromagnetic radiation:

- Frequency
- Intensity
- Plane of polarization

The radiation is described by two perpendicular sinus waves, one representing the oscillation of the electric field, the other representing the oscillation of the magnetic field. Spectroscopy is nearly exclusively relying on the interaction of the E-field with molecules or atoms. The analogous interaction of the B-field is several orders of magnitude weaker, is more difficult to measure and can be safely neglected in most cases.

We have not included the wavelength as basic descriptor because we have already included the frequency. The wavelength of electromagnetic radiation is directly related to the frequency through: n = c/l

Daylight consiste of a random selection of individual waves with different planes of polarization.

Absorption of Electromagnetic Radiation - The Coupling Mechanism

An electromagnetic wave is an oscillating electrical field and interacts only with molecules that can undergo a change in dipole moment

The oscillating dipole can be provided by the rotation of a permanent dipole like for example HCl. This type of interaction leads to microwave spectra

Or by the vibration or bending of a bond. This type of interaction gives rise to vibrational Spectra (IR- and Raman spectra):

Absorption of Radiation - The Energy Treshold

In order to be absorbed, the energy of the radiation must have a minimum energy that is defined by the energy levels of the atom / molecule.

This is a minimum requirement, other criteria are the probability of the individual absorption process (see below).

The energy of a photon of wavelength l is given by the Einstein equation E = h n = h c / l. The energy and the wavelength are thus inversely proportional. Short wavelengths correspond to high energies. This is why you need safet glasses against UV radiation and lead shields against g-radiation.

Q.What is the energy of 1 mol of photons with wavelength 400 nm ?

A.v = c / l

= 3 x 10

^{8}m s^{-1}/ 400x 10^{-9}m= 7.5 x 10

^{14}m

E = hn

= 6.26 x 10

^{-34}j s x 7.5 x 10^{14}s^{-1}= 4.970 x 10

^{-19}J

This is the energy of one individual photon. To obtain the energy of one mol of photons, we have to multiply this value by Avogadro's number:

4.970 x 10

^{-19}J x 6.022 x 10^{23}mol^{-1}

= 299.3 kJ mol^{-1}

The "Wavenumber"

The number of waves that would occupy
1 cm length is called the wavenumber and has the dimension
cm^{-1}: The wavenumber is not an SI unit. One reason for its
continued use is the fact that wavenumbers are a direct measure of
the energy of the respective radiation.

Q.What is the wavenumber corresponding to light with l = 400nm ?

A.we first convert nm into cm and then take the inverse

l = 400nm

= 400 x 10

^{-9}m= 400 x 10

^{-7}cmwavenumber = 1 / (400 x 10

^{-7}cm) =25,000 cm^{-1}

Absoprtion of Radiation - The "Spectrum"

The result of a spectroscopic investigation is typically presented as a 2D chart ("spectrum") with y = intensity of radiation and x = frequency of radiation.

The intensity specifies how much radiation energy has been absorbed (or transmitted) by the sample. However, the molecule may "store" the energy and emit it with certain time delay (fluorescence or phosphorescence) in which case it is advantageous to measure the intensity of the emitted light.

Black Body Radiation

A simple way of generating electromagnetic radiation is to heat a solid, liquid or gas to sufficiently high temperatures.

At about If the temperature is high
enough (> 400 ^{o}C), the broad spectrum of emitted
wavelengths begins to contain some visible (red) light, at higher
temperatures (light bulb: 2800 ^{o}C) the color shifts
towards yellow.

The spectrum (Intensity-frequency or Intensity- wavelength plot) of any hot surface is fully characterized by its temperature. The precise relationship was first given by Max Planck. Planck's law contains two empirical earlier relationships as special cases: Wiens displacement law and Stefan-Boltzmann law.

Wien's Displacement Law

states, that the most "abundant" frequency (the maximumof the intensity-frequency graph) depends only on the temperature T:

T • l_{max}= 0.2878 cm K

Q.What temperature would we need to generate X-rays with l

_{max}= 10 Å ?

A.T • 10• 10

^{-10}m = 0.2878 10^{-2}m KT • 10

^{-7}m = 0.2878 m KT = 2.878 10

^{6}K

Stefan-Boltzmann Law

predicts the Power radiated by "black-body" per unit area:

P = 5.670•10^{-8}W m^{-2}• T^{4}

The law has many interesting applications. One of them is the fact that, due to radiation losses, heating objects to high temperatures becomes more and more difficult with rising temperature.

QHow much more energy would you absorb to if you move from the cosy place in front of a baking oven (450 K ) to a stack furnace (T ~ 1800 K) ? How much more distance would you need ?

A(1800 K)

^{4}/ (450 K)^{4}= 256 times; The radiation drops off with 1 / r^{2 }, you would need (256)^{1/2}= 16 times the distance. 1 m distance to the cookies ~ 16 m distance to the stack furnace.

Spectral Lines and the Quantization of Energy

Despite the apparent simlicity of Wien's displacement law and the Stafan-Boltzman law, attempts to predict these laws with classical mechanisc were ridiculously inaccurate. The classical derivation predicts, that a burning match would emit X-rays !. This disaster of prediciton went down in the history of physics as the so-called "UV-catastrophe"

The breakthrough not only for this problem but for modern Physics in general came 1902, when Max Planck (Nobel Prize 1918) arbitrarily assumed, that the individual atoms can only vibrate in certain fixed frequencies. This means, that they can absorb or radiate energy only in certain discrete amounts ("Quanta").

The resulting folrmula (Planck's radiation law) correctly predicted the radiation behavior for hot bodies at any given temperature.

The idea that energy can only be transmitted in multiples of small quantities was only reluctantly accepted at first but turned out to be the essential concept to describe atomic and molecular physics.

Max Planck (1858-1947)

Erwin Schrödinger (1887 - 1961)

Every confinement of a particle to a discrete space (by potential barriers) creates a quantization of its energy. This quantization of energy levels gives rise to discrete energy levels in molecules and atoms.

A number of other simple cases have relevance for spectroscopy:

- Particle on a ring: internal rotation in molecules
- Particle on a sphere: molecular rotation
- Particle in a box: molecular vibration
- Electron in a spherical potential: hydrogen like atoms

The individual energy levels are obtained by solving the Schrödinger Equation (Erwin Schrödinger: Nobel Prize in Physics 1933). The different solutions arise from the different boundary conditions (spatial confinement etc.

In its most simple form, the Schrödinger equation can be written as

HY= EY

The Hamilton Operator is the operator that generates individual energy from the different wavefunctions. The Hamilton operator is specific for each atom or molecule. Only a small number of functions ("eigenfunctions") solves the equation and each of these functions leads to a specific energy level ("Eigenvalue").

All other physical properties of the system are likewise generated from the wavefunction with specific operators.

A simplistic picture of the whole situation is that of a vending machine. If you throw in the right coins (wavefunctions) you get whatever the vending machine (the operator) sells. Hamilton Inc. sells energies.

Simple Quantum Systems: The Particle In a One Dimensional Box

A simple but highly instructive
quantized systems is the so called particle in a box. Confinement of
a particle of mass m in a one-dimensional box of length L leads to a
series of allow energies E_{n}:

Simple Quantum Systems: The Particle in a Three Dimensional Box

The 3D confinement leads to solutions that are very similar to the 1D solutions above because the 3D wave function can be written as product of 1D wavefunctions.

Y(x,y,z) = Y(x) • Y(y) • Y(z)

The energies are the sums of the 1D energies just that we have 3 independent quantum numbers, one for each dimension.

E(n_{x},n_{y},n_{z}) = E(n_{x}) + E(n_{y}) + E(n_{z})

Simple Quantum Systems: Hydrogen Like Atoms

The first spectra to be studied in detail were the UV-Vis spectra of atomic hydrogen.

Different series of spectral lines were recognized to obey simple numerical rules and were named after the researchers that first studied them (Lyman, Paschen, Bracket, Pfundt).

The unifying principle behind all
these spectral lines was the term diagram of hydrogen which was first
given by Niels Bohr. The description is equally valid for other one
electron systems (for example (He+ or Li^{2+}) but
unfortunately cannot be applied to multi electron systems (for
example He which has two electrons). The reason for this is the so
called electron-electron correlation. The energy levels of such
multi-electron systems cannot be obtained analytically (that is:
through a finite formula) but can still be obtained through
approximation processes (see also other 3-body problems).

The hydrogen atom is historically important as a jump off point for the development of modern quantum physics and atomic spectroscopy. We will not cover atomic spectroscopy in a systematic fashion but will refer to the hydrogen atom and lessons learned from it wherever useful.

The remainder of the Introduction will develop fundamental concepts of spectroscopy.

Line Shapes and the Heisenberg Uncertainty Principle

Even under the best of experimental circumstances, spectral lines are not monochromatic but do cover a certain range of frequencies. This is a consequence of one of the most fundamental laws of physics, the Heisenberg' uncertainty principle.

Werner Heisenberg1901 - 1976Nobel Prize in Physics 1932

"for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen"

This Heisenberg uncertainty principle cannot be derived from other laws of physics . It states, that two so called canonically conjugated variables like impulse p and location (x,y,z) cannot be simultaneously be determined with utmost precision but are "fuzzy". The relationship is:

Dp •Dx > h / 4p~10^{-34}J•s

Where h is the famous Planck constant. Energy and time are also canonically conjugated :

DE •Dt > h / 4p10^{-34}J•s

The latter relationship predicts that
states with a long lifetime (large **
****D****
t **) will have a well defined
energy whereas states with a short lifetime will have ill defined
energy (large energy uncertainty **D
****E **).

For spectroscopic transitions, this uncertainty in energy translates into an uncertainty of frequency which is the natural linewidth. (h dash = h/2p)

The factor
1/2p
is approximately 6. A lifetime of
6•10^{-6} seconds would therefore translate into a
linewidth of 10^{6 }Hz. The linewidth in l is calculated with
the general formula n = c/l.

Excited electronic states have a
typical lifetime of 10^{-8} seconds which leads to an
imprecision of **Dn
**= 1/6•10^{8} Hz.
This seems to be a huge frequency range but because optical
transitions typically have **n
**= 10^{14} -
10^{16} Hz, the relative
imprecision
**Dn
**/
**n
** is ~ 10^{8} /
10^{15} or 10^{-7} - less than 1 ppm!

Note that
**Dn
**/
**n****
**has no unit because the Hz cancel.

For optical spectra (UV / Vis), the quantum chemical line broadening is not only very small but also rarely detectable because Doppler broadening and collisional broadening are much larger.

For other spectroscopic methods the
situation may be reversed. Excited electron spins have a lifetime of
10^{ -7} seconds which leads to
a **Dn
**= 10^{7} Hz. Because
ESR absorptions typically have **n
**= 10^{8} -
10^{9} Hz, the lifetime derived line broadening is quite
prominent.

The intrinsic lifetime of excited
states is ~
**n
**^{3}.
Transitions corresponding to long
wavelengths (microwave transitions in rotational spectroscopy)z have
very small linewidths, those corresponding to short wavelengths (e.
g. UV-VIS) have broad lines. The typical lifetime of an
electronically excited state (UV-VIS) is 10^{-8} sec which
translates into a natural linewidth of about 15 MHz.

The intrinsic linewidth predicted by the Heisenberg uncertainty principle is rarely observed because other mechanisms (see below) lead to even stronger line broadening.

Spectral Lines - Shape and Resolution

The two most important features of
spectral lines aare their height (~ intensity) and their width (
**D****n
**). The precise geometric shape (gaussian line shape, lorentzian
line shape or some other form) becomes important if the spectra are
processed by a computer.

A crucial question that arises with any spectroscopic technique is the question of how well we will be able to resolve two different spectral lines.

Before we blame the spectrometer for not delivering the desired resolution, we have to take a brief look at other factors that might cause the broadening of spectral lines.

Collision Broadening

The collision of molecules causes the excited state to revert to the ground state. The excitation energy is carried away by the collision partner or is emitted as electromagnetic radiation. Collisions thus shorten the lifetimes of excited states and lead to the broadening of the associated spectral lines.

A good rule of thumb is to assume
10^{13} collisions per second in liquids and 10^{9}
collisions in gases at 1 atmosphere. An important difference between
gases.

**In gases**, collision broadening
can largely be suppressed by lowering the pressure - less collisions.

**In liquids**, the density is
typically 1000 times higher than in gases and the collision frequency
goes up by about 10^{6}. This is a large number and spectral
lines in liquids can be quite broad.

**In solids**, the molecules are
packed tightly like in liquids (density of gas ~ 1/1000 of liquid or
solid) but the collisions are reduced due to the confinements imposed
by the crystal lattice. The spectral lines of solids are therefore
often better resolved than those of liquids. Due to the well defined
relative orientation of the molecules in the crystal lattice,
additional splittings (that is: not observed in liquids and gases) of
the spectral lines may occur.

Doppler Broadening

The frequency that an object emits is modified by its speed relative to the observer (detector). The classical example is change in pitch of a train approaching (high pitch) or moving away from an observer.

The phenomenon was first explained by the Austrian Physicist Doppler and is called the Doppler effect. Because molecules are in motion and because some of them move away from the detector while others are moving towards the detector spectral lines will be blurred by the Doppler effect.

Collision broadening is the main cause of line broadening in liquids, Doppler broadening the dominant cause in gases.

The precise formula for the Doppler induced frequency shift is:

n ' = n / (1±v/c)

where **v **is the emitting
frequency (molecule as reference point), **v **= speed of the
molecule relative to observer, and c = speed of light.

It is possible to eliminate Doppler broadening by investigating a molecular beam of the atoms / molecules and placing the detector at a 90 deg angle to the direction of the beam.

Doppler Broadening and Temperature

In molecular spectroscopy, the molecules are not at rest but move in random paths caused by collision with other molecules. Their speed is not uniform but follows a statistical distribution. This speed distribution is determined by only two parameters: the mass of the molecule, m, and the temperature T. The half-height width caused by the thermal motion of the molecules is:

The resulting line shape is of the gaussian type and allows the determination of the temperature if the mass of the particle is known - useful for astrophysics.

Line Shapes: Gaussian vs. Lorentzian

It is useful to have a mathematical description of the measured line shape because it allows to filter the spectroscopic data from background noise or to separate two lines that overlap. It turns out that the mechanisms that lead to line broadening also determine the line shape.

As a result of the speed distribution in gases, the line shape caused by Doppler broadening is a Gaussian function:

The "natural" line shape of spectral
lines was first derived in 1930 with the help of quantum field theory
(Weiskopf and Jeener). They assumed a two level system (states n and
m) with individual lifetimes of
t_{n} and
t_{m}:

This type of line shape is a Lorentzian functions:

The Lorentzian line shape is similar to the Gaussian line shape but falls off more slowly.

In the next sections we will try to understand what determines the intensity of individual spectral lines and how the intensity of absorption and emission might be related.

"Unnatural" Lifetime: The Probability of Stimulated Emission

We have seen that the natural lifetime of an excited state translates into a broadening of the emitted spectral line into a bell shaped profile of different frequencies. This type of emission in the absence of external fields is called spontaneous emission.

As it turns out, the emission process can also be triggered by external radiation ("stimulated emission").

The general relationship between the probability ( = 1/lifetime) of a stimulated emission and spontaneous emission (corresponding to the natural lifetime) was given by Einstein:

A = B • 8phn^{3 }/ c^{2}

With

B= probability of stimulated emission

A= probability spontaneous emission

c= speed of light.

n= frequency of the transition

**A** and **B** are generally
called the Einstein coefficients. One way to read the equation is
that highly excited states (large ** DE **) revert more rapidly to
the ground state through spontaneous emission than excited states
that are closer to the ground state.

While the whole concept may seem somewhat academic, it has led to the development of one of the most useful "machines" of modern science, technology and medicine, the LASER.

Laser Active Materials

The amplification of light with a
material possessing an inverted population became subject of
speculation after
Albert Einstein analyzed the
stimulated emission of light. The idea is simple. Suppose, that you
could somehow generate a system in which the population of the
excited state exceeds that of the connected ground state, that is
N_{e }> N_{g}

Irradiation of such a system with
light matching the gap energy (E_{g}
=
hn
), would trigger a stimulated emission of light which would amount to
a light amplification. Devices that allow such light amplification
are called LASERS (** L**ight

Stimulated emission of Photons Albert Einstein

Einstein laid the foundations for such a light amplification machine by making precise predictions:

- Spontaneous emission rapidly
increases in probability with E
_{g}(~**n**)^{3} - The probability of stimulated absorption and stimulated emission are identical. This means, that it is not possible to generate a population inversion by optical pumping in a two level system.

We can obviously not beat the problem of spontaneous emission but there is a trick around the inability to "pump" a two level system: we build a three level system.

A simple two level system will thus not be able to amplify light because stimulated emission will prevent a build up of excess population in the top level. However, the restriction can be bypassed through the inclusion of a second excited state with the following two characteristics

- filled through thermal relaxation from a higher optically pumped level
- has a low probability of spontaneous emission

The third level is now filled ("populated") indirectly (via 2). Irradiation with light corresponding to the emission 2 -> 1 leads to the stimulated emission of light .

Properties of Laser Light

Laser light differs in many aspects from ordinary light (that is: the light generated by light bulbs or fluorescent tubes). Laser light is

**Monochromatic**The range of frequencies emitted by a laser is very narrow,typically**Dn / n**= 10^{-5}to 10^{-8}.**Low Divergence**Laser beams can not only be focused to a smaller point than any other light source (ca. 1**l**), but the also show very small divergence. Under optimum conditions, the divergence can reach the theoretical limit of**f****= l / d****Coherent**The maxima and minima of all emitted wavelets occur at the same position.**Highly Intense**Pulsed lasers can generate light pulses of extremely high intensity. For example, commercially available Nd:YAG lasers (**l**= 1064 nm) are capable of 10 ns pulses with an energy of 10^{8}W. This corresponds to an energy of 10^{20}W / m2 at the focal point - 10^{17}(!) times more intense than the sunlight at the earth's surface.**Polarized**The E-planes of the individual lightwaves are all parallel to each other.**Precisely Timed**The ability to create incredibly short laser pulses (currently: 4 femtoseconds = 4 fs = 4 x 10^{-15}s) allows the spectroscopic resolution of extremely fast processes. A 4 fs pulse in the visible corresponds to only 2 cycles of the corresponding sin wave.

It goes without saying that such a light source with these exceptional properties is the preferred toy of the spectroscopists.

Lasers in Action

A common three-level laser active
material is ruby, a solid solution of ~ 1 % of
Cr_{2}O_{3} in aluminum oxide
Al_{2}O_{3}. This is the material used to build the
first Laser by Maiman in 1960:

Diagram of Maiman's optically pumped Ruby Laser

Theodor Harold Maiman * July 11 1927 The "coil"

Laser active materials are not restricted to the solid state or inorganic materials. Some lasers work with gases (e.g. the HF laser) and there are many laser active dyes.

A particularly inexpensive way to build a laser employs a special type of semiconductor material (indirect band gap semiconductor). The most commonly used materials are based on gallium arsenide (GaAs), cost around $ 5 and can be found in any CD player, DVD player etc. Silicon, germanium and many others are useless because they have a direct band gap and cannot be made laser active.

Four level lasers use the same principle but the laser emission occurs between two excited states. (3-level: excited state -> ground state).

Most laser active materials give one and only one frequency - they cannot be tuned to different frequencies. Tunable materials are those that contain a large number of individual excited states. Depending on which of these levels we pump (or stimulate) we can obtain a multitude of different wavelengths.

The invention of the laser was in many ways ahead of its time and the laser was originally called "an invention looking for a job". The skepticism was unfounded. Lasers are nearly as pervasive as microelectronics.

They are used in telecommunication (optical fiber networks + LASERs), optical data storage (CD, DVD, high capacity storage systems in hospitals), surgery (microsurgery, eye surgery, neurosurgery....), surveying, etc.... and spectroscopy.

Hunsperger, p316

**Literature**

Construction and use of semiconductor lasers: "R. G. Hunsperger
"Integrated Optics: Theory and Technology" Springer-Verlag,
1991

Spectral Lines (II): Intensity

The intensity of a spectral line is determined by a number of factors:

**Concentration of the emitting
molecule or atom**

**Transition Probability** The
precise calculation of transition probabilities is involved and
requires quantum mechanics. However, it is often possible to predict
if a transition is symmetry allowed or forbidden. We will discuss the
respective **Selection Rules** for each spectroscopic technique
individually.

**Population of States** For
thermal equilibrium, the probability of a state in thermal
equilibrium is given by the Boltzmann Distribution:

N_{upper}/ N_{lower}= exp(-DE/kT)

Where
**D****E**
is the energy difference between
the two states, k is the Boltzmann constant (1.38 •
10^{-23} J / K), T the temperature (absolute temperature,
measured in Kelvin). The probabilities add up to give 1 as long as we
have covered all possible states:

N_{upper}+ N_{lower}= 1

Note that for meaningful temperatures
(T > 0 ) the variable 1/T is also > 0 that means
that
**-****D****E/kT
**is indeed always negative as
the sign suggests. This allows us to draw the graph of the the
Boltzmann distribution as a function of the type y =
exp(-x•constant) with 1/T as the variable.

Because T (and 1/T) is always > 0, we can ignore negative values of x and can now state what the Boltzmann formula predicts.

- As T approaches absolute zero, 1/T (x in the graph) goes to infinity and y approaches zero, but only asymptotically. The physical meaning behind this is that even at very low temperatures we have a small (non zero) probability of the upper state being populated.

- As T approaches infinity, 1/T
approaches zero and y approaches 1. This leaves us with the
somewhat startling prediction that s N
_{upper}can never exceed N_{lower}, the best we can do is N_{upper}= N_{lower}but that requires T to become infinite !

- An inversion of the population
**N**would only be realized for negative temperatures._{upper}> N_{lower}

After we have settled these issues it
is time to worry about the magnitude of **D****E**.

Q.The energy difference between the vibrational ground state and the first excited state of a certain bond was determined to be 2000 cm

^{-1}. What is the ratio of the populations in these two statesa) at room temperature

b) at 1200 K

A.N

_{upper}/N_{lower}= exp(-DE/kT). We could now calculate E from E = h n and n = c/l = c x wavenumber or.....to keep things simple, we could use kT =0.025 eV for T = room temperature and convert this value with the factor 1 eV = 8066 cm

^{-1}to 201.7 cm^{-1}. or, we convert the wavenumbers into electron Volts:2000 cm

^{-1}= 0.248 eVThe result will be the same, namely

a)N_{upper}/N_{lower}= exp(-DE/kT) = exp(-9.92) = 5.9 x 10^{-5}At room temperature, only a minute fraction of the respective bonds would be in the first excited state.

b)At 1000K, we have roughly increased the temperature from 300 to 1200 K. This makes kT 4 times larger => 4 x 0.025 eV = 0.1 eV andN_{upper}/N_{lower}= exp(-DE/kT) = exp(-2.48) =0.084

Q.Silicon, the semiconductor used most widely in computerchips, has a bandgap of E

_{g}~ 1.1 eV between the totally filled valence band (VB) and the empty conduction band (CB). Which percentage of the the electrons in the VB are actually contributing towards conduction at r.t. ?

A.N

_{upper}/N_{lower}= exp(-DE/kT) = exp(-1.1 /0.025) = 7.78 x 10^{-20}= 7.78 x 10^{-18}%.The number is extremely small and pure silicon should be an extremely lousy electric conductor. This is true, but the conductivity can be increased d ramatically by adding certain impurities like boron (B) or phosphorus (P).

Ludwig Boltzmann

Ludwig Boltzmann's influence on on modern physics is so profound that a few biographical remarks are appropriate

"Nothing is more practical than a good theory"

L. Boltzmann

Ludwig Boltzmann 1844-1906

Ludwig Boltzmann was born in 1844 (Vienna, Austro-Hungary). Boltzmann was awarded a doctorate from the University of Vienna in 1866 for a thesis on the kinetic theory of gases supervised by Josef Stefan. Boltzmann taught at Graz, moved to Heidelberg and then to Berlin. In these places he studied under Bunsen, Kirchhoff and Helmholtz. In 1869 Boltzmann was appointed to a chair of theoretical physics at Graz. He held this post for four years then, in 1873, he accepted the chair of mathematics at Vienna. He did not stay very long in any place and after three years he was back in Graz, this time in the chair of experimental physics.

Boltzmann, used to say that the reason he moved around so much was that he was born during the dying hours of a Mardi Gras ball. He did feel that his nature made him subject to rapid swings between happiness and sadness. Although remote diagnosis of mental ailments is notoriously difficult it is fair to say that Boltzmann showed symptoms of manic depressive behavior.

After another three years, in 1894, Boltzmann moved back to Vienna, this time to the chair of theoretical physics which became vacant on the death of his teacher Josef Stefan. However, the following year Ernst Mach was appointed to the chair of history and philosophy of science at Vienna. Boltzmann had many scientific opponents but, to Boltzmann, Mach was more than a scientific opponent as the two were on bad personal terms. In 1900, because of his dislike of working with Mach, Boltzmann moved to Leipzig but here he became a colleague of his strongest scientific opponent Wilhelm Ostwald. Despite their scientific differences Boltzmann and Ostwald were on good personal terms.

In 1901 Mach retired from Vienna due to ill health, and in 1902 Boltzmann returned to Vienna to his chair of theoretical physics. In addition to his teaching in mathematical physics, Boltzmann was given Mach's philosophy course to teach. His philosophy lectures became famous with the audience soon being too large for the biggest lecture hall available.

Boltzmann loved music and even took lessons in composition from Anton Bruckner.

Boltzmann was also an extremely gifted experimentalist. He was the first (15 years ahead of Hertz) to verify Maxwell' theory of electromagnetic radiation by verifying the predicted relationship between refractive indices.

Boltzmann's fame is based on his invention of statistical mechanics. This he did independently of Willard Gibbs. Their theories connected the properties and behavior of atoms and molecules with the large scale properties and behavior of the substances of which they were the building blocks. Boltzmann obtained the Maxwell-Boltzmann distribution in 1871. He was one of the first to recognize the importance of Maxwell's electromagnetic theory.

Boltzmann worked on statistical mechanics using probability to describe how the properties of atoms determine the properties of matter. Due to the success of his work, atoms which were initially only considered as a philosophical speculation became gradually accepted as something "real".

Boltzmann derived the Second Law of Thermodynamics from the principles of mechanics in the 1890s . He gave the atomistic interpretation of the Entropy S that is also on his grave stone:

S = k · ln W

Boltzmann in a way anticipated quantum theory as early as 1891. When Ostwald and Planck tried to convince Boltzmann of the superiority of purely thermodynamic methods over atomism at the Halle Conference in 1891, Boltzmann suddenly said: ``I see no reason why energy shouldn't also be regarded as divided atomically.'' By hindsight a truly profound statement.

In his lectures on Natural Philosophy in 1903 Boltzmann anticipated the equal treatment of space coordinates and time introduced in the theory of special relativity. Furthermore in the lectures by Boltzmann and his successor Fritz Hasenöhrl in Vienna the students learned already about non-Euclidean geometry, so that they could immediately start to work when Einstein's general theory of relativity had been formulated.

Attacks on his work continued and he began to feel that his life's work was about to collapse despite his defense of his theories. Depressed and in bad health, Boltzmann committed suicide just before experiment verified his work. On holiday with his wife and daughter at the Bay of Duino near Trieste, he hanged himself while his wife and daughter were swimming.

The Population of Excited States in Spectroscopy: UV-Vis, IR, Microwaves

The population of the upper level depends in a sensitive way upon the spacing of the energy levels and is readily calculated with the Boltzmann formula above. We will now briefly analyze the populations of excited states for different spectroscopic methods.

- UV-Vis: Optically excited states are typically 1 - 10 eV above the ground state. Compared with the thermal energy of an average molecule at room temperature (0.025 eV), this is a very large figure and excited optical states are therefore empty at room temperature.

- IR Spectra: Vibrational spacings depend on the bond strength and mass of the involved atoms so that general rules are harder to derive. However, most organic molecules have only the ground state populated with some minor population in the first excited state.

- Microwave Spectra: Rotational
states have much smaller energy spacings and the most populated
states are in fact above the ground state.

It must always be remembered that no matter how large the energy spacing is, there is always a non-zero probability of the upper level being populated. The only exception is a system that is at absolute zero. This situation is however hypothetical as absolute zero can be approached but not reached.

For example, pure silicon (spacing**
E _{g}** = 1.1 eV) has enough electrons in its upper level
("conduction level") to conduct electricity even at room temperature.

Selection Rules

Not all possible transitions (absorption or emission processes) are in fact observed. Apart from E = hn, additional requirements must be fulfilled in order for a transition to take place:

- conservation of energy (E = hn)
- conservation of spin
- conservation of "parity" (proper symmetry of the involved states)

Spin Conservation

The absorption of a photon does not
only absorb a certain amount of energy (energy conservation requires
** E = ****h****n**)
but also the spin of the photon. Photons have in general a spin of 1.

As a result, only those transitions will be observed that lead to spin change of one. This implies, that we have to consider not only the energy of a given state but must also have an idea about its spin state.

Symmetry

The second requirement, proper symmetry is readily appreciated if we look at the quantum mechanical description of the transition probability involving an initial state and a final state.

The two states are completely
described by their wave functions
y_{
i},
and
y_{f
}.
The quantum mechanical transition
probability P is (here without proof):

**m**
is the appropriate operator for
the transition.

The proper numerical evaluation of the
integral can be cumbersome to impossible but there is an elegant
shortcut that can save us the trouble of number crunching.: a special
property of the functions y_{
i},
and y_{f
}. their so called "parity"
allows us to determine if **P** is zero ("forbidden transition")
or non-zero. There are only two types of parity, even and odd.

- A function is "even" when F(x) =
F(-x). These are functions that are symmetric with respect to the
y-axis. An example would be y = x
^{2}

- A function is "odd" when F(x) =
-F(-x). These are functions that are symmetric with respect to the
origin of the coordinate system An example would be y =
x
^{3}

In physics, the labels still in use
are "g" for even and 'u" for odd from "**g**" (*German*:
gerade = even) and "**u**" (*German*: **ungerade** = odd).

The integral become zero if the
compound function is of the so called "**u**" and non-zero if it
is "**g**"

They key point is we do not have to know the individual functions, we just have to know if they are even or odd because the parity can be readily determined by three rules:

u x u = g

g x u = u x g = u

g x g = g

It is easy to get confused about these
rules but a simple way to remember them is by picking **y =
x ^{2} **(g-function) and

xtimes^{2}xwhich is a g-function so^{2}= x^{4 }gxg=g

xtimes^{3}xwhich is a u-function so^{2}= x^{5 }uxg=u

xtimes^{3}xwhich is a g-function so^{3}= x^{6 }uxu=g

The dipolemoment operator itself is vector of type y = ax +b and is of type "u"

This means, that P will become zero if the product of the two wavefunctions is "g" because u x g is u and the integral vanishes.

Laporte's Rule

The g x u rule is the so called called "Laporte" rule and explains why certain transitions are "forbidden" although the energy and spin criterion are met.

Laporte's rule explains why transition metal complexes which owe their color to d-d transitions often have comparatively low extinction coefficients:

d-d transitions involve of the same
parity (**g**).

Deeply colored transition metal
complexes owe their color not to d-d transitions but to transitions
involving ligand orbitals and d-orbitals. A good example is
MnO4^{-}

Symmetry Breaking And Vibration

Laporte's rule is not 100 % strict because wave functions are not entirely electronic but contain vibratory and rotatory contributions:

y_{ }= y_{ electronic}• y_{ vibronic}• y_{ rot}

The parity of the vibronic part can water down Laporte's rule and this phenomenon is called "vibronic mixing" . A bad term because nothing is mixed, we just neglected the vibronic part when we derived Laporte's rule.

Atomic Spectra And Selection Rules

Not all transitions between atomic orbitals that are energetically feasible are "allowed": the involved orbitals have to have the correct parity.

the parity of orbitals alternates between g and u. The totally symmetric s-orbitals all have parity g.

p-orbitals have two diffrent lobes which opposite phase, they are "u".

Orbitals

p

d

f

g

Parityg

u

g

u

g

Because the dipol-moment operator's parity is u, the transition probabilty is zero for:

s -> sp -> p

d -> d

and so on.

Allowed transitions are

s - >pp ->d

d ->f

f -> g

s -> f

p - g>

Symmetry thus greatly reduces the number of possible transitions in atomic spectra but the rules remain valid for the electronic spectra of compounds as well.

Grotrian Diagrams and Atomic Spectra

Excited states in vapors of metals or nonmetals emit radiation when they revert to the ground states.

This principle is used in two commercially important light sources, namely the fluorescent tube (excitation of gases) and the sodium lamp (sodium vapor). The excited states are produced by high voltage which ionizes the atoms. Collision and electron captures gives us the atoms back, but as excited states.

There is only one ground state, but many possible excited states which can in principle emit light. It would seem, that the gas discharge idea will give us a fairly wild mixture of diffrent wavelengths.

The term diagram below shows different excited states of lithium atoms.

The individual terms are the wavefunctions (=orbitals) of the lithium atom: 2s, 3s, 4s, 5s...; 2p, 3p, 4p,....; 3d, 4d, 5d, ....; 5g...;

The 1s wavefunction is not shown in the diagram. It houses the two core electrons of lithium, is much lower in energy and its electrons are not readily excited.

Magnetic Transitions

Our derivation of the Laporte rule
also assumed that the transition is exclusively due to the electric
component **E** of the absorbed light. The oscillating magnetic
part of the radiation (**B** field) can cause transitions but the
respective operator has a different parity (g instead of
u).

Magnetically induced transitions are
much weaker (by a factor of 10^{-4}) and can only be observed
with very intense light sources (Lasers). For details see Ira Levine,
Molecular Spectroscopy.

More on the Transition Moment: Charge Transfer Bands

The magnitude of the transition
moment **m**
depends on the molecule. If the
two states are represented by orbitals that are distant from each
other (not on the same atom), the transition
moment **m****
**becomes very large and the
transition become very intense. These transitions are called charge
transfer bands or simply CT transitions. A peculiarity of CT bands is
that their **l**_{max}**
**
depends on the polarity of the
solvent used in the measurement (solvatochromic behavior). This is a
result of the highly polar (compared with the ground state) excited
state.

Band Type

e_{max}[L • mol^{-1}• cm^{-1}]Spin forbidden

< 1

Laporte forbidden

20 - 100

Laporte forbidden - vibronic coupling

100 - 500

Other Symmetry allowed (e.g. CT)

1,000 - 50,000

How to Recognize a UV-VIS Charge Transfer Band: Solvatochromic Effects

The excited states of molecules are usually more polar than the ground state.

The energy of the excited state is
accordingly lowered by polar solvents which reduces the energy
difference between ground state and excite state. The polarity of the
solvent thus influences **D****E**
and
**l**** _{max}**.

The effect is typically small (
**Dl**
,< 10 nm) except for charge
transfer bands (CT bands) for which solvent induced shifts of
**l**** _{max
}**can be 100 nm and
more.

A strong red-shift (~ lower energy) of a band upon change from a nonpolar to a polar solvent identifies the corresponding transition as CT transition.

CT bands are also characterized by
very high extinction coefficients. This is due to the fact
that** m**,
the dipole moment of the
transition is large which gives a high transition probability
**P**:

Lambert-Beer's Law (I)

Lambert-Beer's law is an example for a spectroscopic "law" that was derived by reasoning and some mathematics without Q-mechanics.

The law describes how the sample
length **l** , the incident radiation intensity**
I _{o}** and the transmitted radiation intensity

First, we assume, that the intensity of the transmitted light will proportional to the intensity of the incident light.

In other words, if light has crossed
an infinitesimally small distance dx, the corresponding loss of
intensity **-dI(x) / dx **will be proportional to the incident
intensity ** I(x)** :

-dI(x) / dx = k• I(x)

With k as arbitrary constant. This is equivalent to:

-dI(x) / I(x) = k• dx

Integration over dx in from x = 0 to x = l gets us from the infinitesimal world into the real world of real distances and gives:

-ln[ I/I_{o}] = k • length

The switch of the base of the logarithm to 10 only results in a new constant k' but leaves the law unchanged

-log [ I/I_{o}] = k' • length

The term **log [
I/I _{o}] **is called

We can now collect the bits and pieces and arrive at:

-log [ I/I_{o}] = c •e• length

I= Intensity of transmitted light (Ialways <I)_{o}

I= Intensity of incident light_{o}

c= concentration

e= extinction coefficient

length`= optical pathlength

Note that

-log [ I/I=_{o}]log [ I_{o}/I]

If we can measure the intensities I
and I_{o}, we can determine one of the two variables on the
right side form the two others. For example, we can calculate the
extinction coefficient from the concentration and the length or the
concentration from the extinction coefficient and the
length.

While the Absorbance falls of linearly with x, the intensity of the transmitted light falls off exponentially (compare radioactive decay).

In chemistry, we are usually interested to measure the concentration of a molecule and if we can readily do that trough a calibration curve:

By using the Absorbance** A**, we
arrive at a linear law which made it more easy in the pre-computer
days to determine **k'** which we is simply the slope of the
linear fit. Once k' has been determined from a series of solutions
with known concentrations, unknown concentrations C can be determined
from **c** =** A** /** k'**

The determination of concentration
through absorption is called** Photometry** and allows the
determination of concentration down to 1 part per billion

1 ppb = 0.001 mg / liter solution

1 ppm = mg / liter solution

Lambert-Beer's Law (II) - Physical Meaning of the Extinction Coefficient

Unless noted otherwise, extinction
coefficients refer to the log version of Lambert Beer's law and are
given in 1000 cm^{2} / mol. We have so far not bothered to
find out what the extinction coefficient
**e**
stands for. A good way to do that is a dimension analysis.

The left side of the Lambert Beer equation is a logarithm and hence has no dimension:

-log [ I/I_{o}] = c •e• l

That means that all units in
**c •
****e****
**** • l **cancel which
is only possible if **e
**has the dimension 1 /
concentration•length.

We are not done yet: the
concentration's dimension is mol / volume = mol / length^{3}.
**e**
therefore has the dimension of
length^{2} / mol = area / mol. This
allows us to interpret **e
**as the cross-section of a
molecule , or, more precisely, the capture cross-section for photons
of a given wavelength.

This optical cross-section is usually greater than the geometrical cross section derived from the molecular shape of the respective compound. The interpretation is straightforward: photons do not have to "hit" the molecule head on to be absorbed but are also absorbed if they fly by close enough.

The extinction coefficient is a macroscopic version (mol, not molecules) of the transition probability between two states but can be calculated from the corresponding microscopic value, the transition probability P.

Important Units

A puzzling feature of spectroscopy is the bewildering number of different units. The reasons for this terminology zoo are entirely historic. Spectroscopy did not evolve as a unified discipline but was started by many different schools of scientists, each with their own conception of what makes a unit useful. Bear in mind that computational efficiency was more important in the days when the only computational tool was a slide ruler.

Frequenciesare always expressed in Hz (after Hertz) with the unit 1/second ("cycles" is derived from Hz and only used for cyclic movements. The unit of cycles is 1/2ps ). Spectroscopic methods that developed late (NMR, ESR, Microwave Spectroscopy) are "rational" and stick to frequencies rather than energies or wavelengths.

Wavelengthsare expressed in different units.

Å(Ångstrom, 10^{-10}m) commonly used for high energy radiation with short wavelengths:

- g-rays, X-rays, up short wavelength UV ("vacuum UV")

nm(10^{-9}m) commonly used for UV-Vis (350 nm - 800 nm)

cmcommonly used for IR and microwave spectroscopy.

Energiesare expressed in different units.eV (electron volts) are used atomic spectroscopy, photoelectron spectroscopy, mass spectroscopy ...1 eV is the energy that an electron acquires after passing through a voltage difference of one volt.

Wave numbers (1/l), used in IR spectroscopy and are proportional to the energy of the electromagnetic radiation:

E = hn and n = c/l ~> E = hc/l

Important conversion factors:

- 1 eV = 23 kcal / mol = 8066 cm
^{-1}= 96.5 kJ / mol

- 0.025 eV = thermal energy at room temperature..

The electromagnetic spectrum covers a vast range of different frequencies and this requires the frequent use of prefixes.

NameSymbolExample10

^{12}Tera T THz 10

^{9}Giga G GHz 10

^{6}Mega M MHz 10

^{3}kilo k kbar, kHz 10

^{-3}mili m mL, mmol 10

^{-6}micro m mm, ml 10

^{-9}nano n ng, nm 10

^{-12}pico p pm 10

^{-15}femto f fs

SI Units And Fundamental Constants