2Nd Set of Notes

2nd Set of Notes

The formal way that light interacts with matter should also give insight into the selection rules that govern absorption of radiation

Classically light absorption considers matter as an array of charges that can be set into motion by an oscillating electromagnetic field. For the molecule to be able to interact with the electromagnetic filed and absorb or emit a photon of frequency  it must possess, at least transiently, a dipole oscillating at that frequency. So classically there are electric-dipole oscillators set in motion by the field have certain characteristic or natural frequencies i. When the frequency of the radiation is near the oscillation frequency, absorption occurs and the intensity of the radiation decreases on passing through the substance. The refractive index also undergoes large changes in the same spectral region. The intensity of the interaction is known as the oscillator strength fi, and it can be thought of as characterizing the number of electrons per molecule that oscillate with the characteristic frequency i.

In practice, the oscillator strength is an indicator of the strength of absorption, the intensity of absorption, and is given by: 2 - 1 9 f = (2303 c m)/( N0 e n) * ∫ () dx 10 / n) * ∫ () d (unitless) where  is the molar absorptivity (Remember Beer's Law ( A = [absorber] L) Beer- Lambert Law T = I/Io A = -log10(I/Io) = - logT A =  [absorber] L

A is absorbance, L is pathlength

In base e: ln(I/Io) = Naperian molar absorptivity c is the speed of light; m is the mass of an electron; e is the electron charge, N0 is Avogadro's number, and n is the index of refraction of the medium. The integration is carried out over the frequency range associated with the absorption band. If absorption is strong and fully allowed then the oscillator strength is equal to 1, and for no absorption f = 0.

The most complete description of the absorption of radiation by matter is based on time dependent quantum mechanics which calculates what happens to the molecular wavefunctions in the presence of an external oscillating electromagnetic field. The details of the solution will not be given here, but essentially because the interaction is electromagnetic, there one needs an operator that conveniently describes this time dependent interaction. That is the electric-dipole operator,  = er. This operator is the position operator, r, multiplied by the electronic charge. By Quantum Mechanics then, a transition from one state to another occurs when the radiation field connects the two states. In wave mechanics, the means for making this connection is described by the transition dipole moment:

if = ∫all space if d where i, f are the initial and final energy states (lower and upper states for absorption) d is the volume element, and the integration is over the entire defined space When the transition dipole is not equal to zero, then an absorption (or emission) is photonically allowed. From a consideration of the symmetry of the wavefunctions and the operator, this boils down to the fact that the transition dipole is not zero if the symmetry of the initial and final states differ.

The transition dipole moment is a vector quantity and has both magnitude and direction as characterized by (x)if, (y)if, and (z)if. That is for plane polarized light, it is possible that light polarized say in the x-direction might be absorbed, but light polarized in the y-direction and z-direction might not be absorbed.. Most electronic transitions are polarized meaning that the three vector components are not all equal. For example in the case of a hydrogen-atom transition from s to px, the transition of polarized along x, and in the case of an ethylene  to *, the transition is polarized along the C=C double bond.

The magnitude of the transition is characterized by its absolute value squared, called the 2 - 18 dipole strength: Dif = if  with units Debye squared. 1 Debye = 10 esu cm the dipole strength is also related to the molar absorptivity by 3 - 39 Dif = [(3 h 2302 c) / ( 8  No n)] * ∫ ()/ dx 10 / n) * ∫ ()/ d (esu cm)

The stimulated emission coefficient is related to the transition dipole moment by: 2 2 2 B = 2  if  / (2o h ) Again the rates of stimulated and spontaneous emission are related by: A = B * 8  h 2 / c3 This shows that the rate of spontaneous emission is highly dependent on the transition frequency.

In this class we will be interested in both absorption and emission processes. As mentioned before, lasing is related to the stimulated emission. DOES all of the emission happen at one narrow wavelength?

Emission spectra are seen as transition bands in a plot of intensity vs wavelength

Why aren’t they just stick spectra. Why are they bands with a finite width? b. lifetimes and bandwidth

The band has a width due to several contributions. To minimize the band width, some of these contributions can be minimized by changing experimental conditions. Other contributions are an inherent limitation on the resolution.

A. Doppler Broadening An important broadening process in gases where the radiation is shifted n frequency when the source is moving toward or away from the detection point. If a source emitting the EM radiation is moving at a speed u relative to the detection point, the detection point sees radiation of frequency

1 / 2 1 / 2 receding = [(1- u/c)/(1+ u/c)] and approach = [(1+ u/c)/(1- u/c)]

In a gas, molecules can reach high speeds in all directions and a stationary detector detects the corresponding Doppler-shifted range of frequencies. Some move toward the detector, some away, some are fast and some are slow. The frequency profile then detected in absorption of emission is a compilation of all the resulting Doppler shifts. The Dopper line shape is Gaussian the calculation shows that when the temperature is T and the mass of the moluecul is m, the width of the line at half-height is d = 2/c * [(2kTln2)/m]1 / 2

- 6 For a molecule like N2 at room temp,  = 2.3 x 10

Example For a typical rotational transition of 1 cm- 1 what is the linewidth in terms of d (The conversion between wavenumbers and energy is 8066 cm- 1 = 1.602 x10- 1 9 J.)

B. Lifetime Broadening Even when Doppler Broadening has been somewhat eliminated, spectroscopic band from gas-phase samples are not infinitely sharp. There is a basic principle in Quantum Mechanics governs the linewidths without the Doppler broadening present. It is the Heisenberg Uncertainty Principle which says that: p x ≥ h/(4) also Et ≥ h/(4) That is the Heisenberg Uncertainty Principle provides a lower limit for the product of the time interval during which the energy of a quantum state is observed (t), essentially its lifetime, and the uncertainty in the energy of that state or the energy of the transition to that state. (). Uncertainties in the energy of transition are directly related to the width of a transition peak in the emission spectrum by E = h Thus the greater the lifetime of the state, the small is the uncertainty in the transition energy (the narrower the bandwidth. When the energy spread is expressed as a wavenumber through E = hc d' the relationship is ' ~ 5.3cm- 1 / t(ps) There are two processes responsible for the finite lifetimes if excited states. In the absence of light, the excited state can be deactivated by spontaneously emitting a photon, or by colliding with another molecule or the walls of the container. At low frequencies, collisional deactivation is dominant (Why?)

...... 1) collisional lifetime, coll

This depends on the collisional frequency, z, (from the kinetic theory of gases) coll =

tcoll = 1/z where z is proportional to the pressure. So the collisional linewidth should increase with increasing pressure, as more collisions cause quicker deactivation and a shorter lifetime.

...... 2) natural lifetime, spon Since the rate of spontaneous emission cannot be changed, this is a natural limit to the lifetime of an excited state, and the resulting lifetime broadening is the natural linewidth of the transition. Since A increases at 3 , the natural lifetimes of electronic transitions are much shorter than for rotational transitions, so the natural linewidths of the electronic transitions are much greater.

Example Given that a typical electronic excited state natural lifetime is about 10- 8 s, and that of a rotational state is a kilosecond, calculate the natural linewidths for each of these types of transitions.

b. Scattering a. Elastic

Rayleigh Scattering

Basic Classical Model involves interaction of electromagnetic field with the mobile electrons due to the polarizability of the material. These move with the frequency of the electromagnetic field, creating a dipole moment with in the molecule that can radiate at the same frequency. It can also be modulated by the vibrations of the molecule and reradiate light at frequencies modified by those vibrational frequencies.

Rayleigh Scattering b. Inelastic

Raman Scattering

3. Basic Components for Laser Spectroscopy

detector

source sample

optical components

a. light manipulators, optics 1. mirrors, prisms, lenses, windows

Mirrors, prisms, and lenses are all optical components that will often be found in a setup for laser spectroscopy. As stated before, some of the laser properties include a high-powered, non-divergent, monochromatic, polarized beam. This makes it relatively easy to manipulate the source with optical components such that a sample can be either close to the source and detector, moderately close, or even very far away. (like in the stratosphere). Of course the materials used have to be consistent with the part of the EM that the laser is emitting and the detected signal. The monochromaticity and non-divergence allow ease of turning with either a mirror or prism although the high power can require the use of specialized coatings on the mirror or prism. In fact high-power mirrors are often required to produce a laser. These are dielectric mirrors where the dielectric coatings can be made to specifically reflect one wavelength with very high efficiency > 99.5%. A dielectric mirror consists of multiple thin layers of (usually two) different transparent optical material, (dielectric coatings, thin-film coatings, interference coatings). The reflections from many interfaces can (in a certain wavelength range) constructively interfere to result in a very high overall reflectivity of the device. The simplest and most common design is that of a Bragg mirror, where all optical layer thickness values are just one-quarter of the design wavelength. This design leads to the highest possible reflectivity for a given number of layer pairs and given materials. This condition holds for normal incidence; if the mirror is designed for larger angles of incidence, accordingly thicker layers are needed. The principle of operation can be understood as follows. Each interface between the two materials contributes a reflection. For the design wavelength, the optical path length difference between reflections from subsequent interfaces is half the wavelength; in addition, the amplitude reflection coefficients for the interfaces have alternating signs. Therefore, all reflected components from the interfaces interfere constructively, which results in a strong reflection. The reflectivity achieved is determined by the number of layer pairs and by the refractive index contrast between the layer materials. The reflection bandwidth is determined mainly by the index contrast. It is also possible to design dichroic mirrors with controlled properties for different wavelengths.

2. Wavelength separators We have just mentioned one type of wavelength separator that is important for laser applications. Often after the laser light interacts with the sample we are interested in detecting the signal at a particular wavelength of light or over a range of frequencies, and to do this, the light which heads toward the detector must be filtered to allow the wavelengths of interest to be discriminated from other EM that is reaching the detector. Often this is done using a monochromator which has built into it slits and either a grating or a prism to allow one to discriminate the signal of interest from other sources. These come in a few types arrangements: Prism-based Monochromator

Czerny-Turner Monochromator Fastie-Ebert

Concave Grating

A flat-field spectrograph. The spectrum from 1 to 2 (>1) is shown imaged onto a line.

A flat-field monochromator is an aberration-corrected monochromator with a concave grating. A single concave grating disperses, collimates and refocuses the light from the entrance slit onto the exit slit. Wavelength scanning is obtained through a simple rotation of the grating.

The groove spacing of these gratings is computer-optimized to produce high quality images with a minimum of astigmatism and coma, even at large numerical aperture.

Compared with Czerny-Turner monochromators (equipped with one plane grating, one collimating mirror and one focusing mirror) flat-field aberration corrected monochromator gratings provide much better light collection efficiency and signal-to- noise ratio. Parameters of a diffraction grating

N - grating normal

I - incident beam sr - surface relectance

Grating Equation: m *  = k * (sin  ± sin m - spectral order k - distance between grooves (grooves/mm, grating constant I - incident beam rays, need to come in parallel d - dispersed beams, but different wavelengths leave at different angles angle between the incident beam, I, and N angle between the dispersed beam, d, and N W - ruled width of the grating  and same side, then +

Free Spectral Range The range available without spectral overlay

FSR = /m

Resolution Resolving Power

Grating Efficiency and Polarization

The maximum efficiency occurs at the blaze angle. 3. Detectors and Detection Schemes A detector at its most basic level can be thought of as a transducer that converts a signal, that is embedded in radiation or a flux of photons, into a signal that can be electrically sensed and ultimately converted into a number. The output from the detector is most commonly either a current or a voltage, and this can be measured, sometimes after it is amplified, and recorded. One important for consideration of amplification and general signal readout is to match impedance.

Impedance matching is important both to eliminate unwanted noise and from a point of view of timing. Detectors that put out a current are often low impedance and ones whose output is a voltage are often high impedance

RC time constant

Detection of the signal of interest involves using a sensitive detector that is applicable to the type of radiation you expect with good discrimination of the background noise. There are techniques that can be used to separate the signal from the background providing better signal to noise at the detector. We will discuss a few of these below. Some of the types of noise that contribute are:

White Noise

Flicker Noise or 1/f Noise

Interference Noise

Often separating this noise from the signal can be accomplished temporally and spatially especially if the source is a laser. Use of a Lock in Amplifier One can chop the signal and a reference and use this to discriminate against background noise. The typical signal measured with a lock-in system is a relatively slowly varying signal.

Signal is Ascosst and the chopped carrier is Accosct. The signal and carrier are essentially combined as follows:

Ascosst x Accosct = The output is a modulated waveform of two frequencies. The demodulation of this signal is carried out by the lock-in amplifier by essentially multiplying a sinusoidal signal equal to the carrier, and then use low pass filtering.

2 2 Accosct x (Ascosst x Accosct) = Ac As/2 cosst + Ac As/2 cosst cos2ct

Best to modulate this as soon as possible, for instance, by using a mechanical chopper between the light source and the rest of the experiment or using a pulsed source

However for fast pulsed signals lock-ins cannot be used.

Use of a Boxcar Integrator (Averager) This involves gating to isolate a particular section of the signal waveform and integrating successive gated signals to improve the signal-to-noise ratio. Particularly useful for repetitive short pulse signals and signals that have a slow repetition rate or low duty cycle

Photon Counting Similar in operation except the photons are collected as a function of time. The number of photons collected in a given time interval are counted and put into a bin and then a threshold can be set to discriminate background levels from levels that actually have a signal.

The detector used certainly depends upon the radiation type. Here is a listing of some of the detector types that can be used in each range:

Microwave

Infrared

Visible

Most of the laser spectroscopic applications that we will be discussing use lasers in the visible or ultraviolet region. Lets focus on a few of these detectors. 1-dimensional detectors PMT, photomultiplier tube, based on the photoelectric effect followed by a series of dynodes that amplify the number of electrons (the current). Typically a series of dynodes can amplify the current by a factor of 106 or more. This type of detector has a single signal that is generated. There is no spatial awareness or resolution and so in that sense it is a 1-dimensional detector, and is typically used to detect a narrow wavelength bandwidth making it through narrowed slits. That means that if information on the wavelength dependence of the signal is desired, the monochromator must be scanned across the wavelength range of interest. Simple photodiodes and avalanche photodiodes are also 1-dimensional semiconductor detectors

2-dimensional and multi-dimensional detectors Perhaps the first two dimensional detectors were linear photodiode arrays. These were essentially semiconductor photodiodes that were manufactured to be a fixed distance away from each other in a line providing a spatially resolved signal at each photodiode. Thus here, the exit no longer needed to be a narrow slit, since the light at the various wavelengths could be separated and recorded as the output from the photodiode array. Most recently these types of linear arrays have been replaced by CCD, charge coupled device, cameras. A CCD can provide not just linear resolution say in the x direction, but also resolution in the y direction as well. This is often important for applications like thermal imaging. A CCD basically consists of

Fundamentally, a charge coupled device (CCD) is an integrated circuit etched onto a silicon surface forming light sensitive elements called pixels. Photons incident on this surface generate charge that can be read by electronics and turned into a digital copy of the light patterns falling on the device. CCDs come in a wide variety of sizes and types and are used in many applications from cell phone cameras to high-end scientific applications. The function of a CCD can be visualized as an array of buckets (pixels) collecting rainwater (photons). Each bucket in the array is exposed for the same amount of time to the rain. The buckets fill up with a varying amount of water, and the CCD is then read one bucket at a time. This process is initiated by pouring water into the adjacent empty column. The buckets in this column transfer their ‘water’ down to a final pixel where the electronics of the camera read-out this pixel (the computer measuring the bucket) and turn it into a number that can be understood and stored by a computer.

Of course, this is an oversimplification, all the pixels in a CCD are actually shifted simultaneously, not one column at a time. Look at how an individual simple pixel works. The substrate of a CCD is made of silicon, but this is not where most of the action occurs. Photons coming from above the gate strike the epitaxial layer – essentially silicon with different elements doped into it – and generate photoelectrons. The gate is held at a positive charge in relation to the rest of the device, which attracts the electrons to it. Because of the insulating layer – essentially a layer of glass – the electrons can’t make it through to the gate, and are held in place by the positive charge above them.

b. LASER light source 1. description 2. general methodology