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INTRODUCTION TO SPECTROSCOPY
Literature
[A] P Atkins, J De Paula: Physical Chemistry, Oxford University Press, 2006 [C] RMJ Cotterill: Biophysics. An Introduction, chapter 3, John Wiley & Sons, Chichester, 2002 [CS] CR Cantor, PR Schimmel: Biophysical Chemistry, part II, W H Freeman, NY, 1980 (highly recommended) [CD] ID Campbell, RA Dwek, Biological Spectroscopy, 1984 (highly recommended) [CDW] N B Colthup, L H Daly, S E Wiberley, Introduction to infrared and Raman spectroscopy , 2. Auflage (1975), Academic Press, New York [H] JM Hollas: Modern Spectroscopy, John Wiley, 1992 [HL] W Hoppe, W Lohmann, H Markl, H Ziegler (eds.): Biophysics , Springer Verlag, Berlin, 1983 [HS] FR Hallett, PA Speight, RH Stinson: Introductory Biophysics , Methuen, Toronto, 1977 [Ja] TL James: Fundamentals of NMR , Biophysics Textbook Online, 1998 [M] AG Marshall: Biophysical Chemistry, John Wiley & Sons, New York, 1978
Videos
https://www.youtube.com/watch?v=dNUkHfBpvac&list=PLm9edRZ1r8wWSE3BruFy06q3ckvjzlTfn
How to read my handouts
Some of my handouts contain supplementary information. These sections are indicated by small print. They represent additional information for those who are interested, but are not required for the examination.
Essential knowledge
Essential knowledge question for Introduction to spectroscopy: a) Restate Fermi´s golden rule, explain all the terms and state the significance of the rule. (2 p) b) Define absorbance, or in other words: state how absorbance depends on the measured light intensities. (1 p) c) Write down the Beer-Lambert law and explain all the terms. (1 p)
d) Restate the Boltzmann distribution and explain all the terms and their relevance for the occupancy of the excited state. (2 p)
Examples for general knowledge
- Define spectroscopy
- List spectroscopic techniques according to the energy of the transitions observed from low energy transitions to high energy transitions.
- Name and describe processes that can take place when light interacts with molecules.
- Explain homogeneous and inhomogeneous line broadening.
- Which functions describe homogeneous and inhomogeneous line broadening?
- Give examples for homogeneous and inhomogeneous line broadening
- Which functions describe homogeneous and inhomogeneous line broadening (no need to learn the equations)?
- Qualitative comparison of Lorentz and Gauss functions.
- Restate that RT = 2.5 kJ/mol at room temperature.
- Distinguish between stimulated and spontaneous emission.
- Compare spectroscopic methods.
- Very briefly describe the structure and function of bacteriorhodopsin.
- Appreciate the usefulness of visible spectroscopy for studying bacteriorhodopsin.
Examples for functioning knowledge
- Apply the Boltzmann distribution to explain spectroscopic properties.
- Apply the Beer-Lambert law.
- Predict how the line width of a transition is affected by a given process or change in environment.
What is spectroscopy?
Seeing is spectroscopy: we perceive the world via the interaction of visible light with the light receptors in our eyes. The light is emitted from the sun or from other light sources. It is then reflected from (or transmitted through) the objects in our surroundings. In these processes, the color changes because some of the light is absorbed by the objects. How much and what spectral regions are absorbed depends on the atoms and molecules in these objects. The light not absorbed reaches our eyes. It carries the information of the molecular structure of our surroundings with it. In our eyes its color is analysed by 3 different types of photoreceptors which absorb different light in spectral regions. In this way we perform a spectroscopic experiment every time we look at things. There is a light source, and object that reflects, transmits, scatters and absorbs light and a
Electromagnetic radiation
Electric and magnetic components of an electromagnetic wave. [Cesare Baronio] Molecules and their energy levels are one main ingredient of spectroscopy. The other main ingredient is the electromagnetic radiation that induces transitions between different energy levels. Let us therefore briefly recall what electromagnetic radiation is. There are two general ways of describing electromagnetic radiation: as a wave and as a particle. Some aspects of an experiment are best explained by the wave concept, but others by the particle concept. We will use both views to explain the interaction of matter with radiation. In some experiments and often in the interaction with molecules, electromagnetic radiation behaves particle- like. The particles are called photons. Each photon has a defined energy, which only depends on the frequency (color) of radiation. E = h where h is Planck´s constant ( h = 6.63 × 10 -34^ J s). The intensity (brightness) of radiation depends on the number of photons. If radiation shows its wave face, electromagnetic radiation has two components: and electric field E and a magnetic field B. Both oscillate with the same frequency and are oriented perpendicular to each other and to the direction of propagation at all times. For the phenomena we will describe, it is often sufficient to consider only one of the two components. Light can be polarized, that is the electric and magnetic field oscillate each in one particular direction. In unpolarized light, the electric and magnetic field oscillate in all directions perpendicular to the direction of propagation. Frequency of a wave and wavelength are related by = c / where c is the velocity of propagation of the wave. For electromagnetic radiation in a vacuum, c = 3 × 108 m s-1. Frequency and wavelength are often used to characterize electromagnetic radiation. Another quantity is the wavenumber ν̃ measured in reciprocal centimeters. The wavenumber is the inverse of the wavelength.
ν̃ = 1/ Wavenumber is mainly used in vibrational spectroscopy. Its advantage is that is conveys the information about the wavelength (just calculate the inverse) and is also proportional to the energy or frequency.
Energy levels
We will now turn to molecules and start discussing their energy levels. A system (molecule) can adopt only certain energy values which are the eigenvalues of its Hamilton operator. Therefore the possible energy values, also called energy levels, are discrete and there are gaps between them. The state of lowest energy is named ground state. All states with higher energy are called excited states. Sometimes, two states have the same energy, then they called degenerate. This degeneracy can be lifted by a perturbation, i.e. by interaction with an external influence [CD]. An example are the energy levels of the nuclear and electronic spins. In the absence of an external magnetic field, they are degenerate, i.e. the energy does not depend on the spin orientation. However, when an external magnetic field is applied, the degeneracy is lifted and different spin orientations have different energies. One says that the magnetic field splits the spin energy levels. ENERGY CONTRIBUTIONS Often one can consider a molecule as being composed of several sub-systems (electron orbitals, electron spin, nuclear vibrations, nuclear spin, etc.) that are quite independent from each other. For example, one can consider the electron orbitals separately from the nuclear spin orientation. This is an approximation of the real case and assumes that it does not matter so much to the electrons what the nuclei do and vice versa. Or one can consider the nuclear spin without taking into account the nuclear vibrations and vice versa. Each of the sub-systems contributes to the total energy and the following equation lists the most important contributions [CD]: E total = E electronic + E vibration + E rotation + E electron spin orientation + E nuclear spin orientation + E translation We have: the energy of the electrons in their orbitals ( E electronic), the energy due to the vibrations of the atoms ( E vibration), the energy of molecular rotations ( E rotation), the energy due to the orientation of the spins of the electrons ( E electron spin orientation), the energy due to the orientation of the spins of the nuclei ( E nuclear spin orientation), and the energy due to the translational movement of the molecule in space ( E translation), in other words, the thermal energy. In the above equation, the energy contributions are listed according to the separation between energy levels. Electronic levels have the largest gaps between them and translational levels the smallest. In order for a transition to occur for example from a lower to a higher energy level, energy must be provided. This energy might come from thermal energy but also from the absorption of a photon. The former means that higher energy levels are populated at higher temperatures (see below). The latter means that the photon energy has to match the energy gap between two energy levels. This is one of the fundamental rules (Bohr frequency rule, see below) of spectroscopy. Because the gaps are different for different sub-systems, different photon
In the above illustration, the electronic ground state energy is shown separately from the ground state energy of electrons and nuclear vibrations together. However, in many illustrations these two are combined as shown on the left. Here, the bold line illustrates the energy of the electrons in the electronic ground state plus the energy of the nuclear vibrations in their ground state. In other words, it illustrates the total energy of the molecule in the ground state. The upper bold line illustrates the energy in the electronically excited state and in the vibrational ground state of the electronically excited state. When one considers both the electronic states and the vibrational states together one uses the technical term vibronic state. The difference between these two ways of illustrating energy levels is shown on the left. On the left hand side, we have the case where we consider the electronic and vibrational energy levels separately. The bold line illustrates the energy of the electronic ground state and ignores the energy of the nuclei. The lowest possible total energy is given by the energy of the electrons in their ground state plus the energy of the vibrations in their ground state. This energy corresponds to the lowest thin line. On the right hand side, the energy levels of vibronic states are plotted. Here the bold line corresponds to the lowest energy of electrons and nuclear vibrations together. Thus, it illustrates the ground state energy of the molecule and its energy equals that of the lowest thin line on the left hand side. You can recognize which type of illustration is used by looking at the spacing between the bold line and the first vibrational level shown. On the left side, this spacing is much smaller than the spacing between the subsequent vibrational levels, whereas on the right hand side the spacing to the first vibrational level shown is the same as that to subsequent levels. When the first spacing is smaller than the other spacings, then the first thin line corresponds to the vibrational ground state because the energy of the vibrational ground state is just half of the energy difference to the next vibrational states. In contrast, when the first spacing is the same as the other spacings, as on the right hand side,
the energy of the vibronic states is shown. Then the bold line illustrates the lowest possible energy of the system. POPULATION OF ENERGY LEVELS As mentioned above, thermal energy can raise the energy from that of the ground state to that of an excited state. This depends on the temperature (higher temperature = more thermal energy) and the energy gap between the ground and the excited state. A smaller gap means that less thermal energy is needed to populate (occupy) the excited state. The relative occupancy of two states with different energies is given by the Boltzmann distribution: n upper/ n lower = exp(- E / kT ) where n upper is the number of molecules in the higher energy state, n lower is the number in the lower energy state, E is the energy gap between the two states, and k is the Boltzmann constant (1.38 × 10 -23^ J/K). A special case of the Boltzmann distribution is that calculated for 1 mol of molecules. In that case k has to be replaced by R , the gas constant ( R = 8.31 J mol-1^ K-1, R = N A k ). Remember that RT = 2.5 kJ/mol at room temperature. When E is small or T is large ( E << kT ), exp(- E / kT ) approaches e^0 , which is 1. The number of molecules in the upper and lower levels is then equal. In the opposite case ( E large or T small, i.e. E >> kT ), the exponent is very large and exp(- E / kT ) very small. Then only the ground state is occupied. [CD] As mentioned above, the energy gaps between adjacent energy levels depend on the sub-system considered. For example electronic spin states are very closely spaced, nuclear spin states are even closer. This implies that excited states are considerably populated at room temperature. For nuclear spin states, the populations of excited state and ground state are nearly equal with only 1 out of 20 000 spins more in the ground state [Ja]. In contrast, different electronic orbitals have often quite large separations between their energy levels. As a consequence, only the ground state is populated at room temperature. Intermediate between these extremes are the energy gaps between nuclear vibrations. The gaps between energy levels of the rapidly oscillating vibrations is still larger than the thermal energy, meaning that most molecules are in the vibrational ground state. However, the gaps of slow vibrations are comparable to the thermal energy meaning that a considerable number of molecules is in vibrationally excited states.
The mass on a spring model with driving force (green) and movement of the mass (black) indicated. When a spring with attached weight is extended and released, it will oscillate at its intrinsic frequency 0 which is determined by the spring force constant. However, when it is driven by a sinusoidally time-varying driving force, F 0 cos t , it will eventually oscillate with the external frequency (the frequency of light). When there is friction (= interaction with the environment), then the spring does not immediately follow the driving force, it lags behind. One says that there is a phase delay between driving force and spring movement. The phase delay occurs because friction/resistance has to be overcome. One can separate the resulting spring movement in a part that is in phase (0 or 180 ° phase difference between spring and driving force) and one which is 90° out of phase. The amplitude of these two components depends on the driving frequency and on the intrinsic frequency of the spring (panel e). The components are called dispersion and absorption because their frequency dependencies resemble those for the refractive index and the absorption of energy by the spring respectively. We are interested in the absorption curve in the following.
Decomposition of the mass movement into an in-phase and an out-of-phase movement. The curve for x " (absorption) has Lorentzian line shape x "( ) = [1 + ( 0 - )^2 ^2 ]- where is the angular frequency of the driving force, 0 the intrinsic angular frequency of the weight on the spring in the absence of driving force and the relaxation time which is inversely proportional to the friction coefficient. A Lorentz curve. The interpretation of the curve for x " is as follows: the amplitude of the oscillation that is 90 ° out of phase to the driving force is maximal when the frequency of the driving force matches the intrinsic or natural frequency of the weight on a spring (= frequency of oscillation in the absence of a driving force). This increase in amplitude can be quite dramatic. It is only possible by energy transfer from the driving force to the spring and it turns out that the energy absorbed by the spring is largest when the two frequencies match. Such a phenomenon is called resonance. It gives evidence for the absorption of energy by the spring from the driving force. In spectroscopy, this corresponds to energy absorbed by the atoms or molecules from the electromagnetic radiation.
The Figure on the left shows a spectrum of scattered light. The horizontal axis can be wavelength, wavenumber or frequency. The vertical axis is the intensity of scattered light. The large majority of photons is scattered elastically (Rayleigh scattering, after the English physicist Lord Rayleigh for his discovery of the effect, pronounced ray-lea like in X- ray and in lea ve). An everyday example for scattering is the following: the sky is blue because it is the blue light that is preferentially scattered from small particles in the atmosphere. Thus we see blue light when we do not directly look into the sun. Milk or an oil-water suspension are turbid because light is scattered from small oil droplets. Scattering can be explained as follows: the electric field vector of the electromagnetic wave induces oscillations of the electrons. This creates an oscillating dipole because the nuclei are much less moved because of their larger mass. The oscillation dipole acts like a small antenna and emits radiation in all directions. In the figure below, the oscillating electric field vector is shown and its effect on an atom at different times (phases) of the oscillation. The electron cloud is shown in red and the nuclei in blue. The nuclei are nearly stationary because of their large mass, but the electron cloud moves. Whenever the position of the center of the electron cloud is different from that of the nucleus, a dipole moment is generated. This oscillates with time and emits radiation. If the molecules have a permanent dipole moment an additional effect takes place: the molecules will reorient under the influence of the electric field which again gives rise to an oscillating dipole moment. The emitted radiation from a molecule superimposes with that of all other molecules and with the incident radiation. Therefore, scattering is also the basis of the phenomena reflection and refraction: reflected light is the superposition of the scattered light from all molecules, and refraction is the result of the superposition of the incident wave with all scattered waves.
Photon energy
Intensity
Rayleigh
Raman
Raman
E
Absorption Matter absorbs energy from the electromagnetic wave in specific regions of the electromagnetic spectrum. Leaves are green because light of other colors is absorbed, green light is transmitted and reflected preferentially. The Figure on the left shows an absorption spectrum. The horizontal axis can be wavelength, wavenumber or frequency. The vertical axis is absorbance (see below for the definition of this quantity). The peaks are referred to as absorption bands or absorption lines. Some of the bands are well resolved (the band at the left and in the middle) whereas the bands on the right overlap. The left and the center band of the right band profile are still resolved, but the small component band on the right of the center band is not resolved. It generates a shoulder. Emission Light can be emitted for example from a hot body (light bulb) or after light of one wavelength has been absorbed. In the latter case, the emitted light has lower energy (longer wavelength) than the absorbed light. Example: Whiter than white effect of washing powders: light in the ultraviolet region of the spectrum is absorbed, but emitted is light in the visible region which makes the T-shirt look brighter. Photochemistry Visible and UV light have enough energy to alter a molecule, for example to break covalent bonds, to induce rotations around bonds or to ionize the molecule by abstraction of an electron or a proton. These effects are very important in biology (photosynthesis).
Transitions between energy levels
SPECTROSCOPY IS APPLIED QUANTUM MECHANICS
Spectroscopy is an experimental method which aims at obtaining molecular information on the system under study. The link between observation and information is provided by the theory of the molecular interaction between electromagnetic or particle radiation and matter. In general, this interaction perturbs atoms and molecules which often makes them lose or gain energy. The theoretical challenge is to describe the extent of these effects and why this happens only at certain wavelengths. This can be best done in the framework of quantum mechanics. Fundamental will be the description of energy levels of a molecule and of the interaction between radiation and matter. The following is a summary of important concepts in quantum mechanics. [CS] Absorbance Wavelength
So, if the system is in state | Ψ 0 ñ at a given time, it will remain in that state for all times and no transitions are possible. However, transitions between energy eigenstates are fundamental to understand spectroscopic experiments. How does this fit together?
- In order to induce transitions, the system needs to be perturbed. The system can for example be a molecule. The Hamilton operator of the unperturbed molecule describes the energy contributions and interactions within the molecule, i.e. it does not consider interactions with the environment. A perturbation is for example the interaction of the molecule with an electromagnetic wave. This interaction changes the energy of the system, which means that the original Hamilton operator is no longer a "good" operator to describe the energy of the molecule. Instead the Hamilton operator of the perturbed system consists of the original Hamilton operator H and a perturbation operator V. Since H is no longer an appropriate Hamilton operator, its eigenstates (i. e. the eigenstates of H ) are no longer proper eigenstates of the perturbed molecule. Still they are in many cases good approximations of the real eigenstates and it is easier for us to use them also in the description of the perturbed molecule. If we do so, the perturbation induces transitions between the eigenstates of the unperturbed Hamilton operator, for example between the eigenstates | Ψ 0 ñ and | Ψ 1 ñ considered above. The perturbation V might induce transitions from | Ψ 0 ñ to | Ψ 1 ñ. According to Fermi´s golden rule , the probability for the transition is proportional to | Ψ 1 | V | Ψ 0 ñ|^2. In the position representation this scalar product is written as |∫ Ψ 1 * V Ψ 0 d|^2 where the integration is over all space. An interpretation is easiest in the Dirac notation: The scalar product (or dot product) Ψ 1 | V | Ψ 0 ñ is a projection of vector V | Ψ 0 ñ on vector | Ψ 1 ñ as shown in the figure on the left. It analyses how similar these two vectors are. The projection is zero, if the two vectors are orthogonal, it is maximal if they have the same direction. An analogy is the calculation of the y - component (or y -coordinate) of a vector in 3 dimensional space. This is the same as the projection of this vector on the y -axis. If the perturbation V has no influence on | Ψ 0 ñ then V | Ψ 0 ñ = | Ψ 0 ñ and Ψ 1 | V | Ψ 0 ñ = Ψ 1 | Ψ 0 ñ = 0 since both vectors are eigenvectors of the Hamilton operator and therefore orthogonal to each other. Therefore, a transition will only occur, if the perturbation makes the initial state | Ψ 0 ñ somewhat similar to the final state | Ψ 1 ñ, i.e. if V | Ψ 0 ñ is rotated away from | Ψ 0 ñ towards | Ψ 1 ñ. Summary: transitions between energy eigenstates can only be induced by a perturbation of the system (molecule). The probability of a transition depends on the degree to which the perturbation distorts the initial state of the system so that it becomes more similar to the state after the transition. When we discuss spectroscopic methods, it will be important to identify the perturbation and the relevant eigenstates of the system. We will leave these key concepts for now but discuss them in more detail when we discuss the individual spectroscopic methods.
ABSORPTION
Bohr frequency rule Electromagnetic radiation can induce transitions between energy levels. For example, energy from the electromagnetic wave can be absorbed which takes the system to a state with higher energy. But the system is selective: only radiation with a particular wavelength can induce the transition and can therefore be absorbed. Transition from the ground state E 0 to an excited state induced by absorption of electromagnetic radiation. To understand this selectivity, we have to move from the wave picture for electromagnetic radiation to the particle picture. The transition can take place if the energy of the photons matches the energy gap E between the energy levels. E = h This is known as the Bohr frequency rule and applies to many different types of spectroscopic experiments. The Bohr frequency rule E = h is a consequence of the corpuscular theory of light. Light with low frequency (long wavelength) can only excite low energy transitions, no matter how large the intensity (= number of photons) is. Selection rules Apart from the Bohr frequency rule, other rules exist that are particular for the transitions induced in the experiment. We will encounter them when we discuss transitions between particular energy levels. These rules are called selection rules and they classify transitions into "allowed" and "forbidden" transitions. However, these categories are not absolute. In practice, an allowed transition is one which has a high probability, a forbidden transition is one with a low probability. Definition of selection rule according to IUPAC Gold book: “A rule that states whether a given transition is allowed or forbidden, on the basis of the symmetry or spin of the wavefunctions of the initial and final states. ” (https://goldbook.iupac.org/S05549.html) The reason for the failure of a clear cut distinction between allowed and forbidden transitions is a deficiency of the theory applied to derive the selection rules. The theory often is based on a simplified system which is only an approximation of the real system. An example for such a simplification is the assumption that the phase of the electromagnetic wave is the same across the entire molecule (Douglas, Burrows, Evans, Chapter 1 in Applied Photochemistry edited by Evans 2013). Different ranges of electromagnetic radiation probe different molecular properties We have seen above that the gaps between energy levels can be very large or very small depending on the sub- system. The gaps can be overcome by absorbing a photon with an energy that matches the energy gap between
h
E 0
E 1
Let us now consider a simple absorption experiment: The incident light has intensity I 0 at wavelength . It traverses a sample with a path length d. The light that is not absorbed by the sample emerges with intensity I 1. Consider a sample of molecules in a layer perpendicular to the direction of light propagation, and sufficiently thin (d x ) so that the light intensity within this layer is essentially constant. Then the fraction of light absorbed (-d I / I ) should be simply proportional to the number of absorbing molecules. The resulting equation is -d I / I = C ´d x where C is the concentration of absorbing molecules and ' is a proportionality constant that is proportional to the probability for an individual molecule to absorb a photon. If we integrate this equation over the entire sample (integrating the left-hand side from initial intensity I 0 to final intensity I 1 , and the right-hand side from zero to d we obtain In ( I 0 / I 1 ) = –In ( I 1 / I 0 ) = C ' d Converting to log base 10, we have the common form of the Beer-Lambert law (also known as Beer's law or the Lambert-Beer law or the Beer-Lambert-Bouguer law after those who independently discovered the law in various forms: Bouguer in 1729, Lambert in 1760, and Beer in 1852 [wikipedia.org].): A ( ) = log ( I 0 / I 1 ) = –Iog ( I 1 / I 0 ) = C ( ) d where = '/2.303 is called the molar absorption coefficient and A is called the absorbance or (sometimes) the optical density OD. () is a molecular property that depends on the wavelength of the incoming light. It is proportional to the probability of inducing a transition at that particular wavelength. It was previously called extinction coefficient and this term is still widely used. The reason, why it is recommended to switch from extinction coefficient to absorption coefficient is that extinction includes not only absorption but also intensity losses due to scattering. If only absorption occurs then extinction and absorption are the same. Often only the absorption coefficient (max) at the wavelength of maximal extinction ( max) is of interest. It is used for example to determine the concentration of biomolecules. The value max for electronic transitions of typical chromophores varies over a wide range from as little as 1 M-1cm-1^ to more than 105 M-1cm-1. max depends also on the molecular sub-system. For example, the absorption coefficient for vibrational transitions is typically a factor of 100-1000 lower than that of electronic transitions. can be related to the probability of an individual absorption process (best in [CS]). This probability is abbreviated B and named Einstein coefficient. The Einstein coefficient B is the transition rate per unit energy density of the radiation. The loss of intensity d I (=
loss of energy) will be proportional to B , the energy of the absorbed photons h and the number of incident photons which is proportional to incident intensity. Thus: -d I ~ B I Since -d I = C ´ I d x we get the following proportionality for : ~ B or B ~ / This proportionality holds for a very sharp absorption band only. Real absorption bands have a certain band width (full width at half maximum, see below). Then we have to replace / by the integral over d . This can be approximated by max / max (index "max" indicates values taken at maximum absorption) assuming that the band width is much smaller than the frequency which makes the frequency essentially constant across the absorption band and by approximating d by max . This gives: B ~ max / max The quantity A is called the absorbance or (sometimes) the optical density. Note that "absorption" denotes the physical process and "absorbance" is the quantity that describes how much light is absorbed. One can say both " absorption spectrum " and " absorbance spectrum" but not "the absorption is 0.5" (correct is "the absorbance is 0.5") or " absorbance is a process which attenuates the incoming light" (correct: " absorption is a process which attenuates the incoming light"). Absorbance is a quantity without unit. Some spectroscopy programs label the vertical axis of a spectrum with "absorbance (a.u.)" meaning "absorbance in arbitrary units". This is nonsense. Absorbance has no units, not even arbitrary ones. Most useful is the expression A ( ) = –Iog ( I 1 / I 0 ) because it expressed absorbance in terms of the fraction of incident light that reaches the detector. Note that the relationship is logarithmic. A = 2 means that only 1% of the incoming light transmits the sample and reaches the detector. When A = 3, the transmitted light intensity is 10 fold less, i.e. only 0.1% of the incoming intensity. Somewhere here is the limit for accurate measurements. Higher absorbance values mean that even less light reaches the detector which makes accurate measurements difficult. Equally difficult are measurements at low concentrations. When only very little light is absorbed, the small difference of light intensity with and without sample is difficult to measure. Therefore, the absorption of single molecules cannot be measured. Typical concentrations are in the M range. RELAXATION Definition Systems in excited states have the tendency to return to the ground state. A transition from an excited state to the ground state is called relaxation. One speaks of relaxation processes or says that the system relaxes back to the ground state. Relaxation can involve the emission of a photon or not. Non-radiative relaxation processes [CD] If relaxation does not lead to the emission of a photon one speaks of non-radiative relaxation. We distinguish two non-radiative relaxation processes: thermal relaxation and resonance energy transfer. Thermal relaxation: Here, a molecule looses energy by collisions with other molecules, which transfer energy from one molecule to another. This increases the kinetic energy of many molecules, which means that the temperature of the sample increases.
