Raman scattering

Summary

In this experiment you will use a commercial Raman spectrometer to study various pure liquids and liquid mixtures. Raman scattering spectroscopy detects, in analogy to IR-absorption spectroscopy, energetic transitions relation to molecular vibrations. Resulting spectra therefore show the characteristic “finger-print” allowing to distinguish even molecules with similar chemical structure.

Goals

• Learn how to measure a Raman spectrum with a commercial spectrometer
• Analysis of molecular vibrations of comparatively simple organic molecules
• Quantitative Analysis of the composition of an ethanol/water-mixture
• Theory: Physical principles of Raman scattering, group theory

Theoretical Background

Raman Scattering

Not all molecular vibrations can be excited by absorption of optical radiation: Fermi’s golden rule tells us that the electric dipole momentum of a molecule has to change during the vibrational oscillations. Therefore, homologous bi-atomic molecules like, for example, Cl2, are invisible to IR spectroscopy, and therefore it is also not possible to measure (by IR absorption spectroscopy) the molecular bond strength needed, for example, as input parameter in statistical thermodynamics to calculate the heat capacity as a function of temperature for Cl2. Here, Raman spectroscopy, which is an inelastic scattering technique, provides an adequate solution, and therefore is considered as a method complimentary to IR spectroscopy. Raman scattering also allows to access changes in polarization of incident and scattered light, thereby enabling an accurate assessment of molecular vibrations in conjunction with so-called group theory, a concept most useful for symmetry analysis of molecular vibrations and chemical bonds in molecules or chemical complexes. Therefore, we will also introduce the concept of group theory in a separate chapter of this manuscript.

Raman scattering is based on the irradiation of the sample with a beam of well- defined polarized monochromatic, highly intensive, light. The major part of the scattered light, also called Rayleigh scattering, will then have the same wavelength as the incident light. Note, however, that even the Rayleigh peak shows a slight broadening in wavelength due to the Doppler effect, which may be exploited to determine the velocity of molecules in the gaseous state. Also, the intensity of the Rayleigh peak scales both with the polarizability of the scattering molecules, which is a measure how easy the incident electromagnetic wave may induce an oscillating dipole acting as an emitter, as well as with the frequency of the incident light to the power 4.

In addition to the main quasielastic Rayleigh peak, molecules may also show two inelastic much smaller peaks called Stokes- and Antistokes-peak, respectively. In this case, molecular vibrations are coupled to the emitting oscillating dipole, and the respective vibration energy is either deduced (Stokes-peak, red-shifted compared to the Rayleigh peak) or added to the energy of the emitted light (Antistokes peak, blue- shifted compared to the Rayleigh peak). A Raman spectrum therefore consists of the detected scattered intensity, typically measured at an angle of 90° or even 180° (back scattering) in respect to the incident light beam, plotted vs. the wavelength or wavenumber of the scattered light. Typically, the wavenumber of the incident light or Rayleigh peak is set to zero, the Rayleigh peak because of its high intensity is blocked from the detector, and the weaker Stokes peak is detected as a function of wavenumber shift (Stokes shift). Fig. 1 summarizes the molecular energetic transitions involved in Raman spectroscopy, and sketches the resulting Raman spectrum. Note that the blue-shifted Antistokes peak is usually much lower in intensity than the red- shifted Antistokes peak, reflecting the difference in thermal population of the vibrational ground and the vibrational excited state as defined by the Boltzmann distribution.

As mentioned, the electric field of the incident light induces an oscillating electric dipole momentum, which is directly proportional to the electric field strength, the proportionality given by the polarizability of the molecule \alpha:

with \nu_0 the frequency of the incident light, and with \vec E_0 its electric field vector amplitude. The polarizability itself is a tensor of 2nd rank, indicating that the direction of the oscillating dipole vector may be changed compared to the polarization direction of the incident electric field vector. We will keep this property of the polarizability tensor in mind when we discuss the symmetry of molecular vibrations visible in Raman scattering in chapter 3. An important condition for Raman active vibrations is that the polarizability changes with the atomic coordinates q. As a first approximation, this dependence of the polarizability on the atomic coordinates of a vibrating molecule can be expressed by a Taylor series expansion of \alpha around the equilibrium atomic configuration, i.e.

Here, we are interested only in changes of the atomic coordinates q due to harmonic molecular vibrations, therefore

with \nu_\text M the frequency of the molecular vibration and q_0 the amplitude of the vibration. Note that this frequency is identical to that of IR light needed to incite the molecular vibration directly by light absorption, in case this vibration would show a change in the electric dipole momentum of the molecule. If we insert Eqs. (2), (3) in the expression for the induced dipole momentum (Eq. (1)), we get:

Or, using the theorem for the multiplication of two cosine functions:

Here, the first part of this equation corresponds to the dipolar oscillations responsible for the elastic Rayleigh scattering (\nu_0), and the second part to the oscillations causing the Stokes (\nu_0-\nu_\text M) and the Anti-Stokes (\nu_0+\nu_\text M) peaks, respectively. Eq. (5) shows also directly the important selection rule telling if a molecular vibration is visible in Raman scattering, i.e. the polarizability has to change with the harmonic vibration of the atoms:

The total number of harmonic vibrations of a non-linear molecule consisting of N atoms is given by N-6. For example, NH₃ should have 9 different molecular vibrational modes. Those vibrational modes with a change in the electric dipole momentum should be IR active, whereas those with a change in polarizability should be Raman active. Note that some modes may be neither IR nor Raman active, or both. Whereas it is simple to check these selection rules for biatomic molecules like HCl or Cl₂, it is impossible to address the vibrations of molecules with 3 or more atoms without a suitable concept of symmetry analysis. Group theory here provides a solution, and will be introduced in the next chapter.

Group Theory and Analysis of the Symmetry of Molecular Vibrations

In this chapter, we will introduce the concept of point group theory and character tables. This mathematical concept is essential to analyze fundamental problems in chemistry, like the symmetry of molecular vibrations, or the symmetry of molecular electronic orbitals. Group theory is based on simple symmetry operations like regular rotations, mirror planes, points of inversion, etc., and matrix calculation. Here, it will be shown how group theory is used to predict if molecular vibrations or electronic transitions can be excited by the absorption of light.

The symmetry group C_2v

We start with the character table of the 3-atomic molecule \ce{H_2O} as a simple example:

Character table of symmetry point group C_2v (e.g. \ce{H_2O} molecule)

C_{2v}	E	C_2	\sigma_v(xz)	\sigma'_v(yz)
A_1	1	1	1	1	z, x^2, y^2, z^2
A_2	1	1	-1	-1	R_z, xy
B_1	1	-1	1	-1	x, R_y, xz
B_2	1	-1	1	1	y, R_x, yz

The first row of the character table contains the name of the symmetry point group in the upper left corner. Here, C_{2v} symbolizes a molecule with a C_2 rotation axis (molecule looks identical if rotated by 180° around its principle symmetry axis), and vertical mirror planes \sigma_v. Next, the first row contains the symmetry operations characteristic for the given point group, in this case the identity E (corresponding to rotation by 360°), the C_2 rotation axis already mentioned, and two different vertical mirror planes. The coordinates x, y and z have been defined, in respect to a certain orientation of our molecule in space, as shown in fig. 2:

In the last column of the character table, we find basic operations, like translation along the Cartesian axes (x, y, z), or rotation around an Cartesian axis (R_x, R_y, R_z), which show the same symmetry like the irreducible representations listed in the first column (here: A_1, A_2, B_1, B_2). In addition, we also find products of Cartesian coordinates, like xy, xz etc. These products are necessary to determine if a vibrational mode is detectable in Raman spectroscopy, as we will explain later in more detail.

The irreducible representations are the basic set of the reducible representations of each symmetry operation. To determine these reducible representations, we consider that the 3N degrees of freedom of our molecule are accurately defined by a 3N- dimensional vector, containing the Cartesian coordinates of each of the N atoms. Then, a symmetry operation may be represented by a 3N \cdot 3N dimensional matrix, translating the original vector into a new vector (or atomic configuration of the molecule). The character of this matrix is the sum of the diagonal elements in this case. Note that the 3N \cdot 3N matrix, also called reducible representation, in terms of its character can be rewritten as the sum of irreducible smaller matrices, similar to the concept that a 3d vector can be rewritten as the sum of 3 vectors in directions x, y and z, respectively. The numbers in the character table then represent the characters, or sums of the diagonal elements, of the smaller irreducible matrices. Let us illustrate this principle of reducible and irreducible matrix representation of symmetry operations in more detail:

For example, we consider how the symmetry operation C_2 (= rotation by 180° around the z-axis) transforms the original position vectors of the individual atoms of our \ce{H_2O} molecule, as sketched in fig. 3:

If the vector coordinates of the original molecular configuration are defined as following:

then the symmetry operation C_2 transforms this vector as following:

For example, the y-vector of the left H-atom becomes the y-vector of the right H-atom in negative direction, and therefore q_2 changes to -q_8. In general, the transformation of one vector into another one of the same dimension 3N is mathematically described by the multiplication of the original vector with a matrix of dimension 3N \cdot 3N, i.e.:

For example, the transformation matrix \Gamma for the C_2 rotation of the \ce{H_2O} molecule is given as:

This can easily be confirmed, if we consider that the first component of the new vector \vec q' is given by multiplication of a 3N-dimensional vector, corresponding to the first row of the matrix, with the vector \vec q, and so on..., i.e.

The character of this matrix, or sum of the diagonal elements, then is given as \chi_{C_2} = -1 - 1 + 1 = -1. Here, one should note that the diagonal elements are only non-zero in case an atom does not change its position at the symmetry operation. In case of the C_2 operation, both H atoms swap their positions while the O atom stays at its place and only changes the orientations of its Cartesian position vectors in x- and y-direction. In general, the character of any transformation matrix can easily be determined if we just consider the atoms not changing their position, and consider how the respective Cartesian coordinates change. This simple concept then leads to the following characters of the reducible matrix representations for all symmetry operations belonging to the point group C_{2v}:

These results are summarized in the following table, not to be confused with the character table containing the irreducible representations, only:

Characters of the reducible representations for the symmetry operations of symmetry point group C_{2v} (e.g. \ce{H_2O})

C_{2v}	E	C_2	\sigma_v(xz)	\sigma'_v(yz)
\Gamma	9	-1	1	3

To determine the symmetry of the molecular vibrations, and if they are detectable by IR-spectroscopy, or Raman spectroscopy, or both, we next have to analyze the reducible representation, corresponding to the 3N degrees of freedom (translation plus rotation plus vibration) in respect to its underlying basic irreducible symmetries. This is simply achieved by considering that the reducible representation is the sum of 9 irreducible one-dimensional representations (listed in the character table 1). The contribution of each of the irreducible representations (A_1, A_2, B_1, B_2) is calculated then by taking into account that the sum of the respective characters has to be equal to the characters of the reducible representation given in table 3.2, leading to 4 equations (one for each symmetry operation) with 4 unknown quantities, the respective contributions of each irreducible representation to \Gamma:

Note that the total number of contributions has to equal 3N, i.e. in case of our \ce{H2O} molecule: c_1 + c_2 + c_3 + c_4 = 9. The coefficients c_i in general may be calculated with the formula:

with h the order of the group (or number of basic symmetry operations, here h = 4), h_k the dimension of the irreducible representations (here, h_k = 1 in all cases), R the respective symmetry operations, and \chi_i(R) and \chi(R) the characters of the respective irreducible and reducible representations. This formula then leads to the following results (the corresponding characters are found in tables 1 and 2, respectively):

Therefore, the reducible representation is finally given as:

To identify the symmetry of the vibrational modes, we have to subtract the translational and rotational molecular degrees of freedom first. According to the character table (see table 3.1), the translations x, y and z have the symmetries of A_1, B_1 and B_2, whereas the rotations around the 3 different Cartesian axes R_x, R_y and R_z have the symmetries of A_2, B_1 and B_2. Subtracting these 6 representations in Eq. (3.10) then leads us to the following symmetries for the 3 vibrational modes of the \ce{H_2O} molecule:

We expect therefore two vibrational modes to be totally symmetric (A_1), and one mode to be asymmetric in respect to rotation by 180° around the principle z-axis (B_2). For a simple molecule like H_2O, these vibrations could be visualized as following:

Finally, we have to determine which of these vibrations are detectable by spectroscopic methods. Again, the character table is very helpful: this time, we use its right column to analyze the symmetry of the irreducible representations in respect to the symmetry of an electric dipole momentum, which is a vector with Cartesian components (x, y, z), or the symmetry of the polarizability, which is a tensor and therefore contains the 9 matrix elements (x^2, xy, xz, yx, y^2, yz, zx, zy, z^2).

Character table of symmetry point group C_2v (e.g. \ce{H_2O} molecule). Marked in red are IR active vibrational modes, marked in green are Raman active vibrational modes.

C_{2v}	E	C_2	\sigma_v(xz)	\sigma'_v(yz)
A_1	1	1	1	1	\textcolor{red}{z}, \textcolor{green}{x^2, y^2, z^2}
A_2	1	1	-1	-1	R_z, \textcolor{green}{xy}
B_1	1	-1	1	-1	\textcolor{red}{x}, R_y, \textcolor{green}{xz}
B_2	1	-1	-1	1	\textcolor{red}{y}, R_x, \textcolor{green}{yz}

The character table shows that both the A_1 modes and the B_2 mode correspond in symmetry to Cartesian coordinates as well as to products of Cartesian coordinates. We therefore may conclude that, in case of a sample containing many \ce{H2O} molecules in random orientation in respect to the director of the electric field vector of the incident light, all 3 modes are visible in both IR and Raman spectroscopy. Note, however, if we would prepare a solid sample with all molecules oriented in z-direction, then the vibrations could not be detected with IR light linearly polarized in x-direction (since the transition dipole momentum of the molecular vibrations of \ce{H_2O} has no symmetry component in x-direction perpendicular to the principle molecule axis).

For other molecules with different symmetry the analysis of molecular vibrations always follows the same simple scheme:

1. Determine the symmetry group, using for example a suitable flow chart.
2. Get the corresponding character table from literature (eg. ref. 4)
3. Determine the characters of the reducible representation (see table 2 and Eqs. (12)). Note: consider only atoms which don’t move if the respective symmetry operation is applied to the molecule.
4. Express the reducible representation as sum of irreducible representations (see Eq. (14), and Eqs. (15) for example.
5. Subtract the modes of translation (x, y, z) and rotation (R_x, R_y, R_z) (see character table) to obtain the overall symmetries of all vibrational modes.
6. Identify Raman active vibrational modes using the character table, i.e. look for irreducible representations transforming as x_2, y_2, z_2 or xy, xz and yz. This leads to the maximum number of peaks visible in a Raman spectrum.

Symmetry groups C_3v and T_d

Organic molecules studies in this experiment either belong to the symmetry groups C_{3v} (\ce{CHCl3}, \ce{CDCl3}) or T_d (\ce{CCl4}). We therefore conclude this chapter by providing the respective character tables as well as the reducible representations:

Character table of symmetry point group C_{3v} (e.g. \ce{CHCl3} molecule)

C_{3v}	E	2 C_3	3 \sigma_v
A_1	1	1	1	z; x^2+y^2; z^2
A_2	1	1	-1	R_z
E	2	-1	0	(x, y); (R_x, R_y); (x^2-y^2, xy); (xz, yz)

characters of the reducible representations for the symmetry operations of symmetry point group C_{3v} (e.g. \ce{CHCl3} molecule)

C_{3v}	E	2 C_3	3 \sigma_v
\Gamma	15	0	3

Character table of symmetry point group T_d (e.g. \ce{CCl4} molecule)

T_{d}	E	8 C_3	3 C_2	6 S_4	6 \sigma_d
A_1	1	1	1	1	1	x^2+y^2+z^2
A_2	1	1	1	-1	-1
E	2	-1	2	0	0	(2z^2-x^2-y^2, x^2-y^2)
T_1	3	0	-1	1	-1	(R_x, R_y, R_z)
T_2	3	0	-1	-1	1	(x, y, z); (xy, xz, yz)

Characters of the reducible representations for the symmetry operations of symmetry point group T_d (e.g. \ce{CCl4} molecule)

T_{d}	E	8 C_3	3 C_2	6 S_4	6 \sigma_d
\Gamma	15	0	-1	-1	3

Experimental setup

Experimental procedure

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You will use a laser class 3B. Therefore: never remove the optical guide from the dark chamber, and only open the lid of the dark chamber if the laser shutter is closed!!!

Procedure for measurement of a Raman spectrum:

1. Start the program “BWSpec”
2. Press Button “Dark scan” (1) (only once before you measure the first sample)
3. Make sure the laser shutter is off, open the lid of the dark chamber, and position the sample cell. Organic solvents are measured as original bottles, whereas water/ethanol mixtures are measured in sample vials. Make sure the laser outlet is positioned as close as possible to the sample cell (but never touch the glass walls!).
4. Set “Dark subtracted “in “Display options”, and push start (2) to measure one spectrum.
5. If the spectrum looks too noisy or peak intensities are too high, readjust the laser power (3) and repeat the measurement (iv). If laser power adjustment does not help, reposition the sample (make sure the laser is shut off before you open the lid!).
6. In case the spectrum looks o.k., enable the cursor (see screenshot with markings) and determined peak positions and peak intensities. Eventually readjust the x- axis using “Scales” (see fig.6).

Note: Alternatively you can export your data as “*.txt” (see “File”) and analyze your spectrum with Excel at home.

Raman spectroscopy of pure organic solvents

In this part of the Raman experiment, you will learn how to use group theory and symmetry analysis to interpret the Raman spectra of simple organic molecules. Compare your experimental results with theoretical predictions. Also, investigate the effect of the isotope D (versus H) on the energy of certain molecular vibrations. The organic solvents \ce{CCl4}, \ce{CHCl3} and \ce{CDCl3} are measured as described, using original sample flasks (do not open!!!). Determine the peak positions of all spectra.

1. \ce{CCl4}: determine the peaks in the wavenumber range 200 - 1000 cm-1
2. \ce{CHCl3} determine the peaks in the wavenumber range 200 - 3200 cm-1. To obtain better results, zoom into this wavenumber range using “Scales”.
3. \ce{CDCl3} determine the peaks in the wavenumber range 200 - 3200 cm-1. To obtain better results, zoom into this wavenumber range using “Scales”.

Raman-Spectroscopy of Water/Ethanol-Mixtures

The aim of this part is to determine the ethanol content of unknown commercial liquors using Raman spectroscopy with a calibration obtained from known ethanol concentrations.

Use standard mixtures for calibration. Then, determine the ethanol content of commercial and/or self-distilled samples using the peak intensity of the most prominent peak in the Raman spectrum. First, determine the most prominent peak in the Raman spectrum from the calibration solution with 60% ethanol. Determine the peak intensity, and do not forget to use background subtraction (left and right baseline, background = average intensity). Reposition each sample 3 times to determine the experimental error of this procedure.

1. For calibration: water/ethanol mixtures with 10, 20, 30, 40, 50, 60 vol% EtOH.
2. For analytical ethanol quantification: 3 - 4 unknown samples.

Note that all samples are measured in small glass vials of the same size.

Data Evaluation

Organic Solvents

Determine the wavenumbers of all peaks in the Raman spectra of the organic solvents \ce{CCl4}, \ce{CHCl3} and \ce{CDCl3}.

1. Prepare a table of the peak positions, and compare your results with literature (search yourself for suitable sources!).
2. Assign each peak to its normal vibrational mode (irreducible representation, see also character tables). Briefly discuss your assignments (energy, symmetry, degenerate modes of vibration).
3. Discuss the effect of isotope replacement for the molecules CHCl₃ and CDCl₃ on the respective vibrational modes. Use quantitative calculations if appropriate.

Water/Ethanol Mixtures

1. Determine the calibration line from your data, and the corresponding proportionality constant for Raman peak intensity related to volume% ethanol.
2. Using (i), quantify the ethanol content of various samples. Compare your results to the values given on the original bottles of the commercial spirits.

Literature

Helpful web-links:

Questions for the Oral Entrance Examination

1. What are the differences of IR- and Raman spectroscopy? Draw a scheme showing the respective energy levels and transitions. Give the selection rules for each method.
2. Explain the term “normal modes of vibration”. How many of these normal modes has a molecule consisting of N atoms? What are the symmetries of the normal modes of H₂O, and how do those vibrations look like?
3. Write down the character table of the symmetry group C_{2v}, and explain.
4. Give the characters of all symmetry operations in reducible representation for H₂O.
5. Give the characters of all symmetry operations in reducible representation for NH₃.
6. Discuss practical advantages of the Raman technique compared to IR absorption spectroscopy.

Risk Assessment

During this experiment you will use a diode laser with wavelength 785 nm (red) and laser power < 300 mW (class 3B !). To avoid irreversible eye damage always make sure that the laser beam, or any reflected beams, are never accessible to your eyes. In detail, following these instructions is absolutely mandatory:

1. Only start the experiment after instruction by your supervisor.
2. Never remove the laser outlet from its position at the dark chamber!!
3. Before you open the lid of the dark chamber, always make sure that the laser shutter is closed.
4. Accordingly, only open the laser shutter if the lid of the dark chamber is closed and the laser outlet is attached to the dark chamber.

If you follow instructions (ii) to (iv), there is no risk at all that laser light might accidentally hit your eye. Therefore, the apparatus is practically used as a class 1 laser setup, and no additional safety measures are needed.