2-1

Spectroscopic Studies of HCl and DCl * Butera, R.A., Waldeck, D.H and Wagner, E.P. Revised January 2012 Introduction High-resolution infrared spectroscopy is one of the most useful tools for investigating the structure of small molecules. In this laboratory, you will practice some infrared techniques and analyses for one of the canonical problems in spectroscopy, obtaining the bond length of a diatomic molecule. You will learn about the careful use of glass vacuum lines to prepare gas phase samples for analysis and the operation of a Fourier transform infrared spectrometer. In the analysis you will review the quantum mechanics for a rotating-vibrating molecule and use the data to determine some of the molecular characteristics of HCl and DCl. Theory Vibration-Rotation Spectra: Photons in the infrared range of the electromagnetic spectrum have the correct energy to excite molecular vibrational modes. Because of the selection rules for absorbing radiation, vibrational excitation is usually accompanied by rotational excitation. For energy levels near the bottom of the potential energy well of a diatomic molecule, the energies of the rotational-vibrational levels are given, in units of cm-1, by:

22,

2

, )1()1(2

1

2

1

JJDJJBvXvE veveeeJv (1)

where the first and second terms account for the vibrational energy, and the third and fourth terms account for the rotational energy. The fundamental vibrational frequency of the molecule is νe. The first anharmonic correction to the vibrational frequency is νeXe. Bν is the rotational constant for a given vibrational level, and De is the centrifugal distortion constant. The integers ν and J are the vibrational and rotational quantum numbers respectively. They label the energy level of the molecule. Bν may be obtained from the equilibrium geometry of the molecule using the relationships:

2

1vBB ev (2)

and

)(;;

8)( 2

21

ba

baee

ee mm

mmrI

cI

hcmB

(3)

Here Be is the equilibrium rotation constant, α is the anharmonicity correction factor to the rotational energy, Ie is the equilibrium moment of inertia, μ the reduced mass and ma and mb are the masses of the two atoms that comprise the molecule. The vibrational frequency νe is related to the bond force constant ke by the expression:

2/1

2

1

e

e

k

cv (4)

2-2

Figure 1: Anatomy of a vibration-rotation band showing rotational energy levels in their respective upper and lower vibrational energy levels, along with some allowed transitions. The spectral lines corresponding to these transitions are shown in the spectrum. Note the splitting that arises from the H35Cl and H37Cl isotopic shifts.

2-3 The force constant ke is given by the second derivative of the bond’s potential energy curve with respect to its displacement, evaluated at the equilibrium internuclear distance re. The derivation of these equations is discussed in most elementary physical chemistry texts. Absorption of a photon can only occur when the photon energy is equal to the energy difference between two energy levels of a molecule. A further restriction on the absorption is that selection rules must be followed in the transition. For a diatomic molecule which is modeled as a harmonic oscillator (with a dipole moment), the selection rules are Δν = +1 and ΔJ = ±1. For an anharmonic molecule (that is all real molecules), the selection rule for Δν is not strictly followed, i.e., Δν = +1, +2, …. One of the things you will explore in this lab is how strongly this selection rule (Δν = +1) holds for DCl. Figure 1 illustrates the energy levels for the two lowest vibrational states of the molecule and shows some of the transitions that are allowed between the sublevels. Also shown is a hypothetical IR spectrum. What you should notice is that the spectrum is separated into two branches, with a gap between them. The gap is where the infrared transitions would be if no change in the J value occurred, i.e, ΔJ = 0. This region is referred to as the Q branch and only involves a change ion the vibrational quantum number. The low frequency branch consists of ΔJ = -1 transitions and is called the P branch. The high frequency branch consists of ΔJ = +1 transitions and is called the R branch. Note that the quantum numbers for the lower state in the transition are traditionally labeled as ν” and J” while those for the upper state are labeled ν’ and J’. You will notice that as you count away from the center of the spectrum the intensity of individual lines increases, goes through a maximum and then falls off in the wings. This pattern arises from a combination of two effects, the population of molecules in a quantum state and the number of quantum states at a particular energy. Within a band, the intensities are proportional to the population of molecules in the ground vibrational-rotational level. The population in state J is given approximately by the equation:

kT

JJBJNJN

)1(exp)12()( 0

0 (5)

where k is Boltzmann’s constant and must have the proper units so that the argument of the exponential factor is unitless (you must convert kT to wavenumbers) and N0 is the population in the state ν = 0, J = 0. The (2J + 1) factor is the degeneracy of the rotational energy level and arises from the fact that 2J + 1 values of the m (orientational) quantum number are possible for each value of J. The exponential factor is called a Boltzmann factor and gives the temperature dependence of the distribution. (Here we have neglected the possibility that a few molecules will be vibrationally excited.) As in all quantitative work, using the correct units is vital to avoid extracting nonsense from the spectra. In molecular spectroscopy, it is usually safest to convert everything to cgs units. To make sure you know where to begin, you should decide on consistent units for νe, Be, Ie, re, De, α and ke. The following constants and values may be helpful.

h = 6.626 × 10-27 erg s = 6.6261 × 10-34 Js k = 1.381 × 10-16 erg/K = 1.38066 × 10-23 J/K 1 cm-1 = 1.986 × 10-16 erg = 1.98630 × 10-34 J mH = 1.007825 amu = 1.672623 × 10-27 kg mD = 2.0140 amu = 3.3425 × 10-27 kg 35Cl = 34.968852 amu = 5.803558 × 10-26 kg 37Cl = 36.965903 amu = 6.135000 × 10-26 kg

2-4 The first step in determining the structure of a molecule is to obtain high quality spectra. The following section of this handout will describe how you obtain the spectra of the fundamental transitions for H35Cl, H37Cl, D35Cl, and D37Cl, and the first overtones of the deuterated species. Laboratory Procedure

The laboratory procedure has two main parts. In the first part you prepare the sample. In the second part you record the IR spectrum. Vacuum line and sample preparation Both HCl and DCl can be prepared through a chemical reaction under vacuum. Although you’ll be using D2SO4 to form DCl, sufficient amount hydrogen will be present to allow some HCl to form.

D2SO4(l) + 2KCl(s) → 2DCl(g) + K2SO4(s)

Before trying any of the procedures described here, you should read this entire section while studying the diagram of the vacuum line in Figure 6. When in the laboratory, you should visually examine the vacuum line and you should make sure that you understand the purpose of each valve and the consequences of opening or closing each valve. A glass vacuum line is a very fragile piece of equipment, which is handmade by a skilled glassblower. A sudden, or forced movement can result in the fracture of a joint or tube, and the subsequent release of a large amount of gaseous HCl. Therefore, when working on the vacuum line, every action must be well planned, and every movement slow and deliberate. HCL IS AN EXTREMELY TOXIC GAS, AND CAN CAUSE SEVERE RESPIRATORY DAMAGE. YOU MUST BE WEARING LATEX GLOVES AND EYE PROTECTION BEFORE PROCEEDING.

Figure 6: Vacuum line system.

2-5 Grease the cold trap joint (You only need about the amount equal to the size of a pea for this large joint.) and install the cold trap by twisting it onto the joint. ALWAYS USE TWO HANDS WHEN APPLYING A FORCE ON THE GLASS: ONE FOR TWISTING, THE OTHER TO BRACE THE JOINT. Place the dewar around the trap and adjust the support to hold the dewar.

Check to see that all three manifold valves are closed and that the valve to the trap is open. The vacuum release valve, just above the rubber tube leading to the vacuum pump, should also be closed.

Turn on the vacuum pump. Any time the pump is pumping a large volume of gas, it will

make a loud gurgling sound because it is working hard. When the volume is evacuated, the pump will operate quietly. Therefore, any time you open a valve to evacuate a section of the line and the pump works hard for more than a minute, you have left a valve open or you have a leak. (In this latter case, you may also hear air hissing into the line.) If this occurs, you should immediately close the valve you opened and find the leak.

Once the pump is operating quietly, open the vacuum pump cut off valve. Again, the pump

should make a loud gurgle initially, but run quietly after a few seconds. Also note that the vacuum gauge should indicate a very low pressure (~10 millitorr). If the vacuum gauge reading is high, you’ll need to check for leaks.

Attach the IR gas cell to the first valve on the manifold. Make sure that you do not touch the

cell’s windows. Open the manifold valve first and evacuate the tube leading to the cell. If the vacuum is holding well, install a spring clamp on the ball and socket joint so that the cell does not fall off of the manifold line. Next open cell valve and evacuate the cell. Again, make sure there is a good vacuum. Note: the cell valve is special in that the body of the valve is hollow and needs to be evacuated in order for it to work well. Look closely at the valve barrel and you will see a small pin hole used to evacuate the valve body (Figure 7). Line this hole up to the vacuum line first and evacuate the valve body. After it is evacuated, you can turn the valve back on to the cell.

Figure 7: Valve barrel for IR cells. Put on the face shield over your goggles and fill the dewar with liquid nitrogen.

Pin hole to evacuate the valve body

2-6 NOTE THAT THE LIQUID N2 WILL BOIL WHEN IT HITS THE WARM DEWAR AND TRAP, SO IT MUST BE FILLED SLOWLY OVER A FEW MINUTES. LIQUID NITROGEN WILL INSTANTLY FREEZE YOUR SKIN IF YOU GET A SIGNIFICANT AMOUNT ONTO YOUR HAND (A SIGNIFICANT AMOUNT IS ANY VOLUME GREATER THAN A FEW DROPS). THIS FREEZING MAY CAUSE A VERY PAINFUL “BURN”, SO BE CAREFUL. Weigh out 3g of KCl and place it, along with the magnetic stirring bar, in the 2-neck reaction

flask. Be sure that you do not get any KCl on the flask joints. If you contaminate the joints, you should remove the KCl before proceeding.

Insert a small magnetic stir bar. Install the adapter onto the middle neck. Use enough grease to seal the joint (much less than

the size of a pea) and attach the retainer clamp.

Attach the reaction flask to the third manifold valve using the adapter. To make sure the joint is lubricated, you should rotate it. Attach a spring clamp to the ball and socket joint. Rotate the flask so that you can install the dropping funnel.

A glass dropping funnel will be used to add the D2SO4(l). Lubricate the bottom joint and

install it on the reaction flask using grease and the retainer clamp. Close the valve on the dropping funnel and then open up the manifold valve and evacuate the

entire reaction flask. Make sure it is holding a good seal before continuing. Slowly raise the plate supporting the magnetic stirrer until the stirrer is approximately 2 mm

underneath the reaction flask. When in position, hand-tighten the locking nut.

Turn on the stirrer and make sure the bar in the flask rotates then turn the stirrer off. First make sure the stop cock at the bottom of the dropping funnel is closed (perpendicular to

the vial). Remove a vial of D2SO4 from the desiccator. Using a desposable pipette with great care, withdraw approximately 5 ml of D2SO4 and put it into the dropping funnel. The D2SO4 is very toxic and difficult to work with, so make sure you have gloves on and you handle it carefully.

D2SO4 IS CONCENTRATED SULFURIC ACID, AND IS BOTH DANGEROUS AND VERY VISCOUS. Put the stopper into the top of the funnel.

Place the vial of D2SO4 back into the desiccator.

You are now ready to start the reaction and let the gas product fill the vacuum line and the IR

cell. Before continuing make sure the vacuum line is holding a good vacuum by shutting the

2-7

vacuum pump cut off valve. The line pressure should not increase above 50 millitorr. While you will see some increase in the pressure reading on the meter, it should be a slow increase. If you think it is increasing too quickly, check for leaks and consult your TA.

Verify that the valves leading to the reaction flask and the IR cell are open and that the vacuum pump cut off valve is closed.

Turn on the magnetic stirrer and then slowly open the valve at the bottom of the dropping funnel and allow a couple of drops of D2SO4 to flow into the reaction flask. The reaction should be immediate and vigorous and the pressure in the line should rise.

Add the D2SO4 drop by drop. Do not allow all of the D2SO4 to exit the dropping funnel

because it will introduce air into the vacuum system and convert your DCl to HCl. Stop the reaction after no more than 3 minutes. A good spectrum can be obtained with relatively low pressures, but you should shoot for at least 100 torr of gas collection.

When you have created enough gaseous product, close the valve on the IR cell. Do not add any more D2SO4. Make sure the dropping funnel valve is closed.

Close the valve to the reaction flask, then open the vacuum pump cut off valve so that all the gas in the line is cleared out and condensed in the liquid nitrogen trap.

While the line is clearing you need to remove any excess D2SO4 in the dropping funnel by drawing it out the top using a disposable pipette. Pipette it into a beaker and then transport to the sink. Flush with copious amounts of water.

Fill the dropping funnel with about 10 ml. This water will be dropped into the reaction flask to quench the reaction.

Add the water to the reaction flask. Just like before, make sure you do not add all of it, which would break the vacuum seal.

SLOWLY open the valve to the reaction flask and let the reaction flask clear out any

volatiles. Close the manifold valve leading to the reaction vessel, remove it and clean thoroughly with

plenty of water and rinse with acetone. Be sure to wipe all vacuum grease off of the glassware and joints.

Close the manifold valve leading to the IR cell and remove the IR cell. It is now ready for analysis.

Keep the vacuum pump running. You’ll need it to evacuate your IR cell.

2-8 Recording Spectra You will be taking the IR spectrum using the Nicolet FTIR-200 spectrometer. The fundamental details on how a Fourier transform IR instrument works can be found in Appendix 1. Nicolet FTIR-200 Spectrometer Operation Be sure that the computer is booted up and that the spectrometer is turned on. Double click on the EZ OMNIC icon on the desktop. Look in the upper right to be sure that there is a check mark in the bench status to indicate that the program is communicating with the spectrometer. (If a check is not there, notify your TA.) If the check is there, click on Collect to open the menu and then click on experiment setup. On this window make sure that you have the Collect tab selected. You now need to set the following: Resolution: 1 cm-1 No. of Scans: 64 Final format: Absorbance Correction: None Background Handling: Choose “collect background before every sample”

Now click on the bench tab and a window showing the IR parameters will appear. Change to Max range to 4500 and the Min range to 1500. Click the OK button. This will return you to the spectrum window.

Remove the lid from the spectrometer and click on the Col Sample icon. Type your filename as the spectrum title and click the OK button. A confirmation window will pop up. Click the OK button and wait until the background data collection is completed.

When the background collection is finished, a window will pop up to confirm that you are ready to collect the spectrum on your sample. Insert the sample cell containing your HCl/DCl sample into the spectrometer and then click the OK button on the conformation pop up window. You can observe the data collection progress by looking at the boxes and the bottom left of the screen. When the data collection for the sample is complete, click on Yes to Add to Window 1. You should see a nicely resolved spectrum just below 3000 cm-1 for the HCl species and again a little bit higher than 2000 cm-1 for the DCl species. The overtone spectrum for DCl shows up around 4000 cm-1. Use the mouse to make a box around the spectrum and then click the left mouse button to zoom in. Click on the Analyze menu and select find peaks. Peaks above the movable black line will be assigned and you can also assign more peaks by clicking the mouse to lower the black line. When you have finished with the assignment, click the replace button. Note: You may rescale if needed by using the green arrows at the bottom of the screen. Print this window. Repeat this procedure for the other IR band. When you have completed the procedure for all the bands, close the program.

2-9 Clean-up

Reconnect the IR cell to the vacuum line and evacuate the cell. Wait for two minutes then close the valves on the IR cell and the manifold. Remove the IR cell from the line and place it in the desiccator. Remove the dewar of liquid nitrogen from around the trap. Pour about half of the remain

liquid nitrogen out of the dewar. Open the vacuum release valve (just above the rubber tube leading to the vacuum pump) and

turn off the vacuum pump. You now need to remove the trap before it warms too much, put in the dewar and transport it

to the hood in the prep room. Remember that the trap contains condensed HCl and DCl, so it needs to stay in the liquid nitrogen. Put the dewar and trap in the back corner where it will not be disturbed and pull the hood sash down. Over time, the liquid nitrogen will boil off and the HCL/DCL will vent out through the hood.

Make sure you put everything away clean and organized. It is your responsibility to make sure everything is clean and ready to go for the next group, REGARDLESS OF HOW IT WAS LEFT FOR YOU. Significant lab performance points will be lost if you fail to keep the lab clean, organized and safe.

Data Analysis

Assigning the Spectrum The first step in obtaining spectroscopic constants is to assign your spectrum. Several

methods may be used but the most obvious way for a diatomic molecule is to count outward from the gap between the R and P branches. Remember that the P(0) line is the J” = 0 → J’ = 0 forbidden transition, so you will not see a peak for it. Next you need to tabulate the transition frequencies of each isotope for later manipulation (H35Cl, H37Cl, D35Cl, and D37Cl). Don’t forget the DCl overtone spectrum as well. The accuracy of your results depends on how carefully you record the spectra and read off the center of the line positions, so this step should be performed with great care. Finding Rotational Constants, Bν, using the irspec program. A description of the underlying reasoning that is used to obtain the rotational constants for a particular vibrational level is outlined here. First, consider the ν = 1 level. Note that if you subtract the frequency of the P(J”) line from the that of the R(J”) line, the difference is given by

R(J) - P(J) = B1[(J + 2) (J + 1) – J (J – 1)]-De1 [(J + 2)2 (J + 1)2 – J2 (J – 1)2] (14) Where (J”) is written as (J) for simplicity. This equation may be written as

R(J) – P(J) = 2B1 (2J + 1) – 4 De1 (2J + 1) (J2 + J + 1) (15) and then can be rearranged into a slope-intercept form of the equation

)1(4212

)()( 211

JJDBJ

JPJRe (16)

2-10 So that a plot of [R(J) – P(J)]/(2J + 1) versus (J2 + J + 1) can be use to obtain B1 and De1. To obtain the rotational constants for the ν = 0 level, perform the same type of manipulation for R(J-1) – P(J + 1), whose difference is: R(J – 1) - P(J + 1) = B0(J + 1) (J + 2) - B0J(J – 1)-De0[(J + 1)2 (J + 2)2 - J2(J – 1)] (17)

which simplifies to

R(J – 1) - P(J + 1) = 2B0(2J + 1) - 4De0 (2J + 1) (J2 + J + 1) (18)

These equations apply to both the HCl and DCl species. Obtaining the Rotational Constants using Excel 1. Assign the P and R transitions for a particular vibrational transition. Enter your assignments

of J and the transition energies into an Excel data sheet. We recommend a new data sheet for each vibrational transition.

2. Now you will find the rotational constant B1 and De1 by curve fitting a plot of equation 16. 3. To obtain the values of B0 and De0 again) you will perform a similar rearrangement as in step

2, but you should perform it with the slope intercept form of equation 18. Once again you recast these equations in linear form, transform your raw data appropriately, perform a linear plot and analysis, and obtain the parameters from the slope and intercept.

Finding Molecular Parameters Using the Bν values and equation (2), you can obtain Be for each isotope. Subsequently, you can calculate the equilibrium bond length and the average bond length for each isotope and vibrational level. If the assumptions we have made are all correct, re and α should be independent of isotope. Is this true within the precision of the data? Sample Calculations Given that B0 = Be - α (0 + 1/2) and B1 = Be - α (1 + 1/2), the difference is B0 – B1 = α. Once you evaluate α, substitute it back into the equation and determine Be. Use equations 3 to solve for the equilibrium internuclear distance (re). The Laboratory Report Include the data that you obtained experimentally and the spectra with the line positions labeled. Include a table of B0, B1, De0, De1, Be α, and re for each isotope; H35Cl, H37Cl, D35Cl and D37Cl. Be sure to also include a sample calculation for each type of calculation performed. Compare your values with those given in references 3-5. You should include percent error calculations assuming the literature value to be correct. Items to include in your discussion 1. Do the values of the spectroscopic constants Be, Bν and De,ν vary with isotope and/or with

vibrational level in a way that you would expect? 2. Does re change for each isotope? Explain your answers. 3. In general, what accounts for the uneven spacing on the lines in the P and R branches of a

vibration-rotation spectrum?

2-11 4. The constant B2 for DCl can be obtained by following a procedure analogous to that used to

find B1 but using the overtone spectrum (= 0 2). Determine B2 and De2 for each DCl isotope.

5. Use the Boltzmann distribution equation (5) to model what the intensities of the absorption peaks should be for your experimental data. Does you data match this theory? How would the distribution change at higher temperatures?

6. Manipulate (using equation 1) the R(0) and P(1) line frequencies for both the fundamental and overtone spectra of DCl to obtain νe and νeXe, where νe is the frequency at which the Q branch would appear if it were an allowed transition. You will need to use equation (1) for four different line frequencies:

EA = ΔE(ν″ = 0, J″ = 0 → ν′ = 1, J′ = 1) EB = ΔE(ν″ = 0, J″ = 0 → ν′ = 2, J′ = 1) EC = ΔE(ν″ = 0, J″ = 1 → ν′ = 1, J′ = 0) ED = ΔE(ν″ = 0, J″ = 1 → ν′ = 2, J′ = 0).

7. What additional spectroscopic information could be obtained from your data? Be as detailed as possible.

References

1. Atkins, P. W. and R. S. Friedman. Molecular Quantum Mechanics. 3rd Ed. New York: Oxford, 1997. Reserve QD 462.A84 1997.

2. Harris, D. C. and M. D. Bertolucci. Symmetry and Spectroscopy. New York: Dover, 1978. Reserve QD 96.V53 H37.

3. De Lucia, F.C, Helminger, P., Gordy, W. Submillimeter-Wave Spectra and Equilibrium Structures of the Hydrogen Halides, Phys. Rev. A, 1971, 3, 1849–1857.

4. Rank, D.H., Eastman, D.P., Rao, S., and Wiggins, T.A., Rotational and Vibrational Constants of the HC35 and DCl35 Molecules, J. of the Optical Society of America, 1962, 52(1), 1-7.

5. http://physics.nist.gov/PhysRefData/MolSpec/Diatomic/Html/Tables/HCl.html

6. http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/vibrot.html

7. Pauling, L. and E. B. Wilson. Introduction to Quantum Mechanics. New York: Dover 1935. Reserve QC 174.12.P39 1985.

8. Shoemaker, D.P., Garland, C.W., Nibler, J.W. Experiments in Physical Chemistry, 6th Ed. McGraw Hill, NY, 1996.

9. Sime, R. J. Physical Chemistry: Methods, Techniques and Experiments. New York: Saunders, 1990. Reserve QD 453.2.S524 1990.

10. Steinfeld, J. Molecules and Radiation. Boston: MIT Press, 1985. Reserve QC 454.M6 S83 1985.

11. Willard, H. H., et al. Instrumental Methods of Analysis. 7th Ed. New York: Van Nordstrand, 1988. Reserve QD 79.I5 I52 1988.

*This lab is adapted from Shoemaker et. al., 1996.

2-12 Appendix A: The Fourier Transform Intrared (FTIR) Method This discussion follows that given by J. Moore, et al. in “Building Scientific Apparatus”. Collimated illumination (obtained from a laser, or parallel light obtained by placing a point source at the focal point of a converging lens) is divided into two equal parts at a beamsplitter and sent to two plane mirrors M1 and M2 (Fig. 1). If observations of a broad-band (temporally incoherent0 source are being made, a compensating plate of the same material and thickness as the beamsplitter is included in Arm 2 of the interferometer. This plate compensates for the two additional transits of the beamsplitter substrate made by the beam in Arm 1. Figure 2: Schematic of an interferometer.

Maxima in the detected output occur when 2l = mλ, where 2l is the path difference in the two arms, m is an integer, and λ is the wavelength of light. Assuming monochromatic radiation of angular frequency ω(= 2π λ/c), the fields of the lighwaves in Arms 1 and 2 can be represented as

2(02

1(01 , tti eEEeEE (6)

The output intensity is proportional to the square modules of the electric field

2

21 EEI (7)

2-13 Hence, we find

1220

20

ii20

ti0

i0

ti0

ti0

2

21

cosE2E2

1ee1E

eEeEeEeEEE2121

2121

(8)

or

cos10

2

21 IEE (9)

where Δφ = 2πm and I0 = 20E . Since m = 2l/λ = 2l(ν/c),

c

lvII

4cos10 (10)

Because we are performing spectroscopy, we wish to consider the full frequency dependence of the signal. If the light source has an intensity I(ν) at the frequency ν, then the detected intensity in the interval dν is given by

dvc

lvvIlIv

4cos1)( (11)

where l is the pathlength change. If the mirror is moving at a constant velocity ν, then l = νt where t is the time. The integrated intensity is

dvc

v4cos1vI

2

I

dvc

v4cos1vI)(I

0

0

0v

l

ll

(12)

The second term is the cosine transform of I(ν). Hence, the frequency domain spectrum of the detected light can be determined by taking the inverse transform, namely

ldc

vlIlI

cvI

l

4cos

2)(

2 0

0

(13)

2-14 Figure 3: Schematic of a Fourier transform instrument. For example if a HeNe laser is used as a references source in a FT instrument (figure 3), a signal I(x) as a function of the displacement of the mirror (x), an interference pattern is observed at the detector (Figure 4). If the mirror is moved at a constant rate ν = dx/dt, the detector signal oscillates at the frequency y = 2ν ν , since the path difference changes at twice the rate of mirror movement. Thus, for ν = 1000 cm-1 and ν = 1 mm/s, the resulting oscillation occurs at 200 Hz. A typical ‘real’ interferogram and spectrum are shown in Figure 5.

Figure 4: A plot of the predicted interference pattern.

Figure 5: A plot of an interferogram and its corresponding spectrum.