Using Fourier Transforms to Understand Spectral Line Shapes Ernest Grunwald, Jonathan Herzog and Colin Steel Brandeis University, Waltham, MA 02254 Chemical kinetics a n d molecular spectroscopy a r e taught as separate subjects, and dynamic spectroscopy, in which the rates of chemical or physical processes are deduced from the widths and shapes of spectral lines, is often overlooked. I n such oroblems data in the time domain often stem from wave motions associated with molecules t h a t are members of a statistical ensemble ( I ) . The frequency of these waves can he changed and the phases interrupted by a variety of phenomena including state switching (chemical exchange) and collisions (1,2). The resultant effects for the ensemble in the frequency domain are changes in spectral line shape andlor position. Arepresentative example from NMR spectrosopy is shown in Figure 1(3).Notice how, as the temperature is raised from 27 to 85 "C, the original two sharp lines hroaden, coalesce, and finally become a sharp single line. Agood way to teach dynamic spectroscopy is, first, to formulate the problem and generate data in the time domain of chemical kinetics, and then, by Fourier transformation, to convert the data into the frequency domain of molecular spectroscopy. In a reacting system the lifetimes of the individual molecules are random or stochastic, and the time data representing the ensemhle of molecules must represent this central fact and are consequently quite extensive. However. this oaoer will show that hv ~ r o m a m m i n athe Fast Fourier 6ansform (FFT) algorithm (4y5)for use ( 6 ) with standard oersonal comouters (PC's). one can do calculations that elicidate dynamic spectrosc~pyin well under a n hour. Our discussion will begin with how we generate and then FFT data that incorporate randomness. Such a combmed pmcedllre we can 1;111t:I con\.cnic!ntly as "stochast~F c b T . \Vr shall th6.n ,rpply the mcthnd to simulate fwq u r n q switching in S M R , and shall lind tnar thr trxn.5form ~t,). The data in each such record are then Fourier transformed and the FT's averaged to produce the desired spectrum.
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Time (s) Figure 2. Wave trains when there is switching between two states. - -,VA = 40 s-'; -, VB = 100 s?. In (a)there is no phase-change on switchino. while in ibl , . the .ohase chanoe is random in the ranae 0 to 271radians; in both cases r = 79.2 ms.iourier transforms relevant to these situations are shown in Figures 4 and 5.
Frequency Switching in a Wavetrain For our first example we shall consider a wave train in which the frequency switches stochastically between two . this could be a resofrequencies, v~ and v ~ Physically, nance frequency in a n NMR experiment a s the molecule switches back and forth between two states A and B with rate constants k~ and kg;
The mean lifetimes in states Aand B are l l k and ~ likg, but for simplicity we shall assume that k~ = kg and define a generic rate constant k (= kA= kg), equivalently called the mean switching frequency. Notice that since the populations in A and B do not change with time, concentration does not enter the problem; this corresponds to the amplitude of the wave trains remaining constant. Indeed, the ability to obtain rate parameters for systems in which concentrations are not changing is one of the unique features of dynamic spectroscopy compared to more common kinetic techniques. Sample segments of the typically long wave trains, representing the time data, are shown in Figures 2 and 3. First, note the marked variations of the waiting times between frequency switches in each train. These are generated from the given T (= lik) according to eq 4. Second, notice the mode of frequency shifting. In the (a) part of the figures, a frequency switch involves no phase jump in the wave: the switch is phase-coherent. By contrast, in the (b) part the switches are accompanied by random changes in
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Time ( s ) Figure 3. Wave trains when there is switching between two states. Open squares, vn = I00 s-'; filled circles, vs = 40 s-'. In (a)there is no phase change on switching, while in (b)the phase change is random in the range 0 to 277 radians. In both cases r = 0.9 ms: notice that this is much shorter than in Figure 2. FT's relevant to these situations are shown in Figures 4 and 5. Volume 72 Number 3 March 1995
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eq 4. The results agree with our qualitative expectations: A resolved two-peak spectrum when F = 10.6 and an averaged single-peak spectrum when F = 0.122, with a coalescence point a t F = 1.0. (That coalescence occurs a t F = 1.0 is clear from simulations in whichF= 1.1or 1.2.) Reference to Fieure 1will show that this is iust what havoens . in an ~ ~ K s p e c t r uasmthe rate of chemical exchange increases. Indeed, a separate in-depth examination by the authors has shown that this correspondence is exact (8). Before considering wave trains i n which the frequency switches occur with random phase change, let us consider some im~licationsof the NMR corresvondence. First, i t is clear t h a t the exchange averaging due to chemical exchange in NMR spectra is not a property peculiar to magnetic resonance hut is a g e m r a l wave phenomenon associated with coherent hack-and-forth switching of the wave frequency between two values. Second, because the frequency switching takes place in the time-domain, the descriotion of i t s s~ectroscoviceffect reauires a Fourier transformation, and dynamic spectroscopy; therefore, conforms to mathematical theorems governing Fourier transformations (9,10). For instance, our discussion has implied that the s h a ~ of e the svectrum deveuds on the varameter F a s defined in eq 5 . Accordingly, the spectralLshape depends only on the difference between vAandve, not on their actual values. For a given switching frequency (kj, values of VA = 40 HZ and V B = 100 Hz give exactly the same spectral shape as, say, values of VA = (100 MHz + 40 Hz) and vB = (100 MHz + 100 Hz). Mathematically, this is a consequence of the modulation theorem for Fourier transforms. A uniform change in all frequencies shifts the center (by 100 MHz in the above example) hut does not alter the shape of the spectrum (9). Returning to our FFTb: Figure 5 shows the power spect r a that result when the freauencv switches occur with random phase change. These results can be compared with those in Fimre 4 because thev use the same Darameters fork, VA , and VB. When F = 10.6, the two-peaked spectrum i n F i m r e 5 is verv similar to that in Fieure 4. but as F decrenws, clcar diffiwnct:~:ippr:ir. Thus. there 1s no coulesrenre when F = 1: and whtm (.'= 0.122 thr sinnal I S weak and very hroad. Anticipating the following discussion of collision broadening, when F = 0.122 the resonance consists of two hroad, overlapping bands rather than a single coalesced hand. As a matter of fact, frequency switching due to chemical exchange ( A 2 B) approximates zero phase change (as in Figs. 2a, 3a, and 4) if the transit state (TS) on the dynamical path from A to B is sufficiently short-lived. This condition is met usually in NMR. Exceptions occur when the transit state (TS) is paramagnetic.
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Frequency (c/s) Figure 4. Averaged Fourier transforms of wave trains similar to those shown in Figures 2a and 3a where there is zero phase change on frequencyswitching.va and vs are 40 and I00 s ' . Each time record consisted of 2048 data Doints aathered at a samolina time of 0.45 ms: 2000 such records were averaged in producing'the"transform.
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the phase of the wave. This difference is especially dramatic i n Figure 3, where k is large compared to I VA- VB I . Figure 3a looks like a continuous wave with some frequency noise, while Figure 3h looks pretty chaotic. When h~ = he, i t is useful to classify data in terms of a dimensionless parameter F (2a),defined in eq 5. F=
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.12h
(5)
The key variable here is the ratio of I VA - VB I to the switching frequency k. W h e n F i s substantially > 1(Fig. 21, one can clearly see two frequencies both when the phase change is zero and when it is random. Accordingly, we expect in both cases to find a two-peak spectrum after Fourier transformation. On the other hand, when F < 1 (Fig. 31, the result obtained on Fourier transformation will depend very much on the phase change. When the phase change i s zero and the time domain shows a single averaged wave with frequency noise (Fig. 3a), we expect to find a noise-broadened single-peak spectrum a t a n average frequency. Figure 4 shows the FFT power spectra for zero phase change, based on wave trains with lo4-105 stochastic frequency-switches conforming to 212
Journal of Chemical Education
Collision Broadening of Spectral Lines When we developed our FFT program, we tested i t by simulating known features (I)of the collision broadening of spectral lines. In this section we shall review the relevant results. First let us consider a classical particle oscillating a t a frequency and as a result emitting a sinusoidal wave a t the same frequency, which our apparatus then detects. Also, for simplicity, assume that the lifetime of the oscillator is sufficiently long so that we do not have to worry about the damping of the amplitude of the wave during an ohservation or recording time tR. The recorded wave train then is a long, uninterrupted pure sinusoid, a s shown in Figure 6a. When we carry out the FT of such a signal, we get a sharp spectral line-practically a delta function ( 9 k a t frequency v (Fig. 6h.j. But now suppose the emitting train is interrupted by "collisions", events that discontinuously change the phase
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Figure 6. A long uninterrupted wave train (a)and its Fourier transform (b). Frequency = 45 5.'. In (c) the wave train is interrupted by "collisions", withrcoll= 0.155 S, which are shown by the change from - - - to --, and vice versa. The phase jump at a collision is random in the range 0 to 2a radians. The collision-broadened Fourier transform is shown as dots in (d).Full line is the theoretical Lorentzian envelope.
of the wave. A "hard" collision will change the phase randomly in the range 0 to 2x radians, while a "weak" collision changes i t with a stochastic distribution about a mean value of
