Determination of bandwidth and frequency changes by Raman differencespectroscopyJaan Laane

Citation: The Journal of Chemical Physics 75, 2539 (1981); doi: 10.1063/1.442431 View online: http://dx.doi.org/10.1063/1.442431 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/75/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Detecting Changes in the Cervix with Raman Spectroscopy AIP Conf. Proc. 1267, 441 (2010); 10.1063/1.3482605 Determination of frequency shifts by Raman difference spectroscopy J. Chem. Phys. 72, 5305 (1980); 10.1063/1.439021 Vocaltract determinants of resonance frequencies and bandwidths J. Acoust. Soc. Am. 59, S70 (1976); 10.1121/1.2002852 Simple setup for Raman difference spectroscopy Rev. Sci. Instrum. 46, 1664 (1975); 10.1063/1.1134136 Binaural detection at high frequencies with differing masker bandwidths J. Acoust. Soc. Am. 57, S36 (1975); 10.1121/1.1995197

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Determination of bandwidth and frequency changes by Raman difference spectroscopy

Jaan Laane

Department o/Chemistry. Texas A&M University. College Station. Texas 77843 (Received 4 May 1981; accepted 12 May 1981)

The mathematical expressions necessary for analyzing the Raman difference spectra resulting from bandwidth and/or frequency differences between two samples have been derived for both Lorentzian and Gaussian band shapes. Intermediate bands may be analyzed using interpolation. The results show that the difference spectra have two maxima separated by a minimum near zero intensity whenever bandwidth changes are dominant. If the frequency shift is zero, these two maxima have the same intensity. Nonzero frequency shifts result in two u~equal maxima and their intensity difference may be used to calculate the magnitude of the shift. Three examples of experimental spectra have been analyzed in order to demonstrate the applicability of the technique.

I. INTRODUCTION

The same (or similar) molecules in different chemical environments generally show slightly different vibration­al frequencies. For example, solvent effects are known to shift vibrational frequencies. Recently we have de­tailed how Raman difference spectroscopy (RDS) can be used to determine small frequency differences between corresponding Raman lines for two different samples. 1

We have also described the results for several experi­ments demonstrating some of the applications of this technique. 2

-s In each of these studies the frequency

shift ~ between the two samples was determined from

~ =Br(d/lo) , (1)

where r is the observed bandwidth, 10 is the peak height of the Raman line, and where d is the peak-to-peak height of the derivative-like curve in the difference spectrum. The value of the constant B is a function of both the band shape and the ratio of ~/r. However, most frequency shifts of interest are such that ~/r < 0.2 and under these conditions B = 0.385 for Lorentzian bands and B = O. 350 for Gaussian bands.

For the previous studies it was assumed that the band­width for a particular Raman line was the same for both samples, i. e., r = r A = r B • When small bandwidth dif­ferences occurred, the approximation

(2)

could be satisfactorily applied. It is clear, however, that it is not unusual for a molecule in two different en­vironments to have significant bandwidth differences as well as frequency shifts. It is useful, therefore, to ex­amine the characteristics of Raman difference spec­troscopy ariSing from bandwidth changes. This will be done for situations where no frequency shifts are present and also for those where both bandwidth and frequency changes are evident. The results will not only provide a method for measuring frequency shifts under compli­cated circumstances but will also provide a sensitive method for accurately determining band broadening or narrowing. Several examples of applications using the theoretical expressions will be presented.

II. THEORY

We will examine the nature of the Raman difference spectra resulting from bandwidth changes for Lorentzian and Gaussian Raman lines. The RDS spectra resulting when no frequency shifts are present will first be con­sidered. Then the spectra expected when both band­width and frequency changes are present will be analyzed.

A. Lorentzian bands

The general expression, as previously defined,l for the intensity of the difference spectrum between two Lorentzian Raman lines A and B, which have been nor­malized to have the same peak intensity 10=lt=/L is given by

IL( ) - IOYf loY;' (3) D W - (w-x) +Y! - (w+x) +Y! '

where Y =~ r is one-half the bandwidth, x =~ ~ is one­half of the frequency shift, and w is the frequency with its origin taken as the midpoint between the two Raman bands A and B. That is, the two Raman lines have their maxima at ± x. Equation (3) may be written over a com­mon denominator as follOWS

I$(w) _ (W2+x2)(yi -Y:) + 2x(yi +Y:) 10 - (w2 _X2)2+(W2+X2)(Yi +y~) +yi y~ + 2x(y! -y~)

(4) Using this equation we first examine the case where

the two Raman lines occur at the same frequency (x = 0) but have different bandwidths (y A'" YB)' The Raman band maxima then occur at w = O. Under these conditions, Eq. (4) becomes

I$(w)_ w2(yi-yg) 10 - (w2+yi)(w2+y!)

(5)

Evaluation of alt/aw and setting to zero in order to find the maxima and minima yields the roots

wmax=±(YAYB)1/2=±~(rArB)1/2 , (6)

The intensity of the difference curve at the minimum is zero (since It ==~ at w = 0) and the intensity Devaluated at the maxima is given by

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2540 Jaan Laane: Raman difference spectroscopy

-" c

>-r--C/)

z w r-z

C/)

Q

a:

0.2

0.1

0

0.1

0

-0.1

0.2

0.1

o

- -

T D.

I

_______ J __ _ -30 -20 ·10 o 10 20 30

FIG. 1. Raman difference spectra for Lorentzian Raman bands resulting from bandwidth and frequency changes. (a) Bandwidth change only: r A = 12 cm -I, r B = 8 cm -I; A == o. (b) Frequency shift only: r=rA ==rB =10 cm-I; A==O.5 cm-I• (c) Both band­width and frequency changes: r A == 12 cm -I, r B = 8 cm-I

A=O.5 cm-I .

Therefore,

Q_ rA-rB 10 - r A + r B

(7)

(8)

This is then a useful expression for examining band­width differences between two samples. If we define 6 as the frequency difference between the two maxima in the difference spectrum, we have

6 =(r A r B )1/2"" (r A + r B)/2

and then

(9)

(10)

Figure l(a) presents an example of a calculated Raman difference spectrum which results when a Raman band of one sample is subtracted from a broader Raman band at the same frequency of a second sample.

Previously1 we have derived the expressions relevant to the case where I' =Y A =YB and x*- O. In this situation the Raman difference spectrum re'sults from frequency shifts between the two samples while the bandwidths re­main constant. Equation (4) then becomes

IL(w) 4wxr2

~= (W2 _X2)2+2(W2+X2)y2+y4 • (11)

When the frequency shift is sufficiently small (x« 1'), the minima and maxima occur at W=± 3-1/2 1', where the

intensity of the difference spectrum is given by

!.J2.=±(27)1/2~ • 10 16 Y (12)

The intensity difference d between the maximum and minimum is then

d = 2. 598/0A/r . (13)

Figure l(b) shows a representative difference spectrum resulting from a small frequency difference between two samples.

The most complicated situation for RDS spectra oc­curs when both bandwidth and frequency differences are present between the two Raman lines (x *- 0, I' A*- YB). For most cases of interest we will show that the result­ing difference spectrum will essentially be the sum of the effects from the bandwidth change [Eq. (5)] plus the effects from the frequency shift [Eq. (11)]. As before, we will assume that the frequency shift is small so that 1'» x. We will also assume that the contributions from the bandwidth difference to the RDS spectrum are com­parable or greater than those from the frequency shift. [If this were not the case, the values of r A and r B would be sufficiently similar so that Eq. (2) could be used along with Eq. (13) to determine the frequency shift. ] For the circumstances described above the difference curve will have two maxima in the vicinity of ± (I' A YB )1/2 as given by Eq. (6). Near these maxima then w» x and for the region of the RDS spectrum near the maxima we can approximate Eq. (4) by

It(w) _ w2(yi _1'2) + 4wxr2

10 - (W2+y2)i (W2+y2)2 (14)

where the first term is a bandwidth change contribution and the second term arises from the frequency shift. The value of I' =~ r is given by

Y=~(YA+YB)""(YAYB)1/2. (15)

Evaluation of a/t(w)/aw=O from Eq. (14) shows the maxima to be at

wmax =±Y ,

and the intensities at the maxima D. =/t(+y) and D. =/t(-y) are given by

D 2 2 _z =~±~ 10 41' y'

or from Eq. (15),

Dz=~±~ 10 YA+YB YA+YB

= rA-rB ±~ rA+rB rA+rB

(16)

(17)

(18)

Thus the first term showing the effect of the bandwidth change is identical to Eq. (8). The second term can be obtained from Eq. (11) using w=±y. It differs from Eq. (12) because the maximum for that case occurs at a smaller w value. The minimum for this case with I' A*- YB' x*- 0 will occur not far from 4J = 0 but should not be calculated from Eq. (14) since it had been assumed that w» x. Instead, the assumptions that I' »w and 'Y» x are used in Eq. (4) to give the expression for the

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Jaan Laane: Raman difference spectroscopy 2541

difference curve in the vicinity of the minimum

I$(w) _ (W2+X2)(y! -y;) + 2wx(y! +y;) - 2 2 •

10 YA YB (19)

Differentiating and setting to zero shows the minimum to be at

_ -x(y!+yi) _ -A(r!+ri)",,_ Ar2 wm1n - y! -y! - 2(r! - r!J r! - r~ .

(20)

The value of the difference curve at this point is

1$(wm1n) _ -x(y! +yi)2 ~ _4A2 10 - y! y! (y! - y!) - r! - r!

(21)

Figure l(c) shows a representative Raman difference curve for the case r A * r B and A* O. From Eq. (18) the intensity difference between the two maxima is found to be

D.-D_=~ 10 rA+rB

(22)

and this information can be used to determine the fre­quency shift A. The average intensity of the two maxi-ma is

!2_(D.+DJ_rA-rB 10 - 210 -rA+rB '

(23)

and this result corresponds also to Eq. (8). Equation (23) assumes that D. and D_ are measured relative to the zero of the difference curve. If D. and D_ are mea­sured relative to the minimum of the RDS curve, the small correction given in Eq. (21) must be made.

B. Gaussian bands

The general expression for the difference spectrum between two Gaussian bands [analogous to Eq. (3)] is given by

~(w) =Io{exp[ - C(w -X)2/ r !]- exp[ -C(w+x)2/r;]}, (24)

where C = 4 In 2 = 2.7726. The analysis of the difference curves resulting from Gaussian bands is analogous to that for the Lorentzian bands. We examine first the case where there is no frequency difference between bands A and B so that x = O. Then

~(w)/lo=exp(-Cw2/r~) -exp(-Cw2/r;) . (25)

Evaluation of aPv(w)/aw=o in order to find the maxima

TABLE I. Peak intensities and frequency separations for RDS spectra for various bandwidths.

Bandwidth (cm-I ) Lorentzian bands Gaussian bands

r A r B 15 (cm-I ) D/Io 15 (cm-I ) D/Io

10 5 7.07 0.333 8.16 0.473 10 6 7.74 0.250 9.10 0.360 10 8 8.94 0.111 10.70 0.163 10 9 9.49 0.052 11.40 0.077 10 10 0.000 0.000 10 11 10.49 -0.048 12.59 -0.070 10 12 10.95 -0.091 13.11 - 0.133 10 15 12.25 -0.200 14.51 -0.290 10 20 14.14 - O. 333 16.33 - O. 473

Olio

0.5

0.0 +-----~::.....----------;

0.5

" " "

0.5 1.0 1.5 2.0

FIG. 2. D/Io vs r A/r B curves for Lorentzian (dashed line) and Gaussian (solid line) band shapes.

and minimum gives

wmax

=± 21/2C-1/2 r A r B (r~ - r; tl/2(ln r Air B)1/2. (26)

At the minimum w = 0, Pv(O) = O. The intensity I~( wmax)

at the maxima can be evaluated exactly by substituting the value of wmax from'Eq. (26) into Eq. (25). However, the calculation may be Simplified by expanding In(r A/rB )

in the expression for wmax in a Taylor's series. Then

In(r A/rB) = 2{(rA - rB)/(r A + r B)

+ H(rA - rB)/(rA + r B)]3+···}

(27) and

wmax =± 2C1/2 r A rB/(r A + r B) =± 1. 2011r A rB/(r A + r B) .

(28) Equation (25) then becomes

D/lo=~(wmax)/lo=exp(-r;/r2)-exp(-r~/r2), (29)

where r is the average value of the bandwidths as given by Eq. (2). This result is analogous to Eq. (8) for Lorentzian bands, and it demonstrates that the differ­ence band intensities resulting from bandwidth differ­ences are considerably greater for Gaussian bands than Lorentzian bands. Table I compares the RDS intensities D and peak separations 0 for both Gaussian and Lorentz­ian bands for various values of r A and r B and Fig. 2 shows a graph of D/lo vs rB/rA. This information is useful for determining bandwidth changes from mea­sured D/lo ratios.

As previously shown, 1 when r A = r B and when a fre­quency shift is present (x* 0), then Eq. (24) becomes

When the frequency shift A is sufficiently small, the maximum and minimum occur at w'" =± (2C)- 1/2r = ± O. 425r and the intensity difference d between them is

d = (8C)1/2 e-1/2 Mo/r = 2.856 Mo/r .

This result is analogous to Eq. (13) for Lorentzian bands.

(31)

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2542 Jaan Laane: Raman difference spectroscopy

0.4

0.2

o

-0.2

-10

When both bandwidth and frequency changes are pres­ent (r A * r B and x* 0), Eq. (24) must be used to analyze the RDS spectra. In analogy with the analysis for the Lorentzian spectra we assume that the effect from the bandwidth change dominates and that the band maxima occur near the values given by Eq. (28). For the sake of evaluating the contribution from the frequency shift we use the definition for the average r from Eq. (2), and then Eq. (28) may be approximated by

(32)

and then Eq. (30) gives the contribution from the fre­quency shift as

Ig(wmax) '" 2Ioexp(-1 _cx2/r2) sinh(± 2C 1/2x/r) • (33)

Use of Taylor's series shows that

d '" 2Ig(+ wmax) '" 4e-1 C 1/2 10 ~/r '" 2. 45010 Mr . (34)

It should be noted that the coefficient in the equation is somewhat lower than in Eq. (31) reflecting the fact that the maxima are at higher wmax values than for the case when r A ",rB •

In order to obtain the Gaussian result analogous to Eq. (18) we combine the effects represented by Eqs. (29) and (34). This gives

~- [ -r; J- [ -ri J 2.450~ 10 -exp (rA+rB)2 exp (rA+rB)2 ± rA+rB ' (35)

The intensity difference between the two maxima is

D. -D_ 4. 900~ ---'" (36)

10 r A + r B

and from this equation the frequency shift ~ may be de­termined. The average intensity D "'i (D.+DJ can be determined from Eq. (29) and this value is useful for examining bandwidth changes.

The minimum of the Gaussian difference curve repre­sented by Eq. (24) can be shown (after expansion as a Taylor's series) to have the same value as for the Lorentzian curve and is also given by Eq. (20). The value of the Gaussian difference curve at the minimum is

10

FIG. 3. Calculated RDS spectra: (a) ~=2.0 em-I, r A=12 em-t, r B

= 8 em-I (solid line); (b) sum of con­tributions (c) and (d), (dashed line). (c) ~=2.0 em-I, r=rA =rB =10 em-I; (d) ~=O, r A =12 em-I, r B =8 em-I.

Ig(wm1p) _ _ C~2 _ -2.7726~2 10 - r! - r; - ri - r; (37)

This value, which may be compared to Eq. (21) for Lorentzian difference spectra, will often be near zero since ~ is generally very small.

III. APPLICATIONS

A. Principles

The results of this paper coupled with our previous work l should provide the basis for interpreting the Ra­man difference spectra for any situation where a fre­quency shift and/or a bandwidth change give rise to RDS spectra. Although the results have been derived sepa­rately for Lorentzian and Gaussian bands, the methods are applicable to any intermediate band shape through interpolation. This is especially true since the deter­mination of the frequency shift is only somewhat depen­dent on the band shape.

The following principles are applicable to abstracting frequency shift and bandwidth change information from Raman difference spectroscopy

(a) If the intensity of the RDS spectrum is zero, both the bandwidth and frequency differences are zero, within experimental error.

(b) If the bandwidth change is zero or small, Eq. (2) may be used to calculate r and then Eq. (13) or (31) can be used to find the frequency shift ~ from the experi­mentally determined values of r, d, and 10 , When the bandwidth change is moderate and a maximum and a mini­mum of different magnitudes are produced, the same procedure with somewhat poorer accuracy may be used. The fraction Lorentzian character L can be estimated by measuring ~Vl/Jr (1.732 for Lorentzian bands and 1. 414 for Gaussian bands) or o/r (0.577 for Lorentzian bands and 0.850 for Gaussian bands).

Use of this procedure for a rather extreme case where the frequency shift and bandwidth change are both sig­nificant is demonstrated by Fig. 3. Measurement of the peak to valley height d from this curve gives

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Jaan Laane: Raman difference spectroscopy 2543

TABLE II. Characteristics of Raman difference spectra.

Effect Lorentzian bands Gaussian bands Characteristics

Frequency shift only. ('i= O. 577r 6=0.850r Maximum and minimum of equal magnitude ~= o. 385rd/lo ~= O. 350rd/lo

Bandwidth change only 6=(rAfB)I/2-r

D/lo=(rA -rB)/2r

6=1. 201rA fB/r-1. 201f Two equal maxima and a minimum of 1= 0 at W= O.

D /10 = eXfJ( - r~/ f2) - exp( - ri/r2)

6 -1. 201r Both frequency and bandwidth changes (bandwidth effect dominates).

6",(rAfB)1I2-r

.£;=rA-rB±~ 10 2f r

2 2 D" (- rB) (- r A) 1. 225~ 7; = exp rz - exp rr- ± --r-

Two unequal maxima and a minimum near w = O.

~=(D.-Djr/21o ~= O. 408 (D. -Djf/lo

Both frequency and bandwidth changes (frequency shift dominates).

6 -0. 577r 6_ O. 850f Maximum and minimum of unequal magnitude. ~,- O. 385fd/lo ~- O. 350fd/lo

Definitions: r=(fA +fB)/2",,(rA rB)1/2; 6=differencebetweenbandmaximaorbetween maximum and minimum when only one maximum is present; ~ = frequency difference between two Raman lines; d= intensity difference between maximum and minimum; D = intensity of maximum (D. and D_ are used to represent two unequal maxima); 10 = intensity of Raman band.

d/l0 = 0.552 which for a Lorentzian curve gives ~/r = 0.220. Thus this procedure results in a calculatedfre­quency shift of ~ = 2.20 cm- I as compared to the actual value of 2.00 cm- I

• A 1~ error for most measurements should be satisfactory but this may be further reduced (with considerably more effort) if Eq. (4) is used to simulate the observed RDS curve. The experimental values of Y A and YB would be used as known parameters and then x may be varied to obtain the best fit.

(c) If the frequency shift is zero but a bandwidth change is present, two maxima of equal intensity D result and Eq. (8) or (29) is useful for evaluating the data. The fraction Lorentzian character is determined from ~Vl/4/r or from a/r (-1.0 for Lorentzian bands and - 1. 2 for Gaussian bands).

(d) When the bandwidth change is sizable and a fre­quency shift is present,Eq. (18) or (35) proves useful. Two unequal maxima result and their intenSity difference may be used to find ~ from Eq. (22) or (36). The aver­age intenSity is useful for examining the bandwidth change, see Eq. (23) and (35). The fraction Lorentzian character is determined as in the previous example.

Table II summarizes these results and provides a useful comparison between the RDS spectra from Lorentzian and Gaussian bands.

B. Examples

We previously reported on an experiment5 where in a four-channel experiment the isotropic and ansiotropic Raman spectra of pure benzene and of benzene dissolved in benzene-de were recorded simultaneously. The dif­ference spectra which had perpendicular polarization (anisotropic component) were straightforward to analyze since the bandwidth measured was very nearly constant (r = 4.3 cm-1

) and the band shape was Lorentzian (L = 1. 0). Measurement of d/l0 = 0.142 permitted the calculation of the frequency shift ~ = - O. 23 cm-1 from Eq. (13). The isotropic spectra (I" -i-h), however,

gave a RDS curve similar to Fig. l(c) demonstrating that for this component both bandwidth and frequency changes were present. These bands were mostly Gaussian (L = 0.1) and the bandwidths were measured to be r MX = 3.1 cm-1 for the isotopic mixture and r BZ =4.3 cm-1 for pure benzene (uncorrected for the instru­mental slit effect). The ratio Rl = (D. -DJ/l0 was mea­sured to be 0.13 and R2 = (D.+DJ/2Io was 0.24. By in­terpolation between the Lorentzian and Gaussian limits we find from L = 0.2 that (~/r) = O. 426R 1 and ~ = - 0.21, in fine agreement with the shift from the perpendicular polarized lines. (The negative sign for the shift indi­cated the frequency in the mixture is at a lower value. ) The experimentally determined bandwidths substituted into Eqs. (29) and (8) predict thatR2 should be 0.24 for a Gaussian band shape and O. 16 for a Lorentzian shape. Interpolation using L = O. 1 yields a value of 0.23 in ex­cellent agreement with that experimentally determined. The graph in Fig. 2 may also be utilized to calculate r BZ/r MX from R2 =D/Io = 0.24. After interpolation be­tween the Gaussian and Lorentzian curves r BZ/r MX is calculated as 1. 41. This compares well with the mea­sured value 4.3/3.1 = 1. 39.

Figure 4 presents a second examplee of a Raman dif­ference spectrum resulting when both bandwidth and fre­quency changes occur. Sodium nitrate was dissolved in H20 in one cell compartment and in D20 in the other; the concentration for each was 3.0 M. The figure shows the spectra with the intensities as actually recorded but these were normalized (IH20 set equal to I p20) before the RDS spectrum was obtained. The difference curve (ID20 -IH20 ) shows that the VI NO; symmetric stretching band is broader and at a higher frequency for the D20 solution than for the H20 solution. The bandwidths mea­sured from the Raman bands are r H20 = 8. 9 cm -1 and r D20 = 9.5 cm- l and the ratios Rl = (D. -DJ/Io and R2 = (D.+DJ/2Io from the RDS curve are 0.015 and 0.035, respectively. The fraction Lorentzian character is esti­mated to be L = O. 3. Use of the value of R 1 permits the

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2544 Jaan Laane: Raman difference spectroscopy

calculation of the frequency shift as t::. = 0.06 cm- l and the value of R2 gives rise to the result r Dj!O/r H20 = 1.06. The latter value may be compared to 9.5/8.9 = 1. 07 ob­tained from the Raman bands. While there are no other data which can be used to confirm the frequency shift, the consistency of the bandwidth data gives us further confidence in the overall analysis.

As a third and final example of the effect of bandwidth change on RDS spectra, Fig. 5 shows spectra for a 1 : 1 by volume mixture of chloroform and acetone. Curve (a) is for the mixture (MX), (b) is for pure acetone (AC), and (c) is for RDS spectrum for the mixture minus ace­tone (after appropriate scaling so that IMX =IAC at the band maximum). The RDS curve has been expanded by a factor of 7. O. The measured bandwidths are r MX

= 14. 5 cm- l and r AC = 12.0 cm-l, but it should be noted

that the bands are highly asymmetrical so that their di­rect measurement may not be too meaningful. From the RDS spectrum the ratio D/lo = 0.13 is measured and this corresponds to a r MX/r AC ratio of 1. 30 for a Lorentzian band and 1. 20 for a Gaussian band. The fraction Lorentzian character determined from analyzing the low-frequency half on the bands is L = 0.7 ± 0.2. In­terpolation thus gives r MX/r AC = 1. 27 from the RDS spectrum vs 14.5/12.0 = 1. 21 from the Raman bands. However, if a correction for the asymmetry is made, the measured bandwidths can be estimated to be r MX

= 11. 3 cm- l and r AC = 9.0 cm- l and then r MX/r AC = 1. 26, in much better agreement with the difference curve. What this demonstrates is that the RDS spectrum can be invaluable for analyzing bandwidth changes when the band of interest is overlapped by other bands. The very slight difference in peak heights for the RDS maxima suggests the C=O stretching frequency for acetone in

> ~

CJ)

Z w ~ Z

z < ~ « a:

1070 1050

eM-1

1030

FIG. 4. Raman and RDS spectra for the "1 mode of NO; ion dissolved in H20 (solid line) and in D20 (dashed line) at 3. 0 M concentration. The difference spectrum (lD~ -IH20) has been expanded by a factor of 10.

1 > I-

(J)

Z W I­Z

Z <C ::E <C a:

1740 1720 1700 1680

CM-1 FIG. 5. Raman spectra of the acetone C =0 stretching band a) 1: 1 mixture by volume of chloroform and acetone; and in (b) pure acetone. Spectrum (c) is the RDS spectrum of (a) minus (b) after scaling and expanding by 7. O.

the chloroform-acetone mixture is Slightly higher than in pure acetone. From (D.-DJ/lo=0.017 the frequency shift is calculated to be + O. 08 cm- l

•

IV. SUMMARY

The expressions necessary for analyzing Raman dif­ference spectra which result when bandwidth and fre­quency differences are present in the two samples have been derived for both Lorentzian and Gaussian band shapes. RDS band parameters for intermediate band shapes may be obtained by interpolation. As has been demonstrated, the RDS spectra may be used to obtain the frequency shift and bandwidth broadening (or nar­rowing) data even when the Raman spectra are compli­cated by overlapping bands.

Raman difference spectra can be valuable for studying solvent effects, isotopic shifts, resonant energy trans­fer, and compositional changes. We have presented some examples of such studies in previous papers. 2-. Other workers have also applied this technique to bio­chemical systems. 7•8 In all the previous studies, only frequency shifts have been analyzed and only for those bandS for which bandwidth changes are minor. The re­sults of this paper will permit the analysis of virtually all types of bands even when both bandwidth and fre­quency changes are present.

ACKNOWLEDGMENTS

This research was sponsored by the National Science Foundation. The author is grateful to Professor W.

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Jaan Laane: Raman difference spectroscopy 2545

Kiefer, Universitlit Bayreuth,for introducing him to the experimental techniques of Raman difference spectros­copy.

lJ. Laane and W. Kiefer, J. Chern. Phys. 72, 5304 (1980). 2J. Laane and W. Kiefer, J. Chern. Phys. 73, 4971 (1980).

3J . Laane, H. Eichele, H. P. Hohenberger, and W. Kiefer, J. Mol. Spectrosc. 86, 262 (1981).

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