JOURNAL OF RAMAN SPECTROSCOPYJ. Raman Spectrosc. 2003; 34: 380–387Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jrs.992

Isotope effects on the Raman spectrumof buckminsterfullerene, C60

Jie Ren, John B. Page and Jose Menendez∗

Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85281, USA

Received 28 August 2002; Accepted 15 January 2003

The Raman spectrum of C60 was calculated for molecular samples containing carbon isotopes with naturalabundance. The results confirm experimental results indicating that isotopic perturbations are not animportant source of silent-mode Raman activity. Detailed predictions for the Raman spectrum of C60 in gasphases or isolating matrices are presented. The accuracy of the theoretical calculations was maximized byusing experimental vibrational wavenumbers and bond polarizability parameters fitted to experimentaldata. Copyright 2003 John Wiley & Sons, Ltd.

KEYWORDS: buckminsterfullerene; C60; isotope effects; bond polarizability

INTRODUCTION

Since the discovery of a method for producing buckmin-sterfullerene, C60, in macroscopic quantities,1 vibrationalspectroscopy has played a fundamental role in the studyof this fascinating molecule. The observation of four stronginfrared absorption bands,2 in agreement with group theorypredictions, represented a striking experimental confirma-tion of the soccer-ball structure with icosahedral symmetryproposed in 1985 by Kroto et al.3 In addition to the fourinfrared-active modes, which belong to the T1u representa-tion of the icosahedral point group Ih, group theory predicts4

up to 10 different Raman lines for C60. Two of these belong tothe totally symmetric Ag representation, and the remainingeight modes belong to the five-dimensional Hg representa-tion. The Ag(1) mode (‘breathing’), Ag(2) mode (‘pentagonalpinch’) and Hg(1) mode (‘squashing’) are very prominentin Raman spectra obtained under visible and near-infraredexcitation.5,6 The striking dependence of the position andlineshape of the high-energy Ag(2) line on oxygen expo-sure led to the discovery of photoinduced polymerizationof fullerenes.7 The identification of all Raman modes withHg symmetry turned out to be challenging, owing to theweakness of some of these lines and the possible presenceof other fullerene species in as-grown samples. The prob-lem was solved by Bethune et al.5 by performing Ramanexperiments in chromatographically purified samples. Sub-sequent Raman and IR studies of C60 revealed a much richerspectrum with a complicated fine structure.8 – 15 Most of the

ŁCorrespondence to: Jose Menendez, Department of Physics andAstronomy, Arizona State University, Tempe, AZ 85281, USA.E-mail: jose.menendez@asu.edu

additional features are obviously due to higher order infraredand Raman processes involving two or more vibrationalquanta, but some of the observed peaks have been ascribedto mode fundamentals which are forbidden in an icosahedralmolecule. The observation of such ‘silent’ modes implies theexistence of a perturbation that breaks the icosahedral sym-metry. There are two main candidates for such perturbation:crystal field effects (most experiments are carried out in solidfilms and single crystals) and isotope effects. Despite thefact that 12C is by far the most abundant isotope in natu-ral carbon, the 1.1% relative abundance of 13C is enough toreduce the fraction of pure 12C60 molecules to only 51% if thestarting fabrication material is natural graphite. Therefore,isotope effects cannot be neglected a priori in any detailedanalysis of experimental data. Moreover, since at room tem-perature C60 molecules are nearly free rotators in the solidphases, C60 –C60 interactions must be weak, and this ledto early assumptions that isotope effects must represent thedominant symmetry-breaking perturbation in C60. However,systematic experimental IR studies by Martin et al.16 on sam-ples with different isotopic compositions, and similar Ramanstudies by Rosenberg and Kendziora,17 clearly showed thatisotopic disorder does not play an important role in theactivation of silent modes. Moreover, a comparative high-resolution Raman study18 of single crystals of 12C60, 13C60

and natural C60 showed that aside from a uniform softeningof wavenumbers and somewhat different broadenings, thefine structure derived from each of the unperturbed Raman-active modes is virtually independent of isotopic disorder,with the notable exception of Ag(2) and a weak peak assignedin Ref. 18 to F2g(1) [T3g(1) in our notation]. Therefore, the con-clusion from the experimental evidence is that vibrations in

Copyright 2003 John Wiley & Sons, Ltd.

Isotope effects on the Raman spectrum of C60 381

C60 are much more strongly affected by crystal fields than byisotope effects.

Until now, there has been no theoretical confirmation ofthe experimental results ruling out isotopic perturbationsas an important source of silent-mode Raman activityin C60. A few published calculations report vibrationalwavenumbers or vibrational densities of states in thepresence of isotopes,19 – 22 but the interpretation of Ramanspectra requires realistic calculations of Raman intensitiesas well. This has been done by Guha and co-workers,23,24

but their work was limited to the spectral range near theAg(2) Raman peak at 1470 cm�1. In this study, we extendedthe work of Guha and co-workers to the entire range ofvibrational wavenumbers in C60. Our results confirm theexperimental conclusion that isotope effects are not animportant source of silent mode activation. For those modeswhich are Raman-active in the icosahedral molecule, wepresent detailed calculations of the lineshape distortionsinduced by isotopic disorder. These predictions are subjectto experimental verification using gas-phase C60 or C60

embedded in rare gas matrices.25,26

Guha and co-workers23,24 showed that the predictedRaman spectrum for isotopically mixed C60 depends verysensitively on the wavenumber differences between unper-turbed vibrational modes, to the extent that predictionsbased on calculated wavenumbers are not accurate enoughfor a comparison with experimental spectra. The originalcalculations,23,24 being limited to the spectral region around1470 cm�1, were mainly sensitive to the mode wavenumbersaround this value, but an extension to the entire vibrationalwavenumber range requires an accurate knowledge of all 46unperturbed vibrational wavenumbers in C60. A wavenum-ber set in good agreement with all existing experimental datawas recently proposed by Menendez and Page,4 and this setwas used in the present calculations.

Raman intensities were computed by Guha et al. usinga bond-polarizability model27 whose parameters were fittedto the Raman spectrum of C60. The subsequent removal ofsome experimental inconsistencies led to a slightly differentset of parameters, as presented in Ref. 4. Since our earliercalculations of isotopic effects were limited to the spectralregion near the Ag(2) mode, the predicted spectral lineshapesare largely independent of the fit values for the different bondpolarizability parameters. For the present work, however,our goal is to predict the relative intensities of all isotopic-induced features in the Raman spectrum, so that accuratebond polarizability parameters are important. This hasmotivated a re-evaluation of the previous bond polarizabilityfit to C60 in the light of new experimental results and a betterunderstanding of the Raman spectrum. The result is a newset of parameters which, although not very different from theones in Ref. 4, should enhance the quantitative accuracy ofour predictions. Moreover, the bond polarizability modelhas found a surprisingly wide range of applications toother members of the fullerene family28 – 32 and to carbon

nanotubes,33 in agreement with the transferability hypothesisfor bond polarizabilities. Accordingly, the new set of bondpolarizability parameters should be of interest to workers inthese fields.

The remainder of this paper is organized as follows. Inthe next section we summarize the theoretical expressionsused to compute the Raman spectrum of pure 12C60 andisotopically mixed C60 molecules. We then present a revisedbond polarizability fit to the Raman intensities in C60, and inthe final section we study in detail the effect of isotopes onthe Raman spectrum of this molecule.

THEORETICAL BACKGROUND

Raman intensitiesThe theoretical analysis of the C60 Raman spectrum and,in particular, the study of isotopic effects in this material,requires the calculation of Raman intensities. The approachesthat have been used include ab initio methods,34,35 semiem-pirical quantum chemical models36 and bond polarizabilitymodels.27,37,38 The last models can be complemented byadding a transferability hypothesis, according to which thepolarizability parameters associated with a bond shouldbe the same for all molecules containing such a bond. Inthe particular case of C60, Guha et al.27 showed that thestatic dielectric properties of C60 can be predicted with goodaccuracy using hydrocarbon bond polarizabilities. Ramanintensities depend on bond polarizabilities and their deriva-tives, and hence present a much more stringent test of thetransferability hypothesis. Not surprisingly, the agreementbetween the experimental C60 Raman spectrum and the spec-trum calculated using hydrocarbon polarizability parametersis only semi-quantitative, since the corresponding bonds areof course not identical.27 An alternative approach used byGuha et al. is to fit a bond polarizability model directly toC60, and use the resulting parameters to study isotopic effectsin this system and to predict the Raman spectra of otherfullerenes.27 Since bonds across the fullerene family are moresimilar than those between fullerenes and hydrocarbons, thebond polarizability parameters fit to the Raman spectrumof C60 are more likely to lead to quantitative predictions forthe Raman spectrum of other fullerenes. This approach hasproven to be very effective, especially when applied to poly-meric forms of C60.30,32 In particular, the atomic structure ofthe dimeric odd-numbered fullerene C119 was convincinglyunraveled by analyzing its Raman spectrum using first-principles calculations of vibrations together with the fit C60

bond polarizability parameters.30,32 These parameters havealso been used to predict the off-resonance Raman spectrumof carbon nanotubes.39

Within a bond polarizability model, the observed Ramanintensities depend on the modes’ eigenfrequencies andeigenvectors and on the bond polarizability parameters. Fullderivations in the context of C60 Raman intensities weregiven previously,4,27 and here we will only summarize, for

Copyright 2003 John Wiley & Sons, Ltd. J. Raman Spectrosc. 2003; 34: 380–387

382 J. Ren, J. B. Page and J. Menendez

completeness, the basic equations. For off-resonance Ramanscattering, the Stokes Raman photon differential scatteringcross-section is4

(d2�

d�dω

)��‘

D hωL

2c4

3N∑f D1

�ωL � ωf �3�hn�ωf �i C 1�

ωf

ð∣∣∣∣∣∑˛ˇ

�‘˛�ˇP˛ˇ,f

∣∣∣∣∣2

υ�ω C ωf � �1�

where c is the speed of light, ω D ωS � ωL is the Raman shift,ωS and ωL are the scattering and incident light frequencies,ωf is the normal mode frequency for mode f , P˛ˇ,f isthe polarizability derivative for mode f , � and �0 are thepolarization directions of the incident and scattered light,respectively, and hn�ωf �i is the thermal average occupationnumber of mode f at temperature T. Local field correctionsare not included in Eqn (1), but they are expected to becancelled out in fits to Raman intensity ratios for differentpeaks.

As detailed by Guha et al.,27 if the bond polarizablity isassumed to be an axially symmetric function of the bondlength R, it is given by

˛ˇ� ER� D ˛jj�R�R˛Rˇ/R2 C ˛?�R��υ˛ˇ � R˛Rˇ/R2� �2�

and the mode polarizability derivatives are

P˛ˇ,f D �∑

l

/∑B

[{˛‘jj[R0�lB�]C2˛‘?[R0�lB�]

3

}R0�lB� Ð EX�ljf �υ˛ˇ

C{

˛‘jj[R0�lB�]�˛‘?[R0�lB�]3

}[3 OR0˛�lB� OR0ˇ�lB� � υ˛ˇ]R0�lB� Ð EX�ljf �

C{

˛jj[R0�lB�]�˛?[R0�lB�]R0�lB�

}[ OR0˛�lB��lˇjf � C OR0ˇ�lB��l˛jf �

�2 OR0˛�lB� OR0ˇ�lB�R0�lB� Ð EX�ljf �]]

�3�

where ER0�lB� is the bond vector from atom l to any one of itsbonded neighbors lB in the equilibrium configuration, R0�lB�is a unit vector along the same direction, ˛ and ˇ are cartesiancomponents, X�f � is the f -mode eigenvector, defined as inRef. 4, and ˛‘jj and ˛‘? denote the derivatives of the bondpolarizability parameters with respect to the bond length.The prime on the sum means that only the bonds attached tosite l are summed.

The Raman strength is obtained by inserting Eqn (3) intoEqn (1) and averaging over the possible orientations of themolecule. For C60, the first of the three terms in Eqn (3)determines the scattering intensity of the totally symmetricmodes of Ag symmetry. The remaining two terms control thescattering intensity of Hg modes. In particular, the intensityof the Hg(1) squashing mode depends almost exclusively onthe third term in Eqn (3). A more detailed discussion of thecontribution of the three terms to the Raman signal of C60

is given in Ref. 27. The prefactors in each of the three terms

are the bond polarizabilty parameters that can be fitted toexperiment. In C60 there are two types of bonds, ‘single’and ‘double,’ yielding a total of six independent parameters.This number is reduced to five if one fits only the relativeintensities of the 10 Raman modes.

Molecular vibrations: mode eigenvectorsAn important aspect of Eqn (3) is that bond polarizability fitsrequire a knowledge of the vibrational mode eigenvectorsf�f �g. Comprehensive experimental studies of mode eigen-vectors in C60 are not available, but an analysis of inelasticneutron scattering (INS) intensities by Heid et al.40 showsthat, at least up to 600 cm�1, there is an excellent agree-ment between experiment and eigenvectors obtained fromab initio density functional theory (DFT) within the local den-sity approximation (LDA). These eigenvectors determine theintensities of the inelastic neutron scattering (INS) peaks.Moreover, we have found that eigenvectors resulting fromdifferent implementations of the LDA are often similar, evenwhen there is a significant scatter of the associated predictedmode wavenumbers between these different implementa-tions. This presumably reflects the very high symmetry of Ih

C60. Accordingly, we use ab initio eigenvectors for our bondpolarizability fits.

Molecular vibrations: mode wavenumbersThe calculation of Raman intensities using Eqn (3) alsorequires a knowledge of the modes’ wavenumbers. Forperfectly icosahedral C60, this does not pose a majorproblem because the needed wavenumbers are obtainedunambiguously from an analysis of the strongest lines in theRaman spectrum. Even if theoretical wavenumbers are used,the dependence of the Raman intensity on ωf is rather weak,and the agreement with the experimental intensities shouldbe equally good.

Mode wavenumbers are much more critical for quan-titative predictions of isotopic effects. This is because thewavenumber shifts of Raman-active modes and the intensity‘borrowing’ that activates silent modes depend very sensi-tively on mode wavenumber differences, as pointed out byGuha and co-workers.23,24 This dependence can be betterunderstood if the vibrational problem, including the isotopicperturbation, is expressed in terms of the complete basisof eigensolutions for the isotopically pure (unperturbed)molecule. This results in an eigenvalue equation of the form4

∑f 0

[ω20f υff 0 � ω2

f �υff 0 C Mff 0 �]cff 0 D 0 �4�

where ω0f is the unperturbed frequency of mode f , and thecoefficients cff 0 are the expansion coefficients of the modeeigenvectors in the basis of unperturbed eigenvectors:

X�f � D3N∑

f 0D1

cff 0 X0�f 0� �5�

Copyright 2003 John Wiley & Sons, Ltd. J. Raman Spectrosc. 2003; 34: 380–387

Isotope effects on the Raman spectrum of C60 383

The quantity Mff 0 is given by

Mff 0 D X0�f �MX0�f 0� �6�

where M D fm�l�υll‘υ˛˛‘g is the mass perturbation matrix.Equation (4) shows that the solutions to the isotopic disor-der problem depend on the lineup of unperturbed modefrequenciesfω2

of g. Since the average separation betweenmodes in C60 is only 30 cm�1, calculations based on sim-plified vibrational models are not sufficiently reliable topredict accurately effects such as the isotopic activation ofsilent modes. Moreover, the average wavenumber separa-tion of unperturbed modes is comparable to the averagewavenumber error of even the most accurate DFT–LDAvibrational calculations, so that quantitative predictions ofisotopic effects should preferentially use experimental infor-mation on vibrational modes. An approximate but accurateway of accomplishing this is suggested by the structure ofEqn (4). The idea is to use experimental unperturbed mode fre-quencies fω2

of g combined with matrix elements of the isotopicperturbation computed using theoretical mode eigenvectors.A detailed discussion of the implementation of this methodis given elsewhere4,23,24 for the case of the Ag(2) Raman spec-trum in C60. A complication arises in systems such as C60 forwhich more than one Raman mode belong to the same irre-ducible representation. In such cases, there is more than oneway to pair a given experimental wavenumber with a theo-retical eigenvector. This ambiguity is eliminated by orderinglike-representation eigenvectors according to the theoreti-cal eigenwavenumbers and assigning them to experimentalwavenumbers taken in the same order. This procedure mayfail if the wavenumber separation between modes is onthe order of the theoretical wavenumber errors. In C60, thesmallest wavenumber separation between Raman modes ofthe same symmetry is 63 cm�1 [between the Hg(3) mode at709 cm�1 and the Hg(4) mode at 772 cm�1]. This value islarger than the accuracy of state-of-the-art DFT–LDA cal-culations, so that the risk of making incorrect eigenvectorassignments is small for this molecule.

The complete set of mode frequencies fω2of g is in prac-

tice very difficult to determine, because it includes the 32silent modes which cannot be unambiguously identified infirst-order Raman or IR spectra. Higher order Raman andinfrared spectra are usually consistent with multiple alter-native assignments of silent modes, and other spectroscopictechniques that are sensitive to silent modes usually lack thenecessary wavenumber resolution and/or do not have sim-ple selection rules. Accurate theoretical calculations providea valuable guide, as shown in 1994 by Menendez and Guha41

and Schettino et al.42 In 2000, Menendez and Page4 proposeda set of wavenumbers that is simultaneously consistent withDFT–LDA calculations and with inelastic neutron scattering,high-resolution energy-loss spectroscopy, low-temperaturefluorescence, second-order Raman and infrared spectro-scopies and the isotopic dependence of the Ag(2) Raman

line. More recently, Schettino et al.43 proposed additionalimprovements to the wavenumber set in Ref. 4. However,this latest set worsens the agreement with the inelastic neu-tron scattering spectra of Coulombeau et al.44 and it alsofails to reproduce the experimental isotopic dependence ofAg(2).23 Therefore, we will use the wavenumbers proposedin Ref. 4 in all calculations presented below.

NEW BOND POLARIZABILITY PARAMETERS

Since calculations of isotopic-induced Raman intensities forthe entire range of C60 vibrational wavenumbers dependon the relative values of bond polarizability parameters,we have critically re-examined the bond polarizability fitin Ref. 4. Using newly available experimental data and areinterpretation of the Raman band associated with the Hg(1)mode, we obtain an improved set of parameters.

The experimental data available for bond polarizabilityfits have been obtained from C60 films. The absorptionedge of these films is near 2.7 eV,45 so that Raman spectraexcited with near-infrared 1064 nm (1.16 eV) radiation froman Nd:YAG laser6,46 – 48 can be expected to fulfill the off-resonance condition. The fit in Ref. 4 was based on datafrom Chase et al.6 Here we combine the data of Ref. 6 withadditional 1064 nm spectra obtained by Lynch et al.48

An important aspect of the data obtained by Chase et al.6

is the appearance of a shoulder on the low-energy end ofthe Hg(1) band at 272 cm�1. This shoulder can be modeled asan additional Raman line at 266 cm�1. Since theory does notpredict any vibrational mode with a wavenumber lower thanHg(1), in previous work we interpreted the 266 cm�1 shoulderas arising from an unknown impurity and did not include itsintegrated intensity in the bond polarizability fits. However,the 266 cm�1 peak or shoulder appears consistently in allnear-infrared Raman spectra, including data from single-crystal C60.18 Therefore, a more likely interpretation ofits origin is a splitting of the Hg(1) manifold by a non-icosahedral perturbation, such as C60 –C60 interactions in thesolid phases.18 Since equivalent splittings are not resolvedfor the other Raman modes, we feel that a more reasonableapproach for a bond polarizability fit is to combine theintensities of the 266 and 272 cm�1 lines. In Table 1 we followthis approach and compare the relative intensities of theRaman peaks as obtained by Chase et al.6 and Lynch et al.48

By fitting the data in Table 1, we obtain bond polarizablityparameters and their uncertainties. Table 2 shows the resultsof our fits to the two sets of experimental data. We performedadditional fits to randomly generated sets of data for whichthe intensity of individual peaks was increased or decreasedby an amount equal to the difference in their intensitiesbetween the two experimental sets. The standard deviationfor each parameter obtained from these multiple fits gives theuncertainties listed in Table 2. The differences between ourfitting results and the parameters in Ref. 4 are not dramatic,and therefore they do not affect the main conclusions of our

Copyright 2003 John Wiley & Sons, Ltd. J. Raman Spectrosc. 2003; 34: 380–387

384 J. Ren, J. B. Page and J. Menendez

Table 1. Experimental first-order Raman relative intensities ofChase et al.6 and Lynch et al.48a

Ramanmode

Chase et al.’sdata6

Lynch et al.’sdata48

Hg(1) 1.000 1.000Hg(2) 0.129 0.148Hg(3) 0.027 0.028Hg(4) 0.173 0.189Hg(5) 0.095 0.087Hg(6) 0.118 0.117Hg(7) 0.021 0.032Hg(8) 0.170 0.192Ag(1) 0.693 0.725Ag(2) 0.717 0.996

a Both spectra were obtained at room temperature and withan incident laser line of 1064 nm. The Raman intensities arenormalized to that of the Hg(1) mode. The observed Hg(1)scattering consists of a main peak at 272 cm�1 plus a shoulder at266 cm�1, as explained in the text.

isotopic calculations below. Nevertheless, these differencesare clearly beyond our estimated uncertainties, and futurework in this field should use the new values in Table 2.

ISOTOPIC EFFECTS ON THE RAMANSPECTRUM OF C60

The isotopic perturbation mixes the icosahedral vibrationalmodes. Experimentally, this mixing is expected to manifestitself in the form of shifts and splittings of the allowed Ramanand IR modes as well as in the appearance of Raman and IRsignals associated with the icosahedral silent modes. Thesemodes belong to the T3g, T3u, Gg, Gu and Hu representations.

The mode eigenvectors and eigenwavenumbers in thepresence of isotopic disorder are obtained by solving Eqn (4).We use the wavenumber lineup proposed in Ref. 4. The

(a)

400 600 80010−6

10−4

10−2

100

102

(b)

Hg(

1)H

g(1)

Hg(

2)

Hg(

3)

Hg(

4)

Gg(

3)H

u(4)

Gg(

1)

Gg(

2)/T

1g(1

)

Ag(

1)

T3u

(1)

T1u

(1) H

u(2)

Hu(

3)

T1u

(2)

T1g

(2)G

u(1)

Hg(

2)

Ag(

1)

Hg(

3)

Hg(

4)

Rel

ativ

e In

tens

ity

10−6

10−4

10−2

100

102

Rel

ativ

e In

tens

ity

Wavenumber/cm−1

400 600 800

Wavenumber/cm−1

Figure 1. (a) Calculated first-order low-wavenumber Ramanspectrum for a C60 molecule with a natural isotope distribution.(b) Calculated first-order low-wavenumber Raman spectrum ofpure C60. Both (a) and (b) are Lorentzian broadened with anFWHM of 0.5 cm�1.

matrix elements Mff 0 are computed using DFT–LDAmode eigenvectors for icosahedral C60. The eigenvectorsand eigenwavenumbers so obtained are then inserted intoEqns (3) and (1) to obtain the corresponding Raman spectra.We use the new bond polarizability parameters in Table 2and we assume that these parameters are independent of theisotopic composition, an exact result if zero-point effects on

Table 2. Parameter ratios for our bond-polarizability model fits to the measured first-order C60 Raman spectraa

Present fit

Bond polarizabilityparameter ratio

Fit toChase et al.’s

data6

Fit toLynch et al.’s

data48

Averagedfitting

parameters Hydrocarbons27 Previous fit4

[˛0jj�S� � ˛0

?�S�]/N 2.905 2.895 2.900 š 0.055 2.617 2.80[˛0

jj�S� C 2˛0?�S�]/N 2.232 2.054 2.14 š 0.28 3.546 2.73

[˛0jj�D� � ˛0

?�D�]/N 3.269 3.450 3.36 š 0.12 2.945 3.18[˛0

jj�D� C 2˛0?�D�]/N 7.451 8.085 7.77 š 0.71 7.36 8.98

[˛jj�D� � ˛?�D�]/[NR0�S�] 0.409 0.382 0.396 š 0.022 1.289 0.256

a In the first column of the parameter ratio formulas, S and D denote single and double bonds respectively. The ratios are taken withrespect to N D [˛jj�S� � ˛?�S�]/R0�S�.Here ˛jj�S� � ˛?�S� D 3.03 š 0.036 A3, and the C60 bond lengths are taken as R0�S� D 1.45 A and R0�D� D 1.40 A.4 The error estimatesare discussed in the text.

Copyright 2003 John Wiley & Sons, Ltd. J. Raman Spectrosc. 2003; 34: 380–387

Isotope effects on the Raman spectrum of C60 385

1000 1200 1400 1600

1000 1200 1400 1600

(a)

10−6

10−4

10−2

100

102

(b)

Rel

ativ

e In

tens

ity

10−6

10−4

10−2

100

102

Rel

ativ

e In

tens

ity

Wavenumber/cm−1

Wavenumber/cm−1

Hg(

5)

Hg(

6)

Hg(

7)

Ag(

2)

Hg(

8)

Au

T3u

(3)

T3u

(4)

T1u

(3)

T1g

(3)

T3u

(5)

Gu(

4)

Gu(

5)

Hg(

5)

Hg(

6)

Ag(

2)H

g(7) H

g(8)

Hu(

7)

Hu(

5)

Hu(

6)

Gg(

4)

Gg(

6)

Figure 2. (a) Calculated first-order high-wavenumber Ramanspectrum for a C60 molecule with a natural isotope distribution.(b) Calculated first-order high-wavenumber Raman spectrumof pure C60. Both (a) and (b) are Lorentzian broadened with anFWHM of 0.5 cm�1.

the electronic structure are negligible. Our isotopic Ramanspectra are calculated by averaging, with a statistical weightproportional to their relative abundances, the Raman spectracorresponding to different isotopic distributions. Since allsites in icosahedral C60 are equivalent, only one calculationis needed for 13C1

12C59. For 13C212C58 we include all of the

31 symmetry-inequivalent configurations. The contributionsfrom molecules with three or more isotopes have beenneglected. The final spectrum is Lorentzian-broadened witha full width at half-maximum (FWHM) of 0.5 cm�1.

Figures 1 and 2 show the calculated first-order Ramanspectrum of a sample consisting of C60 molecules witha natural distribution of isotopes. The Raman strengthis plotted in a logarithmic scale to display better therelative magnitude of the weak silent-mode activation. Forcomparison, the first-order Raman spectrum of pure C60

calculated with the same parameter values is also shown inFigs 1 and 2.

We find that if a silent mode is far (>20 cm�1) from aRaman-active vibration in the isotopically pure molecule,its perturbed counterpart has an extremely weak Ramanstrength, between four and five orders of magnitude lessthan the strength of the strongest peak derived from the Hg(1)manifold. Silent modes that are closer to Raman-active modes

become more strongly activated under the isotopic pertur-bation, but even the strongest of those, corresponding to theGg(6) mode, is two orders of magnitude weaker than the mainline derived from Hg(1). Our results thus confirm Rosenberg’scontention that isotopes do not play an important role insilent mode activation.4 As noted in Ref. 4, all Gg-symmetrymodes can be observed in high-sensitivity Raman spectrafrom C60 films and crystals, but the experimental strength ofthese modes8 is much higher than predicted here, confirmingthat they become active via a crystal field perturbation.

Figure 3 shows, in a linear scale, calculated Ramanspectra near the spectral positions of the eight Hg modes inicosahedral 12C60. Figure 4 shows the corresponding spectrafor the two Ag modes. We notice a significant differencebetween the two types of modes: whereas in the calculatedspectrum for the totally symmetric modes there are strongindividual peaks that can be associated with the presenceof 13C1

12C59 and 13C212C58 molecules, it is apparent that the

Raman spectrum from the Hg modes is dominated by themain lines corresponding to pure 12C60. These findings maybe related to the observation18 of very small isotopic effectson the Hg-related vibrations in crystalline C60. The differenteffects of the isotopic perturbation on Ag-like and Hg-likemodes are easily understood. In a C60 sample fabricatedfrom natural graphite, the abundances of 12C60 and 13C1

12C59

1248 1250 1252 1254

0.03

0.06

0.09

0.12

1096 1098 1100 1102

0.02

0.04

0.06

0.08

768 770 772 774 776

0.04

0.08

0.12

0.16

0.20

704 706 708 710 712

0.02

0.04

0.06

0.08

0.10

430 432 434 436

0.02

0.04

0.06

0.08

0.10

268 270 272 274 276

0.20.40.60.81.0

1422 1424 1426 1428

0.01

0.02

0.03

0.04

Hg(7)

Hg(5)

Hg(3)

Hg(1)

Hg(8)

Hg(6)

Hg(4)

Hg(2)

1572 1574 1576 1578

0.04

0.08

0.12

0.16

Rel

ativ

e In

tens

ity

Wavenumber/cm−1

Rel

ativ

e In

tens

ity

Wavenumber/cm−1

Figure 3. Raman spectrum for a C60 molecule with a naturalisotope distribution near the wavenumbers of the eightRaman-active Hg modes. The spectra are Lorentzianbroadened with an FWHM of 0.5 cm�1.

Copyright 2003 John Wiley & Sons, Ltd. J. Raman Spectrosc. 2003; 34: 380–387

386 J. Ren, J. B. Page and J. Menendez

1464 1466 1468 1470 1472 1474

0.2

0.4

0.6

490 492 494 496 498 500

0.2

0.4

0.6Ag(1)

Ag(2)

Rel

ativ

e In

tens

ity

Wavenumber/cm−1

Rel

ativ

e In

tens

ity

Wavenumber/cm−1

Figure 4. Raman spectrum for a C60 molecule with a naturalisotope distribution near the wavenumbers of the twoRaman-active Ag modes. The spectra are Lorentzianbroadened with an FWHM of 0.5 cm�1.

molecules are 51 and 34%, respectively. The unperturbedAg modes are non-degenerate, and their Raman lines areshifted but not split by the isotopic perturbation. Hence theintensity of the Ag-like peaks from 13C1

12C59, molecules isonly ¾ 3/5 D 60% of the intensity of the corresponding linein 12C60. By contrast, the fivefold degeneracy of the Hg modesis broken by the isotopic perturbation, so that on average theintensity of individual Hg-derived Raman lines in 13C1

12C59,should be roughly 12% of the intensity of the corresponding12C60 line. The argument can be extended to the case of aC60 molecule with two 13C isotopes. To first order in theperturbation, the Ag-like Raman spectrum is independent ofthe relative positions of the isotopes, whereas the Hg-likeRaman spectra depend sensitively on the relative positionsof the 13C isotopes. Hence individual Hg-like Raman linesfrom 13C2

12C58 molecules are expected to be scattered over asignificant energy range and contribute a relatively smooth‘background’ to the observed Raman intensity. This isillustrated by the ‘delta function’ spectra plots in Fig. 5, wherewe show, as an example, isotopic effects on the Raman spectrafrom the Hg(4) and Ag(2) modes. As in Figs 1 and 2, therelative Raman intensities are plotted in a logarithmic scaleto show clearly the weak isotope-activated Raman modes.

In their measured Raman spectra for crystalline C60,Horoyski et al.18 found isotope-related structures near aRaman wavenumber of 570 cm�1, which they assigned to

768 770 772 774 776

10−3

10−2

10−1

100

10−2

10−1

100

Hg(4)

1464 1468 1472

Ag(2)

(a)

(b)

Rel

ativ

e In

tens

ityR

elat

ive

Inte

nsity

Wavenumber/cm−1

Wavenumber/cm−1

Figure 5. Calculated ‘delta function’ Raman spectra (i.e. theLorentzian broadening approaching zero) for the Hg(4) andAg(2) modes with a natural C60 isotope distribution. To showbetter the spectral broadening effect on the delta functionspectra, the corresponding Lorentzian broadened spectra arealso shown for each mode.

the F2g(1) mode; in our notation this corresponds to the T3g(1)mode. According to Ref. 4, there are two modes within 3 cm�1

of 570 cm�1, and they were assigned there to T1g(1) and Gg(2).We do not find any anomaly near these wavenumbers in ourisolated molecule calculations. The predicted intensity of theRaman band derived from T1g(1) in the presence of isotopicdisorder is 2 ð 10�5 relative to that of the strongest Hg(1)-derived peak. As just noted, however, the T1g(1) wavenumberis very close to that of Gg(2), whereas the Ag(2) mode, theonly other mode which revealed clear isotopic effects in thecrystalline C60 experiments, is also very close to the Gg(6)mode. In the crystal, the Gg and Ag vibrations are mixedby the crystal field perturbation, and we suggest that thepresence of isotopes may have an observable effect on thismixing.

CONCLUSION

We have calculated the Raman spectrum of C60 moleculeswith isotopic disorder. Our method uses as input exper-imental vibrational wavenumbers and bond polarizabilityparameters fit to experimental data. Our results confirm thatisotopic disorder plays a minor role in the spectroscopicactivation of silent modes. For Raman-active modes we

Copyright 2003 John Wiley & Sons, Ltd. J. Raman Spectrosc. 2003; 34: 380–387

Isotope effects on the Raman spectrum of C60 387

predict the details of the spectral changes in the presenceof isotopes. Aside from the Ag(2) mode results of Guhaand co-workers,23,24 there are still no experimental data forcomparison with these predictions. Successful optical exper-iments on C60 molecules embedded in inert gas matrices25,26

suggest that this sample preparation technique may be idealfor Raman experiments to test our results. Experiments ofthis sort would provide a very sensitive additional test ofthe vibrational mode wavenumber lineup in isotopicallypure C60.

AcknowledgementsWe are extremely grateful to Dr Bruce Chase and Professor W.Brockner for providing us with their data in digital format.

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