ISSN 0021�3640, JETP Letters, 2013, Vol. 98, No. 3, pp. 134–138. © Pleiades Publishing, Inc., 2013.Original Russian Text © B.A. Kolesov, N.B. Egorov, 2013, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2013, Vol. 98, No. 3, pp. 155–159.

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The use of vibrational (Raman) spectra to studyvarious isotopes is based on the well�known notionthat the rms amplitude of zero�point and excitedvibrations is proportional to m–1/2 (where m is the massof the isotope) at an unchanged force coupling con�stant (the condition usually accepted at isotope substi�tution). This leads to change in the structure parame�ters, energy gap, and vibrational spectra of monoiso�topic crystals. It is commonly accepted that zero�point vibrations make real corrections to the vibra�tional spectrum at temperatures below the Debye tem�perature of a crystal. The measurement of the temper�ature dependence of the Raman spectra of differentisotopes of the same element would make it possible todetermine the temperature range where zero�pointvibrations make an efficient contribution to thevibrational spectrum (i.e., to the frequencies andbandwidths of vibrational modes) and to compare itwith the calculated (or estimated) Debye temperature.Furthermore, the amplitudes of vibrations differentfor different isotopes determine the degree of theiranharmonicity. This degree can also be estimated fromthe temperature dependence of the peak position andbandwidth of the vibrational mode. An exhaustivedescription of the expected and observed isotopeeffects for monatomic crystals (silicon, germanium,diamond) was given in [1].

Sulfur (α�S8) strongly differs from Si, Ge, and Cbecause it belongs to molecular crystals in which S8

molecules are coupled to each other by van der Waalsinteraction. Consequently, the temperature depen�dences of Raman spectra of α�S8 would make it possi�ble to estimate the isotope effects for both weak crys�talline (van der Waals) and strong intramolecular

(covalent) interactions. In addition, isotope effectscan be considered on vibrations of various types, intra�and intermolecular, bending, and stretching. Finally, itis very interesting to reveal the role of zero�point vibra�tions in the spectrum and their relative significance forvibrations of various frequencies and types.

These problems are studied in this work.

We used 32S, 33S, and 34S sulfur isotopes producedat the Siberian Chemical Plant with the isotope puri�ties of 99.9, 99.4, and 99.9%, respectively. To obtainthe chemically pure rhombic modification of sulfur(α�S), isotopes were subjected to triple distillation invacuum with their subsequent crystallization from car�bon tetrachloride. The sulfur isotopes thus obtainedhad a chemical purity no less than 99.9%.

Raman spectra were collected using a LabRAMHR (Horiba) spectrometer with the excitation by the488�nm line of the Ar+ laser. The power of lightfocused on a spot with a diameter of about 2 nm was0.8 mW on the surface of the sample. The spectra at alltemperatures were obtained in the backscatteringgeometry using a Raman microscope. The studiedcompound powder was indented into metallic indiumplaced on the cold finger of a helium cryostat (ST�500,Janis Company, United States). The stability of thetemperature better than 0.1 K was ensured in themeasurements by a temperature controller. All mea�surements were performed with a spectral resolutionof 0.7 cm–1.

When measuring the spectra of powders, thestrictly identical conditions of the local heating of asample by laser radiation for all temperatures cannotbe ensured. As a result, the spectral parameters of each

Isotope Effects in Raman Spectra of Crystalline Sulfur α�S8

B. A. Kolesova and N. B. Egorovb

a Nikolaev Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent’eva 3, Novosibirsk, 630090 Russia

e�mail: kolesov@niic.nsc.rub National Research Tomsk Polytechnic University, pr. Lenina 30, Tomsk, 634050 Russia

Received June 7, 2013

The Raman spectra of 32S, 33S, and 34S powders, as well as naturS powder with the natural isotopic composi�tion, have been measured in the temperature range of 5–300 K. The isotope mass dependences of the vibra�tional frequencies have been obtained. It has been shown that such a dependence in the region of mixed trans�lational–librational modes differs from the dependences known for monatomic inorganic crystals. The roleof zero�point vibrations for low� and high�frequency modes has been revealed throughout the entire temper�ature range.

DOI: 10.1134/S0021364013160078

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ISOTOPE EFFECTS IN RAMAN SPECTRA 135

vibrational mode (the position of the maximum,bandwidth, and integral intensity) are reconstructedafter the decomposition of the spectrum into Lorentzcomponents with certain noise. For this reason, tocompare the spectral characteristics of various iso�topes, we used the best fits to the experimental points,rather than the experimental points themselves. Thetemperature dependence of the peak position ω(T)was fitted by anharmonic processes * [2]:

(1)

where x(T) = �ω0/kBT. The sum of the constants Aand B is an anharmonic correction Δanh, which is thedifference of the frequency of the phonon ω0 (har�monic vibration that does not interact with othermodes) from the frequency of the experimentallyobserved phonon. However, this description does notinclude the correction to zero�point vibrations.

The theoretical expression proposed in [2] for thetemperature dependence of the peak position of thephonon under consideration based on the consider�ation of its coupling with two other phonons with alower energy into which the former phonon assuminglydecays is not appropriate for vibrational modes of alltypes and energies. However, we need a good analyticalfit to the experimental data in order to compare themwith each other for various isotopic compositions.

At this stage, it is impossible to accurately deter�mine the dependences of the bandwidths of the vibra�tional modes because the measured bandwidth for

ω T( ) ω0 A 1 2

ex/2 1–��������������+⎝ ⎠

⎛ ⎞+=

+ B 1 3

ex/3 1–�������������� 3

ex/3 1–( )2

�������������������+ +⎝ ⎠⎛ ⎞ ,

almost all isotopes at low temperatures is equal to thespectral resolution of the spectrometer (0.7 cm–1); i.e.,its actual value could be smaller.

The Raman spectrum of α�S8 (Fig. 1) consists ofthree regions separated by significant frequency inter�vals. These regions—30–100, 140–260, and 400–500 cm–1—are reasonably attributed to the external(crystalline), bending, and stretching vibrations of theS8 molecule, respectively. The S8 molecule has theform of a “crown” with an empty center. Therefore,the librational and translational modes should havecomparable frequencies because all eight sulfur atomsare involved in both displacement types. Thus, theregion 30–100 cm–1 is most likely assigned to mixedtranslational–librational modes. The presented sche�matic assignment of vibrations is sufficient for the

Fig. 1. Raman spectra of α�32S8, α�naturS8, α�33S8, and

α�34S8 powders at 5 K.

Fig. 2. Temperature dependences of the (a) position of themaximum, (b) bandwidth, and (c) integral intensity for the45�cm–1 mode of the α�32S8 powder. The solid line inpanel (a) is the fit of the temperature dependence of thefrequency of vibrations by Eq. (1), whereas the solid line inpanel (b) is the fit of the temperature dependence of thebandwidth by a similar expression describing the decay ofthe photon [2]. The solid line in panel (c) is plotted takinginto account the Boltzmann statistics. The arrows markthe features that are attributed to structural changes in thecrystal lattice and are typical of all vibrational modes andall isotopic compositions.

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KOLESOV, EGOROV

aims of this work. A more detailed description of thespectrum can be found in [3, 4].

Temperature dependence of the spectral parame�ters. The spectral parameters of all modes and all iso�topic compositions exhibit two features manifestedusually as narrow dips on the temperature dependenceof the wavenumber and peaks on the temperaturedependences of the bandwidth and intensity. They areobserved at 80 and 190 K for 32S, at 90 and 190 K fornaturS, at 110 and 190 K for 33S, and at 80 and 140 K for34S. The temperatures at which the indicated featuresare manifested depend on the isotopic composition.This means that the phenomenon is caused by changein intermolecular distances, which depend on boththe temperature and the amplitude of zero�pointvibrations. It can be assumed that the appearance ofthe features is due to small rigid rotations of S8 mole�cules with respect to each other in the crystal latticewhen the unit cell parameters change with tempera�ture. The phenomenon of the structural and confor�mational instability is widespread in molecular crys�tals and is associated with (a) the very weak intermo�lecular interaction and (b) the low symmetry ofcrystals, owing to which the lattice parameters varywith the temperature at different rates in differentcrystallographic directions.

Furthermore, certain modes (e.g., 79, 148, 152,183, 215, 219, and 236 cm–1 for 32S) exhibit an anom�alous temperature dependence of the peak position.

The anomaly is a decrease in the frequency of vibra�tions with a decrease in the temperature (beginning atapproximately 200 K) instead of its monotonicincrease characteristic of all other modes (see Fig. 2a).This effect is particularly pronounced for the intense215 and 236 cm–1 bands. Since almost all anomalousmodes are located in the region of bending vibrationsof the S8 molecule, it is reasonable to suppose that thegeometry of the frame of the molecule changes below200 K. (The anomalous temperature dependence forthe 79�cm–1 mode in the region of lattice vibrationsindicates that this mode refers to the rotation of onepart of the molecule with respect to another part; i.e.,it belongs to complex bending and restricted rotation,which were erroneously called [4] torsion vibrations.)

Isotopic�mass dependence of the vibrational fre�quency. Since the mode frequency ω0 in Eq. (1) is pro�portional to m–1/2 and the anharmonic correction Δanh

is proportional to m–1, the frequency ω0i shoulddepend on the isotope mass mi as [5]

(2)

The vibrational frequencies and masses of the 32S iso�tope are taken as reference values. Then, the vibra�tional frequency should follow Eq. (2) (the solidstraight line in Fig. 3, where the ω0 and Δanh are the

ω0i mi( ) ω0m0

mi

����� Δanh

m0

mi

����� .+=

Fig. 3. Isotopic�mass dependences of the of the wavenumbers of the modes at (a) 45, (b) 85, (c) 156, and (d) 478 cm–1 at T = 5 K,which are typical of their frequency ranges. The solid and dashed lines are the m–1/2 and m–1/4 dependences, respectively.

JETP LETTERS Vol. 98 No. 3 2013

ISOTOPE EFFECTS IN RAMAN SPECTRA 137

parameters of fit (1) for the wavenumber of the modeunder consideration). The second term on the right�hand side of Eq. (2) is one and a half or two orders ofmagnitude smaller than the first term. The isotopic�mass dependences of the mode wavenumbers shown inFig. 3 are typical of the vibrational frequency range ofthese modes. It can be seen that the isotopic�massdependences of the wavenumbers of intramolecular(bending and stretching) vibrations strictly followEq. (2). The corresponding dependences for mixedtranslational–librational modes strongly deviate fromEq. (2). The reason for this deviation is just in thenature of the external vibrations of a molecular crystal.The frequency of translational modes, as well as thefrequency of atomic vibrations in an inorganic crystalwithout molecular fragments, is proportional to m–1/2,where m is the mass of the molecule, whereas thefrequency of librations in the molecular crystal is

ω ∝ I ⎯1/2, where I = is the moment of inertia of

the molecule. The distances rj at low temperatures areproportional to the rms amplitude of zero�point vibra�tions Q0. This amplitude is given by the expression

(3)

Hence, the frequency of librations is

(4)

Consequently, the frequency of librations is propor�tional to m–1/4 rather than to m–1/2 as the frequency oftranslations in Eq. (2). The dashed lines in Fig. 3 showthe f(m–1/4) function calculated by the formula similarto Eq. (2). It can be seen that the experimental pointsfor low�frequency modes (30–100 cm–1) lie betweenthe m–1/2 and m–1/4 functions, confirming the assump�tion of the mixed character of low�frequency vibra�tions.

Manifestation of zero�point vibrations. Figure 4ashows the ratios of functions (1) fitting the experimen�tal temperature dependences of the wavenumbers ofvarious modes for two extreme�mass isotopes, 32S and34S. The dashed straight line is the expected ratio of thevibrational frequencies ω(32S)/ω(34S) = (34/32)1/2 =1.0308. The deviation of the experimental ratio fromthe theoretical value is the direct consequence of thedominance of zero�point vibrations at low tempera�tures with amplitude (3) proportional to m–1/4. (In thiscontext, low temperatures are those at which the Bolt�zmann population of this mode is close to zero.) Thepresence of zero�point vibrations and the dependenceof their amplitude on the isotope mass constitute themain isotope effect resulting in the difference in the

mjrj2

∑

Q0�

2mω��������� �

2m k/m������������������ m 1/4–

.∝= =

ω 1/2– I 1/2– 1

mr2���������� 1

mQ02

������������ 1

mm 1/2–�����������������∝ ∝ ∝ ∝ m 1/4–

.=

physical parameters of different pure isotopes (formore detail, see [1]). It can be seen that all presentedcurves approach (within the errors of the experimentand fitting) the expected ratio of the frequencies.However, low�frequency modes approach this valuemuch earlier (in temperature) than high�frequencymodes because the higher the Boltzmann populationof the given oscillator and, consequently, the rmsamplitude of excited vibrations, the smaller the contri�bution of zero�point vibrations in them. The ampli�tude of vibrations of each isotope in the high�temper�ature limit becomes proportional to T [1].

Figure 4a shows another feature that has not yetbeen discussed (as far as we know): the deviation of thefrequency ratio from the theoretical value is muchlarger for low�frequency modes than for high�fre�quency ones. The frequency ratio ω0(

32S)/ω0(34S) at

T = 5 K for various modes, where ω0 is the fittingparameter in Eq. (1) for the corresponding mode, isshown in Fig. 4b, where the solid line is drawn by theequation

(5)

where a and b are constants. The deviation of the ratioω0(

32S)/ω0(34S) from an expected value of (34/32)1/2 =

1.0308 is a result of change in the rms amplitude ofzero�point vibrations. The anomalous deviation forlow�frequency modes is due to the effect of the libra�tional component discussed above. For this reason, thefit of the experimental data by the ω–1/2 term in Eq. (5)most likely reflects the increase in the librational con�tribution to the vibrations in the low�frequencyregion, rather than the mode�wavenumber depen�

ω0 S32( )

ω0 S34( )

��������������� a b

ω������,–=

Fig. 4. (a) Ratio of the temperature dependences of thevibrational frequencies obtained by fitting with Eq. (1) forseveral pairs of vibrational modes of the 32S and 34S iso�topes. (b) Ratio of the vibrational frequencies of variouspairs of vibrational modes in the 32S and 34S isotopes at 5 K.

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KOLESOV, EGOROV

dence of the amplitude of zero�point vibrations, as canbe concluded at first glance.

To conclude, owing to the presence of numerousmodes of various types (low� and high�frequencymodes compared to kT, as well as intra� and intermo�lecular modes) in the vibrational spectrum of molecu�lar crystals, their Raman spectra contain morediverse information than the spectra of monoatomiccompounds. Although only the temperature depen�dence of the mode wavenumber of isotopically purecompounds has been discussed in this work, thebehavior of the bandwidths of vibrational modes andspectra of mixed isotopic compounds can also bestudied.

REFERENCES

1. M. Cardona and M. L. Thewalt, Rev. Mod. Phys. 77,1174 (2005).

2. M. Balkanski, R. F. Wallis, and E. Haro, Phys. Rev. B28, 1928 (1983).

3. M. Becucci, R. Bini, E. Castellucci, et al., J. Phys.Chem. I 101, 2132 (1997).

4. B. Eckert and R. Steudel, Top Curr. Chem. 231, 31(2003).

5. F. Widulle, T. Ruf, M. Konuma, et al., Solid StateCommun. 118, 1 (2001).

Translated by R. Tyapaev