- Email: [email protected]
Low frequency vibrations in solid n-butane and n-hexane by incoherent inelastic neutron scattering
Chemical Physics 51(1980) 197-203 0 North-Holland Publishing Company
LOW FREQUENCY VIBRATIONS IN SOLID n-BUTANE AND n-HEXANE BY INCOHERENT INELASTIC NEUTRON SCATTERING Ii. TAKEUCHI
*, G. ALLEN
Department of Chemical Engineering and Chemical Technology, Impetial College, London SW7 2BY. UK
s. SUZUKI Department of Ciremirrry and Applied Chemistry, University of SaIford, Salford MS 4 WT, UK arid A.J. DIANOUX Institut Max van Lnue-Paul Langevin, 38042 Grenoble Cedex. France Received 25 February
1980
Incoherent inelastic neutron scattering cross sections of solid n-butane and n-hexane have been measured at 10 K, from which the hydrogen-amplitude-weighted density of stat&s functions, G,(W), have been derived. A theoretical GH(w) has been calculated for n-hesane using a model force field. Comparison of the experimental and theoretical GH(LJ) leads to unambiguous assignments of the observed peaks due to methyl and skeletal torsions and a skeletal deformation vibration. For n-butane, ofwhich the crystal structure is not known, vibrational amplitudes of hydrogen atoms calculated for a free molecule have been shown to be helpful to interpret the observed GH(w)_
1. Introduction
of a powder sample. The function is proportional to the product of the sum of the squares of the vibrational amplitudes of hydrogen atoms and the density of states at energy w, and is called hydrogen-amplitude-weighted density of states GH(a). Advantages of deriving experimental G,(W) include: (a) G,(M) is characteristic of the substance and independent of experimental conditions such as scattering angle, incident neutron ener,y etc., (b) spectral band broadening hue to the Debye-Wailer factor and thermal weightings are eliminated in the data processing. Calculations of GH(w) are trivial though tedious if the intra- and inter-molecular force fields are known, and it is thus possible to compare the calculated GH(o) with the observed spectrum not only with respect to peak positions but also the intensity distribution. Vibrational spectra of n-paraffins have been investigated so far mainly by infrared absorption and
Incoherent inelastic neutron scattering (IINS) is a useful technique to investigate molecular motions where hydrogen atoms vibrate with large amplitudes. Neutron scattering events take place due to collisions with atomic nuclei, and the probability is proportional to the scattering cross section of atoms involved. Hydrogen has an incoherent cross section so much greater than the other elements that neutron scattering spectra of compounds containing hydrogen reflect the motions of hydrogen atoms predominantly. On the assumption that the scattering is caused by hydrogen atoms only, a function of transferred energy can be extracted from an experimental IINS spectrum * Present address: Department of Chemistry, Fxulty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan. 197
198
H. Takeuchi et al. /_Neutron scatfeting specrra of
Raman scattering [l--51. A few sets of intramolecular force constants have been obtained by normal coordinate analyses to esiablish the assignments of most infrared nnd FZ.amnnbands [ 1,6,7] _However, low frequency vibrations such as methyl and skeletal torsions have not been weli studied, because they are usually either weak or not ailowed by the selection rules in infrared and Raman spectra. IINS spectra of n-paraffins were observed by Logan et al. [S] to investigate low frequency vibrations in relation to the frequency dispersion of polyethylene vibrations. The observed peaks were assigned by comparing their positions with normal vibration frequencies calculated disregarding intermolecuIar interactions. Little attention was paid to the intensity distribution_ Hudson et al. [9] calculated IINS cross sections for low frequency intramolecular modes of Jz-butane and several branched and ring paraffins. The purpose of their caIcuIations was to reproduce the raw experimental data with respect to peak positions and intensities, and they seemed successful, although the calculations were based on more approsimations than the derivation of an experimental G,(w). The frequency dispersion of intramoIecuIar modes by the intermolecular force field was neglected in their calculations. Taub et al. also reported the IINS spectrum of II-butane at 77 K with an energy resolution of about IO cm-‘, in the course of their study of this molecule adsorbed on graphite [IO]. We have measured IINS cross sections of solid n-butane and ?z-hexane with medium resolution (5 cm-’ ) and statistics good enough to derive reliabIe G&w). The crystal structure ofn-butane is not known, while that of rz-hexane has been determined by X-ray diffraction [ 1 I]. Hence, the calcuIation of G,(w) has been performed for rz-hexane using a model force field and a norma! coordinate analysis of a free Iz-butane moIecule has been carried out to assign some of the observed peaks.
2. Experimental The experiments were carried out on the IN4 time-of-flight spectrometer at the Institut Max von Laue-PauI Langetin, Grenoble, France. The incident energy was 39.8 meV and the Stokes transitions were
n-burane
and n-hexme
measured. The detector boxes which contain six 3He detectors were distributed over an arc of radius 4 m, from 9 to 72’. The analysis of the scattered beam was made by recording the counting rate from the detectors in 512 8-ps time channels. Some of the spectra were eliminated in the data processing when strong reflections from the cryostat were observed. We measured the spectra of n-butane at 10 K (solid) and 150 K (liquid) but the spectrum at 150 K showed no additional features except for the broadening of the elastic peak in the region observed. This is different from the previous results [9] and is due to the fact that we were not able to look at low angles because of the divergence of the incident beam. Hence, the attempt to measure the spectrum of liquid Jz-hexane was abandoned and its spectrum at IO K only was measured.
3. Calculations In the harmonic approximation the hydrogenamplitude-weighted density of states G,(w) is defined by
where AUj(k) is the phonon ener,g of normal modej with a wave vector k, z@) is the eigenvector component for a hydrogen atom in the mass-weighted cartesian coordinate system [ 121. The integrations are carried out over the first Brillouin zone. Contributions of all the hydrogen atoms and normal modes are summed. The crystal structure of n-hexane has been revealed by an X-ray study at 158 K [ 1I] _The crystal belongs to the trichnic system: the space group is Cf -Pi with one molecule per unit cell. The carbon skeleton is pIanar and the centre of symmetry of the molecule coincides with that of the crystal lattice. The X-ray study was not able to determine the precise molecular structure, especiaIIy the positions of the hydrogen atoms. We assumed, therefore, the moIecuiar structura1 parameters: r(C-C) = 0.1539 nm, r(C...C) = 0.2533 nm,r(C-H) = 0.110 nm, and L CCH = 110’40’,
H- Takeuchi et 01. /Neutron
scdrering
which had been determined for n-butane by electron diffraction 1131. The molecular orientation relative to the crystal lattice was derived from the X-ray data. A few sets of intramolecular force constants have been proposed for n-paraffins [ 1,6,7]. We adopted the local symmetry force field obtained in a recent study by Shimanoucbi et al. [7] in the course of normal coordinate analyses of a series of hydrocarbons and related compounds. The molecular structural parameters used in the analyses are the same as we have assumed in the present calculation. For the intermolecular potential, we used the Williams set IV atom-atom potential which was obtained by simultaneous fitting of crystal structures, heats of sublimation, and elastic constants for a number of hydrocarbons [!43 _Although the original potential was based on the interaction centre of hydrogen atoms located at a distance of 0.104 nm from the carbon atom, we assumed the interaction centre at the position of nucleus as assumed in the normal coordinate treatments of longer PI-paraffins [2] and benzene [ 151. The equilibrium molecular orientation calculated with the potential was very close to that obtained from the X-ray data. All the interactions of C---C, C---H, and H---H pairs within 0.6 nm were taken into account. The first and second derivatives of the potential with respect to atom pair distances were first expressed in the Cartesian displacement coordinates and ?hen transformed to elements of the dynamical mat+, which was constructed on the basis coordinate set of three translational and three librational modes and six law frequency internal normal modes. A detailed description of the above transformation procedures has been presented by Taddei et al. [ 1S] _ The integration over k space involving the 6 function was carried out by applying the linear extrapolation method developed by Gilat et al. for the calculation of density of states [16]. The irreducible part of the triclinic Briliouin zone was divided into 500 identical parallelepipeds of the same shape as the Brillouin zone. By diagonalizing the dynamical matrix at the centre ?f the parallelepipeds, Oj(k), its gradient aw,(k)/ak, and u?$)‘were obtained. Throughout each parallelepiped u?(k) was assumed constant and w](k) w3s linearly extrapolated. The number of phonon modes with energy between hw and rZ(w + Aw) is proportional to the volume of the layer confined in
spectra ofn-butane
199
and n-hexane
the parallelepiped by two constant energy planes of Rw and fi(o + nw). A general method of calculating the volume of such layers is described elsewhere [ 171. The channel width, Rhw, was 0.2 cm-’ . As mentioned in section 1, the crystal structure of ?z-butane is not known. We calculated, therefore, only z&O) using Shimanouchi’s force field [7] in order to estimate the relative intensity in C,(w) for intramolecular modes.
4. Results and discussion
Fig. 1 compares the experimentai +(a) with the calculated one. The experimenta! GH(w) was derived
al
CO
100
1M EtiERGY W
200 lori’l
250
Fig. I. GH(W) of rz-hexane: (a) experimental, (b) calculated. Solid and broken vertical bars indicate Raman and infrared active frequencies, respectively.
H. Takuchi er nl. / Neumn scarrering spectra of n-butane and n-hexane
200 Table 1 Observed and c&z&ted
vibrational
Observed RaIna 53 74 87
a)
frequencies
(ii cm-‘) of rr-hexane
Calculated neutron b,
free molecule
(53) 72
137 111 1.50
66 80 100 114 150 161 189
236 2z.s
270 276
==I00
(175) 149 174
182
Z52
(1% (205) 241
crystal
12
mode
liiration LTKitiOIl
liintion c-c torsion (Bu) C-C-C deform. &I,) c-c torsion (3”) C-C torsion (bg)
CH3 torsion (a,) CH, torsion (b,)
a) Solid at 20 K. At 140 R the 53 cm-’ band was not detected and the 74 and 87 cm-’ bands shifted to 68 and 76 cm-l, respectively, ref. [41. b, At 10 K. Frequencies in parentheses indicate shoulder peaks.
from the foltowing equation,
where S(o, p) is the scattering function. The details are given in ref. [ 181. Good agreement is seen in the intensity distribution, though there is some discrepancy in the peak positions. A little modification of the intra- or inter-molecular force constants could yield a better agreement in the peak positions, but would not affect the intensity distribution significantly. The agreement in intensity is sufficient for the assignment of the observed peaks. The frequencies of the observed peaks are listed in table 1 together with those of observed Raman bands and calculated zone centre (X-= 0) modes. Fig. 2 shows the calculated dispersion curves of vibrational frequencies, which il!ustrates the contribution of each vibrational
mode to the calculated peaks in
GH(w).
strongest peak observed at 247 cm-’ is clearly due to the two methyl torsional modes overlapping each other. Because their frequencies are far from those of the other branches and show little dependence on the wave vector, the peak is ascribed to almost pure methyl torsions. The 182 cm-’ peak with a shoulder at 195 cm-’ can be assigned to the b, C-C torsional mode, calculated around 200 cm-‘. The
The peak position is about IO cm-r higher than the corresponding Raman active frequency in both the experimental and the calculated spectra (fig. 1). The 149 cm-r peak, broadened towards the low frequency side, is ascribed to a mixing of three vibrational modes, i.e. the b, C-C-C deformation and two a, C-C skeletal torsions. These three modes mix significantly when the wave vector has a non-zero a* component, where Q* signifies the reciprocal lattice vector perpendicular to both b and c real lattice vectors. Although only a small shoulder
is detected
at 17-5 CIII-~ in the
observed spectrum, the calculation predicts a sharp peak at this frequency, of which origin is the au C-C skeletal torsion. This indicates that more dispersion and mode mixing take place in that frequency region in the real crystal. In the lattice mode region, three peaks are observed in the experimental &(w): a broad peak around IO0 cm-‘, a relatively strong peak at 72 cm-r, and a small shoulder at 53 cm-‘. Although the intensity distribution in this region does not show good agreement between the observed and the calculated r.&(w), the observed peak positions are rather well reproduced in the calculation. By means of the dispersion curves in fig. 2, we tentatively assign the 100 and 72 cm-’ peaks to Iibrations and the 53 cm-’ peak to acoustic phonons.
H. Takeuchi et al. /Neutron scattering spectra of n-butane and n-hexme
201
t fi3glC-C
TOR
iA,l
C-C
TOR
IEl,I
CCC
OEF
(A,l c-c TOR L-1
WAVE
Fig. 2. Calculated
_.
VECTOR
dispersion cuwes of low frequency
Brunei and Dows [4] observed Raman spectra of solid ?z-hexane at various temperatures. The frequencies observed in the lattice mode region were 53,74, and 87 cm-* at 20 K and 68 and 76 cm-’ at 140 K. The 53 cm-’ band, which was weak and broad even at 20 K, was not detected above 110 K because it illcreased in frequency and broadened rapidly as the temperature was raised. In order to interpret the Raman spectra Brunei and Dows calculated the frequencies of three librational modes using the same intermolecular potential function as used in the present calculation but with a different position of the interaction centre for hydrogen atoms. They assumed the interaction centre at a distance of 0.101 nm from the carbon atom, which was significantly shorter than the value (0.110 nm) used here. Their calculated frequencies were, therefore, lower than those of our calcuiation listed in table 1 by about 9 cm-‘, and were close to the observed Raman frequencies at 20 K. Accordingly they assigned the three Raman bands to the librational fundamentals. Their calculation, however, does not seem successful to explain the 100 cm-’ peak in the experimental G,(w). Therefore, the lattice vibrations of n-hexane should be investigated further, especially in connection with the unusual anharmonicity of the 53 cm-’ Raman band and the origin of the 100 cm-’ peak in G&w).
vibrations in crystalline n-hexane.
The experimental GH(w) is given in fig. 3. In table 2, comparison of the peak positions with the observed infrared and Raman frequencies is given
1
0
1
50
I
100
1
150 ENERGY
W
I
200
L
250
kmi’l
Fig. 3. Experimental GH(cJ) of n-butane. Solid and broken vertical bars indicate observed Raman and infrared frequencies. respectively.
H. Takeuchi et al. /Neutron scattering spectra of n-butane and n-hexane
202 Table 2 Observed zmd calculated
vibrational
frequencies (in cm-‘) of n-butane -
Observed
Czdculated
infrazed 3)
Rxnan~)
neutron h)
6s 80 104
(62) 80 93 125 167 236
233 265
free molecule c,
I&d)
mode
122 209 244
0.44 0.97 0.87
CH3 torsion (a,,) CHa torsion (b,)
265
0.39
C-C-C
C-C torsion (au)
265 266.5 3) Solid nt 77 K, ref. [5]. b, At 10 K. The frequency
4 Ref. 171. d) & = x&
in par_enthesesindicates a tioulder.
+_& +5&j.
together with the calculated frequencies
for a free
moIecule. In order to compare the relative intensity of intramolecular vibrational modes, the sup of the squares of hydrogen displacement amplitudes p$ = Z,@(O)\* wns calculated instead of GH(w)- The ratio of p& for the a, CH3 torsion to the sum of those for the b, C-C-C deformation and the b, CH3 torsicn is 0.77, which agrees with the experimentai height ratio, 0.75, of the 236 cm-’ peak to the 265 cm-’ peak. Therefore, the 236 cm-’ peak is assigned to the a, CH3 torsion and the 265 cm-’ peak is ascribed to the b, CH3 torsion and b, C-C-C deformation overlapping each other, in accordance with the assignments proposed by Hudson et al. [9] _ The band width of the methyl torsion peak is relatively narrow compared with the other bands. This indicates that the frequency dispersion of the methyl torsion is small as in the case of n-hexane. The 167 cm-’ peak is due to the a, C-C skeletal torsion. Its intensity is in good agreement with that predicted from ph _As the crystal structure of tz-butane in the stable phase is considered to be isomorphous to that ofrr-hexane [5], three degrees of librational freedom are expected_ These librational modes and acoustic modes
deform. (5,)
compose
a broad
band
with two maxima
at
and a shoulder at 6:! cm-‘. The weak peak at 125 cm-’ might also be due to the librational motion. 93 and ?G cm-’
5. Conclusion We have been able to interpret the IINS spectra of solid n-hexane satisfactorily on the basis of the calculated G”(w). Hydrogen amplitudes of molecular vibrations have been shown to be helpful to assign some of the observed peaks in G,(w) of solid n-butane The CH3 torsional frequencies in these molecules
have been established.
Acknowledgement We thank Dr. S.-B. Suck for his assistance in data processing and Mr. H..Walter for his technical assistance during the experiments.
References
[ 11 3.H. Schachtschneider
and R.G. Snyder, Spectrochim. Acta 19 (1963) 85; R.G. Snyder and J.H. Schachtschneider, Spectrochim. Acta 19 (1963) 117. [Z] H. Takeuchi, T. Shimanouchi, ht. Tasumi, G. Vergoten and G. Fleury, Chem. Phys. Letters 28 (1974) 449. 131 I. Handa, H. Takeuchi, AI. Snkakibnm, ti. ~atsuura and
T. Shimanouchi, BulLChem. Sot. Jpn. SO(1977) 102. [41 L.C. Brunei and D.A. Dows, Spectrochim. (1974) 929.
Acta, 30
H. Takeuchi et [5] ML. Cangeloni and V. Schettino,
~1. /Neutron
scattering spectra ofn-butane and n-hexane
Mol. Cryst. Liquid Crvst. 31 (1975) 219. [6] S. Lifson and A: Warshel, 5. Ctem. Phys. 49 (1968) 5116; A. Warshel and S. Lifson, J. Chem. Phys. 53 (1970) 582. [7] T. Shimanouchi, H. Matsuura, Y. Ogawa and I. Harada, J. Phys. Chem. Ref. Data 7 (1978) 1323. [8] KW. Logan, H.R. Danner, J. D. Gault and H. Kim, J. Chem. Phys. 59 (1973) 2305. [9] B. Hudson, A. Warshel and R.C. Gordon, J. Chem. Phys 61 (1974) 2929. I101_ H. Taub. H.R. Danner. Y.P. Sharma, H.L. hlchIurry and _ R.M. Brugger, Phys. Rev. Letters 39 (1977) 215. [ 1 l] N. Norman and H. Mathisen, Acta Chem. Stand. 15 (1961) 1755.
203
[12] P.A. Reynolds, J.K. Kjems and J.W. White, J. Chem. Phys. 56 (1972) 2928. [13] K. Kuchitsu. BulLChem. Sot. Jpn. 35 (1959) 748. [14] D.E. WiUiams, J. Chem. Phys. 47 (1967) 4680. [ 151 G. Taddei, H. Bonadeo, h1.P. Marzocchi and S. Cahfano, J. Chem. Phys. 58 (1973) 966. [ 161 G. Gilat and L.J. Raubenheimer, Phys. Rev. 114 (1966) 390; L.J. Raubenheimer and C. Gilat, Phys. Rev. 157 (1967) 586. [ 171 H. Takeuchi, Ph.D. Thesis, The University of Tokyo, Japan (1976). [la] P.A. Egelstaff and P. Schofield, J. Nucl. Sci. Eng. 12 (1962) 260.
LOW FREQUENCY VIBRATIONS IN SOLID n-BUTANE AND n-HEXANE BY INCOHERENT INELASTIC NEUTRON SCATTERING Ii. TAKEUCHI
*, G. ALLEN
Department of Chemical Engineering and Chemical Technology, Impetial College, London SW7 2BY. UK
s. SUZUKI Department of Ciremirrry and Applied Chemistry, University of SaIford, Salford MS 4 WT, UK arid A.J. DIANOUX Institut Max van Lnue-Paul Langevin, 38042 Grenoble Cedex. France Received 25 February
1980
Incoherent inelastic neutron scattering cross sections of solid n-butane and n-hexane have been measured at 10 K, from which the hydrogen-amplitude-weighted density of stat&s functions, G,(W), have been derived. A theoretical GH(w) has been calculated for n-hesane using a model force field. Comparison of the experimental and theoretical GH(LJ) leads to unambiguous assignments of the observed peaks due to methyl and skeletal torsions and a skeletal deformation vibration. For n-butane, ofwhich the crystal structure is not known, vibrational amplitudes of hydrogen atoms calculated for a free molecule have been shown to be helpful to interpret the observed GH(w)_
1. Introduction
of a powder sample. The function is proportional to the product of the sum of the squares of the vibrational amplitudes of hydrogen atoms and the density of states at energy w, and is called hydrogen-amplitude-weighted density of states GH(a). Advantages of deriving experimental G,(W) include: (a) G,(M) is characteristic of the substance and independent of experimental conditions such as scattering angle, incident neutron ener,y etc., (b) spectral band broadening hue to the Debye-Wailer factor and thermal weightings are eliminated in the data processing. Calculations of GH(w) are trivial though tedious if the intra- and inter-molecular force fields are known, and it is thus possible to compare the calculated GH(o) with the observed spectrum not only with respect to peak positions but also the intensity distribution. Vibrational spectra of n-paraffins have been investigated so far mainly by infrared absorption and
Incoherent inelastic neutron scattering (IINS) is a useful technique to investigate molecular motions where hydrogen atoms vibrate with large amplitudes. Neutron scattering events take place due to collisions with atomic nuclei, and the probability is proportional to the scattering cross section of atoms involved. Hydrogen has an incoherent cross section so much greater than the other elements that neutron scattering spectra of compounds containing hydrogen reflect the motions of hydrogen atoms predominantly. On the assumption that the scattering is caused by hydrogen atoms only, a function of transferred energy can be extracted from an experimental IINS spectrum * Present address: Department of Chemistry, Fxulty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan. 197
198
H. Takeuchi et al. /_Neutron scatfeting specrra of
Raman scattering [l--51. A few sets of intramolecular force constants have been obtained by normal coordinate analyses to esiablish the assignments of most infrared nnd FZ.amnnbands [ 1,6,7] _However, low frequency vibrations such as methyl and skeletal torsions have not been weli studied, because they are usually either weak or not ailowed by the selection rules in infrared and Raman spectra. IINS spectra of n-paraffins were observed by Logan et al. [S] to investigate low frequency vibrations in relation to the frequency dispersion of polyethylene vibrations. The observed peaks were assigned by comparing their positions with normal vibration frequencies calculated disregarding intermolecuIar interactions. Little attention was paid to the intensity distribution_ Hudson et al. [9] calculated IINS cross sections for low frequency intramolecular modes of Jz-butane and several branched and ring paraffins. The purpose of their caIcuIations was to reproduce the raw experimental data with respect to peak positions and intensities, and they seemed successful, although the calculations were based on more approsimations than the derivation of an experimental G,(w). The frequency dispersion of intramoIecuIar modes by the intermolecular force field was neglected in their calculations. Taub et al. also reported the IINS spectrum of II-butane at 77 K with an energy resolution of about IO cm-‘, in the course of their study of this molecule adsorbed on graphite [IO]. We have measured IINS cross sections of solid n-butane and ?z-hexane with medium resolution (5 cm-’ ) and statistics good enough to derive reliabIe G&w). The crystal structure ofn-butane is not known, while that of rz-hexane has been determined by X-ray diffraction [ 1 I]. Hence, the calcuIation of G,(w) has been performed for rz-hexane using a model force field and a norma! coordinate analysis of a free Iz-butane moIecule has been carried out to assign some of the observed peaks.
2. Experimental The experiments were carried out on the IN4 time-of-flight spectrometer at the Institut Max von Laue-PauI Langetin, Grenoble, France. The incident energy was 39.8 meV and the Stokes transitions were
n-burane
and n-hexme
measured. The detector boxes which contain six 3He detectors were distributed over an arc of radius 4 m, from 9 to 72’. The analysis of the scattered beam was made by recording the counting rate from the detectors in 512 8-ps time channels. Some of the spectra were eliminated in the data processing when strong reflections from the cryostat were observed. We measured the spectra of n-butane at 10 K (solid) and 150 K (liquid) but the spectrum at 150 K showed no additional features except for the broadening of the elastic peak in the region observed. This is different from the previous results [9] and is due to the fact that we were not able to look at low angles because of the divergence of the incident beam. Hence, the attempt to measure the spectrum of liquid Jz-hexane was abandoned and its spectrum at IO K only was measured.
3. Calculations In the harmonic approximation the hydrogenamplitude-weighted density of states G,(w) is defined by
where AUj(k) is the phonon ener,g of normal modej with a wave vector k, z@) is the eigenvector component for a hydrogen atom in the mass-weighted cartesian coordinate system [ 121. The integrations are carried out over the first Brillouin zone. Contributions of all the hydrogen atoms and normal modes are summed. The crystal structure of n-hexane has been revealed by an X-ray study at 158 K [ 1I] _The crystal belongs to the trichnic system: the space group is Cf -Pi with one molecule per unit cell. The carbon skeleton is pIanar and the centre of symmetry of the molecule coincides with that of the crystal lattice. The X-ray study was not able to determine the precise molecular structure, especiaIIy the positions of the hydrogen atoms. We assumed, therefore, the moIecuiar structura1 parameters: r(C-C) = 0.1539 nm, r(C...C) = 0.2533 nm,r(C-H) = 0.110 nm, and L CCH = 110’40’,
H- Takeuchi et 01. /Neutron
scdrering
which had been determined for n-butane by electron diffraction 1131. The molecular orientation relative to the crystal lattice was derived from the X-ray data. A few sets of intramolecular force constants have been proposed for n-paraffins [ 1,6,7]. We adopted the local symmetry force field obtained in a recent study by Shimanoucbi et al. [7] in the course of normal coordinate analyses of a series of hydrocarbons and related compounds. The molecular structural parameters used in the analyses are the same as we have assumed in the present calculation. For the intermolecular potential, we used the Williams set IV atom-atom potential which was obtained by simultaneous fitting of crystal structures, heats of sublimation, and elastic constants for a number of hydrocarbons [!43 _Although the original potential was based on the interaction centre of hydrogen atoms located at a distance of 0.104 nm from the carbon atom, we assumed the interaction centre at the position of nucleus as assumed in the normal coordinate treatments of longer PI-paraffins [2] and benzene [ 151. The equilibrium molecular orientation calculated with the potential was very close to that obtained from the X-ray data. All the interactions of C---C, C---H, and H---H pairs within 0.6 nm were taken into account. The first and second derivatives of the potential with respect to atom pair distances were first expressed in the Cartesian displacement coordinates and ?hen transformed to elements of the dynamical mat+, which was constructed on the basis coordinate set of three translational and three librational modes and six law frequency internal normal modes. A detailed description of the above transformation procedures has been presented by Taddei et al. [ 1S] _ The integration over k space involving the 6 function was carried out by applying the linear extrapolation method developed by Gilat et al. for the calculation of density of states [16]. The irreducible part of the triclinic Briliouin zone was divided into 500 identical parallelepipeds of the same shape as the Brillouin zone. By diagonalizing the dynamical matrix at the centre ?f the parallelepipeds, Oj(k), its gradient aw,(k)/ak, and u?$)‘were obtained. Throughout each parallelepiped u?(k) was assumed constant and w](k) w3s linearly extrapolated. The number of phonon modes with energy between hw and rZ(w + Aw) is proportional to the volume of the layer confined in
spectra ofn-butane
199
and n-hexane
the parallelepiped by two constant energy planes of Rw and fi(o + nw). A general method of calculating the volume of such layers is described elsewhere [ 171. The channel width, Rhw, was 0.2 cm-’ . As mentioned in section 1, the crystal structure of ?z-butane is not known. We calculated, therefore, only z&O) using Shimanouchi’s force field [7] in order to estimate the relative intensity in C,(w) for intramolecular modes.
4. Results and discussion
Fig. 1 compares the experimentai +(a) with the calculated one. The experimenta! GH(w) was derived
al
CO
100
1M EtiERGY W
200 lori’l
250
Fig. I. GH(W) of rz-hexane: (a) experimental, (b) calculated. Solid and broken vertical bars indicate Raman and infrared active frequencies, respectively.
H. Takuchi er nl. / Neumn scarrering spectra of n-butane and n-hexane
200 Table 1 Observed and c&z&ted
vibrational
Observed RaIna 53 74 87
a)
frequencies
(ii cm-‘) of rr-hexane
Calculated neutron b,
free molecule
(53) 72
137 111 1.50
66 80 100 114 150 161 189
236 2z.s
270 276
==I00
(175) 149 174
182
Z52
(1% (205) 241
crystal
12
mode
liiration LTKitiOIl
liintion c-c torsion (Bu) C-C-C deform. &I,) c-c torsion (3”) C-C torsion (bg)
CH3 torsion (a,) CH, torsion (b,)
a) Solid at 20 K. At 140 R the 53 cm-’ band was not detected and the 74 and 87 cm-’ bands shifted to 68 and 76 cm-l, respectively, ref. [41. b, At 10 K. Frequencies in parentheses indicate shoulder peaks.
from the foltowing equation,
where S(o, p) is the scattering function. The details are given in ref. [ 181. Good agreement is seen in the intensity distribution, though there is some discrepancy in the peak positions. A little modification of the intra- or inter-molecular force constants could yield a better agreement in the peak positions, but would not affect the intensity distribution significantly. The agreement in intensity is sufficient for the assignment of the observed peaks. The frequencies of the observed peaks are listed in table 1 together with those of observed Raman bands and calculated zone centre (X-= 0) modes. Fig. 2 shows the calculated dispersion curves of vibrational frequencies, which il!ustrates the contribution of each vibrational
mode to the calculated peaks in
GH(w).
strongest peak observed at 247 cm-’ is clearly due to the two methyl torsional modes overlapping each other. Because their frequencies are far from those of the other branches and show little dependence on the wave vector, the peak is ascribed to almost pure methyl torsions. The 182 cm-’ peak with a shoulder at 195 cm-’ can be assigned to the b, C-C torsional mode, calculated around 200 cm-‘. The
The peak position is about IO cm-r higher than the corresponding Raman active frequency in both the experimental and the calculated spectra (fig. 1). The 149 cm-r peak, broadened towards the low frequency side, is ascribed to a mixing of three vibrational modes, i.e. the b, C-C-C deformation and two a, C-C skeletal torsions. These three modes mix significantly when the wave vector has a non-zero a* component, where Q* signifies the reciprocal lattice vector perpendicular to both b and c real lattice vectors. Although only a small shoulder
is detected
at 17-5 CIII-~ in the
observed spectrum, the calculation predicts a sharp peak at this frequency, of which origin is the au C-C skeletal torsion. This indicates that more dispersion and mode mixing take place in that frequency region in the real crystal. In the lattice mode region, three peaks are observed in the experimental &(w): a broad peak around IO0 cm-‘, a relatively strong peak at 72 cm-r, and a small shoulder at 53 cm-‘. Although the intensity distribution in this region does not show good agreement between the observed and the calculated r.&(w), the observed peak positions are rather well reproduced in the calculation. By means of the dispersion curves in fig. 2, we tentatively assign the 100 and 72 cm-’ peaks to Iibrations and the 53 cm-’ peak to acoustic phonons.
H. Takeuchi et al. /Neutron scattering spectra of n-butane and n-hexme
201
t fi3glC-C
TOR
iA,l
C-C
TOR
IEl,I
CCC
OEF
(A,l c-c TOR L-1
WAVE
Fig. 2. Calculated
_.
VECTOR
dispersion cuwes of low frequency
Brunei and Dows [4] observed Raman spectra of solid ?z-hexane at various temperatures. The frequencies observed in the lattice mode region were 53,74, and 87 cm-* at 20 K and 68 and 76 cm-’ at 140 K. The 53 cm-’ band, which was weak and broad even at 20 K, was not detected above 110 K because it illcreased in frequency and broadened rapidly as the temperature was raised. In order to interpret the Raman spectra Brunei and Dows calculated the frequencies of three librational modes using the same intermolecular potential function as used in the present calculation but with a different position of the interaction centre for hydrogen atoms. They assumed the interaction centre at a distance of 0.101 nm from the carbon atom, which was significantly shorter than the value (0.110 nm) used here. Their calculated frequencies were, therefore, lower than those of our calcuiation listed in table 1 by about 9 cm-‘, and were close to the observed Raman frequencies at 20 K. Accordingly they assigned the three Raman bands to the librational fundamentals. Their calculation, however, does not seem successful to explain the 100 cm-’ peak in the experimental G,(w). Therefore, the lattice vibrations of n-hexane should be investigated further, especially in connection with the unusual anharmonicity of the 53 cm-’ Raman band and the origin of the 100 cm-’ peak in G&w).
vibrations in crystalline n-hexane.
The experimental GH(w) is given in fig. 3. In table 2, comparison of the peak positions with the observed infrared and Raman frequencies is given
1
0
1
50
I
100
1
150 ENERGY
W
I
200
L
250
kmi’l
Fig. 3. Experimental GH(cJ) of n-butane. Solid and broken vertical bars indicate observed Raman and infrared frequencies. respectively.
H. Takeuchi et al. /Neutron scattering spectra of n-butane and n-hexane
202 Table 2 Observed zmd calculated
vibrational
frequencies (in cm-‘) of n-butane -
Observed
Czdculated
infrazed 3)
Rxnan~)
neutron h)
6s 80 104
(62) 80 93 125 167 236
233 265
free molecule c,
I&d)
mode
122 209 244
0.44 0.97 0.87
CH3 torsion (a,,) CHa torsion (b,)
265
0.39
C-C-C
C-C torsion (au)
265 266.5 3) Solid nt 77 K, ref. [5]. b, At 10 K. The frequency
4 Ref. 171. d) & = x&
in par_enthesesindicates a tioulder.
+_& +5&j.
together with the calculated frequencies
for a free
moIecule. In order to compare the relative intensity of intramolecular vibrational modes, the sup of the squares of hydrogen displacement amplitudes p$ = Z,@(O)\* wns calculated instead of GH(w)- The ratio of p& for the a, CH3 torsion to the sum of those for the b, C-C-C deformation and the b, CH3 torsicn is 0.77, which agrees with the experimentai height ratio, 0.75, of the 236 cm-’ peak to the 265 cm-’ peak. Therefore, the 236 cm-’ peak is assigned to the a, CH3 torsion and the 265 cm-’ peak is ascribed to the b, CH3 torsion and b, C-C-C deformation overlapping each other, in accordance with the assignments proposed by Hudson et al. [9] _ The band width of the methyl torsion peak is relatively narrow compared with the other bands. This indicates that the frequency dispersion of the methyl torsion is small as in the case of n-hexane. The 167 cm-’ peak is due to the a, C-C skeletal torsion. Its intensity is in good agreement with that predicted from ph _As the crystal structure of tz-butane in the stable phase is considered to be isomorphous to that ofrr-hexane [5], three degrees of librational freedom are expected_ These librational modes and acoustic modes
deform. (5,)
compose
a broad
band
with two maxima
at
and a shoulder at 6:! cm-‘. The weak peak at 125 cm-’ might also be due to the librational motion. 93 and ?G cm-’
5. Conclusion We have been able to interpret the IINS spectra of solid n-hexane satisfactorily on the basis of the calculated G”(w). Hydrogen amplitudes of molecular vibrations have been shown to be helpful to assign some of the observed peaks in G,(w) of solid n-butane The CH3 torsional frequencies in these molecules
have been established.
Acknowledgement We thank Dr. S.-B. Suck for his assistance in data processing and Mr. H..Walter for his technical assistance during the experiments.
References
[ 11 3.H. Schachtschneider
and R.G. Snyder, Spectrochim. Acta 19 (1963) 85; R.G. Snyder and J.H. Schachtschneider, Spectrochim. Acta 19 (1963) 117. [Z] H. Takeuchi, T. Shimanouchi, ht. Tasumi, G. Vergoten and G. Fleury, Chem. Phys. Letters 28 (1974) 449. 131 I. Handa, H. Takeuchi, AI. Snkakibnm, ti. ~atsuura and
T. Shimanouchi, BulLChem. Sot. Jpn. SO(1977) 102. [41 L.C. Brunei and D.A. Dows, Spectrochim. (1974) 929.
Acta, 30
H. Takeuchi et [5] ML. Cangeloni and V. Schettino,
~1. /Neutron
scattering spectra ofn-butane and n-hexane
Mol. Cryst. Liquid Crvst. 31 (1975) 219. [6] S. Lifson and A: Warshel, 5. Ctem. Phys. 49 (1968) 5116; A. Warshel and S. Lifson, J. Chem. Phys. 53 (1970) 582. [7] T. Shimanouchi, H. Matsuura, Y. Ogawa and I. Harada, J. Phys. Chem. Ref. Data 7 (1978) 1323. [8] KW. Logan, H.R. Danner, J. D. Gault and H. Kim, J. Chem. Phys. 59 (1973) 2305. [9] B. Hudson, A. Warshel and R.C. Gordon, J. Chem. Phys 61 (1974) 2929. I101_ H. Taub. H.R. Danner. Y.P. Sharma, H.L. hlchIurry and _ R.M. Brugger, Phys. Rev. Letters 39 (1977) 215. [ 1 l] N. Norman and H. Mathisen, Acta Chem. Stand. 15 (1961) 1755.
203
[12] P.A. Reynolds, J.K. Kjems and J.W. White, J. Chem. Phys. 56 (1972) 2928. [13] K. Kuchitsu. BulLChem. Sot. Jpn. 35 (1959) 748. [14] D.E. WiUiams, J. Chem. Phys. 47 (1967) 4680. [ 151 G. Taddei, H. Bonadeo, h1.P. Marzocchi and S. Cahfano, J. Chem. Phys. 58 (1973) 966. [ 161 G. Gilat and L.J. Raubenheimer, Phys. Rev. 114 (1966) 390; L.J. Raubenheimer and C. Gilat, Phys. Rev. 157 (1967) 586. [ 171 H. Takeuchi, Ph.D. Thesis, The University of Tokyo, Japan (1976). [la] P.A. Egelstaff and P. Schofield, J. Nucl. Sci. Eng. 12 (1962) 260.