Molecular spectroscopy of n-butane by incoherent inelastic neutron scattering

Nuclear Instruments and Methods in Physics Research A254 (1987) 563-569 North-Holland, Amsterdam

563

M O L E C U L A R S P E C T R O S C O P Y OF n - B U T A N E B Y I N C O H E R E N T INELASTIC NEUTRON SCATYERING W i l l i a m B. N E L L I G A N

a n d D a v i d J. L e P O I R E

Schlumberger- Doll Research, Ridgefield, CT 06877, USA C h u n - K e u n g L O O N G a n d T o r b e n O. B R U N IPNS and MST, Argonne National Laboratory, Argonne, IL 60439, USA S ow H s i n C H E N Nuclear Engineering Department, MIT, Cambridge, MA 02139, USA Received 24 July 1986

The intramolecular modes of vibration of butane have been studied from an energy transfer of 50 to 240 meV using an incident neutron energy of 300 meV from a high resolution chopper spectrometer. A liquid sample just above the melting point (140 K) and a solid sample at 10 K were investigated and the spectra compared. The absolute double-differential cross section was calculated for the solid sample using a force model which was derived from Raman and IR data. The agreement between the calculation and the experimentally measured cross section was good enough to reproduce all the spectral features. In addition, the low frequency inter and intramolecular modes of vibration have also been measured at 10 K by a high resolution crystal analyzer spectrometer covering a range from 1 to 100 meV. Although the cross section is not in an absolute scale the calculation using the same set of force constants with an addition of the two-phonon spectrum reproduces the measured spectrum satisfactorily.

1. Introduction Incoherent inelastic neutron spectroscopy (INS) has been long recognized as a potentially powerful complementary technique to the standard Raman and infrared spectroscopy for testing the force field models in solid molecular substances [1-4]. The necessary ingredients for applicability of INS include a neutron source capable of supplying substantial neutron flux in the energy range spanning the three decades from E 0 = 1 meV to 1 eV, a time-of-flight spectrometer capable of delivering the energy resolution A E / E o of 1-4%, in addition to being able to perform low angle scattering such that the magnitude of the wave-vector transfer Q be maintained at relatively low values, say less than 4 . ~ - l, at the energy transfer range of interest. The requirement of maintaining low Q values in these measurements is so that the cross section can be compared with the onephonon calculation. These experimental conditions are met by the intense pulsed neutron source (IPNS) and the high resolution medium energy chopper spectrometer (HRMECS) at Argonne National Laboratory. In the HRMECS a high energy chopper delivers neutrons with an incident energy in the range of 200 meV to 1 eV, and provides an 0168-9002/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

overall energy resolution A E / E o of the order of 2-4%. The scattered neutrons can be measured at a wide range of scattering angles from 3 ° to 120 ° . Thus by judicious choice of an incident energy and the detection angles, one can maintain reasonably low values of Q for the energy transfer E up to several hundred meV. As part of survey work on molecular vibrations of alkanes and aromatic compounds, we undertook a benchmark experiment on one of the simplest alkanes, butane, in order to establish the feasibility of obtaining a high quality INS spectrum in an absolute scale, for the purpose of comparison with a first principle cross section calculation using an established force field based on Raman and IR spectroscopic measurements. An extensive set of IR measurements by Schachtschneider and Snyder [5,6] on the n-alkanes provided the data for their successful analyses by a set of force constants known as the " S & S force field". Our intention was to use the S&S force field, with appropriate modifications of provide a match to our neutron data, for a first principle calculation of the INS spectra. An earlier neutron scattering measurement on solid polycrystalline n-butane at 106 K by Strong [2] covered an energy transfer range of 0 to 40 meV, and a later analysis of these data was made by Hudson et al. [3]

564

14/.B. Nelligan et al. / Molecular spectroscopy of n-butane

using a consistent force field of Lifson and Warshel [7]. There is also the INS work by Takeuchi et al. [8] covering about the same energy range at T = 10 K. These authors extracted the Q = 0 frequency spectrum G , ( t o ) and compared it with I R and R a m a n spectra. They also calculated the band frequencies and intensifies using the Shimanouchi force field [9]. Most recently Wang et al. [10] performed an INS measurement on n-butane adsorbed on graphite (0001) surfaces, covering an energy range of E = 30 to 400 meV. The energy resolution of the triple axis spectrometer used was not high ( A E / E o - 9%), so that some of the spectral features we shall describe in this paper were not resolved. A calculation based on a force field model due to Logan et al. [11], which was a modified form of the S & S model, reproduced the qualitative features of the measured INS spectra but the comparison was not quantitative. In this paper, we shall report a calculation of the INS spectra in the whole range of the spectrum from the lattice mode region to the intramolecular modes of vibration up to E = 240 meV, which is the same energy range measured in our experiments. The comparison with I N S spectra shows that satisfactory agreement can be obtained using a modified S & S force field, thus establishing a consistency of I R and R a m a n with INS in a quantitative comparison. We believe that this work demonstrates that INS is a useful tool for investigating quantitative spectral features in the energy range of E = 0 to 240 meV. In a separate work [12] one of the authors (S.H.C.) has used the H R M E C S to investigate the O - H stretch vibration of water at supercooled temperature. This result and the results in this paper show that the combination of I P N S and the H R M E C S can be successfully used to investigate quantitative spectral features of intramolecular vibrations up to at least E = 600 meV.

tion (see fig. lb): Q = 0.69512E 0 - E - 2 c o s g ~ E 0 ( E 0 - E ) ] 1 / 2 ~

(1)

In the present measurements we have chosen an incident neutron energy of 300 meV to explore the scattering spectra of liquid and solid butane over a wide range of m o m e n t u m and energy transfer (1 < Q < 5 A - z, 40 < E < 250 meV). The energy resolution, A E / E o, in general depends on the chopper in use and varies with energy transfer (see fig. lc) but is approximately 2-4%. The sample cell was made of 48 thin aluminum tubes assembled in a planar geometry to cover an area of

(a) Proton Pulses

l/ ~

~

U Target

300

12.8 m

P ; y(~tehrYltene ~z 300 K

Chopper

Detectors -20 ° < ~ < 20 ° 85 ° cp < 140 °

-2500

(b) o = 300 meV

_~200 I ~

1oo/~

,,5 10~

\

-50 0 o 10 ° 20 ° 30 °

=

2. Experiment

The experiments at energy transfer greater than 600 c m - 1 were performed with the H R M E C S at the I P N S of the Argonne National Laboratory. A schematic diagram of the H R M E C S instrument is shown in fig. la. Since the details of the I P N S chopper spectrometers have been given elsewhere [13], we shall only give a brief description of the apparatus here. A phased Fermi chopper produces pulses of monoenergetic neutrons which are incident on the sample. Scattering intensities at various energies and m o m e n t u m transfers are measures by neutron time-of-flight techniques using 150 detectors. For each detector the m o m e n t u m transfer h Q and energy transfer E (in meV) are coupled together so that a detector at a particular scattering angle 0 scans a parabolic locus in (Q, E ) space according to the equa-

1.

50°

(c)

-2000

\ \ \\\

,ooo

\, \,

_,oo ==

\ \

'~14

\18

70 °

\ \\o,;',

\22 \\

26

120 °

180 °

90 °

---

E0 - 300 meV

u~ .,~

2 1

0

,

50

~

1 0

~

1 0

i

200

,

250

~

Energy Transfer (meV) Fig. 1. Schematic diagram and the characteristics of the highresolution medium energy spectrometer (HRMECS) at the intense pulsed neutron source (IPNS) of Argonne National Lab. (a) Schematic arrangement. (b) (Q, E) space covered by the spectrometer for detectors at different scattering angles when the incident neutron energy E i = 300 meV. (c) calculated energy resolution with respect to the incident energy E 0 = E i = 300 meV as a function of the energy transfer E = E i - Ef.

W.B. Nelligan et aL / Molecular spectroscopy of n-butane

565

7.5 × 10 cm2. The inner and outer diameter of the tubes were respectively 1.27 and 1.52 mm. The sample holder was connected to a butane gas cylinder by a stainless steel capillary and the gas was condensed in the sample cell by cooling with a helium cyrogenic refrigerator. A total volume of 6.18 cm3 of butane sample was exposed to the beam, giving a neutron transmission of about 85%. The sample temperature was controlled to within 0.5 K. The neutron beam was perpendicular to the plane of the sample cell so that for low scattering angles the beam did not traverse more than 2 mm of the material. Such a geometry minimizes multiple scattering of the neutrons in the sample. A typical run took 60 h followed by 40 h of empty holder run. Measurements of the elastic incoherent scattering from a thin plate of vanadium (12 h) provided detector calibration and intensity normalization. Experiments at energy transfers less than 1000 cm-1 were performed with the crystal analyzer spectrometer (CAS). In this spectrometer the broad spectrum of energies from the IPNS is incident on the sample and the neutrons scattered through 90 ° with a final energy of 3.6 meV are detected. For this spectrometer the energy resolution is A E t / E t = 3% and

Table 1 Williams's potential parameters as used in eq. (93)

Q = 0 . 6 9 5 ( E + 2 E t ) x/z A

where q is the wave vector, r, is the position vector of the lattice site a, and i and j refer to the Cartesian coordinates x, y, z. The potential includes the internal modified S& S potential and the Williams potential. The dynamical matrix was transformed into the massweighted Cartesian system and the corresponding characteristic equation was solved for the frequencies and eigenvectors. A Monte Carlo averaging over the wave vectors in the first Brillouin zone produced a spectrum and could be compared with experiment.

1.

(2)

This measurement is more completely described in an accompanying paper [14].

3. Analysis method Calculations of the frequency and cross section for these energy ranges were made using a modified Schachtsneider and Snyder valence force potential for the internal modes [5,6,14] and a Williams potential for the lattice modes [15]. Geometries for the molecules used are d(C-C) = 1.54 A and d ( C - H ) -- 1.093 ,~ with all bonds formed at tetrahedral angles. The lattice parameters of the crystal were taken from a packing calculation [16]. Solution for the normal modes follows the methods outlined in refs. [17,18] for the internal modes in and in refs. [19,20] for the lattice modes. For the calculation of the molecular vibrations, the S&S force constants were transformed from the internal coordinate system to a mass-weighted Cartesian coordinate system [17] to construct the force constant matrix. Translational invariance was assumed so that the sum over the elements in a column or row is equal to zero. This procedure is similar to the one used by Gwinn [18] except that the transformations between the coordinate systems were computed directly and no numerical approximations were used. The force constant matrix was diagonalized with a computer program [21] to obtain the eigenfrequencies and eigenvectors of the normal modes.

Interaction

A [kcal tool-1 ~-6]

B [kcalmol- 1]

C [~ 1]

H-H H-C C-C

32.3 128.0 505.0

2630.0 11000.0 61900.0

3.74 3.67 3.60

For the lattice dynamics calculations, the Williams potential was added between atoms in different molecules. The Williams potential is expressed as v(r)

= -A/R

6+ B

exp(- CR),

(3)

where the constants A, B, and C for C-C, C - H and H - H interactions were given by Williams [15] and are here reproduced in table 1. The dynamical matrix was constructed between different atoms in the molecules, and u, by summing over the lattice sites,

D~,ij ( q ) = )". exp( - 2 ~riq. r,) Ox~iOxpj,

(4)

a

4. Cross section calculation Zemach and Glauber [1] derived an expression for the differential cross section for a neutron of energy E 0 to be inelastically scattered at an angle 8 from a molecule in thermal equilibrium at temperature T, and simultaneously exciting a set of intemal modes of vibration [nx] in the molecule. Two approximations are appropriate to the experimental conditions of our measurements. The temperature effects are ignored since k T - 7 cm-1 is much smaller than any energy transfer studied. The approximation for averaging over the orientation of the molecules discussed in ref. [3] was used after testing the validity of it by comparing the calculations using the approximation and calculations using numerical averaging. By considering scatterings in which only a single molecular vibrational mode is excited, i.e. n x , = l and n x , x , = 0 , such that E 0 - E f =

W.B. Nelligan et aL / Molecular spectroscopy of n-butane

566

h~0x, = E, the expression for the differential cross section becomes

dEfd$2

= ~.- 2., 4--~YvxD~8(E - h~°x). 1

(5)

v

The Debye-Waller factor for atom v is: D, = x//E-yvx,

(6)

where Y~x= hQ2¢2x//6~°xm,, o, and m y are the bound atom cross section and the mass of atom p, and %x is the mass-weighted normal mode eigenvector of atom in the mode h. The eigenvectors are normalized such that ,~1 c,xl 2 = 1. ~0x is the frequency of model h. The wave vectors of the incident and scattered neutron are k i and kf respectively and Q = k t - k i. For comparison with experiments the delta function in eq. (5) were replaced with Gaussians of width A E / E = 3%.

5. Results Since butane melts at T = 138 K, the HRMECS measurements were made on the solid phase ( T = 10 K) and the liquid phase just above the melting point (T = 140 K). Fig. 2 is a striking illustration of the potential of INS in the study of the change of the molecular vibrations across the melting transition of a molecular substance. We see clearly the smearing and merging of distinct individual molecular modes of vibration in a solid into broad bands in the liquid state. In the low

400

2.0

600

1.5

800 . .

1000 . .

temperature solid state a combined lattice and molecular vibrational analysis was made to compare with the measured neutron cross section. In the liquid state such analyses lose their meaning and a proper analysis method is instead a computer molecular dynamics approach. Such an analysis was done for butane at 140 K by Ullo and Yip [22] and quantitative agreement with our experimental data was obtained. Comparison in absolute units of a normal modes calculation (neglecting the lattice potential) and the HRMECS solid data is shown in figs. 3 and 4 for two different scattering angles. There is a different Q dependence in the energy scale for the different scattering angles. Table 2 lists all the observed INS peak positions and their mode assignments. Table 3 compares the observed and calculated total cross sections in three energy regions. It is seen that the data and the calculation agree to within 20%. A more detailed calculation showed that two-phonon excitations account for most of the structure at energy transfers above 1500 c m - 1. Measurements were also made of the energy dependence of the inelastic neutron single scattering cross section of butane at 10 K in the energy range of 3.8-125 meV (30-1000 cm-1). This range covers both lattice vibrational modes and the lower energy internal vibrational modes. Comparison in arbitrary units of the calculated one- and two-phonon cross sections (neglecting the lattice potential) with the CAS data is shown in fig. 5. The contribution of the two-phonon peaks is indicated by the hatched area. The calculation including the lattice potential is shown in fig. 6 along with the

1200

1600

1800 cm

Ei=300 meV

r,=Solid(1OK)

~

0=18 ° S

1400

,

o = L i q u i d (140K)

°o

1.0 =oo~%o

0.5

£ 40

60

80

100

120

140

160

180

200

220

240

E (meV) Fig. 2. Measured INS cross section S in a unit of S = 4~r

d2o[b] dg~dEf[meV]per butane molecule,

plotted as a function of the energy transfer E taken at HRMECS at two temperatures, below and above the melting point of butane.

W.B. Nelligan et aL / Molecular spectroscopy of n-butane

567

E(meV) 50

2.5

100

200

250

50

2.5 INS Spectrum

2.o

S(E)

E(meV) 150

(HRMECS)

1.5

2.C

A B

150

200

250

(HRMECS)

INS Spectrum

T=10°K ®=4° meV Eo=300

100

f

T=10°K O=18 °

1.5 S(E)

lO

1.0

0.5 0.0

,

~

,

,

E

D

L

~

0.5 i

~

J

0.0

Calculated Spectrum

K

i

I

i

i

I

I

I

Calculated Spectrum

2.0

2.0

1.5

1.5

S(E)

S(E)

1.0

1.(3

0.5 0.0 .

i

0.5 , ~ L j 200 400 600 800 1000 1200 1400 1600 1800 2000

0.0 0

E(cm -1)

I

I'~

I

I

I

200 400 600 800 1000 1200 1400 1600 1800 2000 E(cm -~)

Fig. 3. Measured I N S cross section and calculated spectrum of

Fig. 4. Measured INS cross section and calculated spectrum of n-butane. The measurement was made with HRMECS at a scattering angle of 18 ° . The same cross section units are used as in fig. 2.

n-butane. The measurement was made with HRMECS at a scattering angle of 4 ° . The same cross section units are used as in fig. 2. The calculated peaks were broadened to a resolution comparable with the resolution of the instrument. The labels correspond to the energy regions listed in table 1. I N S d a t a in this region. It was f o u n d t h a t the internal torsional force c o n s t a n t s h a d to be reduced from their values with n o lattice p o t e n t i a l b y a factor of 1.4 to r e p r o d u c e the observed torsional frequencies. T h e low

I

frequency translational a n d librational m o d e s below 100 cm -1 agree f a i r y well with the lattice d y n a m i c s calculations b u t the highest energy librational m o d e a n d the lowest energy internal m o d e s show dispersion which is n o t reproduced in the calculation. This m a y indicate

Table 2 Mode description of peaks observed in INS spectrum of butane Peak INS [cm x]

[meV]

A

1460

183

B

1380

172

C

1300

163

D

1130-1260

140-157

E

930-1100

115-138

F

640-880

80-110

Mode description

Number of modes

Calculated frequencies

Previously observed freq.[5,9]

asym.HCH bend (CH3) HCH bend (CH2) 2 sym. HCH bend (CH3) 1 wagging (CH2) wagging (CH2) twisting (CH2) 2 rocking (CH2) 1 twisting (CH2) skeletal stretch twisting, rocking skeletal stretch twisting, rocking

6 3

1445,1456, 1459 1459,1467,1468 1368,1373,1379

1455, 1459,1460 1462 1462,1468 1360,1375,1378

2

1281,1295

1293,1300

3

1148,1185,1261

1148,1181,1257

4

959, 977, 1004,1048 733,804, 830

944, 965, 1010,1053 733,805,836

3

568

W.B. Nelligan et aL

/ Molecular spectroscopy of n-butane

Table 3 Comparison of measured and calculated total inelastic neutron cross sections in three energy regions Energy region [meV]

Integrated cross section (scattering angle = 4 ° )

Integrated cross section (scattering angle = 18 ° )

Elow

Ehigh

Ie~)p

l~),c

R b)

I~)p

I~) ,:

R b)

75 111 137

111 137 190

7.94 8.29 38.7

6.60 7.45 36.0

1.20 1.11 1.10

13.2 13.0 47.0

13.4 11.7 40.7

0.98 1.17 1.15

Ehigh

t--jrmexnper butane molecule, b) R = lexp/Icalc" f S(E)dE;S=4~-~td~dd2°[blEt

a) I =

EI~

the inadequecy of the Williams potential [15] w h e n used in c o n j u n c t i o n with o u r modified S & S valence force field.

INS Spectrum (CAS) T=10°K 0=90 ° El=29 cm -~

r.o i

6. Conclusion and prospect £ It is clear from this b e n c h m a r k experiment o n b u t a n e t h a t the availability of I P N S together with H R M E C S -

E(meV) 0

25

50

75

100

125

'lW

'

I,

t L

I

I

I

I

I

\l

Calculated One-Phonon Lattice Spectrum

b

tO

tO

60

£

£

_~-T..~~ 20

~" o

Calculated One- and T w o - P h o n o n Spectrum

(/3

9

0 1(30'200 300 400 500 660 700 800 900 1000 E(cm -1) Fig. 5. Measured INS cross section and calculated one- and two-phonon spectrum of n-butane. The measurement was made with CAS at a scattering angle of 90 °. A constant background was subtracted. The two-phonon contribution is either from a combination of two internal modes (slash) or a combination of a lattice mode and an internal mode (cross).

40

60

8'0

100 120 140

160 180

200

E(cm 1) Fig. 6. Fig. 6a is an expanded low energy part of fig. 5a showing the lattice mode spectrum. Fig. 6b is a lattice dynamics calculation. Quadratic smoothing was applied to both experiment and calculation.

a n d CAS-type i n s t r u m e n t s will open up the full potential of I N S as a powerful chemical spectroscopic technique c o m p l e m e n t a r y to the traditional optical spectroscopic techniques. I n this p a p e r we d e m o n strated for the first time the degree of smearing of the i n t r a m o l e c u l a r b e n d i n g a n d wagging m o d e s (see table 3 a n d fig. 2) w h e n solid b u t a n e is melted. I n the low t e m p e r a t u r e solid p h a s e we were able to c o m p u t e the I N S cross section for the lattice m o d e s as well as for the i n t e r n a l m o d e s from a set of self-consistent force field models. It is i m p o r t a n t for the future work of I N S in

IV. B. Nelligan et al. / Molecular spectroscopy of n-butane

molecular solids to demonstrate that, for a simple and well characterized molecular solid like butane, a onep h o n o n plus a two-phonon calculation of the cross section is sufficient to achieve a quantitative agreement with the experiment. It is also clear from these calculations that more accurate and extensive INS measurements in the lattice mode region (0-200 cm 1) should be made to further refine the a t o m - a t o m potentials. A future direction in the force field study may be to find a more consistent way in which to incorporate the a t o m - a t o m potential into the well-established valence force field for the internal modes so that the INS spectrum in the transition region of 100 to 200 c m - 1 can be reproduced more satisfactorily.

Acknowledgements We thank I P N S at Argonne National Laboratory for providing us with sufficient spectrometer time to complete this set of experiments. Assistance from Dr. D.L. Price and R. Kleb of the Argonne National Laboratory in H R M E C S data reduction and sample chamber design and cooling respectively is gratefully acknowledged. This work was partially supported by the U S D e p a r t m e n t of Energy, Office of Basic Energy Scienc e s / M a t e r i a l Sciences, under Contract W-31-109-ENG38.

References [1] A.C. Zemach and R.J. Glauber, Phys. Rev. 101 (1956) 118. [2] D.M. Grant, R.J. Pugmire, R.C. Livingston, K.A. Strong, H.L. McMurry and R.M. Brugger, J. Chem. Phys. 52 (1970) 4424; K.A. Strong, AEC report, Catalogue of Neutron Molecular Spectra, IN1-1237, Idaho Nuclear Corporation.

569

[3] B. Hudson, A. Warshel and R.G. Gordan, J. Chem. Phys. 61 (1974) 2929. [4] S.H. Chen and S. Yip, Phys. Today 29 (1976) 32. [5] J.H. Schachtschneider and R.G. Snyder, Spectrochim. Acta 19 (1963) 117. [6] R.G. Snyder and J.H. Schachtschneider, Spectrochim. Acta 21 (1965) 169. [71 S. Lifson and A. Warshel, J. Chem. Phys 49 (1968) 5116 and 53 (1970) 582. [8] H. Takeuchi, G. Allen, S. Suzuki and A.J. Dianoux, Chem. Phys. 51 (1980) 197. [9] T. Shimanouchi, H. Matsuura, Y. Ogawa and I, Harada, J. Phys. Chem. Ref. Data 7 (1978) 1323. [10] R. Wang, H. Taub, H.J. Lauter, J.P. Biberian and J Suzanne, J. Chem. Phys. 82 (1985) 3465. [11] K.W. Logan, H.R. Danner, J.D. Gault and H. Kim, J. Chem. Phys. 59 (1973) 2305. [12] S.H. Chen, K. Toukan, C.K. Loong, D.L. Price and J Teixeira, Phys. Rev. Lett. 53 (1984) 1360. [13] D.L. Price, J.M. Carpenter, C.A. Pelizzari, S.K. Sinha, I. Bresof and G.E. Ostrowski, Proc. 6th. Meeting Int. Collaboration on Advanced Neutron Sources, ANL 82-80 (1983) p. 207. [14] W. Nelligan, D.J. LePoire, T.O. Brun and R. Kleb, to be published. [15] D.E. Williams, J. Chem. Phys. 47 (1967) 4680. [16] M.L. Cangeloni and V. Schettino, Mol. Cryst. Liq. Cryst. 31 (1975) 219. [17] E.B. Wilson, Molecular Vibrations (Dover, New York, 1955). [18] W.D. Gwinn, J. Chem. Phys. 55 (1971) 477. [19] B.T.M. Willis and A.W. Pryor, Thermal vibrations in Crystallography (Cambridge University Press, 1975) p. 39. [20] L.C. Brunel and D.A. Dows, Spectrochim. Acta 30A (1974) 929. [21] D. LePoire, Codes for the Calculation of Hydrocarbon Vibrational Modes, Schlumberger-Doll Research Internal Note (1986). [22] J.J. Ullo and S. Yip, J. Chem. Phys. 85 (1986) 4056.