methyl libration in propane measured with neutron inelastic scattering

Methyl Libration in Propane Measured with Neutron Inelastic ScatteringDavid M. Grant, Ronald J. Pugmire, Robert C. Livingston, Kenneth A. Strong, Henry L. McMurry, andRobert M. Brugger Citation: The Journal of Chemical Physics 52, 4424 (1970); doi: 10.1063/1.1673668 View online: http://dx.doi.org/10.1063/1.1673668 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/52/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Inelastic neutron scattering study of methyl groups rotation in some methylxanthines J. Chem. Phys. 127, 214509 (2007); 10.1063/1.2803187 Molecular dynamics simulation of inelastic neutron scattering spectra of librational modes of watermolecules in a layered aluminophosphate AIP Conf. Proc. 479, 195 (1999); 10.1063/1.59466 Inelastic incoherent neutron scattering study of the methyl rotation in various methyl halides J. Chem. Phys. 86, 2563 (1987); 10.1063/1.452059 Silicon measurement in a lung phantom by neutron inelastic scattering Med. Phys. 9, 550 (1982); 10.1118/1.595100 Dielectric and Neutron Inelastic Scattering Measurements on Phenanthrene J. Chem. Phys. 54, 2597 (1971); 10.1063/1.1675217

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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 52, NUMBER 9 1 MAY 1970

Methyl Libration in Propane Measured with Neutron Inelastic Scattering

DAVID M. GRANT, RONALD J. PUGMIRE, AND ROBERT C. LIVINGSTON

Department of Chemistry, University of Utah, Salt Lake City, Utah 84112

AND

KENNETH A. STRONG, HENRY L. McMURRY, AND ROBERT M. BRUGGER

Idaho Nuclear Corporation, Idaho Falls, Idaho 83401

(Received 17 November 1969)

The A2 and B2 torsional vibrational frequencies in propane gas have been directly measured at 217 ±8 and 265±8 em-I, respectively, with inelastically scattered neutrons using the" small K" method of neutron molecular spectroscopy. The theoretical basis of the scattering experiment is discussed, and factors affecting resolution and intensity are enumerated. Not subject to the same selection rules as optical spectroscopy, neutron molecular spectroscopy is capable of observing vibrational transitions which are otherwise forbidden by molecular symmetry. The vibrational frequencies are interpreted by numerically solving the two-dimensional wave equation giving librational motion with a two-dimensional Fourier expansion. In this approach, the potential energy is conveniently expanded also in Fourier form and we obtain a V3=3680±190 cal/mole for individual methyl-group librations and a V33 or V6= -280±130 cal/mole term which couples the two methyl tops. Evaluation of additional Fourier constants in the potential-energy expansion must await additional spectroscopic data concerning higher vibrational levels. Intermolecular perturbations of the librational frequencies were observed in both liquid and solid propane. These are sufficiently small, however, that one can conclude, to a first approximation, that the librational motions are not greatly effected in condensed states of propane.

I. INTRODUCTION rotational spectrum which can be observed in the microwave region under high resolution. The microwave

Since the establishment of the existence of a potential spectrum of liquid propane4 has been measured and the barrier hindering rotation of the methyl groups about barrier heights to internal rotation given a lower limit the C-C bond in ethane,! a great deal of effort has been of 2700 caI/mole. Recently, Hirota et al.5 have observed exerted to observe and characterize such barriers to the microwave spectrum of propane in the first excited internal rotation in various organic molecules. Potential states of the CHa torsion. From their data they conclude barriers are presumably caused by the interaction of that the rotational barrier consists of a thre~fold and a two groups of electrons and nuclei and, in principle, it negative sixfold parameter, Va and V6 of magnitudes should be possible to determine the barrier heights from 3325±20 and -170±20 caI/mole, respectively. Hoystraightforward quantum-mechanical calculations. The land6 ha~ analyzed the data of Hirota et al. by means of mathematical complexity of such a treatment, however, a Gaussian basis set Hartree-Fock-Roothaan SCF is so great that rigorous computations were impractical procedure and arrives at values of Va= 3575 and V6= until the recent availability of large, high-speed digital -310 caI/rrLole. computers. These recent computer advances have Weiss and LeroF have reported observation of the spurred increased theoretical interest in the rotation ethane torsional motion in the infrared by using a barrier height in ethane.2 Characterization of the lO-m sample path length at a pressure of 6 atm. These potential barrier height and shape in ethane and other results were followed by a recent attempt to observe simple hydrocarbons is important if one is to understand the infrared-active B2 torsional frequency in propane the nonbonded interactions which play such a signifi- by the high-pressure-Iong-path-Iength technique.8

cant role in molecular conformation and reaction rates. Even though a weak peak was observed at 264 cm- l

The lack of a wide range of accurately measured these authors conclude that the B2 torsional frequency barrier heights of nonpolar molecules has made it cannot be assigned with any certainty from their data. difficult to evaluate theoretical calculations. In the past, Thermal neutrons inelastically scattered from a potential barriers were inferred from heat capacity and sample can complement the ir and Raman measureentropy measurements, ultrasonic relaxation techniques, ments and recently neutron scattering has been successand combination bands in infrared (ir) and Raman fully employed as a sensitive probe for studying atom spectroscopies. Even in the case where symmetry vibrations in molecules and in crystalline solids. A few allows the torsional vibration modes to be ir or Raman representative examples of this type of work are given active, quite frequently the line intensities are so weak in Refs. 9-12. In these experiments both the energy as to make direct observation impossible or subject to exchange and momentum exchange experienced when considerable error. Microwave spectroscopy provides the neutron interacts with the sample are measured to another means for indirect measurement of methyl reveal the frequencies of vibrations in the sample. In librations.a Internal rotation interacts with the over-all these measurements the thermal neutrons have the rotation of a molecule and produces certain effects in its following advantages as compared to electromagnetic

4424


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METHYL LIBRATION IN PROPANE 4425

radiation and charged particle interactions. First, the effective energy range of thermal neutrons is intermediate between the far-infrared and the high-energy side of the microwave region of the spectrum and, hence, they are well suited to investigating vibrational modes which lie within this particular energy range. Within this energy range one finds vibrational modes which reflect perturbations due to important nonbonded interactions between atoms within a molecule. Second, there are no restrictive selection rules arising from symmetry such as exist for optical spectroscopies. Thus, these two factors permit measurement of molecular excitations not readily observed by other spectroscopic techniques.

The use of thermal neutrons to observe torsional motions in gaseous compounds has been atttemptedlO •12

and Straker13 has reported a torsional vibration band in ethane centered at 0.034 eV. Previous slow neutron studies of ethane14 and propane15 yielded reduced partial cross sections but the values of neutron momentum transfer hi K I at energy exchanges near that of the torsional transition were too large to observe the hindered torsional motion. However, by reducing the momentum transfer sufficiently, Strong and Brugger16

observed inelastically scattered neutrons at angles 4.8 and 8.2 deg and reported the ethane torsional frequency of ethane gas to be 0.0344±0.001O eV which corresponds to a barrier of 2,750 cai/mole. This direct observation by the "small K" method17 was next used in studying the symmetric and asymmetric lib rations in liquid neopentane.I8 Frequencies at 0.0260±0.0005 and 0.0330±0.OOOS eV were assigned to the A2 and F2 methyl librational modes, respectively, for transitions between the ground and first excited states in neopentane. The existence of two distinct librational frequencies in neopentane established the importance of a sizable negative sixfold barrier as well as the dominant Va contribution to the potential barrier for methyl rotation in molecules where coupling can occur between the various rotating methyl groups.

In the present work using the small K method, we report the direct observation of librational frequencies of propane in the gas, liquid, and solid states of matter. The directly observed vibrational frequencies in the gas phase are then used to characterize the height and shape of the barrier to internal rotation. A theoretical analysis of the barrier potentials is formulated in terms of a two-dimensional Fourier expansion of the potential. The resulting equation is of the form of Hill's modification of the Mathieu Equation.

II. THE SCATTERING EQUATIONS

A. Theoretical Basis

Comprehensive reviews of slow neutron scattering theory have been given by Lomer and Low and Janik

and Kowalska.9 A recent book by Parks et al. 19 gives a very complete treatment of the theory and applications. The equations employed here are adapted from the basic work of Zemach and Glauber20 whose formalism is particularly suited to treating systems where the particle dynamics is calculable. Simplifications first suggested by Krieger and N elkin21 in which average values are used for functions which vary with molecular orientation are used here. Earlier theory pertinent to the present calculations has been given by McMurry; Marsden; and McMurry, Russell, and Brugger.22

Scattering data provide the double differential scattering cross section u(Eo, E, Qo, Q). This is defined as the number of neutrons emerging per unit solid angle about the direction Q which are found in a unit energy range about E, divided by the flux of incident neutrons of energy Eo and direction Qo, and by the total number of scattering particles per unit volume of sample. Usual units for u are barns/particle· electron volt· steradian, where the term "particle" can mean atom, molecule, or molecular group. For example, the scattering by polyethylene can be given both as per H atom or per CH2 group.

For systems in thermal equilibrium that scatter isotropically (such as gases, liquids, amorphous solids, or powdered crystals), it is convenient to write u in the form2a

u(Eo, E, Qo, Q)=[1+(E/EO)]I!2 exp(-,8/2)S(K,,B). (1)

In Eq. (1) E=E-Eo and ,8=E/kB T. The function S(K, ,8), called the reduced partial differential cross section (RPDC), has the same units as u and is even in ,8. The K= I k- ko I, where ko and k are the initial and final wave vectors of the neutron. The neutron momentum change is xh. A function which appears repeatedly in scattering theory is

')'2= K2h2/2m= Eo+ E- 2J.1.(EoE)1/2. (2)

In Eq. (2) J.I.= Qo' Q. For nonisotropic systems such as single crystals Eq. (1) must be replaced by one in which the function S depends on the direction of x as well as its magnitude.

In the literature it is common to write S in terms of hw= -E=Eo-E rather than E. Here w is the Fourier transform variable customarily used in the equation which relates the cross section to the space-time correlation function G (r, t). 24 In this theoretical discussion the variables ,8 or E are used in order to be more consistent with theoretical work21 •22 which is based on the Z.G. formalism.20 However, the data presentations are in terms of hw whenever this is consistent with Common practice.

It is the function of the theory to show how u and S depend on the dynamics of the scattering systems and, therefore, how the experimental data can reveal physically interesting information about the system.

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4426 G RAN T ETA L .

B. Equation for the Cross Section the Krieger-Nelkin approximation21 (XpO)T is given by

The experiments reported here measure cross sections which depend primarily on energy transfers between levels characterized by CH3 torsional motions. If interference effects are neglected, scattering events which induce a transition from state i to state j of a single torsional mode T give a cross section (J'if,T which is ex "Pressed approximately by

Sij,T= [Eo/EJI/2 exp({3/2)(J'if,T' (3)

In this equation PiT = probability that the scattering molecule will be in the ith state for the T mode, and jp= number of atoms of type I' in the molecule. By "type" is meant environment as well as nucleus. For example, the H atom attached to the methylene carbon on propane is considered to be different from an H on a methyl group. The (a,2) is the appropriate average for the square of the scattering length for the interaction of the neutron with atom 1'.9,19,20,24 The t:..{3T={3/T-{3iT' where 13fT and {3iT are the energies of the final and initial torsional states in units of h/knT. The quantity a p is a measure of the energy transferred to the molecular recoil and is related to K2

(4)

Here M, is the Sachs-Teller25 effective mass for scattering by atom 1'. It is given by9,19,21,22,25

Here M is the total molecular mass, Ix, III, and 1. are moments of inertia about principal axes, and xp, YP, zp are the equilibrium coordinates of atom I' in these axes.

The matrix element MifPT between the initial and final states of the T mode is given by

In this equation \}i iT and \}ifT are the wavefunctions for the T mode in the initial (i) and final (f) states, and t:..~PT is the displacement of atom I' during this vibration, For harmonic vibrations McMurry26 has given a useful formalism for calculating atom displacements t:..~p in terms of normal modes. The factor (XpO)T arises from a product IIl\(LiPil\ [ Miipl\ 12) for all normal modes A whose energies do not change in the interaction. However, the sum takes account of interactions with molecules with these modes in thermally excited state. In

(XpO)T= exp (--y2Bp),

B p= L (a pl\2/El\) coth({3l\/2), l\

apl\2= Hm/m p)c pl\2.

The m p is the mass of atom I' and Cpl\2 is proportional to the square of the amplitude of motion of atom I' in the A normal mode. The El\ is the energy spacing for the A mode and {3l\=El\/kBT.

When K2 (and, therefore, a p) is small (J'if, T will be a maximum when

(6 )

Under these conditions Eq. (6) also gives the value of {3 near the maxima in Sif, .. The relative intensities of the transitions for different values of i andj are then determined by PiT and the [ M ifpT [2. Particularly when a p (or K2) is small Eq. (6) is theoretically well founded and provides a good basis for determining vibrational level spacings from peaks in the (J' vs f curves.

The equations for the [ M ifPT [2 are readily obtained if harmonic-oscillator wavefunctions are used. These relations will be rather qualitative for strongly anharmonic vibrations, but they do provide useful conclusions. In particular, it is found that for small K

the single quantum matrix elements [ M i ,i+1.PT [2 are proportional to K2 (or -y2). It can be shown also that higher quantum transitions of one mode, or simultaneous transitions involving several modes, are proportional to powers of (-y2)n, where the integer n is 2 or more. Therefore, when K2 is small such transitions are weak compared with the single quantum transitions.

C. Basis of the Small K Method

If the values of (3 satisfying Eq. (6) are substituted into Eq. (3) for a situation where scattering by H atoms dominates it is possible to show that the intensity around the peak varies as exp ( - t:..{32/ 4aH ). Here t:..{3 = {3-{3m, where 13m is the value of {3 satisfying Eq. (6). The half-width t:..{31/2 across the resonance is given by

t:..{31/2= 2 (aH In2 )1/2 or

t:..fl/2= (t:..{31/2)kBT=2[(d p In2)knT y2JI/2, (7)

dp=m/M p,

Equation (2) shows that -y2= K2h2/2m is smallest when ,u---tl. As a result, the peaks are sharpest (t:..fl/2 is smallest) for forward scattering. For Eo"'-'kBT measurements near ,1.1 = 1 benefit downscattering more than upscattering of cold neutrons This is because in upscattering Eo is small compared with fm and under these conditions -y2 is insensitive to changes in ,1.1. For downscattering experiments Eo is comparable to fm and the line narrowing on going toward ,1.1=1 may be very significant.


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METHYL LIB RATION IN PROPANE 4427

Line narrowing can be demonstrated quantitatively for the common case where (m/M v)Eo«ET. Then Eq. (6) shows

Under these conditions when IL"-'l the half-width in cold neutron experiments is proportional to (ET )1/2. In a downscattering experiment it is proportional to (EO)I/2X (ET/ Eo). Thus, the cold neutron width is larger than the downscatter width by the ratio (Eo/ ET )1/2. (Here Eo is the energy in the downscattering experiments and Eo>ET.)

In downscattering it is possible also to sharpen the lines by reducing the temperature as far as possible without causing a phase change. However, if this is done in a cold neutron experiment the populations of the thermally excited levels are reduced and a serious intensity reduction may occur.

The ability to locate and separate peaks is determined by the resolution. The width of the peaks arising from the scattering process as described in the preceding paragraphs explains only part of the resolution present in the experiment. This part is referred to as the "recoil resolution" since it is related to the uncertainty in speed and orientation of the entire molecule. The second part, the "energy resolution" will be discussed in Sec. II.D.

The data can be presented as S vs hw or S vs (3. If terms in m/M v are neglected, and if aH is assumed to vary slowly, Eq. (7) also describes the half-width of the S (K, (3) function. As in the case of (J vs E this halfwidth decreases linearly with K.

For a given Eo and Em the smallest K is obtained when IL is near 1, i.e., for forward scattering. However, when Em and IL are fixed there will be one value of Eo which gives the smallest K. This Eo is given by

Eo= - !!;'ET/.N2 (1- IL2)1/2 (1- IL2+ 1 )1/2

';::;!!;'ET/2 (1- IL2)1/2 (8)

when IL~1. In a downscattering experiment It IS necessary to

have Eo> I !!;'ET I. If Eo/I !!;'ET I = 2, then Eq. (8) corresponds to an optimum scattering angle of about 15°. The essence of the small K method lies in the proper selection of Eo, and IL to give good resolution with satisfactory intensity.

Finally, the peak intensities depend on the factors [1+ (E/Eo)]1/2 and on the I Mi.i+l.vT 12 which are largest in upscattering experiments. However, this advantage tends to be offset because (xvO)T involves a negative exponent which is proportional to K2. This factor reduces the intensity of upscattering experiments and cannot be altered greatly by going to forward angles.

D. Energy Resolution

As mentioned in Sec. II.C, the "recoil resolution" which arises from the scattering process is one factor that determines the peakwidth and, thus, the ability of

Uncou pled Mod e

Uncoupled Mode

q

(0)

o u

C. o

+ "0 ~ VI ::> o u «

3 --=:=-1

(b)

FIG. 1. Examples of (a) a dispersion relation for a molecular solid showing three acoustical and three optical modes and (b) the corresponding frequency distribution showing the bands.

the experiment to isolate peaks. The second factor contributing to the peakwidth is the resolution introduced by the mechanical limitations of the experimental equipment. Ideally, one desires a uniquely monoenergetic beam and the ability to precisely determine the energy and angle of the scattered neutrons with high statistical accuracy.

The phased chopper velocity selector falls short of this optimum precision, yet it represents the present state of the art. Uncertainties in energy and angle are introduced both into the incident beam and the scattered beams. The standard deviation of a Gaussian representing the burst passed by the velocity selector has been developed27 and is

52= ({X+[( T/To)3-1JY+aj2M~2

+ {X +[ (T/To)L 1]Yj2!!;.T12) (24a2)-I. (9)

Here, S is the standard deviation, !!;.Tl and !!;.T2 are the open times of the first and second choppers, a is the distance between the two choppers, X is the distance of the chopper near the sample (the second chopper) to the detectors, while Y is the distance from the sample to the detector, T is the reciprocal velocity of neutrons scattered from the sample, and TO is the reciprocal velocity of the neutrons striking the sample. Equation (9) is given in reciprocal velocity or time units because the velocity selector is inherently a timing device. Past experience has shown that the function B sin2(!!;.t)/!!;./2 cutoff at the first minimum is a better representation of the velocity selector neutron burst than is a Gaussian. Therefore, this function, with a width dependence on time of flight assumed to be the same as a Gaussian, was used to resolution broaden the theory to compare with data. This procedure gave theoretical peaks which contained both recoil and energy resolution.


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4428 GRANT ET AL.

TABLE I. Transformation properties of symmetry operations.

Related C2v

Operation elements <>1

E E <>1 P, C2 <>2 P2 u -<>2 Pa u' -<>1 Cal <>1 + 21J"/3 Cal2 <>1+41J"/3 Ca2 <>1 Ca22 <>1

One notes from Eqs. (8) and (9) that while the recoil resolution is improved by going to higher incident energy Eo, the energy resolution deteriorates. Thus, some compromise must be struck between recoil and energy resolution to provide the optimum probability of separating peaks.

In addition to the resolutions which determine the shapes of the peaks, consideration must be given to the statistical resolution. While the MTR (materials testing reactor) is a high flux reactor and the velocity selector has been designed to optimize counting rates, the counting rates are still low, i.e., the order of 1 count/ min· channel in the region of interest. The thermal beam flux distribution in time from the velocity selector peaks near 0.07 eV and drops off rapidly on either side. Attempts to use incident energies greater than 0.1 eV to improve the recoil resolution or less than 0.01 eV to improve the energy resolution have not been satisfactory with the existing equipment. The present experimental arrangement has matched fairly well the scattering angle, and the incident energy for searching for transitions in the lS-60-meV range. Any attempt to improve on the recoil resolution, such as a smaller scattering angle which will then require a larger incident energy, will also require a major increase in incident flux to deliver the necessary counting rates.

E. Free vs Bound Molecules

The discussions of the previous subsections which arrive at Eq. (3) start from the basis of free noninteracting molecules and arrive at a description of the scattering process which allows intramolecular vibrations to be observed. Since the molecules are free, there is no interaction of the nth vibration of one molecule with the nth vibration of another molecule. For liquid and solid samples, however, molecules cannot in general be considered noninteracting, and the possibility of coupling between adjacent molecules must be considered. The result of such coupling is a lattice wave or phonon in a crystal.

In Fig. 1 intramolecular vibrations are represented by optical branches to the dispersion relations. For modes that are prominently coupled to similar modes in

Symmetry angles

<>2 <PI </>2

<>2 </>1 </>2 <>1 </>1 -</>2

-<>1 -</>I </>2 -<>2 -</>I -</>2

<>2 </>1+1J"/3 </>2+1J"/3 <>2 </>1+27r/3 <1>2 + 27r/3 <>2 + 21J"/3 </>1+1J"/3 </>2-1J"/3 <>2+47r/3 </>1+27r/3 </>2-27r/3

neighboring molecules, a pronounced dispersion is evident in the branch. These branches produce bands in the frequency distribution, and, thus in the neutron scattering data. Peaks or cutoffs are evident in the band [Fig. 1 (b) ] corresponding to the energies in the dispersion relation at which a branch has a slope equal to zero. Such neutron scattering spectra28 must be treated as a measurement of a frequency distribution and not as a measurement of an isolated intramolecular state.

For modes of the dispersion relation that have minimal coupling, the intramolecular vibrations are represented by flat branches of the dispersion relations. These modes are essentially free, and are the modes of interest in the experiments discussed in this paper. One should keep in mind the limit of validity discussed above when considering neutron scattering data from liquid and solid samples.

III. EXPERIMENTAL

A. Equipment

The data presented in this study were obtained with the MTR-phased chopper velocity selector in the "small K" mode of operation and the MTR serving as the source of neutrons. The velocity selector, which has been described in detail elsewhere,29 is arranged to deliver monoenergetic neutrons of energy Eo to be scattered from the sample. Subsequent time analysis of the scattered neutron spectra gives the final neutron energy after scattering and the energy change of the neutron can thus be determined. Experimental details have been given by Strong and Bruggerl6 and similar procedures were followed to obtain the gaseous propane data. Recently, however, 18-in. active length 3Re detectors have been employed at angles of 4.1±1.6 and 7.9±2 deg with respect to the incident beam. With 10 such detectors located at the 4.1-deg angle and a bank of 12 detectors at the larger angle, the count rate for small angle scattering has been increased by nearly a factor of 2 over that reported previously.16

B. Sample Preparation Solid and liquid samples were contained in thin

walled (0.OO3-in.) aluminum tubing of small bore


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TABLE II. Characters for species of group I (Ca.XCa.).

E 4Ca 2Ca1Ca2 2CalCa22

Al 1 1 A2 1 1 1 1 BI 1 1 1 1 B2 EI 2 -1 -1 2 E2 2 -1 -1 2 Ea 2 -1 2 -1 E4 2 -1 2 -1 Q 4 -2 -2

(O.022-in. i.d.) separated by strips of gadolinium foil (O.002-in.) to reduce multiple scattering. The samples were introduced in gaseous form and then condensed to the liquid or solid state by means of liquid nitrogen passing around the periphery of the sample holder. Gaseous samples were contained in 2-in. i.d. aluminum cans with lO-mil walls. Cadmium spacers are placed in these containers to limit multiple scattering. The sample assembly consists of two identical sample holders, one empty and one containing the compound to be studied; these are automatically cycled into the beam for alternate lO-min intervals. Data obtained from the empty sample holder are used for background corrections.

Scattered neutron spectra from the sample and from the empty container are stored in separate halves of an 8192 channel time-of-flight analyzer. The time initiation for the analyzer is provided by a magnetic pickup pulse which occurs at a time corresponding to the opening of the second velocity selector chopper. Two beam-monitor fission chambers placed 1 m apart in the forward unscattered beam provide spectra from which the incident neutron energy and inherent electronic delays may be determined. Each scattering angle and the beam-monitor calibration data were allotted lO24 analyzed channels of 2t-JLsec width. Total experimental run time for each set of data was about 24 h.

Research-grade propane (99.99 mole% purity) was obtained from Phillips Petroleum Company and used without further purification.

IV. SPECTROSCOPIC THEORY

A. The Symmetry of Propane

Propane in its equilibrium conformation exhibits C2v symmetry which allows the normal modes to be designated as 9A 1+5A2+7B1+6B2• The two methyl librational modes of interest in this molecule belong to the A2 and the B2 symmetry species. If the threefold rotational symmetry properties of each methyl group are also considered along with symmetry operations similar to those contained in the C2v group, then propane forms the basis for a 36-element group30 which is isomorphous with the Ca.XCa• direct product group.

3PI 6PICalCa2 3P2 6P2CalCa22 9Pa

1 1 1 -1 -1 -1

-1 -1 -1 -1 1 -1 -1 1 1 -1

2 -1 0 0 0 -2 1 0 0 0

0 0 2 -1 0 0 0 -2 1 0 0 0 0 0 0

The essential operations of this group, discussed by Myers and Wilson and referred to by them as group I, are given in Table I for the methyl rotational coordinates designated by al and a2 or in terms of the symmetry coordinates CPl and c/>2, specified by

CPl=t(al+a2); c/>2=t(al-a2)' (lO)

The symmetry coordinates cP and CP2 form bases for irreducible representations of both the C2v and I symmetry groups. The group I is not a point group and as such its operations include but are not restricted to only those transformations which send any given propane configuration into an identical structure. Instead, the group contains all symmetry elements which will convert any propane conformation into all equivalent conformations of the same total energy, thereby leaving the Hamiltonian invariant. As such, group I holds for any propane molecule even though it has been torsionally distorted from its equilibrium configuration. The character table given in Table II for the nine classes in group I is obtained by mUltiplying the character table of the C3• group by itself. The two torsional normal modes associated with symmetry coordinates CPo and c/>2 are still characterized in this group by the irreducible representations labeled A2 and B2• These symmetry species may therefore be used to reduce the secular equation obtained for the wave equation describing the torsional vibrational motion.

B. The Kinetic-Energy Expression for Propane

The kinetic energy of a "rigid molecule"31 having "k" equivalent methyl rotors is given as follows:

T=tCl)·I·CI)+V" L c¥k2+I" L CI)'4ik, (11)

k k

where CI), tik , I, and I" are, respectively, the angular momentum for over-all molecular rotation, methyl rotational angular momentum, the molecular moment of inertia tensor, and the moment of inertia of a single methyl group. Equation (11) may be derived using the method described by Lin and Swalen.3 Attaching a coordinate system to the propane molecule along the principal axis as shown in Fig. 2, the kinetic-energy expression including the appropriate directional cosines


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4430 GRANT ET AL.

z

a2~ ~H ~ H ~ [)p'a l

O'~.: '/cP~)

H/C~ ~C~-.=r- y

/j\\ /H H

x FIG. 2. Coordinate axes for propane. Methyl rotation is des

ignated by the internal rotational coordinates "'1 and "'2.

for assumed tetrahedral bond angles, becomes

T=Y"wx2+YyWy2+YzWz2+Ya(a12+a22)

+ I a (2/3 )1/2Wy (a1-a2)+ la (1/3 )1/2wz (a1+a2)' (12)

Using the symmetry coordinates given in Eq. (10) along with the following Nielsen32 transformations:

wy=wlI'- (21a/[,II) (2/3)1/21>2

Wz=w/- (21a/1z) (1/3)1/21>1. (13)

The kinetic-energy expression is simplified to give:

T= y"wx2+yyWy'2+Yzw/2+rda1>12+rda~2, (14)

where

In this form the torsional coordinates CP1 and tP2 have been separated from the over-all rotational coordinates and we may now limit our discussion solely to the torsional angular momenta of interest which have the form

The total energy may now be written as

E=T+V= (PN4r1I a) + (PN4r2I a) + V, (17)

having the quantum-mechanical Hamiltonian

JC= - a1 (02 /aCP12) -Ih. (02/0tP22)+ V, where

C. Potential-Energy Term for Propane

(18)

The potential energy governing internal rotation of the two methyl groups in propane may be expressed with a two-dimensional Fourier expansion in 3a1 and 3a2. This form of expansion is particularly convenient not only because it exhibits the appropriate threefold dependency which the potential energy must have upon the internal rotational angle; but also because only a limited number of terms in an expansion of this form

are required to characterize the two-dimensional potential-energy surface to a relatively high degree of accuracy. Thus, in trigonometric form the potential energy may be written

00 00

2V = :E :E (Vekl cos3ka1 cos31a2+ V.k1 sin3ka1 sin3la2 k=O 1=0

+ Veksl cos3ka1 sin31a2+ V8kcl sin3ka1 cos31(2). (19)

In exponential form this expansion is more compact and is given as follows:

+00 +00

2V= :E :E vmnexp[3i(ma1+na2)]. (20) m=-co n=-oo

The factor of 2 on the left side of each equation converts all of the Fourier coefficients except the constant term (VeOO= voo) into peak-to-peak energy values thereby conforming to the conventions used in describing rotational barrier heights. The VeOO or Voo term may be considered to be a scaling term and is usually chosen in such a fashion that V = 0 at the minimum point on the energy surface. The relationship existing between the trigonometric and exponential sets of coefficients is

Vmn = [Olml-kOln\-l/ (2-om) (2-on) ]

X (Vek1-O"mO"n Vsk1-iun Veksl-ium Vskcl) , (21)

where the various o's are Kronecker deltas and O"j= j /1 j I for j ~O and O"j= 0 for j = O. For ease in relating to previous work in the literature, the important terms in the trigonometric form of the potential energy will first be designated and then the potential-energy expression converted through the relationship given in Eq. (21) into exponential form to take advantage of simplifications in the mathematical analysis.

Setting a1 = a2= 0 for the equilibrium structure of propane and expanding the Fourier series about this point allows one to simplify Eq. (19) as the odd terms in the expansion vanish because of the symmetry possessed by the Hamiltonian for this molecule both in the distorted as well as the equilibrium forms. The potential-energy term then becomes

2V = :E :E Vekl cos3ka1 cos31a2+ V8k1 sin3ka1 sin3la2 k=O 1=0

(22)

and the exponential coefficients for the expansion m Eq. (20) are now given by

Vmn = [0 Iml-kO I n\-l/ (2-om) (2-on) ] (Vekl- UmO"n V.kl).

(23)

As molecular symmetry requires that Vekl= Velk and V.kl = Vslk, it may be concluded from Eq. (23) that Vmn = Vnm = V-m-n = V-n -m. Expressing the potential energy in the symmetry coordinates tP1 and CP2 transforms Eq. (20) into the following equations:

<Xl <Xl

2V= :E :E .\1'~exp[3i(JLC/>l+1JtP2)J, (24) p.=-oo '1=--00


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where IL=m+n and rJ=m-n, or conversely m= (IL+rJ)/2 and n= (IL-rJ)/2. The relationship between the coefficients in both expansions is given by

or 'Vmn = Am+n,m-n-

(25)

(26)

As Vm,n is zero when either m or n is a half-integer, it is clear from Eqs. (25) that A!,~ is zero whenever the sum or difference of IL and rJ is an odd integer. The potentialenergy expansion in symmetry coordinates, therefore, is restricted to those terms in which IL and rJ are both even or both odd integers. Furthermore, as the potential energy belongs to the Al symmetry species it is evident that A!,,~=A_!,,_~=A_!,,~=A!,,_~. The expansion given in Eq. (24) may therefore be contracted to a summation over positive integer values as follows:

where

00 00

2V= L: L: A!,~F!,~, !'~o ~~o

F !'~= [(2-01') (2-0~)/36] L: R exp[3i (ILc!>I+rJcp.2)] R

(27)

(28)

with R representing the 36 elements of the I group. As the Hamiltonian will belong to the Al irreducible representation of group I, it is evident that F!,~ has the same symmetry.

It has been generally recognized33 that the Fourier expansion of the potential function of two methyl tops has the following trigonometric form:

2V = VO+ VelD COS3al + VeOI COS3a2+ Vell COS3al COS3a2

+ V.ll sin3al sin3a2, (29)

~180 .!!! :0 U Q)

o ~140 c: .~ '0 o

2100 VI I > ~

E 60 o .0

w <! ~,{ 20 (J)

Elastic Peak x_I -\ I

100 j '1

Iii I ! ) J\ .

-~'. { Ii l

200 400

Gaseous Propane

T= 23 0 C 8 = 7.9 0

Eo=50.75meV

; •• :: .:.. 1 1··\ l, .,._. .... I •• .:: ..... ,,:

.... ~.. j 1 1 \ .d'" ~ .. . Y...j~'" i \ ~ ...... y ..... :.: ...

L----1--_--:4"0---'---c_ 2i'i;0---'"-"'-c0;l;-~' 2'0 4'0' .6E (meV)

FIG. 3. Neutron inelastic scattering spectra of propane gas. The large elastic peak corresponds to no energy exchange of the neutrons with the sample. The envelope due to inelastic scattering contains the unresolved A2 and B2 torsional modes as well as the Al skeletal bending mode. Because of the absence of data and poor statistics above 40 meV in the energy loss region, only the energy gain data was analyzed. The lines indicate the positions of the torsional peaks.

c.E(cm- l )

~180r--4~0=0_.~_-~2~0~0~ __ ~0r-~~2~0~0 ____ ~4~0~0--,

1,40 ;IO_l~_~ ,oo'j J c: I 11 ~ i .1

~100 i

\

i ·i i

\

Liquid Propane

T=-181°C 8 = 7.9°

VI i >. ! ~ i

.\

~ 60 f ~ .l

1\ .~/ J'-": ~r-

w

~20 (j) ... ;'"

-40 -20

i It 1 l

.; .....

o c.E(meV)

i .. \ \. ..

~~~~.::~ ... -0 ..

20 40

FIG. 4. Neutron inelastic scattering spectra of cold liquid propane. The torsional peaks are clearly resolved in the energy loss region but the first-excited-state population is too low to yield precise data in the energy gain region.

where for convenience we take

The relationship between Vell and V.ll is admittedly an approximation but theoretical estimates support34 its validity. Until data on higher vibrational states are obtained, the small difference in Vell and - Vsll and magnitudes for Ve21 cannot be discussed as only two terms, V3 and V 6 ,35 may be evaluated from the two torsional frequencies (A2 and B2) obtained in this study. The constant term Vo in Eq. (29) is a scale factor as indicated above and has the value Vo= 2Va+ V6. Thus we use the potential function

2V= Va(2- cos3al- COS3a2)+ V6(1- cos3al COs3a2

+ sin3al sin3a2)' (30)

In exponential form the coefficients are obtained from Eqs. (22) and (26) and are given by

A1I= - Va/2,

All other A'S are taken as zero under the approximations which are made in this treatment.

D. The Torsional Wave Equation and Its Solution

The following wave equation in two dimensions, characterizing the torsional motion of two coupled methyl tops, may now be obtained from Eqs. (18) and (27) :

JC.y = - al (a2.y I a<P12) - a2 (a2.y I a<P22) 00

+! L: A!,~F!,~.y=.E;i¥. (32)

The solutions of this equation describing the torsional motions in propane may be expressed in terms of a two-dimensional expansion in <PI and <P2 of the following


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4432 GRANT ET AL.

FIG. 5. Neutron inelastic scattering spectra of solid propane. The increased resolution over that in the gas is due to both temperature and recoil mass effects. Also visible in the spectra are low-energy transitions which are probably due to lattice mode vibrations but no attempt has been made to interpret these.

general form36 :

-1.0.+1 +"" q;= L L c •• t· T exp[(3s+CT)icf>1+(3t+T)icf>2].

fI,r 8,t=-OO

(33)

The stipulation that q; be single valued requires that CT, T, S, and t be integer, and redundancy is avoided by restricting the set of values taken by CT and T either to -1,0, and +1 as in Eq. (33) or to -2,0, and +2. Interaction between the nine different types of terms with different CT and T values gives rise to the quantummechanical concept of tunneling. In the very high

15

'" ~ 10 :;,

.. o 5 H

o

r-

-

I I

• •

\ / -

• •

\ / \ / -

• • \.1

~ I 230 235 240

Torsional Frequency (em-I)

FIG. 6. Plot of the least-squares fit of the data to theory for several values of the B2 librational mode in solid propane. Similar plots were employed to extract the frequency of the A2 mode as well as the torsional frequencies in the liquid state. Q is the difference between the experimental values of S (K, .) and those calculated by means of Eq. (3).

barrier approximation Eq. (33) in the lower region of the potential well yields solutions to the energy which have a ninefold degeneracy. In the upper energy regions of a high barrier case or for a low barrier approximation, tunneling splits the nine otherwise degenerate levels into four levels of different symmetry having one single (0,0), two double (±1,0) amd (0, ±1), and one quadruple (±1, ±1) degeneracies. The value for CT and T are given in parentheses for each irreducible symmetry. As propane belongs to the high barrier approximation (i.e., tunneling splittings, ~1 cm-I, are less than the spectroscopic resolution of our technique, ~8 cm-l ) it becomes both convenient and sufficient to consider only those solutions for which CT= r= O. Equation (33) now becomes

+"" q;= L C •• t exp[3i(scf>t+tc/>2)]. (34)

8.t=-OO

Further symmetry factoring into the AI, A 2 , BI , and B2

TABLE III. Methyl torsional frequencies in propane.

Neutron values Literature values (cm-I) (em-I)

"'Mode ------------State'" A2 B2 A2 B2

Gas 217±S 265±S 20S,' 216b 223,' 271b

264c

Liquid 226±3 271±3 Solid 234±2 276±2

a Gayles and King. Ref. 41. These values are estimated from combination bands.

b Hirota et al .. Ref. 5. These values were obtained from microwave splitting in the first-excited state of the CH, torsion.

c Weiss and Leroi. Ref. 8. This transition was observed but not assigned to B,.

irreducible representations is possible allowing the sum over sand t to be contracted to 0 to + 00 from - 00

to + 00 thereby yielding the following expression:

8.t=O R

X exp[3i(scf>t+tc/>2)], (35)

where R represents the 36 group elements and xr(R) are the characters of the R elements in the four singly degenerate irreducible representations.

The individual terms in q;r given by Eq. (35) constitute an orthonormal set and the coefficients c.1 may be obtained along with the corresponding energies by solving the following secular equation:

(36)

where g: is the unitary matrix for the individual Fourier terms. Equation (36) may be solved numerically to any desired accuracy by choosing the requisite number of terms in the expansion given in Eq. (35). The matrix


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METHYL LIBRATION IN PROPANE 4433

TABLE IV. Potential barriers for methyl rotation in propane.

"'-Barrier State"'- Va

Neutron values Literature values

Gas 3680±190a -280±130a 3325±2Qc 2945d

348Qc

-170±2Qc

Liquid Solid

3800b

3960b

_200b -l60b

8 The error limits are due to the uncertainties in the A2 and B2 torsional freQ uencies.

b The calculations were made assuming no change in geometry or effective moments of inertia in the condensed state as compared to the gas.

e Hirota el al .• Ref. 5. These calculations were made by treating the top-top interaction by perturbation theory.

d Weiss and Leroi. Ref. 8. These values were obtained by assuming a

elements can be shown to have the following form:

X s'I'.8l= Ts'I'.sl+ Vs'I'.sl, (37)

Ts'I,.J = 08 '-,0:'_1 (9als2+9a2/2) , (38)

Vs'I' .st= !NI/2{ xr (E)Als'-..I.1 t'_tl+xr (PI )Als'-sl.t'+t

+Xr (P2)As'+s.lt'_tl+Xr (P3 )As'+8.t,+d, (39) where

N = (1-!os' )(l-!os)(l-!ot' )(l-!ot). (40)

Whenever any term in ..yr vanishes ion Eq. (35) because of symmetry this term in the expansion, of course, disappears from the corresponding secular equation. Thus, all Fourier terms are included in X s't'.8tA1 as all XA1 (R) = 1. If s' or s are zero, corresponding matrix elements do not appear either in the A2 or Bl secular determinant and in the Bl and B2 secular equation no Fourier terms appear in the expansion for I' or t equal to zero.

Under the conditions of symmetry which obtain when Eq. (35) is expanded in <PI and </>2 it is sufficient to select terms for which (s+/) is even. Justification of this restriction on the indices follows along the same lines considered for the potential energy term. Thus, transformation of Eq. (35) into an equivalent expansion in al and a2 using Eq. (10) yields exponential terms with whole integers (s+l, even) and with half integers (s+t, odd). As these two series are not linearly independent, the redundancy was eliminated in this work by excluding those terms for which s+1 is odd.

v. RESULTS AND DISCUSSION

The neutron scattering data on propane in the three states of matter are presented in Figs. 3-5 in terms of reduced partial differential cross sections, S(I(, E) 37 VS E.

These spectra are characterized by the dominant elastic peak corresponding to a zero energy exchange and the smaller peaks arising from inelastically scattered neutrons which have gained or lost energy to the

-283d

-357" 2400,1 3200g

harmonic oscillator to approximate the barrier height and top-top inter· action.

e Hoyland. Ref. 6. The calculations were carried out using a Gaussian basis set and Hartree--Fock-Roothaan self-consistent field formalism to obtain Va and V •.

I Reference 44. This is a thermodynamic value. g Reference 43. This is a thermodynamic value.

molecules in the sample. Intensities of typical inelastic peaks are about 5 X 10-3 the intensity of the elastic peak.

To extract the molecular transition energies E T = liw from the S(I(, E) VS E scattering data, theoretical curves are calculated with Eq. (3) for various ET values until the best fit of the experimental data is obtained. These calculations require values for the effective mass M v

which for gases may be calculated using Eq. (5) (for the gas this value is 12.8). In the analysis of data obtained for liquids or solids, the M v values for gases are no longer appropriate because intermolecular interactions restrict molecular motion. The result of such interactions is to increase the effective value of M v, thereby decreasing the width of the inelastic peak. In these cases an empirical value for M v is used to fit the half-width of the inelastic peak in question.38 The increased resolution observed in the liquid and solid samples (see Figs. 4 and 5) arise primarily from the increase in effective mass. However, the reduction in temperature is also of importance in producing sharper peaks as predicted by Eq. (7). Thus it is seen that both temperature and effective mass increases combine to give the dramatic improvement in the resolution shown in Figs. 4 and 5. In the data on liquid and soid propane, the peak widths are sufficiently narrow to clearly delineate both the A2 and B2 torsional vibrations. Precise values for the torsional vibrational frequencies are then obtained by the curve-fitting techniques previously described. Figure 6 portrays the results of a least-squares fit of the data to the theory, using Eq. (3), for various values of E T • In the case of propane gas, however, the resolution is limited by the lower recoil mass and increased temperature and it is not possible to observe two distinct torsional peaks. In addition, spectral resolution for the room temperature gas data is further complicated by the presence of transitions between excited energy levels.

As the first-excited torsional state in propane has a


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4434 GRANT RT AL.

PROPANE TORSIONAL ENERGY LEVELS

o 2 1 1 2 0

Energy (meV)

147.5' V3 + V6

________ A~I~----~957 ______ ~B~I _____ ~BB5

At ~B42

o FIG. 7. Potential-energy[diagram for the methyl torsional

vibrations in propane. The torsional quantum numbers for the A2 and B2 modes are represented IA2 and 1m , respectively. The top of the energy barrier is represented as the sum of Va and V6 given in millielectron volts which is equivalent to 3400 cal/mole. Higher energy states were not included because of the neglect of tunneling which will perturb the higher states.

measurable population at room temperature, one observes both upscattering (neutron energy gain), as well as downscattering (neutron energy loss) for the gaseous samples. Some up scattering is also observed in the cold liquid and solid data but the intensities are so low that only the down scattering peaks have been analyzed. Even though the torsional peaks are not resolved in the gas phase, estimated ET values and intensities for both A2 and B2 transitions obtained from the condensed data can be used as first approximations in the curve-fitting process. Using the first approximate values for the Az and B2 torsional modes, Eq. (36) is solved to obtain the energies of the one-quantum excited-state transitions.39 These excited-state transitions are weighted (with respect to the ground-state transition) by the Boltzmann factor and the transition normalization factor and then included in the curvefitting process to approximate the experimental envelope. This process is repeated in a cyclic manner until the best fit to the data is obtained. In the curve-fitting process one must also consider the contribution made by the CCC bending mode at 371 cm-I as the peak is very broad and overlaps the region of interest. No attempts were made in the analysis to include any of the excitedstate modes from this transition. While it is recognized that the preceding analysis of the data is only approximate in treating the torsional transition, the error arising from this source is still considered to be less than ±1 meV (8 em-I). The principle error involved in the determination of the transition frequencies arises in the experimental determination of the final neutron energy E. The initial neutron energy is determined to ±0.1 meV by time analysis of the incident neutron beam.

The A2 and B2 librational frequencies for propane in the three states of matter are given in Table III.40 It is to be noted that these frequencies increase gradually as one moves from gas to solid state, indicating the existence of increased intermolecular forces. N evertheless, both frequencies vary by less than 10% indicating that intramolecular forces dominate the librational motion. The differential frequency shifts in the gasliquid and liquid-solid phase changes are indistinguishable for the respective A2 and B2 frequencies. The propane gas frequencies from Gayles and Kintl and Hirota et al.,s respectively, are given in Table III for comparison purposes. Estimates of the torsional transition energies from ir combination bands are in reasonable agreement for the A2 mode but the Bz value of 223 cm-I is apparently in error as compared to the neutron and microwave values. Weiss and Lerois have observed a band with ir techniques at 264 cm-1 which agrees well with our Bz mode at 265 em-I. The values of Hirota et al.s are within the experimental error of the neutron data and, hence, the neutron molecular spectroscopy data provides a direct verification of the earlier indirect determination.

The greater frequency change (9 and 8 em-I) of the A2 motion between the liquid and gaseous states and the solid and liquid states, respectively, when compared to the Bz mode (6 and 5 cm-r, respectively) might well be anticipated on the basis of the larger microscopic rotational viscosity coefficient expected for rotation about the z axis (C2v axis) than about the y axis. As the Az mode couples with over-all rotation about the z axis, intermolecular interactions in the condensed states can be expected, therefore, to hinder this motion, thereby increasing the librational frequency. The greater cylindrical symmetry of propane about its y axis (consider Fig. 2) can be expected to result in a smaller microscopic rotational viscosity coefficient and a decreased effect of intermolecular interactions upon the B2 lib ration which couples with over-all rotation about the y axis.

With the aid of Eqs. (31) and (36) and the values for the Az and Bz torsional frequencies, the three- and sixfold contributions to the potential barrier were calculated and the results are presented in Table IVY The value of 3680±190 caI/mole for V3 is somewhat higher than the thermodynamic value (3200 caI/mole,43 3400 caI/mole),44 where the assumption is made that the barrier is due only to a threefold contribution. However, the sum of V3 and V6 represents the total barrier height and our theoretical values agree with the thermodynamic value (3400 caI/mole). Lide's analysis of the microwave data for propane gas could only fix a lower limit of 2700 caI/mole for the threefold barrier while the microwave data reported by Hirota et al.5

is V3=3325 and V6= -170 caI/mole. The latter potential values were obtained by treating the top-top interaction by perturbation theory. Hoyland6 has used a Gaussian basis set and the Hartree-Fock-Roothaan


131.94.16.10 On: Sat, 20 Dec 2014 13:55:37


TABLE V. Eigenvalues and eigenvectors for the first six eigenstates of the torsional vibrations in propane gas.

Symmetry Al A2 B2 Al BI Al

Energy·.b 30.77 57.69 63.69 84.22 88.49 95.70 (248.2) (465.3) (513.7) (679.3) (713.7) (771. 9)

0.382 0.471 0.344 0.412 0.606 0.242 0.335 0.500 0.091 0.399 0.249 0.145 0.307 0.040 0.237 0.141 0.083 0.215 0.102 0.188 0.043 0.089 0.084 0.020 0.103 0.003 0.078 0.015 0.038 0.037 0.009 0.059 0.000 0.052 0.003 0.027 0.010 0.008 0.019 0.001 0.023 0.000 0.017 0.001 0.008 0.002 0.003 0.006 0.000 0.009

a. In millielectron volts.

SCF formalism to obtain 3450 and -357 caI/mole for what we designate as the three- and sixfold barrier, respectively. Within the experimental errors involved, the values 3680±190 and -280±130 caI/mole obtained in this study agree with this reliable theoretical calculation.

It is interesting to note at this point that the authors have employed the torsional frequencies reported in Ref. 5 in calculating V3 and V6 by means of Eqs. (31) and (36). The resulting values of 3815 and - 369 caI/mole for the three- and sixfold barriers, respectively (as compared to the values of 3325 and -170 caI/mole obtained using the perturbation approach), are indicative of the inadequacy of treating the top-top interaction by means of perturbation theory.

The energy levels calculated for the first six torsional eigenstates in propane are presented in Fig. 7 while the corresponding eigenvalues and eigenvectors are given in Table V. Higher energy levels are not portrayed as tunneling effects would be expected to distort significantly the energy of these states and, furthermore, anharmonicity not included in the two-parameter fit of the observed transition frequencies may well effect the higher levels to a greater extent.

The inelastic neutron scattering results offer the first direct measurement of the torsional frequencies

0.429 -0.317 0.325 -0.276 0.474 0.190 0.344 -0.255 0.158 -0.265 0.586 -0.214 0.412 -0.209 0.323 0.260 0.486 0.393 0.078 -0.080 0.222 0.535 0.267 -0.126 0.022 0.177 0.381 0.174 0.097 -0.035 0.246 0.564 0.192 -0.012 0.067 0.380 0.202 0.100 0.006 -0.013 0.123 0.293 0.037 -0.037 0.033 0.336 0.099 0.003 0.001 0.195 0.133 0.173 0.009 0.055 0.090 0.321 0.030 0.011 0.031 0.261 0.054 0.003 0.005 0.083 0.052 0.007 0.000 -0.001 0.031 0.040 0.002 -0.005 0.011 0.087 0.009 -0.007 0.002 0.108 0.022 0.011 0.000 0.074 0.029 0.062 0.000 0.028 0.020 0.097 0.002 0.007 0.008 0.073 0.006 0.002 0.002 0.032 0.009 0.000 0.000 0.006

bIn cm-1 in parentheses.

in the ground state of the molecule. Future improvements in the "small K" technique should permit resolution of excited-state transitions and thus allow the evaluation of additional Fourier coefficients to give a refined estimate of the shape of the potential barrier.

ACKNOWLEDGMENTS

Professor Austin L. Wahrhaftig is acknowledged for helpful discussions on the group theoretical considerations. The authors also acknowledge the assistance of Associated Western Universities and Idaho Nuclear Corporation for the financial and technical support which made this work possible.

1 J. D. Kemp and K. S. Pitzer, J. Chern. Phys. 4,749 (1936). 2 For recent work see: L. Pedersen and K. Morokuman, J.

Chern. Phys. 46, 3941 (1967); W. H. Fink and L. C. Allen, ibid. 46,2261 (1967); J. Goddisman, ibid. 45, 4689 (1966); E. Clementi and D. R. Davis, ibid. 45, 2593 (1966); J. P. Lowe and R. G. Parr, ibid. 44, 3001 (1966); J. Goddisman, ibid. 44,2085 (1966); and R. E. Wyatt and R. G. Parr, ibid. 44, 1529 (1966). Earlier references are cited therein.

3 For an excellent review of microwave spectroscopy see: C. C. Lin and J. D. Swalen, Rev. Mod. Phys. 31,841 (1959).

4 D. R. Lide, Jr., J. Chern. Phys. 33, 1514 (1960). 6 E. Hirota, C. Matsumura, and Y. Morino, Bull. Chern. Soc.

Japan 40, 1124 (1967). 6 J. R. Hoyland, J. Chern. Phys. 49,1908 (1968).


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4436 GRANT ET AL.

7 S. Weiss and G. E. Leroi, J. Chern. Phys. 48,962 (1968). 8 S. Weiss and G. E. Leroi (private communication). We are

indebted to the authors for releasing their data prior to publication. 9 Thermal Neutral Scattering, P. A. Egelstaff, Ed. (Academic

Press Inc., London and New York, 1965). 10 J. J. Rush and T. I. Taylor, J. Chern. Phys. 44,2749 (1966);

J. J. Rush, ibid. 46,2285 (1967); 47, 2936 (1967). 11 G. J. Safford, P. C. Schaffer, P. S. Leung, G. F. Doebbler,

G. W. Brady, and E. F. X. Lyden, J. Chern. Phys. 50, 2140 (1969) .

12 J. M. Carpenter and N. A. Lurie, Neutron Inelastic Scattering (International Atomic Energy Agency, Vienna, 1968), Vol. 2, p.205.

13 E. A. Straker, J. Chern. Phys. 43, 4134 (1965). 14 Phillips Petroleum Co., Quarterly Progress Report, IDO-

16633, First Quarter, 1960. 15 K. A. Strong, G. D. Marshall, R. M. Brugger, and P. D.

Randolph, Phys. Rev. 215, 933 (1962). 16 K. A. Strong and R. M. Brugger, J. Chern. Phys. 47, 421

(1967). 17 R. M. Brugger, K. A. Strong, and D. M. Grant, "The Small K

Method of Neutron Molecular Spectroscopy," in Neutron Inelastic Scattering, Proceedings of the Copenhagen Symposium (International Atomic Energy Agency, Vienna 1968).

18 D. M. Grant, R. M. Brugger, and K. A. Strong, Phys. Rev. Letters 20,983 (1968).

19 D. E. Parks, M. S. Nelkin, J. R. Beyster, and N. F. Wikner, Slow Neutron Scattering and Thermalization (W. A. Benjamin, Inc., New York, 1970).

20 A. C. Zemach and R. J. Glauber, Phys. Rev. 101, 118 (1956). 21 T. J. Krieger and M. S. Nelkin, Phys. Rev. 106, 290 (1957). '2 H. L. McMurry, Nucl. Sci. Eng. 15, 429 (1963). R. S.

Marsden, U.S. At. Energy Rept. IN-1129 (1968). Also, H. L. McMurry, G. J. Russell, and R. M. Brugger, Nucl. Sci. Eng. 25, 248 (1966).

23 For a discussion of the properties of the function S (K, (3) see articles by P. Schofield, in Inelastic Scattering of Neutrons in Solids and Liqltids (International Atomic Energy Agency, Vienna, 1961), p. 39 and Phys. Rev. Letters 4,239 (1960).

24 L. Van Hove, Phys. Rev. 95, 249 (1954). 25 R. G. Sachs and E. Teller, Phys. Rev. 60, 18 (1941). '6 H. L. McMurry, Spectrochim. Acta 21,2091 (1965). 27 R. J. Royston, Nucl. Instr. Methods 30, 184 (1964). 28 W. R. Myers and P. D. Randolph, J. Chern. Phys. 49, 1043

(1968) . 29 R. M. Brugger and J. E. Evans, Nucl. Instr. 12, 75 (1961). 30 R. J. Myers and E. B. Wilson, J. Chern. Phys. 33, 186

(1960). 31 This restriction presumes negligible coupling between vibra

tion and librational motion of the same symmetry and is a reasonable assumption for propane as non torsional vibrational motions of A2 and B2 symmetry have frequencies which exceed the librational values by more than three fold in the least favorable case. See H. L. McMurry and D. Speas, Spectrochim. Acta 21,2105 (1965). The C-C-C angle deformation has a vibrational frequency which is near the torsional frequencies, but this mode is of a different symmetry (AI) and will not perturb the A2 and B, librational energy levels.

32 H. H. Nielsen, Phys. Rev. 40, 445 (1932). 33 J. D. Swalen and C. C. Costain, J. Chern. Phys. 31, 1562

(1959); R. J. Myers and E. B. Silson, Jr., J. Chern. Phys. 33, 186 (1960); L. Pierce, J. Chern. Phys. 34, 498 (1961); E. Hirota, C. Matsumura, and Y. Morino, Bull. Chern. Soc. Japan 40, 1124 (1967) .

34 Expansion of any steep proton-proton potential function in Fourier form leads one to the conclusion that V.u = - Vcu [e.g., using Eq. (4) from D. M. Grant and B. V. Cheney, J. Am. Chern. Soc. 89,5315 (1967), it was found that Vcu =-1.06 (V,u)]. Neglect of terms of the type VC21 (cos6al cos3a,+cos3al cos6a2) also introduces an error of a similar magnitUde to the Vcu~- V.u approximation.

35 Vc11 and - V,l1 are designated by - V6 on the basis of the trigonometric identity (COS3al cos3a2-sin3al sin3a,) = cos6,/>J, The subscript designates the coefficient of the </>1 symmetry coordinate.

36 See E. T. Whittaker and G. N. Watson, Mathematical Analysis (Cambridge University Press, Cambridge, England, 1940), Chap. 19, for a discussion of the Hill's modification of the Mathieu Equation. Equation (32) is of the same form with a dimensionality of two and lends itself to analogous solutions.

37 For a discussion of the relationship between reduced partial differential cross section [Eq. (1) ] and the double differential scattering cross section see P. D. Randolph, Phys. Rev. 123, A1238 (1964) .

38 While the value of Mv has physical significance for the condensed states of matter as well as the gas, the liquid- and solidstate theory has not yet progressed to the point where one can derive quantitative information from the effective mass values. Qualitatively, the empirically determined mass values reflect the intermolecular interactions associated with the liquid and solid states of matter. The averages of the empirically determined values of M, for the torsional modes in the liquid and solid states are 75±20 and 405±25, respectively. Although the magnitudes of these values may not be particularly significant, they do suggest a relationship to the intermolecular interactions associated with a solid as compared to the liquid. A detailed theoretical examination of M, is considered beyond the scope of the present paper.

39 Although two-, three-, etc., quantum transitions are possible, the probabilities are small as compared to the one-quantum jumps, and hence, have been neglected in this work. Two-quantum transitions have been studied by J. M. Carpenter and N. A. Lurie, Neutron Inelastic Scattering (International Atomic Energy Agency, Vienna, 1968), Vol. 2, p. 205.

40 The assignments, in each case, of the low frequency to the A2 mode is consistent with the proposed assignments derived from infrared work (see, for instance, Gayles and King. Ref. 41) as well as microwave data (see Ref. 5). Second, the use of any steep proton-proton potential function leads to a negative V6 contribution to the potential barrier. The A2 librational mode experiences the V6 contribution to the potential barrier and, hence, one would predict its frequency at a lower value than that of the B2 mode. This analysis is consistent with similar arguments employed in the analysis of the librational motion in neopentane (see Ref. 18) and isobutane where the torsional mode assignments can be made from the peak intensities.

41 J. N. Gayles, Jr. and W. T. King, Spectrochim. Acta 21, 543 (1965) .

42 It is worth noting at this point that the results calculated with Eq. (36) were obtained by employing a relatively small number of terms in the wavefunction (in the present case 25 terms) and addition of more terms did not significantly change the eigenvalues. Hence, notwithstanding the experimental uncertainties of the A, and B2 frequencies and the assumptions made in arriving at the kinetic- and potential-energy terms in the Hamiltonian, Eq. (36) can be solved rather exactly with a minimal number of terms.

43 G. B. Kistiakowsky and W. W. Rice, J. Chern. Phys. 8, 610 (1940) .

44 K. S. Pitzer, Discussions Faraday Soc. 10,66 (1951).


IP: 131.94.16.10 On: Sat, 20 Dec 2014 13:55:37

methyl libration in propane measured with neutron inelastic scattering

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