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Phase transitions and molecular dynamics of n-hexadecanol confined in silicon nanochannels

We present a combined x-ray diffraction and infrared spectroscopy study on the phase behavior and molecular dynamics of n-hexadecanol in its bulk state and confined in an array of aligned nanochannels of 8 nm diameter in mesoporous silicon. Under confinement the transition temperatures between the liquid, the rotator RII and the crystalline C phase are lowered by approximately 20K. While bulk n-hexadecanol exhibits at low temperatures a polycrystalline mixture of orthorhombic beta- and monoclinic gamma-forms, geometrical confinement favors the more simple beta-form: only crystallites are formed, where the chain axis are parallel to the layer normal. However, the gamma-form, in which the chain axis are tilted with respect to the layer normal, is entirely suppressed. The beta-crystallites form bi-layers, that are not randomly orientated in the pores. The molecules are arranged with their long axis perpendicular to the long channel axis. With regard to the molecular dynamics, we were able to show that confinement does not affect the inner-molecular dynamics of the CH_2 scissor vibration and to evaluate the inter-molecular force constants in the C phase.


I. INTRODUCTION
The physical properties of condensed matter spatially confined in pores or channels of a few nanometer in diameter can differ markedly from the behavior in the bulk state. In particular, phase transitions can be entirely suppressed or significantly altered in comparison to their bulk counterparts 1,2,3,4 . Also the dynamics of condensed matter confined in mesopores, most prominently in the vicinity of glass transitions 5,6,7,8,9,10,11,12,13,14 , can be affected markedly.
Intimately related to these changes in the phase transition phenomenology the architectural principles of molecular solids can substantially differ in the spatially confined state from the bulk state. This depends, however, sensitively on the complexity of the building blocks. For simple van-der-Waals systems, such as Ar and N 2 , a remarkable robustness of the bulk structures has been found for the solid state under confinement 15,16,17 . By contrast, the structural properties of pore fillings built out of more complex building blocks, such as linear hydrocarbons 18,19,20,21,22 or liquid crystals 23,24 are very susceptible to confinement on the meso-and nanoscale. For example, a quenching of the lamellar ordering of molecular crystals of n-alkanes has been observed in tortuous silica mesopores of Vycor 25 . However, in tubular channels of mesoporous silicon this building principle of hydrocarbon molecular crystals survives, albeit a peculiar texture has been observed for the pore confined solids 19 : The long axes of the molecules and thus the stacking direction of the lamellae are oriented perpendicular to the long axis of the pores.
Here we present an experimental study on a mediumlength, linear alcohol C 16 H 33 OH, a representative of the 1-alcohol series, imbibed in mesoporous silicon. We explore the phase behavior of the confined alcohol by a com-bination of x-ray diffraction and infrared spectroscopy measurements. As we shall demonstrate, we profit in those experiments both from the parallel alignment of the silicon channels and from the transparency of the silicon host in the infrared region.

II. EXPERIMENTAL
The porous silicon samples used in this study were prepared by electrochemical etching of a heavily p-doped (100) silicon wafer 1 with a current density of 13 mA cm 2 in a solution composed of HF, ethanol and H 2 O (1:3:1 per volume) 26,27,28 . These conditions led to a parallel arrangement of non-interconnected channels oriented with their long axes along the <100> crystallographic direction of silicon, which coincides with the normal of the wafer surface. After the porous layer had reached the desired thickness of 70 microns, the anodization current was increased by a factor of ten with the result that the porous layer was released from the bulk wafer underneath. Using nitrogen sorption isotherms at T = 77 K, we determined a porosity of 60% and a mean channel diameter of 8 nm. The single crystalline character of the matrix was checked by x-ray diffraction. Transmission electron micrographs of channel cross sections indicate polygonal, rough channel perimeters rather than circular, smooth circumferences 29 .
The matrix both for the infrared spectroscopy and the x-ray measurements were filled completely via capillary action (spontaneous imbibition) with liquefied C 16 H 33 OH 30 . Bulk excess material at the surface was removed by paper tissues.
Infrared spectra in a range of wavenumbers ν from 4000 to 800 cm −1 with a resolution of 1 cm −1 were measured with a Fourier Transform Spectrometer (FTIR Perkin Elmer System 2000). This range corresponds to frequencies from 3 · 10 13 Hz to 1.2 · 10 14 Hz (wavelengths from 10 µm to 2.5 µm). For both the bulk material and the filled porous samples the same sample holder was used, i. e. a copper cell with two transparent KBr windows. In the confinement experiments the long channel axes were oriented parallel to the beam axis, i. e. perpendicular to the electric field vector. The sample holder was placed into a cryostat (a closed cycle refrigerator CTI cryogenics, Model 22) allowing us to vary the temperature from 50 to 340 K. The temperature was controlled with a LakeShore 340 temperature controller with a precision of ±0.25 K. All IR-spectra that we show in the following were measured during cooling (typical cooling rates were of the order of 0.5 K/min). Heating scans show the same behavior except for the transition temperatures, which are some degrees higher (see below).
For the x-ray measurements the sample was mounted on a frame in a sample cell consisting of a Peltier cooled base plate and a Be cap. The cell was filled with He gas for better thermal contact. The Be cap sits in a vacuum chamber, the outer jacket of which has Mylar windows allowing the passage of the x-rays over a wide range of scattering angles θ within the scattering plane (see Fig. 1). But the set-up allowed practically no tilt with respect to the scattering plane. The temperature was controlled by a LakeShore 330 over an accessible range from 245 K up to 370 K. The measurements were carried out on a twocircle x-ray diffractometer with graphite monochromatized CuK α radiation emanating from a rotating anode. The porous sheet was mounted perpendicular to the scattering plane. The two angles that could be varied were the detector angle 2θ and the rotation angle ω about the normal of the scattering plane. The samples were studied as a function of temperature by performing several Φ-scans. In this paper we concentrate on radial 2θ-ω-Scans in reflection geometry, i.e. along q p with Φ=0 • , and in transmission geometry, i.e. along q s with Φ=90 • (see Fig. 1

).
A. Structure of bulk n-hexadecanol n-Hexadecanol, C 16 H 33 OH, is an almost rod-like molecule with a length of 22Å and a width of 4Å. The C-atoms of the backbone are in an all-trans-configuration so that they are located in a plane 25 .
At low temperatures n-alcohols form bi-layered crystals in two possible modifications: the so-called γ-form, i. e. a monoclinic structure as sketched in Fig. 2  A2/a 31,32 ), or the so-called β-form, i. e. an orthorhombic structure as sketched in Fig. 3 33 . In the γ-form, the molecules include an angle of 122 • with the layer plane. Within the layers, they are close-packed in a quasi-hexagonal 2D array, described by the rectangular in-plane lattice parameters a and b (according to Ref. 32 a = 7.42Å and b = 4.93Å holds, so that a/b = 1.5).
There are two different alternating orientations for the C-C-plane of the backbone leading to a herringbone structure (see Fig. 2b). The β-form exhibits an identical orientational order of the backbone, but the molecules' axes remain perpendicular to the layers as sketched in Fig. 3 33 .
In addition, gauche-and trans-conformation of the CObond alternate with molecules in this phase, while they are in an all-trans configuration in the γ-form. In general, the γ-form dominates at low temperatures for the even alcohols, while the β-form is more frequent in odd n-alcohols 33,34 . For n-hexadecanol both the orthorhombic β-form 33 and the monoclinic γ-form 31,32 are reported. Depending on the preparation conditions it is possible to obtain a polycristalline mixture of the monoclinic γ-and the orthorhombic β-form 34 . Upon heating, the crystalline phase undergoes a transition into a so-called Rotator-(II)-phase R II , which is schematically depicted in Fig. 4 2 . This phase has a hexagonal in-plane arrangement with the c-direction perpendicular to the cell base. The hexagonal arrangement can be indexed with an orthorhombic cell with a ratio of rectangular basal lattice parameters of a/b = √ 3 35 . On a microscopic level the change in the center of mass lattice from the low-temperature crystalline phase to the rotator phase can be attributed to the onset of hindered rotations of the molecules about their long axes between six equivalent positions (the stars in Fig. 4b). Further heating above 322 K leads to the liquid state 35 .

A. Structure of confined n-hexadecanol
We have determined structures, phase sequences and transition temperatures of n-C 16 H 33 OH confined in mesoporous silicon by x-ray diffractometry. The upper panel in Fig. 5 shows diffraction patterns along q p at se- Confined C16H33OH exhibits the same structure in its rotator phase, but in a different temperature range (see below, Table II).
lected temperatures while cooling. The appearance of a broad Bragg peak at 2θ ≃ 21 • indicates solidifaction. Its position is compatible with the leading hexagonal inplane reflection of the R II phase. Upon further cooling a second peak at 2θ ≃ 24 • shows up. This change in the diffraction pattern indicates an uniaxial deformation of the hexagonal lattice. Both reflections can be mapped on a 2D rectangular mesh characteristic of an uniaxially deformed hexagonal cell. The overall resulting pattern is, however, incompatible with the monoclinic structure of the low temperature bulk crystalline phase.
Additionally to the q p -scans, we performed also scans for a variety of additional orientations of the scattering vector with regard to the long axis of the channels. These patterns differ markedly, which is indicative of a strong texture of the pore confined cystallized alcohol. It is no powder in the crystallographic sense. In particular, there are strong in-plane reflections and no layering reflections for scans along q p , while the q s -scans for the same sample show at least very weak reflections characteristic of a bilayer stacking and only very weak leading in-plane reflections (see Fig. 5). An analysis of the width of the layering reflections yields a coherence length of 7 (±1.5) nm.
As discussed in more detail in Refs. 19 and 36 , the overall picture which emerges from these results can be summed up as follows: the alcohol molecules form orthorhombic structures with a bilayer-stacking direction along the c-direction. Within the bilayers (the a-b-plane), the molecules' backbones are untilted with regard to the stacking direction and the backbones are orientationally either fully ordered (in a herringbone fashion) or partially ordered, as known from the R I phase of n-alkanes. The superlattice reflection characteristic of the full, herringbone type orientational ordering has been searched for and could weakly be detected at low temperatures. The degree of uniaxial deformation of the hexagonal center of mass cell, quantified by the deviation of the ratio a/b from its value in the hexagonal phase ( √ 3), also in-dicates a full orientational ordered state (see Table I, 32 ). Thus, the diffraction data are compatible with the bulk β modification discussed above. This conclusion is also supported by an analysis of the infrared spectroscopy data sets presented below. More importantly, the peculiar dependency of the diffraction patterns on the orientation of the q-vector with regard to the silicon host indicate that the bi-layer stacking direction is perpendicular to the long axis of the channels and, consequently, that the long axis of the molecules is oriented perpendicular to the long axis of the channels (see Fig. 6). At first glance, this finding may appear somewhat counter-intuitive. Albeit it can be understood as resulting from the crystallization process in a strongly anistropic, capillary-like confined liquid 19,37 . It is a well established principle in single crystal growth that in narrow capillaries the fastest growing crystallization direction prevails over other directions and propagates along the long axes of capillaries 38 . For layered molecular crystals of rod-like building blocks this direction is an in-plane direction, which is perpendicular to the long axis of the rods. If this direction is aligned parallel to the silicon nanochannels due to the crystallization process, it dictates a perpendicular arrangement of the molecules' long axes with regard to the long channel axis, in agreement with the diffraction results presented here. The temperature dependent diffraction study allows us to gain additional information on the relative stability of the different nanochannel confined phases. In Table II we display the phase transition temperatures of confined C 16 H 33 OH as inferred from the appearance or disappaerance of characteristic Bragg peaks. There is a hysteresis between heating and cooling for both the fluid-R II -and the R II -C-transition (8 K and 3 K, respectively). Compared to the bulk data (see also Tab. II), the transition temperatures of pore confined C 16 H 33 OH are lowered. On cooling, the lowering is of the order of ∆T = 18 K for the fluid-R II -transition and ∆T = 26 K for the R II -Ctransition. This observation is analogous to phase transitions shifts reported for other pore condensates 2,3 .
Furthermore, the temperature range of the confined R II phase, 14 K upon cooling and 19 K upon heating, is larger than that of the bulk material (12 K). Obviously, confinement stabilizes the orientational disordered R II phase, similarly as has been found for n-alkanes 19 and for other orientational disordered, plastic phases under spatial confinement 4 .
Since the pores were completely filled at higher temperatures, when hexadecanol is in its liquid state, the pore filling at low temperatures does not consist only of bi-layer crystals: the change of volume at the R II -C phase transition is about 10% (see Fig. 7), so that there are voids and/or molecules that are not part of a bi-layer crystal. However, our experiments do not give us information about their spatial arrangement.   6: Sketch of the orientation of pore condensed C16H33OH in the porous sheet. Left pore: view on the rectangular in-plane arrangement of the molecules. Right pore: bi-layered crystal; the long molecule axis are oriented perpendicular to the pore axes. At maximum two bi-layers fit into a pore.

B. Molecular dynamics
The dynamics of bulk-C 16 H 33 OH has already been investigated in IR-measurements in the past 31,33 . In order to show later on how the molecular dynamics is affected by spatial confinement on the nm-scale, we display some of our bulk spectra in the following. Here we focus on two characteristic vibrations, the OH-stretching and the CH 2 -scissoring vibration.
Figs. 8a) and 9a) show the bulk spectra of the OHstretching-band in the respective phases (compare with Figs. [2][3][4]. In the liquid state (above 322 K) the peak maximum is located at about 3345 cm −1 . A decrease of temperature below 321 K yields a shift of the peak position to about 3325 cm −1 indicating the molecular rearrangement in the R II phase. A further decrease of temperature below 310 K results in a splitting into two peaks at approximately 3310 cm −1 and 3220 cm −1 . Confined C 16 H 33 OH shows a different behavior. There is only one peak in the whole temperature range, the position of which changes reflecting the transition between liquid phase and R II phase as well as between R II phase and C phase (see Figs. 8b and 9b).
The fact that the OH-band of bulk C 16 H 33 OH splits at low temperatures while no splitting is observed under confinement confirms the structural differences already observed in the x-ray experiment. For example, Tasumi et. al have studied bulk alcohols C n H 2n+1 OH from n = 11 − 37 using infrared spectroscopy 33 , Ventòla et al. alcohols with n = 17 − 20 34 . Those alcohols showing at low temperatures (C phase) the monoclinic γ-form, such as C 16 H 33 OH, exhibit the splitting of the OH-band, while those that take the orthorhombic β-form show a single peak. This is due to differences in the spatial arrangement of the hydrogen bonds as well as in the distances of neighboring O-atoms: in the crystalline γ-form, where the molecule axis are tilted (see Fig. 2), the molecules show an all trans conformation, and the intra-layer O-distance (≃ 2.74Å) differs from the interlayer O-distance (≃ 2.69Å). However, in the orthorhombic β-form (Fig. 3) trans-and gauche-molecules alternate and the intra-layer O-distance (2.73Å) nearly equals the inter-layer O-distance (2.72Å), so that the splitting is suppressed 33 . Therefore, the observed OH-band splitting for bulk C 16 H 33 OH shows the presence of the γ-form. Either the whole bulk material exhibits the γ-form or there is a mixture of γ-and β-crystallites. The latter case is frequently observed 33,34 : in fact, in the range of wavenumbers from 1150 cm −1 to 950 cm −1 , where C-C stretching vibrations are visible, we see indications for a superposition of both forms (not shown). On the other hand, pore confined C 16 H 33 OH shows no OH-band-splitting at low temperatures. This reflects that the molecular arrangement doesn't transform in the monoclinic γ-form but remains in an orthorhombic structure, i. e. only the β-form is present (compare Figs. 2 and 3). This result is in agreement with the x-ray data presented above. Upon cooling, both the bulk and the confined hexadecanol pass from an hexagonal R II -phase into a crystalline phase. The bulk material undergoes a stronger structural change, i. e. there is a mixture of the orthorombic βand the monoclinic γ-form. The latter one is suppressed under confinement, so that only the β-form remains, which is quite similar to the hexagonal structure of the R II -phase: the fact that the crystallites have to fit into nanopores of irregular shape might favor the geometrically more simple β-form 3,39 (see Fig. 6). Now let us turn towards the scissor-vibration of the CH 2 groups (bending mode) that will give us information about inner-molecular and inter-molecular force constants. The spectra are shown in Fig. 10. At first, we want to discuss the bulk material. At high temperatures (liquid state) a superposition of two peaks at 1467 cm −1 and 1460 cm −1 is observed. In the intermediate temperature range (R II phase; see Fig. 4) the intensity of the peak labeld "1" increases strongly. At low temperatures (C phase; see Fig. 2) this band splits up into two peaks. The latter transition can be clearly seen in Figs. 11a) and 12a), where we display the peak positions and intensities as a function of temperature. The results are similar to those obtained for the bulk state of n-paraffines, that apart from the missing OH-group are similar in their structure, i. e. that have the same CH 2 -backbone 40 . In IR-spectra only one CH 2 -scissoring-band is observed at high temperatures, i. e. intra-molecular interactions of the CH 2 -groups are too small to lead to a series of distinct  Fig. 8a)]. Three different phases are visible: a) above 321 K, b) from 310 to 321 K, where the peak position appears at lower wavenumbers and c) below 310 K where the peak splits up into two peaks. (b) Wavenumber of the OH -stretching peak for confined C16H33OH (compare with Fig. 8b). The transition between the C and the RII phase seems to be smeared in a range around T = 291 ± 5 K. The RII -liquid transition does not affect the OH-stretching.
peaks. The band splitting at low temperatures has been attributed to inter-molecular interactions (see Ref. 40 and text below).
Qualitatively, a behavior similar to that of the bulk state is observed for confined C 16 H 33 OH (see Fig. 10b). In the high-temperature liquid phase two overlapping peaks are visible. The stronger one, i. e. that at higher wavenumbers, undergoes an increase in intensity at about 304 K (see Fig. 12b), indicating the transition from the liquid phase to the R II phase, while the secondary peak at lower wavenumbers gets weaker and finally disappears. At the second transition temperature of T = 291 K the remaining strong peak splits (see also Fig. 11b). The separation is not as distinct as for bulk material. These transition temperatures, T = 304 K and T = 291 K (see Figs. 11b and 12b), agree well with those obtained via x-ray measurements (compare with  Table II). In the following we want to analyze the dynamics of the CH 2 -groups in order to check whether it is affected by the geometric confinement, e. g. by an interaction with the pore surfaces, by the limited number of neighboring molecules (finite-size-effects) or by structural changes. In a first approximation we can assume that it is not affected by the stretching of the OH -groups. On the one hand, there is the scissor vibration, where the angle α between the two CH-bonds oscillates around its equilibrium value α = 109.47 • (see Fig. 13). In addition, symmetric and asymmetric stretching vibrations of the CH-bonds are observable (for the values see Table III). Let f α and f d denote the respective force constants. These can be calculated from the measured vibration frequencies using Eqs. (26)-(28) (see Appendix A; the difference in calculating f α via Eq. (27) or Eq. (28) is below 3.5% confirming that the innermolecular coupling terms can be neglected). Table III shows the results for the liquid and the R II phase. Neither the phase transition liquid → R II nor geometrical confinement does markely affect the innermolecular constants.
In the R II phase the molecules rotate about their long axis, so that the primitive cell consists of only one molecule per layer (see Fig. 4). Therefore, no splitting is observed. But in the C phase (below 310 K for bulk and below 291 K for confined C 16 H 33 OH), where the molecules are arranged in a herringbone structure, there are two molecules per layer in the primitive cell. So the symmetry of the arrangement allows a splitting of the scissoring band and obviously the molecular interactions are sufficiently strong that we are able to observe a double peak (see above, Figs. 10 and 11). The strength of interaction depends on the distances between neighboring H-atoms of adjacent chains and can be analyzed using a formalism developed by Snyder (see Ref. 40 and Appendix B). In Fig. 14 we have sketched the orthorhombic lattice of the crystalline C 16 H 33 OH subcell (a view on the a-b-plane perpendicular to the molecules axis). In what follows we restrict ourselves to this β- form, that is characteristic for confined C 16 H 33 OH (a quantitative analysis of bulk C 16 H 33 OH is difficult due to the superposition of β-and γ-form). Assuming that the inner force constant f α does not change at the phase transition, the intermolecular force constants f 3,j can be evaluated from the observed splitting of the scissor band as described in Appendix B (see Eq. (30)). The values needed are the lattice parameters (see Table I) and the herringbone angle ζ between the projection of the backbone and the a-axis (see Fig. 14). The latter one is determined via Eq. (29) and the measured intensities of the two CH 2 -scissoring-peaks. For confined C 16 H 33 OH we have I a = 13.29 and I b = 22.23 yielding an angle of ζ = 37.7 • (see Fig. 12b for T = 245 K). We display the intermolecular force constants in Table IV. For comparison, we also list literature values for an alkane, C 23 H 48 at 90 K, which have been evaluated in the same way 40 . This alkane and C 16 H 33 OH exhibit a similar structure: The backbones of the molecules consist of the same CH 2units and both take the β-form at low-temperatures. In addition, also the values of the lattice constants for    Table IV are similar, however, the respective force constants differ slightly by 10 to 20%. This is mainly due to the orientation of CH 2 -groups (the projection of the backbones on the ab-plane) characterized by the herringbone angle ζ. For C 16 H 33 OH ζ = 37.7 • holds, for the alkane ζ = 42 • . This difference is probably due to the presence of polar OH-groups in C 16 H 33 OH that are strongly interacting and thus have an impact on the molecular orientation. The above comparison confirms once again that confined C 16 H 33 OH takes the β-form in contrast to the bulk material (γ-and β-form).
In order to assess the validity of our analysis, we  Table IV as well as Eqs. (33) and (34), we get a theoretical value of ∆ν calc = 8.1 cm −1 for confined C 16 H 33 OH at T = 245 K. The measured band splitting is ∆ν meas = 7.8 cm −1 . Therefore, the experimental data is in good agreement with the theory.  IV: Distances between two neighboring hydrogen atoms and the resulting interaction force constants in the C phase confined C16H33OH (β-form). ζ denotes the herringbone angle between the projection of the backbone and the a-axis (see Fig. 14). For comparison, we also list literature values for an alkane, C23H48 at 90 K, that also takes the βform 40 . The CH2-backbones of both molecules are similar, but C16H33OH exhibits an additional polar OH-group that affects the orientation angle ζ.

IV. SUMMARY
We have studied the structure and molecular dynamics of n-hexadecanol confined in nanochannels of mesoporous silicon and of bulk n-hexadecanol in their respective phases (in the order of decreasing temperature: liquid, rotator R II and C). For this purpose we have performed x-ray and infrared-measurements.
The transition-temperatures for confined C 16 H 33 OH are lower than for bulk C 16 H 33 OH (∆T ≃ 20 K, see Table II). In addition, under confinement the phase transitions are smeared, probably due to a distribution of pore diameters. Geometrical confinement does not affect the innermolecular force constants of the CH 2 -scissoring vibration (see Table III) but has an impact on the molecular arrangement. The R II phase of both bulk and confined hexadecanol is characterized by an orthorhombic subcell, where the chain axis are parallel to the layer normal (see Fig. 4). However, in the low-temperature C phase there is a fundamental structural difference. While bulk C 16 H 33 OH exhibits a polycrystalline mixture of βand γ-forms (see Figs. 2 and 3), geometrical confinement favors a phase closely related to the β-form: only crystallites with an orthorhombic subcell are formed, where the chain axes are parallel to the bi-layer normal. However, the γ-form having a monoclinic subcell, in which the chain axis are tilted with respect to the layer normal, is suppressed. A reason for this might be the irregular shape of the nanochannels, into which the crystallites have to fit, favoring the formation of the geometrically more simple and less bulky form 3,39 (see Fig. 6). Since only the pure β-form is present under confinement, we were able to evaluate the inter-molecular force constants of the CH 2 -scissor vibration. Also the orientation of the β-crystallites has been determined: the molecules are arranged with their long axis perpendicular to the pore axis.

Appendix A
In this section we show how the innermolecular force constants of the CH 2 -groups can be evaluated using three characteristic vibration frequencies, that are easily measured: the scissor vibration as well as the symmetric and asymmetric CH-bond stretching. For this purpose we apply the Wilson FG -matrix method 42 . We use the notation of Meister and Cleveland for the similar H 2 O molecule 41 and perform the calculations in the same way. Fig. 13 shows a single CH 2 -molecule. In the following we will neglect the influence of the neighboring molecules on this one. d = 1.09Å is the length of the C -H bond and α = 109.47 • the angle between the two C -H bonds 32 . This kind of molecule belongs to the C 2ν point group. This means, there are two vibrations of type A 1 (symmetric stretching and bending vibration) and one vibration of type B 2 (asymmetric stretching vibration). The internal coordinates of this molecule are ∆d 1 , ∆d 2 and ∆α. ∆d 1 and ∆d 2 mean changes in the bond length of the two C -H -bonds and ∆α changes in the angle between the two bonds. Therefore we get three symmetry coordinates, two for A 1 and one for B 2 . If we assume d being the equilibrium C -H bond length, then we obtain for the three symmetry coordinates: Now, we have to calculate the F matrix, related to the potential energy, and the G matrix related to the kinetic energy. The potential energy can be written as and with the internal coordinates Now we set d 1 = d 2 = d and write Eq. (5) as with F jl = F lj . In matrix form, Eqs. (4) and (5) become and r ′ and R ′ are the transposes of r and R. With Eqs. (7) + (8) The R i 's are linear combinations of the r i 's Since the R i 's are orthogonal and normalized, then This means with Eqn. (10) The F matrix is So, for the type A 1 the F matrix is and for the B 2 type The exact derivation of the G matrix shouldn't be shown here. It can be gleaned by Meister and Cleveland 41 . Only the most important steps shall be explained here.
If only non-degenerate vibrations are present, the elements of the kinetic energy matrix can be written as where j and l refer to symmetry coordinates used in determining the S vector, p refer to a set of equivalent atoms, a typical one of the set being t. µ p is the reciprocal of the mass of the typical atom t p and g p is the number of equivalent atoms in the pth set. The S vector is given by where j, U j k and k have the same meaning as above. s kt can be expressed in terms of unit vectors along the chemical bonds and depends on the changes in the bond length or the angle between the bonds. So, the G matrix for the A 1 vibration type has the form and for the B 2 vibration type To determine the frequencies, one has to solve the equation where λ = ω 2 = (ν2πc) 2 denotes the square of the angular frequency. For the A 1 type one gets the equation and for the B 2 type Eq. (22) can be separated with the Vieta expression 43 . Inserting the terms for the F ij and G ij , we obtain For Eq. (23) one obtains Neglecting the coupling constants f dd and f dα allows to evaluate the innermolecular force constants using the measured wave numbers, ν d,sym = √ λ 1 /(2πc), ν α = √ λ 2 /(2πc) and ν d,asym = √ λ 3 /(2πc). Then Eq. (25) yields Inserting this result into Eq. (24) yields There is a second possibility to evaluate f α , i. e. by inserting Eq. (26) into Eq. (24). This yields Taking the measured wavenumbers listed in Table III and the average angle between the CH-bonds, α = 109.4 • , as well as the masses of the atoms, 1/µ C = 12u and 1/µ H = 1u (u = 1.6606 · 10 −27 kg), Eqs. (26)-(28) yield the force constants listed in Table III. The difference in calculating f α via Eq. (27) or Eq. (28) is below 3.1% confirming that the inner-molecular coupling terms can be neglected.

Appendix B
What follows is a summary of Snyder's derivation of the intermolecular force constants between the CH 2 groups of neighboring molecules that gives rise to a splitting of the scissor band at low temperatures 40 . We show how this formalism can be applied to C 16 H 33 OH. An alternative description can be found in Ref. 44 .
In Fig. 14 we display a hexagonal subcell of C 16 H 33 OH. While Stein 45 has taken only one pair of neighboring CH 2 into account to calculate the splitting of rocking and scissoring bands, Snyder has shown that more pairs have to be included. When we consider the distances of H-atoms from the H-atom no. 3 (see Fig. 14), then all atoms except no. 2 ′ , 6 ′ , 5 and 6 have distances larger then 3.7Å. The internal coordinates α i are always half of the angle between the C -H bonds of a CH 2 molecule. Solid circles are H-atoms in the same plane, dashed circles H-atoms in a plane above or below. with a and b being the lattice constants of the crystalline phase. We get the distances between the atom 3 and the other ones (see Fig. 14) with where j = 2 ′ , 6 ′ , 5, 6 is. The herringbone angle ζ (see the upper left corner of Fig. 14) can be determined with the relation 40 where I a is the integrated intensity of the scissoring mode, which is polarized in the a direction (higher mode at 1473 cm −1 ) and I b the one of the mode, which is polarized in the b direction (lower mode at 1462 cm −1 ).