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Methyl rotational potentials of trimethyl metal compounds studied by inelastic and quasielastic neutron scattering
Chemical Physics 292 (2003) 161–169 www.elsevier.com/locate/chemphys
Methyl rotational potentials of trimethyl metal compounds studied by inelastic and quasielastic neutron scattering M. Prager a,*, H. Grimm a, A. Desmedt b, R.E. Lechner b a
Institut fu€r Festk€ orperforschung, Forschungzzentrum J€ ulich, D-52425 J€ ulich, Germany b Hahn Meitner Institut, Glienickerstr. 100, D-14109 Berlin, Germany Received 8 October 2002; in final form 16 April 2003
Abstract Neutron scattering data on classical and quantum rotation of methyl groups in the organometallic molecular crystal In(CH3 )3 are presented. The tunneling spectra are analyzed using the mean field single particle model of rotational tunneling. The presence of six inequivalent rotors of equal occurrence probabilities in InðCH3 Þ3 is reflected in the number and intensities of the transitions. In a similar way classical spectra are interpreted as superposition of quasielastic Lorentzians. A consistent combination of tunneling and quasielastic spectroscopies is based on equal relative intensities of the respective spectral components and allows a determination of the coefficients of a Fourier expansion of the rotational potentials up to second order with only a few percent of uncertainty. The results are compared to published data on the homologues Al and Ga compounds on the basis of the molecular and crystal structures. Intraand intermolecular interactions are found to be of similar importance. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 6112E; 6320; 6630
1. Introduction One of the most fundamental and most simple dynamical processes is the rotation of a methyl group into a new equilibrium orientation. The potential which describes this dynamics is determined by the intermolecular interactions with the neighboring atoms which may be parameterized as
*
Corresponding author. Fax: +49-2461-612610. E-mail address: [email protected] (M. Prager).
universal force fields (UFF), pair interaction potentials or in ab initio program codes. Neutron scattering has developed two techniques to explore molecular rotation in solids. On the one hand quasielastic spectroscopy reveals the classical motion of an individual atom [1]. If this atom is bound to a molecule or a molecular side group it monitors its rotational dynamics. The spectra are analyzed with the concept of the elastic incoherent structure factor. The output from this method are the jump geometry including jump distances, the related
0301-0104/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00207-6
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jump times, phase transitions as a discontinuity in the evolution of the dynamics with increasing temperature, etc. On the other hand molecules show a quantum behavior at low temperatures. The quantum excitations of a rotor are observed by rotational tunneling [2,3] and vibrational spectroscopies. Both theories represent in their standard forms single particle descriptions. The rotor is characterized by its momentum of inertia and the environment is represented by a potential. It is this potential which links the results of the two techniques to each other. The topology of the potential/environment determines the jump geometry; the barrier height determines the classical jump rate at a given temperature. The shape and height of the same potential determine also the eigenstates and eigenvalues of a rotor which – vice versa – allow to determine the potential parameters. An important property of the single particle model is its transferability. The symmetry of the rotational potential is only determined by the rotor and is the same in any material. In cases of coupling a second degree of freedom has to be considered. The mathematical description has to be formulated newly for each new type of coupling. This is done for very few cases only like direct rotor–rotor coupling or the so-called rotation–translation coupling [4]. Even here a new symmetry already requires the development of a new mathematical theory. Thus, in spite of weak points the single particle model remains the standard description of tunneling methyl groups. The following discussion is restricted to the one-dimensional rotation of a methyl group present in the InðCH3 Þ3 compound. InðCH3 Þ3 like many other materials contains inequivalent rotors. Their number is determined by the crystal structure and eventually disorder. By combining quasielastic and tunneling spectroscopies rotational potentials of all inequivalent rotors can be obtained with high accuracy and confidence. A comparison to results from the homologues Ga and Al compounds will allow an evaluation of the relative importance of intra- and intermolecular interactions.
2. Single particle dynamics of methyl groups 2.1. Rotational tunneling The methyl rotational potential in the single particle model [2,3] shows the threefold symmetry of the molecule and is expressed in form of a Fourier expansion V ðuÞ ¼
N X V3n ð1 cosð3nu u3n ÞÞ: 2 n¼1
ð1Þ
Arbitrarily one can chose u3 ¼ 0. The potential determines the excitations of the hindered rotor through the eigenvalues of the single particle Schr€ odinger equation [2] o2 B 2 þ V ðuÞ Wi ¼ Ei Wi : ð2Þ ou The more eigenvalues are measured the better the shape of the potential – including phase factors – can be determined. u6 ¼ 0 or p is often applied to limit the number of free parameters to be deduced from the few observables like tunneling splitting and activation energy. In the tables shown in the following we (must) adopt this restriction. A positive (negative) sign of V6 corresponds to u6 ¼ 0ðpÞ. In general, unrestricted phase factors are only obtained when the potential is derived from ab initio or force-field calculations using the crystal structure. The scattering of neutrons by a methyl group is dominated by the large incoherent scattering cross-section of the hydrogen atoms. The powder averaged scattering function of a single tunneling rotor is [2] 5 4 Si ðQ; xÞ ¼ þ j0 ðQdÞ dðxÞ 3 3 2 2 þ j0 ðQdÞ fdðx þ xti Þ 3 3 þ dðx xti Þg:
ð3Þ
Here xti means the tunnel frequency. The Q-dependent intensity coefficients are called structure factors. Its expression in Eq. (3) is based on dlocalized methyl protons. This approximation is 1 . good up to momentum transfers of Q 1:6 A The spherical Bessel function j0 ðQdÞ contains
M. Prager et al. / Chemical Physics 292 (2003) 161–169
in the the proton–proton distance d ¼ 1:76 A molecule. The single particle model yields the total scattering function of a sample with M inequivalent methyl groups as a sum of scattering functions of independent rotors weighted by their occurrence probabilities pi SðQ; xÞ ¼
M X
pi Si ðQ; xÞ:
ð4Þ
i¼1
The occurrence probabilities pi of all species i sum up to 1. From this scattering function, one derives the ratios of the inelastic intensity of an individual methyl group to the total elastic intensity Ri;el and the ratio Ri;j of two different tunneling transitions Iinel pi ð2 2j0 Þ ¼ ; 5 þ 4j0 Iel pi Ri;j ¼ : pj
Ri;el ¼
ð5Þ
These expressions are very useful for the interpretation of a multiline spectrum but in the given form only valid if all tunneling transitions are resolved. If this is not the case obvious modifications have to be applied. Interaction of a rotor with a phonon bath at finite sample temperature introduces stochastic forces and therefrom fluctuating potential terms. As a result the tunnel lines shift by Dhxt and broaden by Ct with increasing temperature [5] ES D hxt exp ; kT ð6Þ EC : Ct ¼ C0 exp kT The activation energy of line broadening is determined by resonant coupling to phonons with an energy equal to the methyl libration. Thus it is possible to get the librational energy E01 of the respective unperturbed rotor from EC ¼ E01 . 2.2. Classical reorientational motion According to Eqs. (6) a tunnel transitions merges into the elastic line and broadens. As required by BohrÕs correspondence principle the resulting quasielastic Lorentzian is the response of a
163
rotor reorienting classically under the influence of a stochastic force by random jumps over the rotational barrier. The increase of the jump rate with temperature follows an Arrhenius function Ea exp ; ð7Þ s1 ¼ s1 0 kT where the activation energy Ea is the difference of the potential maximum and the rotational ground state. The prefactor C0 ¼ s1 represents a line0 width for infinite temperature and is about 5 meV for methyl groups. A data description which yields significantly different values of C0 is very likely based on a wrong model. For a single methyl group rotating in a threefold potential the scattering function is composed of a dfunction and a Lorentzian LðC; xÞ. Inequivalency leads to a superposition of Lorentzians in complete analogy to the superposition of tunneling spectra SðQ; xÞ ¼ ð1 þ 2j0 ðQdÞÞdðxÞ M X pi ð2 2j0 ðQdÞÞLðCi ; xÞ: þ
ð8Þ
i¼1
The occurrence probabilities pi sum up to 1 and are the same as in Eq. (3), the scattering function for tunneling. The Q-dependence of the purely elastic intensity is characteristic of the jump geometry and is called the elastic incoherent structure factor (EISF). Eq. (8) shows the simplest EISF of a methyl group, namely the Fourier transform of an equilateral triangle of d-localized mass points which is sufficiently good for mo1 . The jump distance mentum transfers Q K 1:6 A d of a rotational jump is equal to the proton– in the molecule (cf. Eq. proton distance d ¼ 1:76 A (3)). If one can decompose the quasielastic scattering into its M components, a measurement of the temperature-dependent linewidths yields all M activation energies Ea ðiÞ. Often significant correlations of the fitted linewidths are found. Such correlations can be largely reduced if the line broadenings are forced to follow a sum of Arrhenius functions. By this technique which requires the simultaneous fit of all spectra the number of fit parameters is largely reduced. The nontrivial ones represent directly the various barrier heights.
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2.3. Comparison of tunneling and quasielastic spectroscopy A look on the outlined equations shows that tunneling spectra (TS) involves two characteristic excitations of a rotor, hxt and E01 , while quasielastic scattering (QNS) involves just one, Ea . This means, that TS can allow a refinement of rotational potentials Eq. (1) to second order while QNS alone can reveal a single parameter potential only. Especially for multirotor systems the combined use of TS and QNS is recommended. In such systems the scattering functions of tunneling and jump reorientation are linked by identical occurrence probabilities pi . This allows to combine the complementary informations of the two techniques into more precisely defined shapes of the rotational potentials which is widely freed from arbitrariness. It is helpful to use the knowledge from a systematic analysis of many systems by tunneling spectroscopy that the leading term of rotational potentials is threefold [6]. The energy resolution of a neutron spectrometer determines the regime of potentials which can be explored. The very best energy resolution in neutron spectroscopy of 0.3 leV FWHM limits tunneling spectroscopy of methyl group to potentials smaller than V3 64 meV. For QNS higher barriers are accessible by measuring at higher sample temperatures. At intermediate temperatures around (30–40) K the dynamics is neither weakly perturbed quantum mechanical nor purely classical and cannot be analyzed quantitatively. Furthermore the phase behavior of materials may limit the applicability of these techniques. For example, phases of methane [7], stable below about 40 K only, may not be studied by QNS.
trimethyl compounds, AlðCH3 Þ3 . A room temperature X-ray crystal structure determination shows a tetragonal space group P 42 =nðZ ¼ 8Þ with three inequivalent methyl groups [9]. Tunneling spectra of InðCH3 Þ3 were measured using the backscattering spectrometer BSS1 at FZJ, J€ ulich, with its Si/Ge offset monochromator [10]. With this setup the energy regime 32 < hx½leV < 3 is accessible with an energy resolution of dE 1:5 leV FWHM. Spectra were taken at various temperatures in the range 1:8 < T ½K < 36. A selection of spectra is shown in Fig. 1. Four tunneling bands with intensity ratios of 1:1:1:3 are resolved at the lowest sample temperature (Table 1). The most intense line at about 1.7 leV is not visible in Fig. 1 but was clearly resolved as a shoulder in spectra taken at higher energy resolution. From the total intensities and the ratio of the inelastic intensities, Eq. (5), it is clear that all tunnel transitions have been observed. The overlapping lines show on the one hand that the potentials are very similar in heights, on the other hand this fact prevents the extraction of temperature-dependent linewidths.
3. Trimethylindium In(CH3 )3 InðCH3 Þ3 is very similar to the recently studied GaðCH3 Þ3 [8]. Both materials are used for doping semiconductors by thermal decomposition of the molecules. Since the isolated molecules form flat equilateral triangles with In at the center and methyl groups at the corners there is no tendency to form dimers as the first member of group IIIa
Fig. 1. Tunneling spectra of In(CH3 )3 at temperatures 4.2 K (M), 21.2 K (), 24.4 K ( ), 26.3 K (s). Instrument: BSS1, 1 . FZJ, J€ ulich. Average momentum transfer Q ¼ 1:64 A
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Table 1 exp Rotational potentials derived from tunnel frequencies hxt , the measured librational modes E01 and the activation energies Eaexp xt h [leV]
Relative intensity
exp E01 [meV]
Eaexp [K]
1.7 4.7 8.7 15.9
1 2 1 6 1 6 1 6
16.2 12.7 10.3 10.3
305 305 305 210
Relative intensity 5 6 1 6
V3 [meV]
V6 [meV]
calc E01 [meV]
Eacalc [K]
37.8 34.8 36.7 25.8
10.7 1.8 – 1.7
17.2 13.3 10.3 11.2
320 310 310 220
calc Extracted potentials show consistent first librational levels E01 and activation energies Eacalc . Some excitations are multiple times reproduced to allow for the assignment.
The four tunneling transitions found are not consistent with the crystal structure at room temperature [11]. Thus the system must have undergone a phase transition with cooling. The similarity of the pattern of tunneling lines with that of GaðCH3 Þ3 makes us suppose that the detected phase transition leads into the same low temperature phase C2=c(Z ¼ 16) with six inequivalent methyl groups found for GaðCH3 Þ3 below a temperature T ¼ 130 K [12]. 1 Quasielastic neutron spectra were measured at the cold multichopper time-of-flight spectrometer NEAT [14] of the HMI, Berlin, in the temperature regime 45 < T ½K < 98 using a wavelength of 7 A and an elastic energy resolution of 50 leV. The data were prepared for the fit by subtracting the measured background and using the vanadium resolution function for numerical convolution. The number of Lorentzians to be fitted into the quasielastic intensity and their relative weights are based on the tunneling spectra and the suggested crystal structure. It was impossible to extract six independent activation energies. The fitting procedure resulted in a collapse of the lines in an unsystematic fashion. A fit by a single Lorentzian was also impossible when the total quasielastic intensity was imposed to be fixed for all temperatures. A promising generalization of this simplest model was obtained from a guess based on combining tunnel transitions with peaks of the measured vibrational density of states such that the resulting rotational potentials were almost three-
1
In the meantime an X-ray diffraction experiment has confirmed a phase transition at a temperature of about 65 K [13].
fold: the extracted barrier heights differed by less than 8% for 5 of the 6 methyl groups. One group showed a clearly weaker rotational potential. Thus a new fit model contained two Lorentzians of an intensity ratio 5:1. Further the ratio of inelastic to elastic intensity was determined at a high temperature, where all quasielastic scattering is fully resolved, and kept constant for all temperatures. The final fit had just three nontrivial free parameters, two linewidths and one intensity factor. A consistent fit with this model was possible but up to a temperature T ¼ 55 K only. In this range the final fit (solid lines of Fig. 2) is very good. At higher
Fig. 2. Selected quasielastic spectra of In(CH3 )3 taken at temperatures 45.0 K (s), 50 K ( ) and 55.0 K (). Instrument: NEAT, HMI, Berlin, Germany. Average momentum transfer 1 . Solid lines: fit with the same two Arrhenius Q ¼ 1:4 A functions of intensity ratio 5:1 at all temperatures.
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temperatures 70 6 T ½K 6 98 a single Lorentzian describes the data rather well. This discontinuity is the fingerprint of the postulated phase transition. Due to the unforeseen phase transition at 65 K there were only three temperatures left in each phase to extract activation energies by linear regression (Fig. 3). The activation energies are 210 and 305 K in the LT phase for the components with relative intensities 1 and 5, respectively, and 215 K in the HT phase (Fig. 3). The prefactors are equal within a factor 1.5 and of the right magnitude. (Almost the same result is obtained using the second fit algorithm which fits all spectra simultaneously and assumes the validity of two Arrhenius laws.) It must be kept in mind that the stronger component of the LT phase contains more than one rotor system. Thus the fit optimizes an average barrier height. The same is true for the HT phase which still contains the contributions from three inequivalent methyl species. Due to accidental degeneracies neither the number of resolved Lorentzian nor the number of the tunnel transitions corresponds to the number of crystallographically inequivalent methyl groups. In the present example, it was a combination of tunnel transition with estimated librational energies which yielded a good starting point. Now the librational modes are – and are allowed to be –
Fig. 3. Arrhenius plot of the quasielastic linewidths of the two Lorentzians describing the low temperature spectra at T 6 55 K. Weight factors are 16 ðsÞ and 56 ðÞ. Similar plot for the one linewidth required at T P 70 K. Weight factor 1 (M).
slightly adjusted within the width of dispersed librational bands. Similarly, uncertainties are contained in the mean activation energy. The interpolation between these values allows some little freedom in the choice of the final potential parameters. By combining tunneling transitions and quasielastic Lorentzians of equal relative intensities the procedure automatically contains occurrence probabilities in a consistent form only. The potentials of Table 1 deviate by no more than 20% from pure cosð3uÞ. The rotational potentials in the high temperature (HT) phase show an activation energy very close to the lower one of the low temperature (LT) phase.
4. Potentials of other trimethyl metal compounds 4.1. Trimethylaluminum Al(CH3)3 Trimethylaluminum is of special interest. At first this is due to its technical importance as starting material for doping semiconductors and as catalyst. Secondly trimethylaluminum is the prototype of an electron deficiency compound involving methyl groups. The monomer resembles a tetrahedron with one corner cut away. The free electron density at this corner leads already in the gas phase to the formation of dimers of the shape of two tetrahedra connected by a common edge. The coordination number of Al in the center of each terahedron changes to four, that of the bridging carbon atoms on the common edge to five and bond lengths deviate significantly from standard values. A recent neutron diffraction study [15] has confirmed the space group C2=c with dimers forming the constituents also in the solid state. Due to a center of inversion linking the two halve of the dimer there are three inequivalent methyl groups. Bridging methyl groups are symmetrically bound to the two aluminum atoms. Terminal methyl groups are – as usual – bound by a central r-bond. A recent comprehensive neutron spectroscopic study [16] led to the rotational potentials shown in Table 2. The potential of the most interesting bridging methyl group can be seen only in the QNS spectra and thus can be refined only to first
M. Prager et al. / Chemical Physics 292 (2003) 161–169
167
Table 2 Parameters of rotational potentials (columns 6 and 7) are derived from tunnel splittings hxt and activation energies Ea xt h [leV]
Relative intensity
exp E01 [meV]
Ea [K]
Relative intensity
V3 [meV]
V6 [meV]
calc E01 [meV]
Assignment
(0.33) 12.63 16.24
1 3 1 3 1 3
16–19 12–14 12–14
620 205 170
1 3 1 3 1 3
63 25.0 20.7
– 5.5 9.7
17.6 12.4 12.7
Bridging (b) Terminal (t) Terminal (t)
exp calc Librational modes E01 calculated therefrom agree reasonably well with E01 derived from the measured densities of states on the basis of the isotope effect. hxt shown in brackets is a calculated value.
order. However, the pure cosð3uÞ potential accounts well for a measured librational band extending from 16 to 19 meV. The strength of this potential is due to the need of breaking the two weak bonds to the aluminum atoms with rotation and is thus mainly of intramolecular origin. Therefore it is found nicely in the GAUSSIAN calculation of the isolated dimer at 18.8 meV [16]. 4.2. Trimethylgallium High resolution neutron spectra show four tunneling transitions of different intensities between 4.5 and 19 leV. The spectrum can be explained within the single particle model on the basis of the monoclinic C2=c(Z ¼ 16) low temperature crystal structure of GaðCH3 Þ3 with six inequivalent methyl groups in the unit cell [12]. Overlapping tunneling lines prevent the extraction of temperature-dependent linewidths. Classical rotational motion was studied by quasielastic neutron scattering and led to three distinct activation energies. Methyl librations, tunneling energies and barrier heights are combined with consistent intensities into rotational potentials (Table 3).
5. Structure–potential relation Due to the lack of precise knowledge of the atomic positions in the low temperature phases of GaðCH3 Þ3 and InðCH3 Þ3 the discussion will be based on properties of the molecules and general parameters of the known high temperature structures [9,11,15]. The relevant information is presented in Table 4. We first discuss the pair Ga and In. Table 4 shows an average increase of the In–C bond length compared to the Ga–C bond by about 0.17 A length. This leads to a reduction of the intramolecular contribution to the rotational potential in InðCH3 Þ3 . However, at the same time the intermolecular distances characterized by the In–In distance decreases compared to Ga, which leads to an increase of the intermolecular contribution to the rotational potential. The experiment shows rotational potentials of very similar magnitude for the two materials with somewhat stronger barriers in InðCH3 Þ3 (Tables 1 and 3). Thus the two effects almost cancel. For this reason a systematic trend with the size of the molecule as found for the more spherical tetramethyl metal compounds is absent [17].
Table 3 Parameters of rotational potentials derived from tunnel frequencies hxt and the activation energies Eaexp such that the first librational exp level E01 assigned to the strongest peaks of the DOS are reproduced xt h [leV]
Relative intensity
exp E01 [meV]
Eaexp [K]
Relative intensity
V3 [meV]
V6 [meV]
Eacalc [K]
4.5–6 4.5–6 10.06 17.02 19.05
1 2
11.8 15.0 11.5 12.0 11.6
310 240 240 170 170
1 6 1 2
34.1 29.0 30.4 22.0 22.0
2.5 9.7 )0.6 6.2 5.5
302 235 238 168 170
1 6 1 6 1 6
1 3
The almost degenerate tunneling mode around 5 leV is shown to contain different rotor systems. Some excitations are multiple times reproduced to allow for the assignment.
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M. Prager et al. / Chemical Physics 292 (2003) 161–169
Table 4 Characteristic parameters of the structures of three trimethyl metal compounds Me(CH3 )3 Metal atom Me Al Ga In
Me–C distance 1.926–1.945(t) 2.145(b) 1.952–1.962 2.121–2.179
Me–Me distance a
2.7
3.314–3.647 3.083–3.558
C–Me–C angle
Molecular shape
126 109 120 120
Edge-connected tetrahedra Flat triangle Flat triangle
The Al compound contains terminal (t) and bridging (b) methyl groups due to dimerization [16]. For precise information see the references [9,11,15]. a Within the dimer.
The Al compound differs qualitatively from the two others by the shape of the building unit. Upon dimer formation the angle between the terminal methyl groups widens to 126°. The related increase of the distance between the two terminal methyl groups reduces the intramolecular interaction and the rotational potential weakens. The bridging methyl groups play a minor role because they are out-of-plane and more distant due to the elongated Al–C(b) bond. By chance, the rotational potentials of terminal CH3 groups have similar strengths as methyl groups in InðCH3 Þ3 and GaðCH3 Þ3 . Thus, with the exception of the strongly hindered CH3 group in AlðCH3 Þ3 all rotational potentials fall into the range of 170-310K. The uniquely bond bridging CH3 group in AlðCH3 Þ3 does and cannot fit into the outlined scenario. The loss of a unique rotation axis upon formation of dimers breaks any systematic tendencies anticipated on the basis of the molecular structure of monomers. From the shape of the molecules it must be concluded that intermolecular interaction is mediated via direct methyl–methyl interaction. It is astonishing but a fact that the single particle model describes the rotational potentials well. A discussion beyond these general tendencies will require the not yet existing knowledge of the low temperature structures of InðCH3 Þ3 and GaðCH3 Þ3 . 6. General conclusions Quasielastic spectra and tunneling splittings of rotational groundstates of methyl groups in group
IIIa trimethyl compounds are consistently interpreted by the single particle model of rotation. This shows that even for molecules which interact with each other via their shell of methyl groups the potential energy due to interaction with the static lattice dominates effects of coupling. The similarities of rotational potentials in the three homologue compounds with different bondlengths and intermolecular distances is due to an equal importance of intra- and intermolecular interactions. The results from quasielastic and tunneling spectroscopies are used to determine rotational potentials. This is only possible if the system shows the same crystal structure in the two distinct temperature regimes studied by the two techniques. If, furthermore, the assignment problem of multirotor systems can be solved – for example via consistent relative intensities – potential shapes may safely be determined to second order of a Fourier expansion. In case of trimethylindium only a narrow temperature window is available to study the low temperature phase by quasielastic scattering. In multirotor systems, problems may arise from accidental degeneracies and assignment of characteristic energies. Since sharp inelastic lines are easier to distinguish than overlapping Lorentzians, tunneling spectroscopy shows the higher power of discrimination. The risk that parameters average over different subsystems is significantly larger in case of quasielastic spectra. Accidental degeneracies may often be identified experimentally. Since the rotational potentials include the second-order term such degeneracies are unlikely to occur simultaneously for tunneling and activation energies. While this is true for trimethylgallium, trimethylindium is a counter-example.
M. Prager et al. / Chemical Physics 292 (2003) 161–169
As a rule of thumb the ratio between the threeand the sixfold term as well as their sum may vary within 5% from the proposed potentials due to uncertainties of activation energies arising from averaged Lorentzians or broad librational bands affected by dispersion. For the latter reason one should use librational energies only as control parameters. Finally, it is the consistency of all spectral features and accompanying calculations – if possible – which creates confidence into a proposed assignment.
References [1] M. Bee, Quasielastic Neutron Scattering, Adam Hilger, Bristol, 1988. [2] W. Press, Single Particle Rotations in Molecular Crystals, Springer Tracts in Modern Physics, vol. 81, Springer, Berlin, 1981. [3] M. Prager, A. Heidemann, Chem. Rev. 97 (1997) 2933.
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[4] P. Schiebel, A. Hoser, W. Prandl, G. Heger, W. Paulus, P. Schweiss, J. Phys.: Condens. Mat. 6 (1994) 10989. [5] A.C. Hewson, J. Phys. C 15 (1982) 3841 and 3855. [6] S. Clough, A. Heidemann, A. Horsewill, J.D. Lewis, M.N.J. Paley, J. Phys. C 15 (1982) 2495. [7] W. Press, J. Chem. Phys. 56 (1972) 2597. [8] M. Prager, J. Combet, S.F. Parker, A. Desmedt, R. Lechner, J. Phys.: Condens. Mat. 14 (2002) 10145. [9] A.J. Blake, S. Cradock, J. Chem. Soc. Dalton Trans. (1990) 2393. [10] http://www.fz-juelich.de/iff/Institute/ins/Broschuere_NSE/ bss.shtml. [11] N.W. Mitzel, C. Lustig, R.J.F. Berger, N. Runeberg, Angew. Chem. Int. Ed. 41 (2002) 2519. [12] R. Boese, W. Bl€aser, M. Prager, unpublished. [13] B. Amann, J. Ihringer, M. Prager, unpublished. [14] R.E. Lechner, R. Melzer, J. Fitter, Phys. B 226 (1996) 86. [15] G.S. McGrady, J.F.C. Turner, R.M. Ibberson, M. Prager, Organometallics 19 (2000) 4398. [16] M. Prager, H. Grimm, S.F. Parker, R. Lechner, A. Desmedt, S. McGrady, E. Koglin, J. Phys.: Condens. Mat. 14 (2002) 1833. [17] W. M€ uller-Warmuth, K.H. Dupree, M. Prager, Z. Naturforsch. 39a (1983) 66.
Methyl rotational potentials of trimethyl metal compounds studied by inelastic and quasielastic neutron scattering M. Prager a,*, H. Grimm a, A. Desmedt b, R.E. Lechner b a
Institut fu€r Festk€ orperforschung, Forschungzzentrum J€ ulich, D-52425 J€ ulich, Germany b Hahn Meitner Institut, Glienickerstr. 100, D-14109 Berlin, Germany Received 8 October 2002; in final form 16 April 2003
Abstract Neutron scattering data on classical and quantum rotation of methyl groups in the organometallic molecular crystal In(CH3 )3 are presented. The tunneling spectra are analyzed using the mean field single particle model of rotational tunneling. The presence of six inequivalent rotors of equal occurrence probabilities in InðCH3 Þ3 is reflected in the number and intensities of the transitions. In a similar way classical spectra are interpreted as superposition of quasielastic Lorentzians. A consistent combination of tunneling and quasielastic spectroscopies is based on equal relative intensities of the respective spectral components and allows a determination of the coefficients of a Fourier expansion of the rotational potentials up to second order with only a few percent of uncertainty. The results are compared to published data on the homologues Al and Ga compounds on the basis of the molecular and crystal structures. Intraand intermolecular interactions are found to be of similar importance. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 6112E; 6320; 6630
1. Introduction One of the most fundamental and most simple dynamical processes is the rotation of a methyl group into a new equilibrium orientation. The potential which describes this dynamics is determined by the intermolecular interactions with the neighboring atoms which may be parameterized as
*
Corresponding author. Fax: +49-2461-612610. E-mail address: [email protected] (M. Prager).
universal force fields (UFF), pair interaction potentials or in ab initio program codes. Neutron scattering has developed two techniques to explore molecular rotation in solids. On the one hand quasielastic spectroscopy reveals the classical motion of an individual atom [1]. If this atom is bound to a molecule or a molecular side group it monitors its rotational dynamics. The spectra are analyzed with the concept of the elastic incoherent structure factor. The output from this method are the jump geometry including jump distances, the related
0301-0104/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00207-6
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jump times, phase transitions as a discontinuity in the evolution of the dynamics with increasing temperature, etc. On the other hand molecules show a quantum behavior at low temperatures. The quantum excitations of a rotor are observed by rotational tunneling [2,3] and vibrational spectroscopies. Both theories represent in their standard forms single particle descriptions. The rotor is characterized by its momentum of inertia and the environment is represented by a potential. It is this potential which links the results of the two techniques to each other. The topology of the potential/environment determines the jump geometry; the barrier height determines the classical jump rate at a given temperature. The shape and height of the same potential determine also the eigenstates and eigenvalues of a rotor which – vice versa – allow to determine the potential parameters. An important property of the single particle model is its transferability. The symmetry of the rotational potential is only determined by the rotor and is the same in any material. In cases of coupling a second degree of freedom has to be considered. The mathematical description has to be formulated newly for each new type of coupling. This is done for very few cases only like direct rotor–rotor coupling or the so-called rotation–translation coupling [4]. Even here a new symmetry already requires the development of a new mathematical theory. Thus, in spite of weak points the single particle model remains the standard description of tunneling methyl groups. The following discussion is restricted to the one-dimensional rotation of a methyl group present in the InðCH3 Þ3 compound. InðCH3 Þ3 like many other materials contains inequivalent rotors. Their number is determined by the crystal structure and eventually disorder. By combining quasielastic and tunneling spectroscopies rotational potentials of all inequivalent rotors can be obtained with high accuracy and confidence. A comparison to results from the homologues Ga and Al compounds will allow an evaluation of the relative importance of intra- and intermolecular interactions.
2. Single particle dynamics of methyl groups 2.1. Rotational tunneling The methyl rotational potential in the single particle model [2,3] shows the threefold symmetry of the molecule and is expressed in form of a Fourier expansion V ðuÞ ¼
N X V3n ð1 cosð3nu u3n ÞÞ: 2 n¼1
ð1Þ
Arbitrarily one can chose u3 ¼ 0. The potential determines the excitations of the hindered rotor through the eigenvalues of the single particle Schr€ odinger equation [2] o2 B 2 þ V ðuÞ Wi ¼ Ei Wi : ð2Þ ou The more eigenvalues are measured the better the shape of the potential – including phase factors – can be determined. u6 ¼ 0 or p is often applied to limit the number of free parameters to be deduced from the few observables like tunneling splitting and activation energy. In the tables shown in the following we (must) adopt this restriction. A positive (negative) sign of V6 corresponds to u6 ¼ 0ðpÞ. In general, unrestricted phase factors are only obtained when the potential is derived from ab initio or force-field calculations using the crystal structure. The scattering of neutrons by a methyl group is dominated by the large incoherent scattering cross-section of the hydrogen atoms. The powder averaged scattering function of a single tunneling rotor is [2] 5 4 Si ðQ; xÞ ¼ þ j0 ðQdÞ dðxÞ 3 3 2 2 þ j0 ðQdÞ fdðx þ xti Þ 3 3 þ dðx xti Þg:
ð3Þ
Here xti means the tunnel frequency. The Q-dependent intensity coefficients are called structure factors. Its expression in Eq. (3) is based on dlocalized methyl protons. This approximation is 1 . good up to momentum transfers of Q 1:6 A The spherical Bessel function j0 ðQdÞ contains
M. Prager et al. / Chemical Physics 292 (2003) 161–169
in the the proton–proton distance d ¼ 1:76 A molecule. The single particle model yields the total scattering function of a sample with M inequivalent methyl groups as a sum of scattering functions of independent rotors weighted by their occurrence probabilities pi SðQ; xÞ ¼
M X
pi Si ðQ; xÞ:
ð4Þ
i¼1
The occurrence probabilities pi of all species i sum up to 1. From this scattering function, one derives the ratios of the inelastic intensity of an individual methyl group to the total elastic intensity Ri;el and the ratio Ri;j of two different tunneling transitions Iinel pi ð2 2j0 Þ ¼ ; 5 þ 4j0 Iel pi Ri;j ¼ : pj
Ri;el ¼
ð5Þ
These expressions are very useful for the interpretation of a multiline spectrum but in the given form only valid if all tunneling transitions are resolved. If this is not the case obvious modifications have to be applied. Interaction of a rotor with a phonon bath at finite sample temperature introduces stochastic forces and therefrom fluctuating potential terms. As a result the tunnel lines shift by Dhxt and broaden by Ct with increasing temperature [5] ES D hxt exp ; kT ð6Þ EC : Ct ¼ C0 exp kT The activation energy of line broadening is determined by resonant coupling to phonons with an energy equal to the methyl libration. Thus it is possible to get the librational energy E01 of the respective unperturbed rotor from EC ¼ E01 . 2.2. Classical reorientational motion According to Eqs. (6) a tunnel transitions merges into the elastic line and broadens. As required by BohrÕs correspondence principle the resulting quasielastic Lorentzian is the response of a
163
rotor reorienting classically under the influence of a stochastic force by random jumps over the rotational barrier. The increase of the jump rate with temperature follows an Arrhenius function Ea exp ; ð7Þ s1 ¼ s1 0 kT where the activation energy Ea is the difference of the potential maximum and the rotational ground state. The prefactor C0 ¼ s1 represents a line0 width for infinite temperature and is about 5 meV for methyl groups. A data description which yields significantly different values of C0 is very likely based on a wrong model. For a single methyl group rotating in a threefold potential the scattering function is composed of a dfunction and a Lorentzian LðC; xÞ. Inequivalency leads to a superposition of Lorentzians in complete analogy to the superposition of tunneling spectra SðQ; xÞ ¼ ð1 þ 2j0 ðQdÞÞdðxÞ M X pi ð2 2j0 ðQdÞÞLðCi ; xÞ: þ
ð8Þ
i¼1
The occurrence probabilities pi sum up to 1 and are the same as in Eq. (3), the scattering function for tunneling. The Q-dependence of the purely elastic intensity is characteristic of the jump geometry and is called the elastic incoherent structure factor (EISF). Eq. (8) shows the simplest EISF of a methyl group, namely the Fourier transform of an equilateral triangle of d-localized mass points which is sufficiently good for mo1 . The jump distance mentum transfers Q K 1:6 A d of a rotational jump is equal to the proton– in the molecule (cf. Eq. proton distance d ¼ 1:76 A (3)). If one can decompose the quasielastic scattering into its M components, a measurement of the temperature-dependent linewidths yields all M activation energies Ea ðiÞ. Often significant correlations of the fitted linewidths are found. Such correlations can be largely reduced if the line broadenings are forced to follow a sum of Arrhenius functions. By this technique which requires the simultaneous fit of all spectra the number of fit parameters is largely reduced. The nontrivial ones represent directly the various barrier heights.
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2.3. Comparison of tunneling and quasielastic spectroscopy A look on the outlined equations shows that tunneling spectra (TS) involves two characteristic excitations of a rotor, hxt and E01 , while quasielastic scattering (QNS) involves just one, Ea . This means, that TS can allow a refinement of rotational potentials Eq. (1) to second order while QNS alone can reveal a single parameter potential only. Especially for multirotor systems the combined use of TS and QNS is recommended. In such systems the scattering functions of tunneling and jump reorientation are linked by identical occurrence probabilities pi . This allows to combine the complementary informations of the two techniques into more precisely defined shapes of the rotational potentials which is widely freed from arbitrariness. It is helpful to use the knowledge from a systematic analysis of many systems by tunneling spectroscopy that the leading term of rotational potentials is threefold [6]. The energy resolution of a neutron spectrometer determines the regime of potentials which can be explored. The very best energy resolution in neutron spectroscopy of 0.3 leV FWHM limits tunneling spectroscopy of methyl group to potentials smaller than V3 64 meV. For QNS higher barriers are accessible by measuring at higher sample temperatures. At intermediate temperatures around (30–40) K the dynamics is neither weakly perturbed quantum mechanical nor purely classical and cannot be analyzed quantitatively. Furthermore the phase behavior of materials may limit the applicability of these techniques. For example, phases of methane [7], stable below about 40 K only, may not be studied by QNS.
trimethyl compounds, AlðCH3 Þ3 . A room temperature X-ray crystal structure determination shows a tetragonal space group P 42 =nðZ ¼ 8Þ with three inequivalent methyl groups [9]. Tunneling spectra of InðCH3 Þ3 were measured using the backscattering spectrometer BSS1 at FZJ, J€ ulich, with its Si/Ge offset monochromator [10]. With this setup the energy regime 32 < hx½leV < 3 is accessible with an energy resolution of dE 1:5 leV FWHM. Spectra were taken at various temperatures in the range 1:8 < T ½K < 36. A selection of spectra is shown in Fig. 1. Four tunneling bands with intensity ratios of 1:1:1:3 are resolved at the lowest sample temperature (Table 1). The most intense line at about 1.7 leV is not visible in Fig. 1 but was clearly resolved as a shoulder in spectra taken at higher energy resolution. From the total intensities and the ratio of the inelastic intensities, Eq. (5), it is clear that all tunnel transitions have been observed. The overlapping lines show on the one hand that the potentials are very similar in heights, on the other hand this fact prevents the extraction of temperature-dependent linewidths.
3. Trimethylindium In(CH3 )3 InðCH3 Þ3 is very similar to the recently studied GaðCH3 Þ3 [8]. Both materials are used for doping semiconductors by thermal decomposition of the molecules. Since the isolated molecules form flat equilateral triangles with In at the center and methyl groups at the corners there is no tendency to form dimers as the first member of group IIIa
Fig. 1. Tunneling spectra of In(CH3 )3 at temperatures 4.2 K (M), 21.2 K (), 24.4 K ( ), 26.3 K (s). Instrument: BSS1, 1 . FZJ, J€ ulich. Average momentum transfer Q ¼ 1:64 A
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165
Table 1 exp Rotational potentials derived from tunnel frequencies hxt , the measured librational modes E01 and the activation energies Eaexp xt h [leV]
Relative intensity
exp E01 [meV]
Eaexp [K]
1.7 4.7 8.7 15.9
1 2 1 6 1 6 1 6
16.2 12.7 10.3 10.3
305 305 305 210
Relative intensity 5 6 1 6
V3 [meV]
V6 [meV]
calc E01 [meV]
Eacalc [K]
37.8 34.8 36.7 25.8
10.7 1.8 – 1.7
17.2 13.3 10.3 11.2
320 310 310 220
calc Extracted potentials show consistent first librational levels E01 and activation energies Eacalc . Some excitations are multiple times reproduced to allow for the assignment.
The four tunneling transitions found are not consistent with the crystal structure at room temperature [11]. Thus the system must have undergone a phase transition with cooling. The similarity of the pattern of tunneling lines with that of GaðCH3 Þ3 makes us suppose that the detected phase transition leads into the same low temperature phase C2=c(Z ¼ 16) with six inequivalent methyl groups found for GaðCH3 Þ3 below a temperature T ¼ 130 K [12]. 1 Quasielastic neutron spectra were measured at the cold multichopper time-of-flight spectrometer NEAT [14] of the HMI, Berlin, in the temperature regime 45 < T ½K < 98 using a wavelength of 7 A and an elastic energy resolution of 50 leV. The data were prepared for the fit by subtracting the measured background and using the vanadium resolution function for numerical convolution. The number of Lorentzians to be fitted into the quasielastic intensity and their relative weights are based on the tunneling spectra and the suggested crystal structure. It was impossible to extract six independent activation energies. The fitting procedure resulted in a collapse of the lines in an unsystematic fashion. A fit by a single Lorentzian was also impossible when the total quasielastic intensity was imposed to be fixed for all temperatures. A promising generalization of this simplest model was obtained from a guess based on combining tunnel transitions with peaks of the measured vibrational density of states such that the resulting rotational potentials were almost three-
1
In the meantime an X-ray diffraction experiment has confirmed a phase transition at a temperature of about 65 K [13].
fold: the extracted barrier heights differed by less than 8% for 5 of the 6 methyl groups. One group showed a clearly weaker rotational potential. Thus a new fit model contained two Lorentzians of an intensity ratio 5:1. Further the ratio of inelastic to elastic intensity was determined at a high temperature, where all quasielastic scattering is fully resolved, and kept constant for all temperatures. The final fit had just three nontrivial free parameters, two linewidths and one intensity factor. A consistent fit with this model was possible but up to a temperature T ¼ 55 K only. In this range the final fit (solid lines of Fig. 2) is very good. At higher
Fig. 2. Selected quasielastic spectra of In(CH3 )3 taken at temperatures 45.0 K (s), 50 K ( ) and 55.0 K (). Instrument: NEAT, HMI, Berlin, Germany. Average momentum transfer 1 . Solid lines: fit with the same two Arrhenius Q ¼ 1:4 A functions of intensity ratio 5:1 at all temperatures.
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temperatures 70 6 T ½K 6 98 a single Lorentzian describes the data rather well. This discontinuity is the fingerprint of the postulated phase transition. Due to the unforeseen phase transition at 65 K there were only three temperatures left in each phase to extract activation energies by linear regression (Fig. 3). The activation energies are 210 and 305 K in the LT phase for the components with relative intensities 1 and 5, respectively, and 215 K in the HT phase (Fig. 3). The prefactors are equal within a factor 1.5 and of the right magnitude. (Almost the same result is obtained using the second fit algorithm which fits all spectra simultaneously and assumes the validity of two Arrhenius laws.) It must be kept in mind that the stronger component of the LT phase contains more than one rotor system. Thus the fit optimizes an average barrier height. The same is true for the HT phase which still contains the contributions from three inequivalent methyl species. Due to accidental degeneracies neither the number of resolved Lorentzian nor the number of the tunnel transitions corresponds to the number of crystallographically inequivalent methyl groups. In the present example, it was a combination of tunnel transition with estimated librational energies which yielded a good starting point. Now the librational modes are – and are allowed to be –
Fig. 3. Arrhenius plot of the quasielastic linewidths of the two Lorentzians describing the low temperature spectra at T 6 55 K. Weight factors are 16 ðsÞ and 56 ðÞ. Similar plot for the one linewidth required at T P 70 K. Weight factor 1 (M).
slightly adjusted within the width of dispersed librational bands. Similarly, uncertainties are contained in the mean activation energy. The interpolation between these values allows some little freedom in the choice of the final potential parameters. By combining tunneling transitions and quasielastic Lorentzians of equal relative intensities the procedure automatically contains occurrence probabilities in a consistent form only. The potentials of Table 1 deviate by no more than 20% from pure cosð3uÞ. The rotational potentials in the high temperature (HT) phase show an activation energy very close to the lower one of the low temperature (LT) phase.
4. Potentials of other trimethyl metal compounds 4.1. Trimethylaluminum Al(CH3)3 Trimethylaluminum is of special interest. At first this is due to its technical importance as starting material for doping semiconductors and as catalyst. Secondly trimethylaluminum is the prototype of an electron deficiency compound involving methyl groups. The monomer resembles a tetrahedron with one corner cut away. The free electron density at this corner leads already in the gas phase to the formation of dimers of the shape of two tetrahedra connected by a common edge. The coordination number of Al in the center of each terahedron changes to four, that of the bridging carbon atoms on the common edge to five and bond lengths deviate significantly from standard values. A recent neutron diffraction study [15] has confirmed the space group C2=c with dimers forming the constituents also in the solid state. Due to a center of inversion linking the two halve of the dimer there are three inequivalent methyl groups. Bridging methyl groups are symmetrically bound to the two aluminum atoms. Terminal methyl groups are – as usual – bound by a central r-bond. A recent comprehensive neutron spectroscopic study [16] led to the rotational potentials shown in Table 2. The potential of the most interesting bridging methyl group can be seen only in the QNS spectra and thus can be refined only to first
M. Prager et al. / Chemical Physics 292 (2003) 161–169
167
Table 2 Parameters of rotational potentials (columns 6 and 7) are derived from tunnel splittings hxt and activation energies Ea xt h [leV]
Relative intensity
exp E01 [meV]
Ea [K]
Relative intensity
V3 [meV]
V6 [meV]
calc E01 [meV]
Assignment
(0.33) 12.63 16.24
1 3 1 3 1 3
16–19 12–14 12–14
620 205 170
1 3 1 3 1 3
63 25.0 20.7
– 5.5 9.7
17.6 12.4 12.7
Bridging (b) Terminal (t) Terminal (t)
exp calc Librational modes E01 calculated therefrom agree reasonably well with E01 derived from the measured densities of states on the basis of the isotope effect. hxt shown in brackets is a calculated value.
order. However, the pure cosð3uÞ potential accounts well for a measured librational band extending from 16 to 19 meV. The strength of this potential is due to the need of breaking the two weak bonds to the aluminum atoms with rotation and is thus mainly of intramolecular origin. Therefore it is found nicely in the GAUSSIAN calculation of the isolated dimer at 18.8 meV [16]. 4.2. Trimethylgallium High resolution neutron spectra show four tunneling transitions of different intensities between 4.5 and 19 leV. The spectrum can be explained within the single particle model on the basis of the monoclinic C2=c(Z ¼ 16) low temperature crystal structure of GaðCH3 Þ3 with six inequivalent methyl groups in the unit cell [12]. Overlapping tunneling lines prevent the extraction of temperature-dependent linewidths. Classical rotational motion was studied by quasielastic neutron scattering and led to three distinct activation energies. Methyl librations, tunneling energies and barrier heights are combined with consistent intensities into rotational potentials (Table 3).
5. Structure–potential relation Due to the lack of precise knowledge of the atomic positions in the low temperature phases of GaðCH3 Þ3 and InðCH3 Þ3 the discussion will be based on properties of the molecules and general parameters of the known high temperature structures [9,11,15]. The relevant information is presented in Table 4. We first discuss the pair Ga and In. Table 4 shows an average increase of the In–C bond length compared to the Ga–C bond by about 0.17 A length. This leads to a reduction of the intramolecular contribution to the rotational potential in InðCH3 Þ3 . However, at the same time the intermolecular distances characterized by the In–In distance decreases compared to Ga, which leads to an increase of the intermolecular contribution to the rotational potential. The experiment shows rotational potentials of very similar magnitude for the two materials with somewhat stronger barriers in InðCH3 Þ3 (Tables 1 and 3). Thus the two effects almost cancel. For this reason a systematic trend with the size of the molecule as found for the more spherical tetramethyl metal compounds is absent [17].
Table 3 Parameters of rotational potentials derived from tunnel frequencies hxt and the activation energies Eaexp such that the first librational exp level E01 assigned to the strongest peaks of the DOS are reproduced xt h [leV]
Relative intensity
exp E01 [meV]
Eaexp [K]
Relative intensity
V3 [meV]
V6 [meV]
Eacalc [K]
4.5–6 4.5–6 10.06 17.02 19.05
1 2
11.8 15.0 11.5 12.0 11.6
310 240 240 170 170
1 6 1 2
34.1 29.0 30.4 22.0 22.0
2.5 9.7 )0.6 6.2 5.5
302 235 238 168 170
1 6 1 6 1 6
1 3
The almost degenerate tunneling mode around 5 leV is shown to contain different rotor systems. Some excitations are multiple times reproduced to allow for the assignment.
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M. Prager et al. / Chemical Physics 292 (2003) 161–169
Table 4 Characteristic parameters of the structures of three trimethyl metal compounds Me(CH3 )3 Metal atom Me Al Ga In
Me–C distance 1.926–1.945(t) 2.145(b) 1.952–1.962 2.121–2.179
Me–Me distance a
2.7
3.314–3.647 3.083–3.558
C–Me–C angle
Molecular shape
126 109 120 120
Edge-connected tetrahedra Flat triangle Flat triangle
The Al compound contains terminal (t) and bridging (b) methyl groups due to dimerization [16]. For precise information see the references [9,11,15]. a Within the dimer.
The Al compound differs qualitatively from the two others by the shape of the building unit. Upon dimer formation the angle between the terminal methyl groups widens to 126°. The related increase of the distance between the two terminal methyl groups reduces the intramolecular interaction and the rotational potential weakens. The bridging methyl groups play a minor role because they are out-of-plane and more distant due to the elongated Al–C(b) bond. By chance, the rotational potentials of terminal CH3 groups have similar strengths as methyl groups in InðCH3 Þ3 and GaðCH3 Þ3 . Thus, with the exception of the strongly hindered CH3 group in AlðCH3 Þ3 all rotational potentials fall into the range of 170-310K. The uniquely bond bridging CH3 group in AlðCH3 Þ3 does and cannot fit into the outlined scenario. The loss of a unique rotation axis upon formation of dimers breaks any systematic tendencies anticipated on the basis of the molecular structure of monomers. From the shape of the molecules it must be concluded that intermolecular interaction is mediated via direct methyl–methyl interaction. It is astonishing but a fact that the single particle model describes the rotational potentials well. A discussion beyond these general tendencies will require the not yet existing knowledge of the low temperature structures of InðCH3 Þ3 and GaðCH3 Þ3 . 6. General conclusions Quasielastic spectra and tunneling splittings of rotational groundstates of methyl groups in group
IIIa trimethyl compounds are consistently interpreted by the single particle model of rotation. This shows that even for molecules which interact with each other via their shell of methyl groups the potential energy due to interaction with the static lattice dominates effects of coupling. The similarities of rotational potentials in the three homologue compounds with different bondlengths and intermolecular distances is due to an equal importance of intra- and intermolecular interactions. The results from quasielastic and tunneling spectroscopies are used to determine rotational potentials. This is only possible if the system shows the same crystal structure in the two distinct temperature regimes studied by the two techniques. If, furthermore, the assignment problem of multirotor systems can be solved – for example via consistent relative intensities – potential shapes may safely be determined to second order of a Fourier expansion. In case of trimethylindium only a narrow temperature window is available to study the low temperature phase by quasielastic scattering. In multirotor systems, problems may arise from accidental degeneracies and assignment of characteristic energies. Since sharp inelastic lines are easier to distinguish than overlapping Lorentzians, tunneling spectroscopy shows the higher power of discrimination. The risk that parameters average over different subsystems is significantly larger in case of quasielastic spectra. Accidental degeneracies may often be identified experimentally. Since the rotational potentials include the second-order term such degeneracies are unlikely to occur simultaneously for tunneling and activation energies. While this is true for trimethylgallium, trimethylindium is a counter-example.
M. Prager et al. / Chemical Physics 292 (2003) 161–169
As a rule of thumb the ratio between the threeand the sixfold term as well as their sum may vary within 5% from the proposed potentials due to uncertainties of activation energies arising from averaged Lorentzians or broad librational bands affected by dispersion. For the latter reason one should use librational energies only as control parameters. Finally, it is the consistency of all spectral features and accompanying calculations – if possible – which creates confidence into a proposed assignment.
References [1] M. Bee, Quasielastic Neutron Scattering, Adam Hilger, Bristol, 1988. [2] W. Press, Single Particle Rotations in Molecular Crystals, Springer Tracts in Modern Physics, vol. 81, Springer, Berlin, 1981. [3] M. Prager, A. Heidemann, Chem. Rev. 97 (1997) 2933.
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[4] P. Schiebel, A. Hoser, W. Prandl, G. Heger, W. Paulus, P. Schweiss, J. Phys.: Condens. Mat. 6 (1994) 10989. [5] A.C. Hewson, J. Phys. C 15 (1982) 3841 and 3855. [6] S. Clough, A. Heidemann, A. Horsewill, J.D. Lewis, M.N.J. Paley, J. Phys. C 15 (1982) 2495. [7] W. Press, J. Chem. Phys. 56 (1972) 2597. [8] M. Prager, J. Combet, S.F. Parker, A. Desmedt, R. Lechner, J. Phys.: Condens. Mat. 14 (2002) 10145. [9] A.J. Blake, S. Cradock, J. Chem. Soc. Dalton Trans. (1990) 2393. [10] http://www.fz-juelich.de/iff/Institute/ins/Broschuere_NSE/ bss.shtml. [11] N.W. Mitzel, C. Lustig, R.J.F. Berger, N. Runeberg, Angew. Chem. Int. Ed. 41 (2002) 2519. [12] R. Boese, W. Bl€aser, M. Prager, unpublished. [13] B. Amann, J. Ihringer, M. Prager, unpublished. [14] R.E. Lechner, R. Melzer, J. Fitter, Phys. B 226 (1996) 86. [15] G.S. McGrady, J.F.C. Turner, R.M. Ibberson, M. Prager, Organometallics 19 (2000) 4398. [16] M. Prager, H. Grimm, S.F. Parker, R. Lechner, A. Desmedt, S. McGrady, E. Koglin, J. Phys.: Condens. Mat. 14 (2002) 1833. [17] W. M€ uller-Warmuth, K.H. Dupree, M. Prager, Z. Naturforsch. 39a (1983) 66.