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Diffusive dynamics in polymer gel electrolytes investigated by quasi-elastic neutron scattering
Physica B 301 (2001) 44–48
Diffusive dynamics in polymer gel electrolytes investigated by quasi-elastic neutron scattering b . D. Anderssona,*, C. Svanberga, J. Swensonb, W.S. Howellsc, L. Borjesson a
. Department of Experimental Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden b . Department of Applied Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden c Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK
Abstract Quasi-elastic neutron scattering has been performed on a polymer gel electrolyte consisting of lithium perchlorate dissolved in ethylene carbonate=propylene carbonate and stabilized with poly(methyl methacrylate). Two relaxational processes are observed, and their momentum transfer dependencies are studied. From this we attribute the faster process to a rotational motion and the slower process to diffusive motion of the solvent. The diffusive process was modelled with a jump diffusion model, giving a diffusion constant of 1:6 1010 m2 =s, a mean residence time of 34 ps ( A jump rotation model analysis of the rotational motion yields a rotation radius of and mean jump length of 1:8 A. ( which is compatible with the size of the solvent molecules. # 2001 Published by Elsevier Science B.V. 1:5 A, PACS: 66.10.x; 83.70.Hq; 29.30.Hs Keywords: Polymer gel; Electrolyte; Quasi-elastic scattering; Diffusion
1. Introduction The increasing demand for battery-powered products, and the limited battery capacities hampering potential applications, has boosted an intensive research aiming at developing new battery concepts. Regarding the electrolyte, a common choice, e.g. in lithium batteries, is a salt in various organic solvents with high dielectric constants. Safety, construction and mechanical aspects would favour the use of solid electrolytes. This has fed a large interest in polymer electrolytes *Corresponding author. Tel.: +46-31-772-3352; fax: +4631-772-3177. E-mail address: [email protected] (D. Andersson).
[1], in which alkali salts are dissolved in polyethers. Unfortunately, despite intensive research, the conductivity, which in these systems are related to the polymer segmental mobility [2], is still too low to be commercially viable. An alternative way to obtain a solid electrolyte is to start from the organic electrolyte and immobilize it in a polymer gel. Such a polymer gel electrolyte based on poly(methyl methacrylate) (PMMA) has a conductivity close to that of the liquid electrolyte [3]. To understand the high conductivity of polymer gel electrolytes, the microscopic dynamics of these systems should be studied. Previous photon correlation spectroscopy (PCS) studies on polymer gel electrolytes have shown that they have a complicated dynamical
0921-4526/01/$ - see front matter # 2001 Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 5 0 9 - 9
D. Andersson et al. / Physica B 301 (2001) 44–48
response [4]. A diffusive motion of low molecular weight compounds was shown to be closely coupled to the conductivity mechanism, while the segmental motion of the polymer is almost completely decoupled from the conductivity properties. This suggests that the polymer gel electrolyte resembles liquid electrolytes concerning the conductivity properties while the polymer matrix gives the mechanical stability. In the present study, the dynamics in the picosecond range at molecular length scales is studied using quasi-elastic neutron scattering (QENS). Two processes are identified and characterised from their momentum transfer (Q) dependencies. Their molecular origins are briefly discussed as is the influence of the polymer matrix on the dynamics. This is the first QENS study of a polymer gel electrolyte.
2. Experimental The polymer gel electrolyte was prepared by dissolving lithium perchlorate ðLiClO4 Þ in a mixture of ethylene carbonate (EC) and propylene carbonate (PC) to form a liquid electrolyte. The liquid electrolyte was then immobilized into a polymer gel with high molecular weight PMMA. For the sample of the present study deuterated PMMA (Polymer Source Inc., MW ¼ 1 587 000) has been used. The preparation follows the standard gel preparation procedures described in Ref. [3]. The initial materials used were of battery grade i.e. of high purity and the water content was less than 50 ppm. The obtained sample was optically transparent and sealed under argon atmosphere. The average mole ratio of LiClO4 : EC : PC : PMMA in the sample was 4:5 : 46:5 : 19 : 30, where the PMMA ratio refers to the repeating unit. After completed gelation the sample was placed in a flat air-tight aluminium cell with a sample thickness of 360 mm. The thickness was chosen to have 10% of the neutrons scattered by the sample when its angle to the beam is 458, the angle used in the present study. The level of scattering was chosen to keep down multiple scattering to the extent that it can be ignored in the data analysis. The measurements were performed
45
on the IRIS time of flight (back-scattering) spectrometer at the ISIS pulsed spallation source, Rutherford Appleton Laboratory, UK. The pyrolythic graphite 002 (PG002) analyser reflection used in the present study has a resolution (FWHM) of 15 meV over an energy ( 1 . window 0:4 meV and a Q-range 0.4–1:85 A The measurement was performed at 293 K, which is well above the glass transition temperature of the liquid electrolyte (Tg 160 K) but below Tg of bulk PMMA. Due to the high abundance of 1 H in the sample (25%) and the large incoherent scattering cross section of 1 H, the incoherent scattering from the hydrogens of the solvent amounts to 81% of the total scattering. The scattering from PMMA amounts to only 10% of the total scattering due to the use of deuterated PMMA. Since the PMMA scattering is mainly coherent, this part is considerably higher at peaks of the static structure factor of PMMA. The scattering from PMMA will mainly contribute to the elastic scattering as the measurement is made well below Tg of PMMA and methyl group dynamics does not give any coherent scattering. The quasielastic scattering is thus likely to be predominantly incoherent. The 51 detectors were grouped into 16 groups to give a good coverage of the available Q-window. Data corrections for absorption, background and can scattering were performed using the on-site software package GUIDE [5]. Data from detectors ( 1 ), which in the plane of the sample (Q 1:7 A suffer from severe absorption and multiple scattering effects, were discarded.
3. Results The obtained SðQ; oÞ for the studied polymer ( 1 is gel electrolyte at 293 K and Q ¼ 1:27 A shown in Fig. 1. The data were fitted with an elastic peak and Lorentzian peaks, all convoluted with the resolution from a vanadium reference. Two Lorentzians were necessary to properly describe the data. The fit of the data in Fig. 1 and the components used are included in the figure together with the errors of the fit. No significant
46
D. Andersson et al. / Physica B 301 (2001) 44–48
( 1 ; oÞ. Experimental data plus the two Fig. 1. IðQ ¼ 1:27 A curve-fitted processes, static scattering and background are shown on a log-scale. Underneath the difference between data and curve-fit is shown.
improvement was achieved using additional Lorentzians. For comparison the data were also analysed with a stretched exponential relaxation process, fðtÞ ¼ expððt=tKWW Þb Þ for the quasi-elastic scattering. The fits were of similar quality, but the uncertainties for the stretched exponential were larger with no clear trends for neither t nor b. Additionally there is no physical process with stretched behaviour to be expected using QENS at the present temperature, which is far above Tg of the solvent. Hence this function will not be further considered. The widths (FWHM) of the two processes are shown in Fig. 2. As seen in the graph the faster process has no clear momentum transfer dependence, in contrast to the slower process. The relative integrated intensities of the two processes and the elastic scattering from the fits are shown in Fig. 3. One should note that the elastic part may contain some Q-dependence reflecting the coherent scattering, mainly from PMMA. The elastic line decreases in intensity with momentum transfer, whereas the faster process shows an increase. The Q-dependence of the slower process is more complex.
Fig. 2. The widths (FWHM) of the Lorentzians obtained from the curve-fits described in the text. The upper graph shows the fast process (&), while the lower graph shows the slow process (*). A curve-fit to the Gaussian jump length distribution model of diffusion (}) is also inserted.
Fig. 3. Q-dependence of the integrated intensities, normalised to the total integrated scattering intensity. Fast process ð&Þ, slow process (*) and elastic line ( * ). A fit of the fast process to the expected intensity of a jump rotation process (}) is also included.
D. Andersson et al. / Physica B 301 (2001) 44–48
4. Discussion The polymer gel electrolytes are complex materials with many possible dynamical processes. Since we are substantially above Tg of the solvent, translational diffussion processes are plausible, as well as (jump) rotational diffusion of solvent molecules or part thereof. To distinguish these processes it is valuable to study the Q-dependence of the width as the momentum transfer approaches zero. Diffusive processes have a Q2 -dependence of the width for low Q, whereas homogenous rotational processes should have Q-independent widths. The slower process shows clear diffusive features and we show in Fig. 2 that the wave vector dependence can be fitted to the Gaussian jump length distribution model [6]: DE ¼ ð2h=tÞ ½1 expðQ2 hr2 i=2Þ . From the curve-fit we obtain a harmonic mean residence time t of 34ð5Þ ps and a rms jump length hr2 i1=2 of ( Using Dtrans ¼ hr2 i=ð6tÞ gives a 1:8ð0:2Þ A. translational diffusion constant of 1:6ð0:3Þ 1010 m2 =s. The width of the faster process has larger uncertainty, but indicates a nonzero width as the momentum transfer approaches zero, incompatible with diffusion. The slight increase with Q, might well be compatible with the identification of this process as rotational, considering the complexity of this system, and the different molecules in the solvent. More insight can be obtained from the Qdependencies of the relative intensities. From simple models, diffusion is expected to have constant intensities, whereas the intensity of rotation processes increases with momentum transfer for Q lower than those corresponding to the size of the rotating moieties. The fast process can be fitted to p a ffiffimodel for jump rotation [7]: ffi IðQÞ ¼ A23ð1 j0 ð 3Qrrot ÞÞ where A is the fraction of atoms involved in the process, rrot is the radius of the circle along which the scattering atoms rotate and j0 is the zeroth order spherical Bessel function of first kind. We obtain a radius of ( which is comparable with rotation of 1:5ð0:2Þ A, the radius of the solvent molecules. The nontrivial Q-dependence of the slower process, can be understood from a consideration of the decrease in the elastic scattering with Q, and by assuming
47
that (part of) the diffusing molecules are also rotating. Such an analysis along with a study of the temperature dependence will be covered in a forthcoming publication [8]. Finally, it is instructive to see if there are any indications of influence from the polymer matrix on the dynamics. Such effects are expected at Q-values corresponding to the length scale of the confinement. According to a recent SANS study of the polymer gel electrolyte of the present ( [9]. PCS, which study, the mesh size is 10 A probes density fluctuations on an optical length scale, yields a diffusion constant of 7ð2Þ 1012 m2 =s for the polymer gel electrolyte of the present study [4]. This is roughly 20 times lower than the diffusion constant of the present study obtained from a fit of the Q-dependence of the width of the slower process to a jump diffusion model. A further uncertainty is the problem to obtain the diffusion constant from the limited energy window covered in the QENS study. If the different diffusivities reflect the different length scales probed by PCS and QENS, confinement effects could be the cause. A more thorough discussion of the issue is referred to the forthcoming publication [8].
5. Conclusions We have shown that quasi-elastic neutron scattering on a polymer gel electrolyte reveals two dynamical processes at molecular distances on the experimental timescale. We can convincingly attribute the processes to a rotational motion of the solvent, and a slower diffusive process of the solvent within the polymer matrix. The diffusion constant we obtain from a curve-fit to the Gaussian jump length distribution model is 1:6 1010 m2 =s.
Acknowledgements Financial support from the Swedish Natural Science Research Council and the Swedish Foundation for Strategic Research is gratefully acknowledged.
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D. Andersson et al. / Physica B 301 (2001) 44–48
References [1] F.M. Gray, Solid Polymer Electrolytes, Fundamentals and Technological Applications, VCH, New York, 1991. [2] L.M. Torell, C.A. Angell, Bri. Polym. J. 20 (1988) 173. [3] G.B. Appetecchi, F. Croce, B. Scrosati, Electrochim. Acta 40 (1995) 991. [4] C. Svanberg, G.B. Appetecchi, J. Adebahr, H. Ericsson, . L.M. Torell, L. Borjesson, B. Scrosati, J. Chem. Phys. 111 (1999) 11 216.
[5] M.T.F. Telling, W.S. Howells, GUIDE}IRIS Data analysis, RAL Technical Report RAL-TR-2000-004, January 2000. [6] P.L. Hall, D.K. Ross, Mol. Phys. 42 (1981) 673. [7] A.J. Dianoux, F. Volino, H. Hervet, Mol. Phys. 30 (1975) 1181. [8] D. Andersson, C. Svanberg, J. Swenson, W.S. Howells, L. . Borjesson, Manuscript, 2001. . [9] C. Svanberg, W. Pyckhout-Hintzen, L. Borjesson, Manuscript, 2001.
Diffusive dynamics in polymer gel electrolytes investigated by quasi-elastic neutron scattering b . D. Anderssona,*, C. Svanberga, J. Swensonb, W.S. Howellsc, L. Borjesson a
. Department of Experimental Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden b . Department of Applied Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden c Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK
Abstract Quasi-elastic neutron scattering has been performed on a polymer gel electrolyte consisting of lithium perchlorate dissolved in ethylene carbonate=propylene carbonate and stabilized with poly(methyl methacrylate). Two relaxational processes are observed, and their momentum transfer dependencies are studied. From this we attribute the faster process to a rotational motion and the slower process to diffusive motion of the solvent. The diffusive process was modelled with a jump diffusion model, giving a diffusion constant of 1:6 1010 m2 =s, a mean residence time of 34 ps ( A jump rotation model analysis of the rotational motion yields a rotation radius of and mean jump length of 1:8 A. ( which is compatible with the size of the solvent molecules. # 2001 Published by Elsevier Science B.V. 1:5 A, PACS: 66.10.x; 83.70.Hq; 29.30.Hs Keywords: Polymer gel; Electrolyte; Quasi-elastic scattering; Diffusion
1. Introduction The increasing demand for battery-powered products, and the limited battery capacities hampering potential applications, has boosted an intensive research aiming at developing new battery concepts. Regarding the electrolyte, a common choice, e.g. in lithium batteries, is a salt in various organic solvents with high dielectric constants. Safety, construction and mechanical aspects would favour the use of solid electrolytes. This has fed a large interest in polymer electrolytes *Corresponding author. Tel.: +46-31-772-3352; fax: +4631-772-3177. E-mail address: [email protected] (D. Andersson).
[1], in which alkali salts are dissolved in polyethers. Unfortunately, despite intensive research, the conductivity, which in these systems are related to the polymer segmental mobility [2], is still too low to be commercially viable. An alternative way to obtain a solid electrolyte is to start from the organic electrolyte and immobilize it in a polymer gel. Such a polymer gel electrolyte based on poly(methyl methacrylate) (PMMA) has a conductivity close to that of the liquid electrolyte [3]. To understand the high conductivity of polymer gel electrolytes, the microscopic dynamics of these systems should be studied. Previous photon correlation spectroscopy (PCS) studies on polymer gel electrolytes have shown that they have a complicated dynamical
0921-4526/01/$ - see front matter # 2001 Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 5 0 9 - 9
D. Andersson et al. / Physica B 301 (2001) 44–48
response [4]. A diffusive motion of low molecular weight compounds was shown to be closely coupled to the conductivity mechanism, while the segmental motion of the polymer is almost completely decoupled from the conductivity properties. This suggests that the polymer gel electrolyte resembles liquid electrolytes concerning the conductivity properties while the polymer matrix gives the mechanical stability. In the present study, the dynamics in the picosecond range at molecular length scales is studied using quasi-elastic neutron scattering (QENS). Two processes are identified and characterised from their momentum transfer (Q) dependencies. Their molecular origins are briefly discussed as is the influence of the polymer matrix on the dynamics. This is the first QENS study of a polymer gel electrolyte.
2. Experimental The polymer gel electrolyte was prepared by dissolving lithium perchlorate ðLiClO4 Þ in a mixture of ethylene carbonate (EC) and propylene carbonate (PC) to form a liquid electrolyte. The liquid electrolyte was then immobilized into a polymer gel with high molecular weight PMMA. For the sample of the present study deuterated PMMA (Polymer Source Inc., MW ¼ 1 587 000) has been used. The preparation follows the standard gel preparation procedures described in Ref. [3]. The initial materials used were of battery grade i.e. of high purity and the water content was less than 50 ppm. The obtained sample was optically transparent and sealed under argon atmosphere. The average mole ratio of LiClO4 : EC : PC : PMMA in the sample was 4:5 : 46:5 : 19 : 30, where the PMMA ratio refers to the repeating unit. After completed gelation the sample was placed in a flat air-tight aluminium cell with a sample thickness of 360 mm. The thickness was chosen to have 10% of the neutrons scattered by the sample when its angle to the beam is 458, the angle used in the present study. The level of scattering was chosen to keep down multiple scattering to the extent that it can be ignored in the data analysis. The measurements were performed
45
on the IRIS time of flight (back-scattering) spectrometer at the ISIS pulsed spallation source, Rutherford Appleton Laboratory, UK. The pyrolythic graphite 002 (PG002) analyser reflection used in the present study has a resolution (FWHM) of 15 meV over an energy ( 1 . window 0:4 meV and a Q-range 0.4–1:85 A The measurement was performed at 293 K, which is well above the glass transition temperature of the liquid electrolyte (Tg 160 K) but below Tg of bulk PMMA. Due to the high abundance of 1 H in the sample (25%) and the large incoherent scattering cross section of 1 H, the incoherent scattering from the hydrogens of the solvent amounts to 81% of the total scattering. The scattering from PMMA amounts to only 10% of the total scattering due to the use of deuterated PMMA. Since the PMMA scattering is mainly coherent, this part is considerably higher at peaks of the static structure factor of PMMA. The scattering from PMMA will mainly contribute to the elastic scattering as the measurement is made well below Tg of PMMA and methyl group dynamics does not give any coherent scattering. The quasielastic scattering is thus likely to be predominantly incoherent. The 51 detectors were grouped into 16 groups to give a good coverage of the available Q-window. Data corrections for absorption, background and can scattering were performed using the on-site software package GUIDE [5]. Data from detectors ( 1 ), which in the plane of the sample (Q 1:7 A suffer from severe absorption and multiple scattering effects, were discarded.
3. Results The obtained SðQ; oÞ for the studied polymer ( 1 is gel electrolyte at 293 K and Q ¼ 1:27 A shown in Fig. 1. The data were fitted with an elastic peak and Lorentzian peaks, all convoluted with the resolution from a vanadium reference. Two Lorentzians were necessary to properly describe the data. The fit of the data in Fig. 1 and the components used are included in the figure together with the errors of the fit. No significant
46
D. Andersson et al. / Physica B 301 (2001) 44–48
( 1 ; oÞ. Experimental data plus the two Fig. 1. IðQ ¼ 1:27 A curve-fitted processes, static scattering and background are shown on a log-scale. Underneath the difference between data and curve-fit is shown.
improvement was achieved using additional Lorentzians. For comparison the data were also analysed with a stretched exponential relaxation process, fðtÞ ¼ expððt=tKWW Þb Þ for the quasi-elastic scattering. The fits were of similar quality, but the uncertainties for the stretched exponential were larger with no clear trends for neither t nor b. Additionally there is no physical process with stretched behaviour to be expected using QENS at the present temperature, which is far above Tg of the solvent. Hence this function will not be further considered. The widths (FWHM) of the two processes are shown in Fig. 2. As seen in the graph the faster process has no clear momentum transfer dependence, in contrast to the slower process. The relative integrated intensities of the two processes and the elastic scattering from the fits are shown in Fig. 3. One should note that the elastic part may contain some Q-dependence reflecting the coherent scattering, mainly from PMMA. The elastic line decreases in intensity with momentum transfer, whereas the faster process shows an increase. The Q-dependence of the slower process is more complex.
Fig. 2. The widths (FWHM) of the Lorentzians obtained from the curve-fits described in the text. The upper graph shows the fast process (&), while the lower graph shows the slow process (*). A curve-fit to the Gaussian jump length distribution model of diffusion (}) is also inserted.
Fig. 3. Q-dependence of the integrated intensities, normalised to the total integrated scattering intensity. Fast process ð&Þ, slow process (*) and elastic line ( * ). A fit of the fast process to the expected intensity of a jump rotation process (}) is also included.
D. Andersson et al. / Physica B 301 (2001) 44–48
4. Discussion The polymer gel electrolytes are complex materials with many possible dynamical processes. Since we are substantially above Tg of the solvent, translational diffussion processes are plausible, as well as (jump) rotational diffusion of solvent molecules or part thereof. To distinguish these processes it is valuable to study the Q-dependence of the width as the momentum transfer approaches zero. Diffusive processes have a Q2 -dependence of the width for low Q, whereas homogenous rotational processes should have Q-independent widths. The slower process shows clear diffusive features and we show in Fig. 2 that the wave vector dependence can be fitted to the Gaussian jump length distribution model [6]: DE ¼ ð2h=tÞ ½1 expðQ2 hr2 i=2Þ . From the curve-fit we obtain a harmonic mean residence time t of 34ð5Þ ps and a rms jump length hr2 i1=2 of ( Using Dtrans ¼ hr2 i=ð6tÞ gives a 1:8ð0:2Þ A. translational diffusion constant of 1:6ð0:3Þ 1010 m2 =s. The width of the faster process has larger uncertainty, but indicates a nonzero width as the momentum transfer approaches zero, incompatible with diffusion. The slight increase with Q, might well be compatible with the identification of this process as rotational, considering the complexity of this system, and the different molecules in the solvent. More insight can be obtained from the Qdependencies of the relative intensities. From simple models, diffusion is expected to have constant intensities, whereas the intensity of rotation processes increases with momentum transfer for Q lower than those corresponding to the size of the rotating moieties. The fast process can be fitted to p a ffiffimodel for jump rotation [7]: ffi IðQÞ ¼ A23ð1 j0 ð 3Qrrot ÞÞ where A is the fraction of atoms involved in the process, rrot is the radius of the circle along which the scattering atoms rotate and j0 is the zeroth order spherical Bessel function of first kind. We obtain a radius of ( which is comparable with rotation of 1:5ð0:2Þ A, the radius of the solvent molecules. The nontrivial Q-dependence of the slower process, can be understood from a consideration of the decrease in the elastic scattering with Q, and by assuming
47
that (part of) the diffusing molecules are also rotating. Such an analysis along with a study of the temperature dependence will be covered in a forthcoming publication [8]. Finally, it is instructive to see if there are any indications of influence from the polymer matrix on the dynamics. Such effects are expected at Q-values corresponding to the length scale of the confinement. According to a recent SANS study of the polymer gel electrolyte of the present ( [9]. PCS, which study, the mesh size is 10 A probes density fluctuations on an optical length scale, yields a diffusion constant of 7ð2Þ 1012 m2 =s for the polymer gel electrolyte of the present study [4]. This is roughly 20 times lower than the diffusion constant of the present study obtained from a fit of the Q-dependence of the width of the slower process to a jump diffusion model. A further uncertainty is the problem to obtain the diffusion constant from the limited energy window covered in the QENS study. If the different diffusivities reflect the different length scales probed by PCS and QENS, confinement effects could be the cause. A more thorough discussion of the issue is referred to the forthcoming publication [8].
5. Conclusions We have shown that quasi-elastic neutron scattering on a polymer gel electrolyte reveals two dynamical processes at molecular distances on the experimental timescale. We can convincingly attribute the processes to a rotational motion of the solvent, and a slower diffusive process of the solvent within the polymer matrix. The diffusion constant we obtain from a curve-fit to the Gaussian jump length distribution model is 1:6 1010 m2 =s.
Acknowledgements Financial support from the Swedish Natural Science Research Council and the Swedish Foundation for Strategic Research is gratefully acknowledged.
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D. Andersson et al. / Physica B 301 (2001) 44–48
References [1] F.M. Gray, Solid Polymer Electrolytes, Fundamentals and Technological Applications, VCH, New York, 1991. [2] L.M. Torell, C.A. Angell, Bri. Polym. J. 20 (1988) 173. [3] G.B. Appetecchi, F. Croce, B. Scrosati, Electrochim. Acta 40 (1995) 991. [4] C. Svanberg, G.B. Appetecchi, J. Adebahr, H. Ericsson, . L.M. Torell, L. Borjesson, B. Scrosati, J. Chem. Phys. 111 (1999) 11 216.
[5] M.T.F. Telling, W.S. Howells, GUIDE}IRIS Data analysis, RAL Technical Report RAL-TR-2000-004, January 2000. [6] P.L. Hall, D.K. Ross, Mol. Phys. 42 (1981) 673. [7] A.J. Dianoux, F. Volino, H. Hervet, Mol. Phys. 30 (1975) 1181. [8] D. Andersson, C. Svanberg, J. Swenson, W.S. Howells, L. . Borjesson, Manuscript, 2001. . [9] C. Svanberg, W. Pyckhout-Hintzen, L. Borjesson, Manuscript, 2001.