VOLUME 48, NUMBER 16 PHYSICAL REVIEW LETTERS 19 Aprit 1982 In this Letter, we have investigated the way in which electron motion in the field of an obliquely propagating Langmuir wave feeds back on the wave to affect its evolution. In the trapping regime, we found that the amplitude oscillations disappear as the angle of propagation increases. When a transition is made between the trapping and stochastic regimes, we found a significant decrease in the asymptotic (¢=~) wave amplitude. Simulations verify this decrease and indicate that it is observable at any time greater than five bounce periods. The methods we used can be applied to a variety of other problems in which there is a transition from regular to stochastic behavior and the field evolution is of interest. This work has been supported by U. S. Depart- ment of Energy Contracts No, DE-AM03-76- SF00010 PA 26, Tasks I and III, by National Science Foundation Contracts No. PHY-77-12873 and No, PHY-80-14189, and by the Fannie and John Hertz Foundation. Igee, e.g., R. C. Davidson, Methods in Nonlinear Plasma Theory (Academic, New York, 1972), Chaps. 4 and 5. 2G. R. Smith and A. N. Kaufman, Phys. Fluids 21, 2230 (1978). 3gee, e.g., “Long-Time Prediction in Dynamics,” ed- ited by Wendell Horton, Linda Reichl, and Victor Sze- behely (Wiley, New York, to be published). 4T, O'Neil, Phys. Fluids 8, 2225 (1965). 5G. J. Morales and T. M. O’Neil, Phys. Rev. Lett. 28, 417 (1972). 6, H. Malmberg and C. B. Wharton, Phys. Rev. Lett. 19, 775 (1967). "See, e.g., T. H. Stix, The Theory of Plasma Waves (McGraw-Hill, New York, 1962), Chap. 8. 8¢,(x) —1 is plotted in Fig. 1 of T. T. Tsai, J. Plasma Phys. 11, 213 (1974). °See Tsai, Rev. 8, and references cited therein. Smectic-A Order at the Surface of a Nematic Liquid Crystal: Synchrotron X-Ray Diffraction J. Als-Nielsen and F. Christensen Ris¢g National Laboratory, DK-4000 Roskilde, Denmark and P. S. Pershan Gordon McKay Laboratory, Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138 (Received 25 January 1982) A novel geometry in which it is possible to do x-ray diffraction from a horizontal sur- face of fluids is applied to liquid crystals. A large-diameter drop of octyloxycyanobi- phenyl (SOCB) on a glass plate treated for homeotropic alignment yields perfect align- ment of the smectic-A layers at the top surface over an area of several square millime- ters. The surface in the bulk nematic as well as in the isotropic phase was found to con- sist of smectic-A layers with a penetration depth equal to the longitudinal smectic-A cor- relation length &, ~ (T —Ty,)"”" determined previously. ° PACS numbers: 61.30.Eb, 61.10.Fr, 61.30.Jf, 64.70.Ew In spite of the large number of recent experi- mental! and theoretical studies” the nematic to smectic-A (N-A) phase transition is not well understood. The smectic-A phase has the inter- esting property that positional correlations of the layers do not have true long-range order but rather exhibit algebraic decay with a tempera- ture-dependent exponent 7.° Recent x-ray-dif- fraction studies have confirmed this result*; however, difficulties in controlling the mosaic spread of the smectic layers precluded meas - urements that are required for quantitative evalu- © 1982 The American Physical Society ation of current proposals regarding the behavior of nas T~ Tan.” Some years ago one of us (P.S.P.) showed that if the effects of surface tension can be neglected the combination of a free surface and a parallel planar rigid surface with homeotropic alignment would naturally expel mobile defects.® This geometry thus suggested the possibility of a practical solution to the mosaicity problem. We report here the first x-ray-diffraction study of the horizontal free surface of a nematic liquid crystal.® In addition to sharp peaks, with unde- 1107 VOLUME 48, NUMBER 16 PHYSICAL REVIEW LETTERS 19 AprRIL 1982 tectably small mosaic spread for TTan. The experiment was carried out at the synchro- tron radiation laboratory HASYLAB at Deutsches Elektronen Synchrotron DESY in Hamburg. Fig- ure 1 displays the experimental geometry neces- sitated by the requirement that the fluid sample be kept horizontal. A monochromatic beam of wave vector e& is extracted from the polychro- matic synchrotron spectrum by Bragg reflection from Si(111) planes of a triple-bounce channel- cut crystal. The Bragg angle is 6,,, and7,,, =2ksin6@,,, The monochromatic beam is bent downwards through an angle @, by tilting the monochromator by an angle y=(k/T,,,)0,. Hori- zontal smectic layers with layer spacing d will give Bragg reflection of this beam when Q, =(27/ a) = 2k sin@, =27,,,~. Note that in this particular geometry the Bragg condition is independent of wavelength in the small-angle approximation. An analyzer system symmetric with the monochro- mator obtains a horizontal beam that is detected in the position-sensitive detector (PSD). Fora given setting successive channels of the PSD probe a line of reciprocal space making an angle y/cos6@,,, with the longitudinal, vertical direction. By scanning the monochromator tilt an entire plane in wave-vector transfer space is probed. The transverse wave-vector transfer component perpendicular to this plane is probed by rotating the analyzer crystal around a vertical axis. We now discuss the scattering as obtained from the liquid crystal octyloxycyanobiphenyl (80CB) Lig. X tal FIG. 1. Schematic illustration of the geometry for x-ray diffraction from a horizontal fluid surface. The slit S is 0.1x0.8 mm’, l= 575 mm, J,= 620 mm, the wave vector k= 4.0786 A~', and the Bragg vector for $i(111) 74,;= 2.0039 A~!. Source-to-slit distance is 20 m. 1108 held 0.020 °C above the nematic to smectic-A transition temperature for bulk 80CB, Tan =67.350°C. Results for six different monochro- mator tilts are shown in Fig. 2(a). For each monochromator tilt setting, the intensity versus analyzer rotation was measured to be a sharp peak, less than 0.001° broader than the combined Darwin widths of the monochromator and analyzer (~0.0023°), This implies a very sharply peaked cross section in the transverse wave-vector transfer Q a The narrow width of each PSD spectrum in Fig. 2(a) is indeed consistent with this statement. If we take the cross section as a delta function in Q, [the intrinsic width, AQ, < (30000 A)~*], the peak intensity J,(~) of each PSD spectrum represents the dependence of the cross section on the longitudinal wave-vector transfer a,, folded of course with the longitudinal resolution function, Such data are shown in Fig. 2(c). The width w of the peak, 0.0116°, is at this temperature only slightly larger than the resolution of 0.0090°. Finally, the data in Fig. 2(b) of the peak channel versus tilt shows that the scattered beam peaks in the direction of specular reflection (line s) rather than that ex- g0c _ rr re | OCB 2 J a | 67.370°C ® RAIA 680| 680 | 680| 680] 680| 680| 690 690 690 690 690 690 CHANNEL NO. 2.798 5000 |- 2.807 2.794 2.812 INTENSITY 5000 690 mM PEAK CHANNEL PEAK INTENSITY Jy J L 1 1 2.79 280 2.819 279 280 2.81° MONOCHROMATOR TILT FIG. 2. (a) PSD spectra for different monochromator tilt angles. (b) Comparison between observed peak channel vs tilt and expected value if wave-vector trans- fer is normal to surface, that is, specular reflection (full line), and along the surface as in a mosaic crystal (dashed line). (c) Peak intensity vs tilt. Correction for resolution width (0. 0090°) yields a longitudinal correla- tion range of £ ~6000 A at a reduced temperature of 5.9 x107*, VOLUME 48, NUMBER 16 PHYSICAL REVIEW LETTERS 19 APRIL 1982 pécted from a mosaic crystal (line m), again indicating an extremely sharply peaked cross section in Q 1- With increasing temperature, the following features were observed: (i) J,(y) de- creases and the width w increases but a peak re- mains even into the isotropic phase. Data are shown in Figs. 3(a) and 3(b). (ii) The width of each PSD spectrum remains as narrow as those in Fig. 2(a) and the widths of analyzer rotation scans remain Darwin limited. The free surface of 80CB acts as a.temperature-dependent optical- ly flat mirror for incident angles around 1.4°, In obtaining these data the height h was adjusted to h =1,(1,,,/k) at each monochromator tilt set- ting y and also the analyzer rotation was opti- mized at each setting. We shall now discuss and analyze these observations. Item (ii) implies that the delta-function charac- ter in Q 1. is maintained at all temperatures where- as (i) implies a peak in the longitudinal direction dz =(Q-Q) -2 which we take to be of Lorentzian form with a temperature-dependent width &,7': o(Q) = ao,(6t)[1 +(E,92)7]*0(Q,), (1) < 10° QT rr er q Z oF \, 80CB 3 rs [ © *, J T,y- The solid line is calculated by convolution of the resolution function with a Lorentzian with reciprocal of half width at half maximum of é)(T ) as measured in the bulk nematic, Ref. 1. a being a constant and of the reduced tempera- ture (T-Tan)/Tan- The Lorentzian line shape and the width parameter £, may be derived from the phenomenological free energy, F=A(50)|~? +>[(2/8, -7Q))p]?+..., (2) of the smectic order parameter y.° In the Landau or mean-field approximation A(6t) is taken to vary linearly with 6f, yielding a spontaneous smectic phase below T ay and critical fluctuations above Tan With a longitudinal correlation range Ey =(/A)*1? «< 6t7-/?, More generally €,=£ (5t)"™" and for bulk 80CB previous x-ray scattering stud- ies have determined & =3.3 A and y,=0.71.! Un- der the assumption that the free surface is equiv- alent to a boundary condition of p= %, at z= 0, the z dependence of is found by minimizing F: v= yoexpl —2(iQ,+&,)), E,=(y/A)?, (3) i.e., the penetration depth is simply identical to the correlation range of the bulk nematic phase.°® The scattering cross section, Eq. (1), is obtained from the Fourier transform of Eq. (3), and we infer 0,(d5t) « &?<(d5¢)~***, The cross section must be folded with the instrumental resolution before comparison with experimental data. In Fig. 3(b) the full line shows the inverse width of a Lorentzian peak of width ¢,~' folded with a tri- angular resolution function corresponding to a tilt width of 0.0090°; c.f. Fig. 2(c). If we note that the comparison between this theoretical ex- pectation and the observed widths does not in- volve any adjustable parameters the agreement must be said to be strikingly good. It was there- fore somewhat surprising to find that the unfolded peak intensities, Fig. 3(a), do not vary as (6t)~?”" =(df)7!-# but rather as (5#)""°. One possibility, pointed out by B. Halperin, is that the boundary condition might be so strong, i.e., % so large, that the linear theory represented by Eq. (2) is not applicable. In that case ~ would be expected to decrease from its surface value to a value within the linear range of the theory, in a dis- tance much less than é,. This would have the effect of adding a broadened background to the line shape. We have not yet carried out detailed measurements of the tails of the peak. In summary, we have demonstrated the feasibil- ity of using synchrotron radiation, in a novel geometry, to study x-ray diffraction from the horizontal surface of fluid samples. Structure observed in specular reflection can be correlated with the propagation into the bulk of positional molecular order imposed by the surface.’ In 1109 VOLUME 48, NUMBER 16 PHYSICAL REVIEW LETTERS 19 ApRIL 1982 the nematic phase of 80CB the propagation length coincides precisely with the correlation length previously measured in bulk samples. We have also demonstrated that smectic-A samples with vanishingly small mosaic spreads can be obtained with the free-surface technique. Subjects for future research include (a) detailed line-shape studies of both melting from the smectic-A to the nematic phase and the temperature-dependent intensity in the nematic phase; (b) quantitative comparison between the intensities due to smectic layering at the free surface and specular reflec- tion due to the small but finite value of the x-ray index of refraction mismatch between the liquid crystal and air—this should be observable at smaller 6, and would provide an absolute meas- urement of the smectic-A order parameter; and (c) in the vicinity of the smectic-A to smectic-B phase, measurements in the specular direction along z, but at finite values of Q 1, Should obtain information on structural correlations within the surface layers.'! The excellent research conditions provided by HASYLAB and the competent assistance of Risg technical staff members E. Dahl Petersen, S. Jérgensen, J. Linderholm, and J. Munck are gratefully acknowledged. This work was support- ed in part by grants from the Danish National Science Foundation, by the Ris¢ National Labora- tory, by the National Science Foundation under Grant No. DRM-79-19479, and by the Joint Ser- vices Electronics Program (U. S, Army, Navy, and Air Force) under Grant No. 14-75-C-0648. (®Work partly performed as a visiting scientist at Risé National Laboratory, DK-4000 Roskilde, Denmark. 14 review of x-ray and light scattering results in given in J. D. Litster, J. Als-Nielsen, R. J. Birgeneau, S.S. Dana, D. Davidov, F. Garcia-Golding, M. Kaplan, 1110 C. R. Safinya, and R. Schaetzing, J. Phys. (Paris), Collog. 40, C3-339 (1979). Specific heat of 80CB has been reported by D. L. Johnson, C. F. Hayes, R. Jd. de Hoff, and C. A. Schantz, Phys. Rev. B18, 4902 (1978); C. W. Garland, G. B. Kasting, and K. J. Lush- ington, Phys. Rev. Lett. 43, 1420 (1979); J. D. Le- Grange and J. M. Mochel, Phys. Rev. Lett. 45, 35 (1980). For a recent discussion of the N-A transition, see R. J. Birgeneau, C. W. Garland, G. B. Kasting, and B, M. Ocko, Phys. Rev. A 24, 2624 (1981). 2D, R. Nelson and J. Toner, Phys. Rev. B 24, 363 (1981); G. Grinstein and R. B. Pelcovitz, Phys. Rev. Lett. 47, 856 (1981); S. G. Dunn and T. C. Lubensky, J. Phys. (Paris) 42, 1201 (1981); C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47, 1556 (1981). 8A. Caillé, C. R. Acad. Sci., Ser. B 274, 891 (1972). 43, Als-Nielsen, J. D. Litster, R. J. Birgeneau, M. Kaplan, C. R. Safinya, A. Lindegaard- Andersen, and 8S. Mathiesen, Phys. Rev. B 22, 312 (1980). 5p, S. Pershan, J. Appl. Phys. 45, 1590 (1974); P.S. Pershan and J. Prost, J. Appl. Phys. 46, 2343 (1975). ®Previous phenomenological studies involving free surface samples include D. Langevin, Phys. Lett. 56A, 61 (1976); C. H. Sohl, K. Miyano, J. B. Ketterson, and G. Wong, Phys. Rev. A 22, 1256 (1980); M. R. Fisch, L. B. Sorensen, and P, S. Pershan, Phys. Rev. Lett. 47, 43 (1981); M. G. J. Gannon and T. W. Farber, Philos. Mag. A 37. TSurface-induced smectic ordering at temperatures for which the bulk phase is nematic was previously ob- served in freely suspended thin films by C. Rosenblatt and N. M. Amer, Appl. Phys. Lett. 36, 432 (1980). It was also predicted on the basis of a lattice model by C. Rosenblatt and D. Ronis, Phys. Rev. A 23, 305 (1981). 8p, G. de Gennes, Solid State Commun. 10, 783 (1972), and The Physics of Liquid Crystals (Clarendon, Oxford, 1974). °An identical analysis was independently given by H. v. Kanel, J. D. Litster, J. Melngailis, and H. I. Smith, Phys. Rev. A 24, 2713 (1981). 10This technique may also be applied to study the liquid-metal—vapor interface, c.f., M. P. D’Evelyn and S. A. Rice, Phys. Rev. Lett. 47, 1844 (1981). ‘lp, Eisenberger and W. C. Marra, Phys. Rev. Lett. 46, 1081 (1981). Lig. X tal FIG. 1. Schematic illustration of the geometry for x-ray diffraction from a horizontal fluid surface. The slit S is 0.1x 0.8 mm’, 1)= 575 mm, /,= 620 mm, the wave vector k= 4.0786 A~', and the Bragg vector for $i(111) 71;;= 2.0039 A~!. Source-to-slit distance is 20 m.