Abstract

In X-ray photon correlation spectroscopy (XPCS) the degree of coherence of the X-ray beam determines the contrast of the observed intensity correlation function. In this article, we present XPCS measurements of smectic liquid crystal membranes in a reflectivity geometry showing that both coherence and resolution can influence the time dependence of the correlation function. Variation of the pre-detector slits as well as of the projected coherence length on the membrane induce a time dependence of the intensity correlation function. We also treat several practical aspects and limitations we encountered during our XPCS studies. Finally the conditions for heterodyne detection at the specular ridge and homodyne detection at off-specular conditions are discussed.

Keywords: X-ray photon correlation spectroscopy; Partially coherent radiation; Speckle; Smectic liquid crystal membranes

PACS: 61.30.−v; 61.10.Kw; 42.25.Kb

Article Outline

1. Introduction

2. Theoretical background

2.1. Coherence of X-ray radiation and XPCS

2.2. Dynamics of fluctuations in smectic membranes

3. Experimental

4. Results and discussion

4.1. Resolution effects from the pre-detector slits

4.2. Dependence of the relaxation on q_z

4.3. Homodyne/heterodyne detection

5. Conclusions

Acknowledgements

References

(14K)
Fig. 1. Rocking curve of a 13-layer FPP membrane at the Bragg position. The almost perfect uniformity is reflected in the FWHM of 0.7 mdeg.

(16K)
Fig. 2. Dispersion curve of the relaxations of a smectic membrane; q is the wave vector of the fluctuations.

(27K)
Fig. 3. Scattering geometry.

(42K)
Fig. 4. Correlation functions from a 1.7-μm vertical FPP membrane (10-μm pinhole). Left: vertical slit fixed at 0.02 mm, horizontal slit (from top to bottom): 0.03, 0.06, 0.1, 0.2 mm. Right: horizontal slit fixed at 0.02 mm, vertical slit (from above): 0.01, 0.03, 0.06, 0.1, 0.2 mm. Fit parameters in Table 1.

(46K)
Fig. 5. Correlation functions from a 1.4-μm vertical FPP membrane (100-μm pinhole). Left: vertical slit fixed at 0.02 mm, horizontal slit (from top to bottom): 0.01, 0.03, 0.06, 0.1, 0.2 mm. Right: horizontal slit fixed at 0.02 mm, vertical slit (from top to bottom): 0.01, 0.03, 0.06, 0.1, 0.2 mm. Fit parameters in Table 2.

(80K)
Fig. 6. Results for a 13-layer 4O.8 membrane at different specular positions: (a) reflectivity curve; (b) correlation functions (10-μm pinhole) taken at the maxima of the Kiessig fringes (shifted along the vertical axis with values of q_z indicated); (c) dependence of the fitting parameters τ (triangles) and ω (circles) on the position along the specular ridge; (d) ibid for the contrast.

(14K)
Fig. 7. Correlation functions of a 13.2-μm FPP membrane (10-μm pinhole) at the first and the second Bragg position.

(43K)
Fig. 8. XPCS measurements of a 3.8-μm 8 CB membrane at specular and off-specular positions: (a) correlation functions at the positions indicated above each curve; (b) variation of the relaxation times as a function of the offset. Inset: rocking curve with arrows indicating the measurement positions.

Table 1.
Fitting parameters to Eq. (19) for the correlation functions of the vertical FPP membrane of Fig. 4 (10 μm pinhole)

In-plane slit/mm	A ± 0.01	(τ ± 0.1)/μs	Out-of-plane slit/mm	A ± 0.01	(τ ± 0.1)/μs
0.03	0.29	3.8	0.01	0.33	3.9
0.06	0.22	4.6	0.03	0.31	3.9
0.1	0.16	5.2	0.06	0.29	3.9
0.2	0.08	5.7	0.1	0.26	4.0
			0.2	0.24	4.0

Left: vertical slit fixed at 0.02 mm. Right: horizontal slit fixed at 0.02 mm; ω = 0.30 ± 0.02 μs⁻¹,  = 1.9 ± 0.1.

Table 2.
Fitting parameters to Eq. (19) for the correlation functions of the vertical FPP membrane of Fig. 5 (100 μm pinhole)

In-plane slit/mm	A ± 0.01	(τ ± 0.1)/μs	Out-of-plane slit/mm	A ± 0.01	(τ ± 0.1)/μs
0.01	0.16	4.6	0.01	0.22	4.3
0.03	0.14	5.3	0.03	0.13	5.3
0.06	0.1	6.5	0.06	0.08	6.0
0.1	0.07	7.1	0.1	0.05	6.1
0.2	0.04	7.8	0.2	0.04	6.5

Left: vertical slit fixed at 0.02 mm. Right: horizontal slit fixed at 0.02 mm; ω = 0.29 ± 0.02 μs⁻¹,  = 1.8 ± 0.1.

References

[1] S. Dierker, NSLS Newsletter (July) (1995) 1.

[2] D.L. Abernathy, G. Grübel, S. Brauer, I. McNulty, G.B. Stephenson, S.G.J. Mochrie, A.R. Sandy, N. Mulders and N. Sutton, J. Synchrotron Radiat. 5 (1998), p. 37. Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[3] D. Lumma, L.B. Lurio, S.G.J. Mochrie and M. Sutton, Rev. Sci. Inst. 71 (2000), p. 3274. Abstract-INSPEC   | $Order Document | OJPS full text | Full Text via CrossRef

[4] T. Seydel, A. Madsen, M. Sprung, M. Tolan, G. Grübel and W. Press, Rev. Sci. Inst. 74 (2003), p. 4033. Abstract-Compendex   | $Order Document | Full Text via CrossRef

[5] M. Sutton, K. Laaziri, F. Livet and F. Bley, Opt. Express 11 (2003), p. 2268. Abstract-Compendex   | $Order Document

[6] G. Grübel and F. Zontone, J. Alloys Compd. 362 (2004), p. 3.

[7] S.G.J. Mochrie, A.M. Mayes, A.R. Sandy, M. Sutton, S. Brauer, G.B. Stephenson, D.L. Abernathy and G. Grübel, Phys. Rev. Lett. 78 (1997), p. 1275. Abstract-Compendex | Abstract-INSPEC   | $Order Document | APS full text | Full Text via CrossRef

[8] T. Thurn-Albrecht, G. Meier, P. Müller-Buschbaum, A. Patkowski, W. Steffen, G. Grübel, D.L. Abernathy, O. Diat, M. Winter, M.G. Koch and M.T. Reetz, Phys. Rev. E 59 (1999), p. 642. Abstract-INSPEC   | $Order Document | APS full text | Full Text via CrossRef

[9] L.B. Lurio, D. Lumma, A.R. Sandy, M.A. Borthwick, P. Falus, S.G.J. Mochrie, J.F. Pelletier, M. Sutton, L. Regan, A. Malik and G.B. Stephenson, Phys. Rev. Lett. 84 (2000), p. 785. Abstract-INSPEC | Abstract-MEDLINE   | $Order Document | APS full text | Full Text via CrossRef

[10] D.O. Riese, W.L. Vos, G.H. Wegdam, F.J. Poelwijk, D.L. Abernathy and G. Grübel, Phys. Rev. E 61 (2000), p. 1676. Abstract-INSPEC   | $Order Document | APS full text | Full Text via CrossRef

[11] G. Grübel, D.L. Abernathy, D.O. Riese, W.L. Vos and G.H. Wegdam, J. Appl. Crystallogr. 33 (2000), p. 424.

[12] D. Lumma, L.B. Lurio, M.A. Borthwick, P. Falus and S.G.J. Mochrie, Phys. Rev. E 62 (2000), p. 8258. Abstract-Compendex | Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[13] T. Seydel, A. Madsen, M. Tolan, G. Grübel and W. Press, Phys. Rev. B 63 (2001), p. 3409.

[14] J. Lal, D.L. Abernathy, L. Auvray, O. Diat and G. Grübel, Eur. Phys. J. E 4 (2001), p. 263. Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[15] M. Tolan, T. Seydel, A. Madsen, G. Grübel, W. Press and S.K. Sinha, Appl. Surf. Sci. 182 (2001), p. 236. SummaryPlus | Full Text + Links | PDF (149 K)

[16] H. Kim, A. Rühm, L.B. Lurio, J.K. Basu, J. Lal, D. Lumma, S.G.J. Mochrie and S.K. Sinha, Phys. Rev. Lett. 90 (2003), p. 068302. Full Text via CrossRef

[17] A. Madsen, J. Als-Nielsen and G. Grübel, Phys. Rev. Lett. 90 (2003), p. 085701. Full Text via CrossRef

[18] S.G.J. Mochrie, L.B. Lurio, A. Rühm, D. Lumma, M. Borthwick, P. Falus, H.J. Kim, J.K. Basu, J. Lal and S.K. Sinha, Phys. B 336 (2003), p. 173. SummaryPlus | Full Text + Links | PDF (269 K)

[19] A. Madsen, T. Seydel, M. Sprung, C. Gutt, M. Tolan and G. Grübel, Phys. Rev. Lett. 92 (2004), p. 096104. Full Text via CrossRef

[20] B. Lengeler, Naturwissenschaften 88 (2001), p. 249. Abstract-EMBASE | Abstract-GEOBASE | Abstract-MEDLINE   | $Order Document

[21] I.A. Vartanyants and I.K. Robinson, Opt. Commun. 222 (2003), p. 29. SummaryPlus | Full Text + Links | PDF (410 K)

[22] W.H. de Jeu, B.I. Ostrovskii and A.N. Shalaginov, Rev. Mod. Phys. 75 (2003), p. 181 and references therein. Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[23] A.C. Price, L.B. Sorensen, S.D. Kevan, J. Toner, A. Poniewierski and R. Hołyst, Phys. Rev. Lett. 82 (1999), p. 755. Abstract-INSPEC   | $Order Document | APS full text | Full Text via CrossRef

[24] A. Fera, I.P. Dolbnya, G. Grübel, H.-G. Muller, B.I. Ostrovskii, A.N. Shalaginov and W.H. de Jeu, Phys. Rev. Lett. 85 (2000), p. 2316. Abstract-Compendex | Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[25] I. Sikharulidze, B. Farago, I.P. Dolbnya, A. Madsen and W.H. de Jeu, Phys. Rev. Lett. 91 (2003), p. 165504. Full Text via CrossRef

[26] I. Sikharulidze, I.P. Dolbnya, A. Fera, A. Madsen, B.I. Ostrovskii and W.H. de Jeu, Phys. Rev. Lett. 88 (2002), p. 115503. Full Text via CrossRef

[27] See, for exampleP.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge (1995).

[28] R. Hołyst, Phys. Rev. A 44 (1991), p. 3692.

[29] See, for exampleB. Chu, Laser Light Scattering: Basic Principles and Practice, Academic Press, San Diego (1991).

[30] B. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology and Physics, Dover Publications Inc., New York (2000).

[31] D. Langevin, Experimental realization. In: D. Langevin, Editor, Light Scattering by Liquid Surfaces and Complementary Techniques, Dekker, New York (1992).

[32] C. Gutt, T. Ghaderi, V. Chamard, A. Madsen, T. Seydel, M. Tolan, M. Sprung, G. Grübel and S.K. Sinha, Phys. Rev. Lett. 91 (2003), p. 076104. Full Text via CrossRef

[33] S.K. Sinha, M. Tolan and A. Gibaud, Phys. Rev. B 57 (1998), p. 2740. Abstract-INSPEC   | $Order Document | APS full text | Full Text via CrossRef

[34] D. Sentenac, A.N. Shalaginov, A. Fera and W.H. de Jeu, Appl. Cryst. 33 (2000), p. 130. Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[35] A.N. Shalaginov and D.E. Sullivan, Phys. Rev. E 62 (2000), p. 699. Abstract-Compendex | Abstract-INSPEC   | $Order Document | APS full text | Full Text via CrossRef

[36] A.Q.R. Baron, Hyperfine Interactions 125 (2000), p. 29. Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[37] C. Rosenblatt and N. Amer, Appl. Phys. Lett. 36 (1980), p. 432. Abstract-INSPEC   | $Order Document | Full Text via CrossRef

[38] P. Pieranski, L. Beliard, J.-Ph. Tournellec, X. Leoncini, C. Furtlehner, H. Dumoulin, E. Riou, B. Jouvin, J.-P. Fénerol, Ph. Palaric, J. Heuving, B. Cartier and I. Kraus, Phys. A 194 (1993), p. 364. Abstract | Abstract + References | PDF (1350 K)

[39] W.H. de Jeu, A. Madsen, I. Sikharulidze, S. Sprunt, Phys. B (in press) doi:10.1016/j.physb.2004.11.016.

Corresponding author. Tel.: +31 20 608 1234; fax: +31 20 668 4106