Meyerheim, Holger L.

Moritz, Wolfgang

Robinson, Ian K.

Schamper, Christian

Schulz, H.

Zabel, Hartmut

English

Universität München: Elektronischen Publikationen

Spr inger 
P r o c e e d i n g s in P h y s i c s 61 
S p r i n g e r P r o c e e d i n g s i n P h y s i c s 
Managing Editor: H . K . V. Lotsch 
44 Optical Fiber Sensors 
Editors: H. J. Arditty, J. P. Dakin, 
and R. Th. Kersten 
45 Computer S i m u l a t i o n Studies in Condensed 
Matter Physics II: New Directions 
Editors: D. P. Landau, K. K. Mon, 
andH.-B. Schüttler 
46 Cellular Automata and Modeling of Complex 
Physical Systems 
Editors: P. Manneville, N. Boccara, 
G. Y. Vichniac, and R. Bidaux 
54 Polycrystalline Semiconductors II 
Editors: J. H. Werner and H. P. Strunk 
55 Nonlinear Dynamics and Q u a n t u m Phenomena 
in Optical Systems 
Editors: R. Vüaseca and R. Corbalän 
56 Amorphous and Crystalline 
Silicon Carbide III and Other G r o u p 
I V - I V Materials 
Editors: G. L. Harris, M. G. Spencer, 
andC. Y-W.Yang 
47 Number Theory and Physics 
Editors: J.-M. Luck, P. Moussa, 
and M. Waldschmidt 
48 M a n y - A t o m Interactions in Solids 
Editors: R .M. Nieminen, M. J. Puska, 
and M. J. Manninen 
57 Evolutionary Trends in the Physical Sciences 
Editors: M. Suzuki and R. Kubo 
58 New Trends in Nuclear Collective Dynamics 
Editors: Y Abe, H. Horiuchi, 
and K. Matsuyanagi 
49 Ultrafast Phenomena in Spectroscopy 
Editors: E. Klose and B. Wilhelmi 
50 Magnetic Properties of L o w - D i m e n s i o n a l 
Systems II: New Developments 
Editors: L. M. Falicov, F. Mejia-Lira, 
and J. L. Morän-Löpez 
51 The Physics and Chemistry of Organic 
Superconductors 
Editors: G. Saito and S. Kagoshima 
52 Dynamics and Patterns in Complex Fluids: 
New Aspects ofthe Physics-Chemistry 
Interface 
Editors: A. Onuki and K. Kawasaki 
53 Computer S i m u l a t i o n Studies in Condensed 
Matter Physics III 
Editors: D. P. Landau, K. K. Mon, 
and H.-B. Schüttler 
59 Exotic Atoms in Condensed Matter 
Editors: G. Benedek and H. Schneuwly 
60 The Physics and Chemistry of 
Oxide Superconductors 
Editors: Y Iye and H. Yasuoka 
61 Surface X-Ray and Neutron Scattering 
Editors: H. Zabel and I.K. Robinson 
62 Surface Science: Lectures on Basic Concepts 
and Applications 
Editors: E A. Ponce and M. Cardona 
63 Coherent R a m a n Spectroscopy: Recent Advances 
Editors: G. Marowsky and V. V. Smirnov 
64 Superconducting Devices and Their Applications 
Editors: H. Koch and H. Lübbig 
Volumes 1-43 are listed on the back inside cover 
H. Zabel I. K. Robinson (Eds.) 
Surface X-Ray 
and Neutron Scattering 
Proceedings of the 2nd International Conference, 
Physik Zentrum, Bad Honnef, Fed. Rep. of Germany, 
June 25-28,1991 
With 120 Figures 
Springer-Verlag 
Berlin Heidelberg New York 
London Paris Tokyo 
HongKong Barcelona 
Budapest 
Contents 
Conference Summary 
By S.K. Sinha 1 
Part I Surface Crystallography and Phase Transitions 
Surface X-Ray Crystallography and STM Images 
By R. Feidenhans'l (With 3 Figures) 5 
Determination of Metal Adsorbed Surfaces by X-Ray Diffraction 
By T. Takahashi (With 5 Figures) 11 
Au Adsorption on S i ( l l l ) Studied 
by Grazing Incidence X-Ray Diffraction 
By C. Schamper, D. Dornisch, W. Moritz, H. Schulz, R. Feidenhans'l, 
M . Nielsen, F. Grey, and R.L. Johnson (With 3 Figures) 17 
Grazing Incidence X-Ray Scattering Study of Staircases 
of Steps on Si(OOl) Surfaces 
By G. Renaud, P.H. Fuoss, J. Bevk, and B.S. Freer 
(With 2 Figures) 21 
Structure and Phase Transitions of Ge( l l l ) and S i ( l l l ) Surfaces 
at High Temperatures 
By K.L Blum, D.Y. Noh, A. Mak, K.W. Evans-Lutterodt, J.D. Brock, 
G.A. Held, and R J . Birgeneau (With 3 Figures) 27 
Anomalous Scattering Applied to Co/Si(ll l) Interface Structure 
By N . Jedrecy, A. Waldhauer, M . Sauvage-Simkin, R. Pinchaux, 
and V.H. Etgens (With 2 Figures) 33 
X-Ray Reflectivity Studies of Au Surfaces 
By D.L. Abernathy, D. Gibbs, G. Grübel, K.G. Huang, S.GJ. Mochrie, 
B . M . Ocko, A.R. Sandy, and D.M. Zehner (With 5 Figures) 37 
Crystal Truncation Rod as a Convolution 
of Three-Dimensional Bravais Lattice with X-Ray Reflectivity 
By Hoydoo You 47 
Extended X-Ray Reflectivity Analysis of Si(l 11)7x7 
B y I.K. Robinson and E. Vlieg (With 3 Figures) 51 
VII 
Critical Phenomena at Surfaces and Interfaces 
By R. Lipowsky 57 
Surface-Induced Order Observed on a Cu3Au(001) Surface 
By K.S. Liang, H.H. Hung, S.L. Chang, Z. Fu, S.C. Moss, 
and K. Oshima (With 2 Figures) 65 
Thermal Dynamics of (110) fcc Metal Surfaces 
B y K . Kern 69 
Facet Coexistence in the Roughening Transition of Ag(110) 
By H.G. Hörnis, E.H. Conrad, E. Vlieg, and I.K. Robinson 
(With 5 Figures) 73 
Kinetics of Ordering with Random Lnpurities: Pb on Ni(001) 
By P.W. Stephens, P J . Eng, and T. Tse (With 3 Figures) 79 
Part H Reflectivity 
Anomalous Reflectivity: 
A New Method for Determining Density Profiles of Thin Films 
By S.K. Sinha, M.K. Sanyal, K.G. Huang, A. Gibaud, M . Rafailovich, 
J. Sokolov, X . Zhao, and W. Zhao (With 3 Figures) 85 
Specular and Diffuse Scattering Studies of Multilayer Interfaces 
By M.K. Sanyal, S.K. Sinha, A. Gibaud, S.K. Satija, C.F. Majkrzak, 
and H. Homa (With 3 Figures) 91 
Scattering Cross-Section of X-Rays and Neutrons 
for Grazing Incidence onto Thin Films 
By A. Haase and S. Dietrich 95 
Total Neutron Reflection: Experiments and Analysis 
By G.P. Felcher, W.D. Dozier, Y . Y . Huang, and X.L . Zhou 
(With 3 Figures) 99 
Profile Refinement in Neutron Reflectivity 
and Grazing Angle Diffraction 
By J.F. Ankner (With 1 Figure) 105 
Part m Surface X-Ray Standing Waves 
X-Ray Standing Wave Studies of the Liquid/Solid Interface 
and Ultrathin Organic Films 
By M.J. Bedzyk (With 3 Figures) 113 
Glancing-Incidence X-Ray Analysis of Layered Materials 
By D.K.G. de Boer and W.W. van den Hoogenhof 
(With 4 Figures) 119 
VIII 
Investigation of the Heavy-Atom Distribution 
in a Langmuir-Blodgett Film by an X-Ray Total Extemal Reflection 
and Fluorescence Study 
By S.I. Zheludeva, M.V. Kovalchuk, N.N. Novikova, I.V. Bashelhanov, 
T. Ishikawa, and K. Izumi (With 2 Figures) 125 
A Structural Investigation of an Ultra-Thin Langmuir-Blodgett Film 
by an X-Ray Standing Wave Excited in a L S M Substrate 
Under the Bragg Diffraction Condition 
By M.V. Kovalchuk, S.I. Zheludeva, N.N. Novikova, I.V. Bashelhanov, 
M J . Bedzyk, and G.M. Bommarito (With 1 Figure) 129 
Part IV Liquid Surfaces 
The Structure of Self-Assembled Monolayers 
By P. Eisenberger, P. Ferner, and K.S. Liang (With 1 Figure) 135 
Behenic Acid as a Structural Model for Fatty Acid Monolayers 
at the Air/Water Interface: An X-Ray Diffraction Study 
By R.M. Kenn, C. Böhm, H. Möhwald, K. Kjaer, and J. Ais-Nielsen 
(With 3 Figures) 139 
X-Ray Scattering Studies of Organic Monolayers 
on Electrolytic Solutions: Arachidic Acid on CdCh 
By K . Kjaer, J. Ais-Nielsen, R.M. Kenn, C. Böhm, P. Tippmann-Krayer, 
C A . Helm, H. Möhwald, F. Leveiller, D. Jacquemain, M . Lahav, 
L. Leiserowitz, and M . Deutsch (With 3 Figures) 143 
The Phases of Phosphatidyl Ethanolamine Monolayers 
By C A . Helm, P. Tippmann-Krayer, R.M. Kenn, H. Möhwald, 
J. Ais-Nielsen, and K. Kjaer (With 2 Figures) 147 
X-Ray Diffraction Studies of Fatty Acid Monolayers 
on the Surface of Water 
By M . C Shih, T.M. Bohanon, J.M. Mikrut, P. Zschack, and P. Dutta 
(With 2 Figures) 151 
Protein Recognition Processes at Functionalized Lipid Surfaces: 
A Neutron Reflectivity Study 
By D. Vaknin, J. Ais-Nielsen, M . Piepenstock, and M . Lösche 
(With 1 Figure) 155 
Neutron Reflection from Liquid/Liquid Interfaces 
By T. Cosgrove, A. Eaglesham, D. Hörne, J.S. Phipps, 
and R.M. Richardson (With 5 Figures) 159 
Polymer Interfaces Analysed on a Nanometer Scale: 
X-Ray and Neutron Reflectometry 
By M . Stamm 167 
IX 
Neutron Reflection from Polymers Adsorbed 
at the Solid/Liquid Interface 
By T. Cosgrove, J.S. Phipps, and R.M. Richardson 
(With 3 Figures) 169 
PartV Electrochemistry 
Electrochemical Roughening of Au(l 10) Single Crystal Electrodes 
By K . M . Robinson, W.E. O'Grady, and I.K. Robinson 
(With 1 Figure) 175 
Part VI Thin Films and Multilayers 
Reflectivity Studies of Thin Au Films and Au Bicrystals 
with Grain Boundaries 
By E. Burkel, M . Fitzsimmons, and M . Müller-Stach 
(With 3 Figures) 181 
Depth Resolved Diffuse Scattering from Buried CoSi2 Layers 
in Silicon 
By D. Bahr, B. Burandt, M . Tolan, W. Press, R. Jebasinski, 
and S. Mantl (With 2 Figures) 187 
Glancing Angle X-Ray Techniques for the Analysis 
of Ion Beam Modified Surfaces 
By T.A. Crabb and P.N. Gibson (With 3 Figures) 191 
Surface Analysis of Borkron Glass for Neutron Applications 
By M . Maaza, C. Sella, B. Farnoux, F. Samuel, and P. Trocellier 
(With 3 Figures) 195 
X-Ray Bragg Reflectivity of ErAs Epitaxial Films 
By P.F. Miceli, C J . Palmstr0m, and K.W. Moyers 
(With 4 Figures) 203 
Measurement of Magnetic Field Penetration Depth in Niobium 
Polycrystalline Films by the Polarized Neutron Reflection Method 
By L.P. Chernenko, D.A. Korneev, A.V. Petrenko, N.I. Balalykin, 
and A.V. Skripnik (With 1 Figure) 209 
Neutron Reflectivity Studies on Superconducting, Magnetic 
and Absorbing Thin Films on the Polarized Neutron Spectrometer 
at the Pulsed Reactor IBR-2 
By D.A. Korneev, V.V. Pasyuk, A.V. Petrenko, and E.B. Dokukin 
(With 3 Figures) 213 
x 
Magnetic Properties of Ultrathin Co/Ag Films Investigated 
by Polarised Neutron Reflection 
By H J . Lauter, J.A.C. Bland, R.D. Bateson, and A.D. Johnson 
(With 3 Figures) 219 
Depth Selective Real Structure Analysis of Semiconductor Superlattices 
Using Grazing Incidence X-Ray Diffraction 
By U. Pietsch (With 2 Figures) 223 
Investigation of Interfaces with Grazing Incidence Neutron Radiation 
By V . l . Mikerov, A.V. Vinogradov, I.V. Kozhevnikov, F.A. Pudonin, 
V.A. Tukarev, and M.P. Yakovlev (With 5 Figures) 227 
Roughness Characterization of the Surface and Interface 
of MBE-Grown Thin Films 
By A. Stierte, A. Abromeit, K. Brohl, N . Metoki, and H. Zabel 
(With 3 Figures) 233 
Part V E Instrumentation and Methods 
Neutron Diffraction Under Grazing Incidence: Recent Results 
from the Evanescent Wave Diffractometer 
By K. AI Usta, H. Dosch, A. Lied, and J. Peisl (With 5 Figures) 239 
Analytical Calculation of the Resolution Correction Function 
for X-Ray Surface Structure Analysis at High Exit Angles 
By C. Schamper, H.L. Meyerheim, W. Moritz, and H. Schulz 
(With 2 Figures) 247 
Neutron Double Crystal Diffractometry - A Precise Method 
for Surface Investigations 
By F. Eichhorn, K . M . Podurets, S.Sh. Shilstein, and Z.N. Soroko 
(With 4 Figures) 251 
Index of Contributors 255 
XI 
Analytical Calculation of the Resolution Correction Function 
for X-Ray Surface Structure Analysis 
at High Exit Angles 
C. Schamper, H . L . M e y e r h e i m , W. M o r i t z , and H. Schulz 
Institute of Crystallography and Mineralogy, University of Munich, 
Theresienstr. 41, W-8000 München 2, Fed.Rep.of Germany 
Abstract. The analysis of rod scans in surface x-ray diffraction provides information 
about structure parameters normal to the sample surface. In order to achieve high 
resolution the measurements have to extend to momentum transfers q as large as 
possible. The proper correction of the measured intensity data for the resolution 
function of the detector is a prerequisite for obtaining reliable structure information. 
We have developed an analytical expression for the resolution correction of rod scan 
intensity data which take into account an anisotropic detector resolution T ( Q , $ ) f the 
domain size of the sample, f, and the primary beam divergence parallel to the sample 
surface Ar. 
1. Introducüon 
Glancing incidence angle x-ray scattering is now a well established technique for 
surface structure analysis [1,2]. So far work has been concentrated on measurements 
of 'in plane' data providing only the projection of the structure. In order to obtain 
infomation about the structural parameters normal to the sample surface, measurments 
along the reciprocal lattice rods up to high momentum transfer qz=f-c* have to be 
performed. However, in all diffraction experiments the measured intensities have to 
be corrected for the instrumental resolution. For in plane structure analysis this is 
simply done by multiplying the intensity by a factor Aq z corresponding to the length 
of the rod accepted by the detector [1,2]. 
The correction factors become complicated for rod scans, when the scattering 
plane is no longer parallel to the sample surface. In the following we shortly outline 
the intensity correction due to the detector resolution for our z-axis diffractometer 
with two independent detector angles. Subsequently we discuss the results by 
considering different detector arrangements, where the primary beam divergence and 
the finite domain size of the sample are taken into account. 
2. Calculation 
Experimentally, the integrated intensity is measured by rotating the sample around the 
surface normal keeping the detector at a fixed position out of plane. In Fig.l this 
procedura is shown in projection to the £=0 plane when the reciprocal lattice rod is 
swept through the detector' s aperture. The integrated intensity is given by the 
convolution of the reflectivity with the transmission function: 
Springer Proceedings in Physics, Vol. 61 Surface X-Ray and Neutron Scattering 
Eds.: H. Zabel and IX. Robinson © Springer-Verlag Berlin Heidelberg 1992 
247 
Fig. 1: Projection of the diffrac-
tion geometry on the £=0 plane , 
illustrating the measurement of 
the integrated intensity. The 
sample is rotated around the 
surface normal. Diffraction is 
obtained when the rod intersects 
the detector aperture. The angular 
ränge of <p where the rod remains 
within the detector changes at 
different l values. The measured 
intensities must therefore be 
corrected due to the different 
shapes of the resolution ellipse. 
E ( Q A ) i ( k r k i 0 ) . | F ( q ) | 2 . 
• T ( q - Q 0 - A Q 1 ( k r k i 0 ) - A Q 2 ( ^ ) ) d q 3 d k i 3 d ^ . ( i ) 
In Eq. 1 i(kj—k|0> describes the intensity distribution in the primary beam around the 
incident beam KJQ. Gaussian functions are assumed for the primary beam intensity 
distribution i , the transmission function T and the rod profde parallel to the surface 
I F(q|) | 2 . The parameters q and Q A represent vectors in reciprocal space, the latter 
indicates the reflection under consideration. The primary beam divergence and the 
rotation angle <p of the crystal are taken into account by the arguments AQj(k-— k^p 
and AQ2(<p)- The calulation is based on three assumptions: (1) a grazing incidence 
geometry is used, (2) the radiation is strictly monochromatic and (3) the lateral 
extensions of the reciprocal lattice rod and the detector acceptance are small as 
compared to the radius of the Ewald sphere. The analytic Solution of Eq. (1) is simple 
but tedious and will be described elsewhere [3]. Finally we are left with an expression 
E(QA) OC I F C Q A ) ! 2 • LCQQ), where 
X • A G • A * 
L(QA) = { 1 + wA^'T2 + m / r 2 } ' m (2) 
d l / 2 - a -b . | Q 0 , | 
indicates the Lorentz factor which describes the dependence of the integrated intensity 
on pure geometric parameters, particularly on the acceptance angles of the detector 
AO and A$. The parameters d, m and T are given by 
c = cos(ß-e) • cos(ß~e) • (1/a2 - 1/b2) (3) 
d = ( sin(/?-€) ) 2 / a 2 + ( cos(ß-e) ) 2 / b 2 (4) 
e = ( cos(/3-€) ) 2 / a 2 + ( sm(ß-e) ) 2 / b 2 (5) 
248 
m = X 2 - ( e - c 2 / d ) [ A 2 ] (6) 
r = ( c o s / i ) / X - | Q Q | I cos(/i+€), [Ä- 1 ] (7) 
where a/X and b/X [ A " 1 ] are the short and the long axis of the projected resolution 
ellipse, respectively (Fig. 1) and fi is the projection of the scattering angle to the 
surface. The angles ß and € are explained in Fig. 1. The Lorentz factor factor given 
by Eq. (2) is composed of two terms, the first representing the intensity correction 
assuming Ö-runction like lattice rods and strictly parallel incident radiation, the second 
(under the square root) takes account for the domain size, f, ^ d the primary beam 
divergence Ar . In the low exit angle regime (l <* 1) the correction calculated from Eq. 
(1) agrees with the published formula [1,2], provided that the domain size f is large 
and the primary beam is parallel (Ar=0). 
3. Results and Discussion 
In order to elucidate the different factors contributing to the intensity corrections we 
have calculated L(£) for the (1/2 1/2) and (3/2 5/2) superlattice reflection rod of the 
Ge(001)(2xl) structure assuming a wavelength of X= 1.54 A . The results are shown in 
Fig. 2. In order to maximize the intensity at low £ it is evident that the detector has to 
be aligned with a high acceptance parallel to the rods. However, the Situation changes 
at intermediate and high £, where the intersection of the lattice rod with the Ewald 
sphere crosses the detector transmission increasingly oblique to the long axis. Under 
these conditions it is preferable to rotate the detector by 90°, as displayed in Fig.L 
Additionally, the dependence on Q Q | results in an about four times larger intensity of 
the (1/2 1/2 £) as compared with the (3/2 5/2 £) reflections for £ £ 1. 
0 1 2 3 4 0 1 2 3 4 
Fig. 2: Lorentz factor as a function of £ for both detector settings (solid and dashed 
line) assuming parallel primary beam and Mike rods (left panel). The detector 
acceptance angles are 0.4° and 2.0°. The dotted line indicates the resolution 
correction for the (1/2 1/2) rod as given by ref. [2]. The right panel shows the £-
dependence of the correction term in Eq. 2 for a domain size f of 500 A and a beam 
divergence Ar of 0.23°. 
249 


Analytical calculation of the resolution correction function for X-ray surface structure analysis

Open Access LMU

Spr inger P r o c e e d i n g s in P h y s i c s 61 S p r i n g e r P r o c e e d i n g s i n P h y s i c s Managing Editor: H . K . V. Lotsch 44 Optical Fiber Sensors Editors: H. J. Arditty, J. P. Dakin, and R. Th. Kersten 45 Computer S i m u l a t i o n Studies in Condensed Matter Physics II: New Directions Editors: D. P. Landau, K. K. Mon, andH.-B. Schüttler 46 Cellular Automata and Modeling of Complex Physical Systems Editors: P. Manneville, N. Boccara, G. Y. Vichniac, and R. Bidaux 54 Polycrystalline Semiconductors II Editors: J. H. Werner and H. P. Strunk 55 Nonlinear Dynamics and Q u a n t u m Phenomena in Optical Systems Editors: R. Vüaseca and R. Corbalän 56 Amorphous and Crystalline Silicon Carbide III and Other G r o u p I V - I V Materials Editors: G. L. Harris, M. G. Spencer, andC. Y-W.Yang 47 Number Theory and Physics Editors: J.-M. Luck, P. Moussa, and M. Waldschmidt 48 M a n y - A t o m Interactions in Solids Editors: R .M. Nieminen, M. J. Puska, and M. J. Manninen 57 Evolutionary Trends in the Physical Sciences Editors: M. Suzuki and R. Kubo 58 New Trends in Nuclear Collective Dynamics Editors: Y Abe, H. Horiuchi, and K. Matsuyanagi 49 Ultrafast Phenomena in Spectroscopy Editors: E. Klose and B. Wilhelmi 50 Magnetic Properties of L o w - D i m e n s i o n a l Systems II: New Developments Editors: L. M. Falicov, F. Mejia-Lira, and J. L. Morän-Löpez 51 The Physics and Chemistry of Organic Superconductors Editors: G. Saito and S. Kagoshima 52 Dynamics and Patterns in Complex Fluids: New Aspects ofthe Physics-Chemistry Interface Editors: A. Onuki and K. Kawasaki 53 Computer S i m u l a t i o n Studies in Condensed Matter Physics III Editors: D. P. Landau, K. K. Mon, and H.-B. Schüttler 59 Exotic Atoms in Condensed Matter Editors: G. Benedek and H. Schneuwly 60 The Physics and Chemistry of Oxide Superconductors Editors: Y Iye and H. Yasuoka 61 Surface X-Ray and Neutron Scattering Editors: H. Zabel and I.K. Robinson 62 Surface Science: Lectures on Basic Concepts and Applications Editors: E A. Ponce and M. Cardona 63 Coherent R a m a n Spectroscopy: Recent Advances Editors: G. Marowsky and V. V. Smirnov 64 Superconducting Devices and Their Applications Editors: H. Koch and H. Lübbig Volumes 1-43 are listed on the back inside cover H. Zabel I. K. Robinson (Eds.) Surface X-Ray and Neutron Scattering Proceedings of the 2nd International Conference, Physik Zentrum, Bad Honnef, Fed. Rep. of Germany, June 25-28,1991 With 120 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Contents Conference Summary By S.K. Sinha 1 Part I Surface Crystallography and Phase Transitions Surface X-Ray Crystallography and STM Images By R. Feidenhans'l (With 3 Figures) 5 Determination of Metal Adsorbed Surfaces by X-Ray Diffraction By T. Takahashi (With 5 Figures) 11 Au Adsorption on S i ( l l l ) Studied by Grazing Incidence X-Ray Diffraction By C. Schamper, D. Dornisch, W. Moritz, H. Schulz, R. Feidenhans'l, M . Nielsen, F. Grey, and R.L. Johnson (With 3 Figures) 17 Grazing Incidence X-Ray Scattering Study of Staircases of Steps on Si(OOl) Surfaces By G. Renaud, P.H. Fuoss, J. Bevk, and B.S. Freer (With 2 Figures) 21 Structure and Phase Transitions of Ge( l l l ) and S i ( l l l ) Surfaces at High Temperatures By K.L Blum, D.Y. Noh, A. Mak, K.W. Evans-Lutterodt, J.D. Brock, G.A. Held, and R J . Birgeneau (With 3 Figures) 27 Anomalous Scattering Applied to Co/Si(ll l) Interface Structure By N . Jedrecy, A. Waldhauer, M . Sauvage-Simkin, R. Pinchaux, and V.H. Etgens (With 2 Figures) 33 X-Ray Reflectivity Studies of Au Surfaces By D.L. Abernathy, D. Gibbs, G. Grübel, K.G. Huang, S.GJ. Mochrie, B . M . Ocko, A.R. Sandy, and D.M. Zehner (With 5 Figures) 37 Crystal Truncation Rod as a Convolution of Three-Dimensional Bravais Lattice with X-Ray Reflectivity By Hoydoo You 47 Extended X-Ray Reflectivity Analysis of Si(l 11)7x7 B y I.K. Robinson and E. Vlieg (With 3 Figures) 51 VII Critical Phenomena at Surfaces and Interfaces By R. Lipowsky 57 Surface-Induced Order Observed on a Cu3Au(001) Surface By K.S. Liang, H.H. Hung, S.L. Chang, Z. Fu, S.C. Moss, and K. Oshima (With 2 Figures) 65 Thermal Dynamics of (110) fcc Metal Surfaces B y K . Kern 69 Facet Coexistence in the Roughening Transition of Ag(110) By H.G. Hörnis, E.H. Conrad, E. Vlieg, and I.K. Robinson (With 5 Figures) 73 Kinetics of Ordering with Random Lnpurities: Pb on Ni(001) By P.W. Stephens, P J . Eng, and T. Tse (With 3 Figures) 79 Part H Reflectivity Anomalous Reflectivity: A New Method for Determining Density Profiles of Thin Films By S.K. Sinha, M.K. Sanyal, K.G. Huang, A. Gibaud, M . Rafailovich, J. Sokolov, X . Zhao, and W. Zhao (With 3 Figures) 85 Specular and Diffuse Scattering Studies of Multilayer Interfaces By M.K. Sanyal, S.K. Sinha, A. Gibaud, S.K. Satija, C.F. Majkrzak, and H. Homa (With 3 Figures) 91 Scattering Cross-Section of X-Rays and Neutrons for Grazing Incidence onto Thin Films By A. Haase and S. Dietrich 95 Total Neutron Reflection: Experiments and Analysis By G.P. Felcher, W.D. Dozier, Y . Y . Huang, and X.L . Zhou (With 3 Figures) 99 Profile Refinement in Neutron Reflectivity and Grazing Angle Diffraction By J.F. Ankner (With 1 Figure) 105 Part m Surface X-Ray Standing Waves X-Ray Standing Wave Studies of the Liquid/Solid Interface and Ultrathin Organic Films By M.J. Bedzyk (With 3 Figures) 113 Glancing-Incidence X-Ray Analysis of Layered Materials By D.K.G. de Boer and W.W. van den Hoogenhof (With 4 Figures) 119 VIII Investigation of the Heavy-Atom Distribution in a Langmuir-Blodgett Film by an X-Ray Total Extemal Reflection and Fluorescence Study By S.I. Zheludeva, M.V. Kovalchuk, N.N. Novikova, I.V. Bashelhanov, T. Ishikawa, and K. Izumi (With 2 Figures) 125 A Structural Investigation of an Ultra-Thin Langmuir-Blodgett Film by an X-Ray Standing Wave Excited in a L S M Substrate Under the Bragg Diffraction Condition By M.V. Kovalchuk, S.I. Zheludeva, N.N. Novikova, I.V. Bashelhanov, M J . Bedzyk, and G.M. Bommarito (With 1 Figure) 129 Part IV Liquid Surfaces The Structure of Self-Assembled Monolayers By P. Eisenberger, P. Ferner, and K.S. Liang (With 1 Figure) 135 Behenic Acid as a Structural Model for Fatty Acid Monolayers at the Air/Water Interface: An X-Ray Diffraction Study By R.M. Kenn, C. Böhm, H. Möhwald, K. Kjaer, and J. Ais-Nielsen (With 3 Figures) 139 X-Ray Scattering Studies of Organic Monolayers on Electrolytic Solutions: Arachidic Acid on CdCh By K . Kjaer, J. Ais-Nielsen, R.M. Kenn, C. Böhm, P. Tippmann-Krayer, C A . Helm, H. Möhwald, F. Leveiller, D. Jacquemain, M . Lahav, L. Leiserowitz, and M . Deutsch (With 3 Figures) 143 The Phases of Phosphatidyl Ethanolamine Monolayers By C A . Helm, P. Tippmann-Krayer, R.M. Kenn, H. Möhwald, J. Ais-Nielsen, and K. Kjaer (With 2 Figures) 147 X-Ray Diffraction Studies of Fatty Acid Monolayers on the Surface of Water By M . C Shih, T.M. Bohanon, J.M. Mikrut, P. Zschack, and P. Dutta (With 2 Figures) 151 Protein Recognition Processes at Functionalized Lipid Surfaces: A Neutron Reflectivity Study By D. Vaknin, J. Ais-Nielsen, M . Piepenstock, and M . Lösche (With 1 Figure) 155 Neutron Reflection from Liquid/Liquid Interfaces By T. Cosgrove, A. Eaglesham, D. Hörne, J.S. Phipps, and R.M. Richardson (With 5 Figures) 159 Polymer Interfaces Analysed on a Nanometer Scale: X-Ray and Neutron Reflectometry By M . Stamm 167 IX Neutron Reflection from Polymers Adsorbed at the Solid/Liquid Interface By T. Cosgrove, J.S. Phipps, and R.M. Richardson (With 3 Figures) 169 PartV Electrochemistry Electrochemical Roughening of Au(l 10) Single Crystal Electrodes By K . M . Robinson, W.E. O'Grady, and I.K. Robinson (With 1 Figure) 175 Part VI Thin Films and Multilayers Reflectivity Studies of Thin Au Films and Au Bicrystals with Grain Boundaries By E. Burkel, M . Fitzsimmons, and M . Müller-Stach (With 3 Figures) 181 Depth Resolved Diffuse Scattering from Buried CoSi2 Layers in Silicon By D. Bahr, B. Burandt, M . Tolan, W. Press, R. Jebasinski, and S. Mantl (With 2 Figures) 187 Glancing Angle X-Ray Techniques for the Analysis of Ion Beam Modified Surfaces By T.A. Crabb and P.N. Gibson (With 3 Figures) 191 Surface Analysis of Borkron Glass for Neutron Applications By M . Maaza, C. Sella, B. Farnoux, F. Samuel, and P. Trocellier (With 3 Figures) 195 X-Ray Bragg Reflectivity of ErAs Epitaxial Films By P.F. Miceli, C J . Palmstr0m, and K.W. Moyers (With 4 Figures) 203 Measurement of Magnetic Field Penetration Depth in Niobium Polycrystalline Films by the Polarized Neutron Reflection Method By L.P. Chernenko, D.A. Korneev, A.V. Petrenko, N.I. Balalykin, and A.V. Skripnik (With 1 Figure) 209 Neutron Reflectivity Studies on Superconducting, Magnetic and Absorbing Thin Films on the Polarized Neutron Spectrometer at the Pulsed Reactor IBR-2 By D.A. Korneev, V.V. Pasyuk, A.V. Petrenko, and E.B. Dokukin (With 3 Figures) 213 x Magnetic Properties of Ultrathin Co/Ag Films Investigated by Polarised Neutron Reflection By H J . Lauter, J.A.C. Bland, R.D. Bateson, and A.D. Johnson (With 3 Figures) 219 Depth Selective Real Structure Analysis of Semiconductor Superlattices Using Grazing Incidence X-Ray Diffraction By U. Pietsch (With 2 Figures) 223 Investigation of Interfaces with Grazing Incidence Neutron Radiation By V . l . Mikerov, A.V. Vinogradov, I.V. Kozhevnikov, F.A. Pudonin, V.A. Tukarev, and M.P. Yakovlev (With 5 Figures) 227 Roughness Characterization of the Surface and Interface of MBE-Grown Thin Films By A. Stierte, A. Abromeit, K. Brohl, N . Metoki, and H. Zabel (With 3 Figures) 233 Part V E Instrumentation and Methods Neutron Diffraction Under Grazing Incidence: Recent Results from the Evanescent Wave Diffractometer By K. AI Usta, H. Dosch, A. Lied, and J. Peisl (With 5 Figures) 239 Analytical Calculation of the Resolution Correction Function for X-Ray Surface Structure Analysis at High Exit Angles By C. Schamper, H.L. Meyerheim, W. Moritz, and H. Schulz (With 2 Figures) 247 Neutron Double Crystal Diffractometry - A Precise Method for Surface Investigations By F. Eichhorn, K . M . Podurets, S.Sh. Shilstein, and Z.N. Soroko (With 4 Figures) 251 Index of Contributors 255 XI Analytical Calculation of the Resolution Correction Function for X-Ray Surface Structure Analysis at High Exit Angles C. Schamper, H . L . M e y e r h e i m , W. M o r i t z , and H. Schulz Institute of Crystallography and Mineralogy, University of Munich, Theresienstr. 41, W-8000 München 2, Fed.Rep.of Germany Abstract. The analysis of rod scans in surface x-ray diffraction provides information about structure parameters normal to the sample surface. In order to achieve high resolution the measurements have to extend to momentum transfers q as large as possible. The proper correction of the measured intensity data for the resolution function of the detector is a prerequisite for obtaining reliable structure information. We have developed an analytical expression for the resolution correction of rod scan intensity data which take into account an anisotropic detector resolution T ( Q , $ ) f the domain size of the sample, f, and the primary beam divergence parallel to the sample surface Ar. 1. Introducüon Glancing incidence angle x-ray scattering is now a well established technique for surface structure analysis [1,2]. So far work has been concentrated on measurements of 'in plane' data providing only the projection of the structure. In order to obtain infomation about the structural parameters normal to the sample surface, measurments along the reciprocal lattice rods up to high momentum transfer qz=f-c* have to be performed. However, in all diffraction experiments the measured intensities have to be corrected for the instrumental resolution. For in plane structure analysis this is simply done by multiplying the intensity by a factor Aq z corresponding to the length of the rod accepted by the detector [1,2]. The correction factors become complicated for rod scans, when the scattering plane is no longer parallel to the sample surface. In the following we shortly outline the intensity correction due to the detector resolution for our z-axis diffractometer with two independent detector angles. Subsequently we discuss the results by considering different detector arrangements, where the primary beam divergence and the finite domain size of the sample are taken into account. 2. Calculation Experimentally, the integrated intensity is measured by rotating the sample around the surface normal keeping the detector at a fixed position out of plane. In Fig.l this procedura is shown in projection to the £=0 plane when the reciprocal lattice rod is swept through the detector' s aperture. The integrated intensity is given by the convolution of the reflectivity with the transmission function: Springer Proceedings in Physics, Vol. 61 Surface X-Ray and Neutron Scattering Eds.: H. Zabel and IX. Robinson © Springer-Verlag Berlin Heidelberg 1992 247 Fig. 1: Projection of the diffrac-tion geometry on the £=0 plane , illustrating the measurement of the integrated intensity. The sample is rotated around the surface normal. Diffraction is obtained when the rod intersects the detector aperture. The angular ränge of <p where the rod remains within the detector changes at different l values. The measured intensities must therefore be corrected due to the different shapes of the resolution ellipse. E ( Q A ) i ( k r k i 0 ) . | F ( q ) | 2 . • T ( q - Q 0 - A Q 1 ( k r k i 0 ) - A Q 2 ( ^ ) ) d q 3 d k i 3 d ^ . ( i ) In Eq. 1 i(kj—k|0> describes the intensity distribution in the primary beam around the incident beam KJQ. Gaussian functions are assumed for the primary beam intensity distribution i , the transmission function T and the rod profde parallel to the surface I F(q|) | 2 . The parameters q and Q A represent vectors in reciprocal space, the latter indicates the reflection under consideration. The primary beam divergence and the rotation angle <p of the crystal are taken into account by the arguments AQj(k-— k^p and AQ2(<p)- The calulation is based on three assumptions: (1) a grazing incidence geometry is used, (2) the radiation is strictly monochromatic and (3) the lateral extensions of the reciprocal lattice rod and the detector acceptance are small as compared to the radius of the Ewald sphere. The analytic Solution of Eq. (1) is simple but tedious and will be described elsewhere [3]. Finally we are left with an expression E(QA) OC I F C Q A ) ! 2 • LCQQ), where X • A G • A * L(QA) = { 1 + wA^'T2 + m / r 2 } ' m (2) d l / 2 - a -b . | Q 0 , | indicates the Lorentz factor which describes the dependence of the integrated intensity on pure geometric parameters, particularly on the acceptance angles of the detector AO and A$. The parameters d, m and T are given by c = cos(ß-e) • cos(ß~e) • (1/a2 - 1/b2) (3) d = ( sin(/?-€) ) 2 / a 2 + ( cos(ß-e) ) 2 / b 2 (4) e = ( cos(/3-€) ) 2 / a 2 + ( sm(ß-e) ) 2 / b 2 (5) 248 m = X 2 - ( e - c 2 / d ) [ A 2 ] (6) r = ( c o s / i ) / X - | Q Q | I cos(/i+€), [Ä- 1 ] (7) where a/X and b/X [ A " 1 ] are the short and the long axis of the projected resolution ellipse, respectively (Fig. 1) and fi is the projection of the scattering angle to the surface. The angles ß and € are explained in Fig. 1. The Lorentz factor factor given by Eq. (2) is composed of two terms, the first representing the intensity correction assuming Ö-runction like lattice rods and strictly parallel incident radiation, the second (under the square root) takes account for the domain size, f, ^ d the primary beam divergence Ar . In the low exit angle regime (l <* 1) the correction calculated from Eq. (1) agrees with the published formula [1,2], provided that the domain size f is large and the primary beam is parallel (Ar=0). 3. Results and Discussion In order to elucidate the different factors contributing to the intensity corrections we have calculated L(£) for the (1/2 1/2) and (3/2 5/2) superlattice reflection rod of the Ge(001)(2xl) structure assuming a wavelength of X= 1.54 A . The results are shown in Fig. 2. In order to maximize the intensity at low £ it is evident that the detector has to be aligned with a high acceptance parallel to the rods. However, the Situation changes at intermediate and high £, where the intersection of the lattice rod with the Ewald sphere crosses the detector transmission increasingly oblique to the long axis. Under these conditions it is preferable to rotate the detector by 90°, as displayed in Fig.L Additionally, the dependence on Q Q | results in an about four times larger intensity of the (1/2 1/2 £) as compared with the (3/2 5/2 £) reflections for £ £ 1. 0 1 2 3 4 0 1 2 3 4 Fig. 2: Lorentz factor as a function of £ for both detector settings (solid and dashed line) assuming parallel primary beam and Mike rods (left panel). The detector acceptance angles are 0.4° and 2.0°. The dotted line indicates the resolution correction for the (1/2 1/2) rod as given by ref. [2]. The right panel shows the £-dependence of the correction term in Eq. 2 for a domain size f of 500 A and a beam divergence Ar of 0.23°. 249 

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Analytical calculation of the resolution correction function for X-ray surface structure analysis

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