X-ray Diffraction Techniques to Characterize Liquid Surfaces and Monomolecular Layers at Gas-Liquid Interfaces
David Vaknin
Liqsurf_eprint.TEX
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\title{X-ray Diffraction Techniques to Characterize Liquid Surfaces and Monomolecular Layers at Gas-Liquid Interfaces}
% repeat the \author\address pair as needed
\author{ David Vaknin}
\address{Ames Laboratory and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011, USA}
\date{\today}
\maketitle
%\twocolumn
\section{Introduction}

X-ray and neutron scattering techniques
are probably the most effective tools to be employed in order to determine
the structure of liquid interfaces on molecular length scales.
These  are not different in principle from  conventional X-ray 
diffraction techniques that are usually applied to three dimensional 
crystals, liquids, solid surfaces etc.  However,  special
diffractometers that enable  scattering from fixed horizontal surfaces are
required to carry out the experiments.
Indeed, systematic studies of liquid surfaces had not begun until
the introduction of the first liquid surface
reflectometer.\cite{Als-Nielsen83}.

A basic property of a liquid-gas interface is the length scale over
which the molecular density changes from the bulk value to that of
the homogeneous gaseous medium.  Molecular size and capillary waves,
that depend on surface tension and gravity, are among the most 
important factors that shape  the density profile
across the interface and the planar correlations
\cite{Evans79,Braslau88,Sinha88}.
In some instances the topmost layers
of liquids are packed differently than in the bulk, giving rise
to layering phenomena at the interface.
Monolayers of compounds different than the liquid can be spread
at the gas-liquid  at interface, and are termed Langmuir monolayers
\cite{Gaines66,Swalen87}.
The spread compound might {\it wet} the liquid surface to form a film
of homogeneous thickness or cluster to form an inhomogeneous {\it rough}
surface.  The X-ray reflectivity (XR) technique allows one  
to determine the electron density across such interfaces from which
the molecular density and the total thickness can be extracted.
The grazing angle diffraction (GID) technique is commonly used to determine
lateral arrangements and correlations of the topmost layers at interfaces.
GID is especially efficient in cases where surface crystallization
of the liquid or spread monolayers occurs.
Both techniques (XR and GID) provide structural information that is
averaged over macroscopic areas in contrast to scanning probe
microscopies (SPM's) where local arrangements are probed.
For an inhomogeneous interface the reflectivity
is an incoherent sum of reflectivities,
accompanied by strong diffuse scattering which in general is difficult
to interpret with definitive answers and often requires
complementary techniques to support the X-ray analysis.
Therefore, preparation of well-defined homogeneous interfaces is a
key to a more definitive and straightforward interpretation.

\subsection{Competitive and Related Techniques}
Although modern scanning probe microscopies (SPM's)
such as scanning tunneling microscopy (STM) \cite{Binnig83} and
atomic force microscopy (AFM)\cite {Binnig92}
rival X-ray scattering techniques
in probing atomic arrangements of solid surfaces,
they have not yet become suitable techniques for free liquid surfaces.
The large fluctuations due to the two-dimensional nature of the
liquid interface, high molecular mobility, and the lack of
electron conductivity (which is necessary for STM) are among
some of the main obstacles that make it difficult to  apply 
these techniques to gas-liquid interfaces.   
In dealing with volatile liquids, inadvertent
deposition or wetting of the probe by the liquid can occur which may
obscure the measurements.  In addition, the relatively strong
interaction of the probe with the surface might alter its
pristine properties.                      
For similar and other reasons, electron microscopy and electron
diffraction techniques, which are among the best choices for probing
solid surfaces, are not suitable
to most liquid surfaces, and in particular aqueous interfaces.
On the other hand, visible light microscopy techniques
such as Brewster angle microscopy (BAM)\cite{Azzam77,Henon91,Honig91}
or fluorescence microscopy\cite{Losche83} have been used
very successfully in providing  morphological pictures of the interface 
on the micrometer length scale. This information in general, is complementary to 
that extracted from
X-ray scattering.  These techniques are very useful for characterizing 
inhomogeneous surfaces with two or more distinct domains
for which XR and GID results are usually
difficult to interpret. However, it is impossible to determine the
position  of the domains with respect to the liquid
interface, their thicknesses or their chemical nature.  
Ellipsometry is another technique that
exploits visible light, to allow determination of film
thickness on a molecular length scale and assumes that one knows the
refractive index of the substrate and of the film \cite{Azzam77,Ducharme90};
either might be different from their bulk values and difficult to determine.

In the following sections theoretical background to the X-ray
techniques is presented together with  experimental procedures
and data analysis concerning liquid surfaces.  It is intended to provide
 a basic formulation which can be developed for further specific applications.
Several examples  of these techniques applied  to a variety
of problems are presented briefly to  demonstrate the strengths
and limitations of the techniques.
It should be borne in mind that the derivations and  procedures
described below are mostly general and can be applied to solid surfaces,
and {\it vice-versa}, many results applicable
to solid surfaces can be used for liquid surfaces.
X-ray reflectivity from surfaces and GID have been treated in recent
reviews\cite{Als-Nielsen89,Russell90,Zhou95}.

\section{Theoretical Background}
We assume that a plane harmonic wave of frequency $\omega$ and 
wave-vector $\bbox{k}_0$ (with electric field, 
${\bbox E}={\bbox E_0}{\rm e}^{i\omega t - i\bbox{k}_0\cdot \bbox{r}}$) is scattered from a 
distribution of free electrons, with a number density $N_e(\bbox{r})$. 
 Due to the interaction with the electric field of the 
X-ray wave, each free electron experiences a displacement proportional to the electric field, 
 $\bbox{X} = -\frac{e}{m\omega^2}\bbox{E}$.  This displacement gives rise to a 
polarization $\bbox{P}(\bbox{r})$ distribution vector  
\begin{eqnarray}
\bbox{P}(\bbox{r}) = N_e(\bbox{r})e{\bbox X}  
\end{eqnarray} in the medium. 
For the sake of convenience we define the scattering length density (SLD) 
$\rho({\bbox r})$ in terms of the classical radius of the electron 
$ r_0=\frac{e^2}{4\pi\epsilon_0 m c^2} = 2.82\times 10^{-13}$ cm as follows
\begin{eqnarray}
\rho({\bbox r}) = N_e({\bbox r})r_0  \label{eq2} 
\end{eqnarray}
The polarization then can be written as 
\begin{eqnarray}
\bbox{P}(\bbox{r}) = -\frac{N_e(\bbox{r})e^2}{\omega^2m_e}\bbox{E} 
= -\frac{4\pi\epsilon_0}{k_0^2}\rho({\bbox r})\bbox{E} 
\end{eqnarray} 
The scattering length density (or the electron density) is what we wish to extract 
from reflectivity and GID experiments and relate it to atomic or molecular positions at 
liquid interfaces.  
The  displacement vector $\bbox{D}$ can now be constructed 
as follows:
\begin{eqnarray}
\bbox{D} = \epsilon_0\bbox{E}+ \bbox{P}(\bbox{r})=\epsilon(\bbox{r})\bbox{E} 
\end{eqnarray}
where $\epsilon(\bbox{r})$ is the permittivity of the medium usually associated with the 
refractive index $ n(\bbox{r})=\sqrt{\epsilon(\bbox{r})}$ . 
To account for absorption by the medium we introduce a phenomenological factor $\beta$ 
that we calculate from the linear absorption coefficient $\mu$, 
(given in tables\cite{ITC_volC}) as follows $\beta = \mu/(2k_0)$.
Then the most general permittivity for X-rays becomes
\begin{eqnarray}
  \epsilon(\bbox{r}) 
        = \epsilon_0 [ {1 - \frac{4\pi}{k_0^2}}\rho({\bbox r})] + 2i\beta 
\end{eqnarray}
 Typical values of the SLD ($\rho$) and the absorption term ($\beta$) 
for water and liquid mercury are
listed in Table\ \ref{tb1}.
In the absence of true charges in the scattering medium  
(i.e., a neutral medium) and under the assumption that the medium is nonmagnetic
(magnetic permeability $\mu=1$) the wave equations that need to be solved to predict 
the scattering from a known SLD 
can be derived from the following Maxwell equations \cite{Panofsky62},
\begin{eqnarray}
\nabla\cdot\bbox{D} = 0 &
\nabla\cdot{\bbox H} = 0  \nonumber \\                          
\nabla\times{\bbox E} = -\partial{\bbox H}/\partial t &
\qquad \nabla\times{\bbox H} = \partial{\bbox D}/\partial t.                  
\label{Maxwell1}
\end{eqnarray}
Under the assumption of harmonic plane waves,
${\bbox E}={\bbox E_0}{\rm e}^{i\omega t - i{\bbox k\cdot r}}$,
the following general equations are obtained from Eq.\ \ref{Maxwell1}:
\begin{eqnarray}
\nabla^2{\bbox E} + [k^2_{0} - 4\pi\rho({\bbox r})]{\bbox E} =
-\nabla\left(\nabla \ln \epsilon(\bbox{r}) \cdot {\bbox E}\right), \label{WaveEq1}
\\
\nabla^2{\bbox H} + [k^2_{0} - 4\pi\rho({\bbox r})]{\bbox H} =
-\nabla \ln \epsilon(\bbox{r}) \times\left(\nabla\times  {\bbox H}\right).
\label{WaveEq2}
\end{eqnarray}
In some particular cases the right hand side of Equations\ \ref{WaveEq1} and \ref{WaveEq2} 
is zero.  We notice then, that the term $4\pi\rho(\bbox{r})$ in the equation plays a 
role similar to that of a potential $V(\bbox{r})$ in wave mechanics.  
 In  fact, for most practical cases the right hand side of Equations\ \ref{WaveEq1} and
\ref{WaveEq2} can be approximated to zero, thus the equation for each component of the fields 
resembles a stationary wave equation.  In those cases general mathematical tools, 
such as the Born approximation (BA) and the distorted wave Born approximation (DWBA) 
can be used\cite{Schiff68}.  

\subsection{Reflectivity}

In reflectivity experiments a monochromatic X-ray beam
of wavelength $\lambda$ [wavevector ${k_0} = 2\pi/\lambda
\mbox{ and } \bbox{k_i} = (0,k_y,k_z)$]
is incident at an angle $\alpha_i$ on a liquid
surface and is detected at an outgoing angle $\alpha_r$ such that
$\alpha_i = \alpha_r $, as shown in Figure~\ref{fig1}, with a final wave
vector $\bbox{k}_f$.
The momentum transfer is defined in terms of the incident
and reflected beam as follows,
\begin{equation}
\bbox{Q} = \bbox{k}_i - \bbox{k}_f,
\end{equation}
where in the reflectivity case ${\bbox Q}$ is strictly
along the surface normal, with $Q_z = 2k_0\sin\alpha = 2k_z$.


\subsubsection{Single, Ideally Sharp Interface - Fresnel Reflectivity}

Solving the scattering problem exactly for the ideally sharp interface
although simple, is very useful for the derivation of 
more complicated electron density profiles across interfaces.
The wavefunctions employed are also essential for
inclusion of dynamical effects when dealing with 
non-specular scattering i.e., GID and diffuse scattering.

\centerline{\it 1a. s-polarized X-ray beam}

For a stratified medium with an electron density that varies along one
direction, $z$, ($\rho({\bbox r}) = \rho({z})$ assuming no
absorption i.e., $\beta = 0$), an s-type polarized
X-ray beam with  the electric field parallel to the surface (along the 
x-axis, see Fig.\ \ref{fig1})
obeys the stationary wave equation as derived from Eq.\ \ref{WaveEq1} and is
simplified as follows,
\begin{equation}
\nabla^2E_x + [k^2_{0} - V({z})]E_x = 0,
\label{eq4}
\end{equation}
with an effective potential $V(z) = 4\pi\rho(z)$.
The general solution to Eq.\ \ref{eq2} is then given by,
\begin{equation}
E_x = E(z) {\rm e}^{ik_yy}
\label{eq6}
\end{equation}
where the momentum transfer along $y$ is conserved when the wave travels
through the medium leading to the well-known Snell's rule for refraction.
Inserting Eq.\ \ref{eq6} in Eq.\ \ref{eq4}  leads to a one dimensional 
wave equation through a potential $V(z)$,
\begin{equation}
{ {\rm d}^2E\over {{\rm d}z}^2 } + [k^2_{z} - V(z)]E = 0.
\label{eq7}
\end{equation}
The simplest case of Eq.\ \ref{eq7} is that of an ideally
flat interface, separating the vapor phase and the bulk scattering
length density $\rho_s$, at $z=0$.
The solution of Eq.\ \ref{eq7} is then given by
\begin{equation}
E(z)  = \left\{
\begin{array}{ll}
{\rm e}^{ik_{z,0}z} + r(k_{z,s}){\rm e}^{-ik_{z,0}z}&  z \geq  0
\mbox{ in  gas}   \\
t(k_{z,s}) {\rm e}^{ik_{z,s}z} &  z \leq 0
\mbox{ in liquid}
\end{array}
\right.
\label{eq8}
\end{equation}
where
\begin{equation}
k_{z,s} = \sqrt{k_{z,0}^2 - 4\pi\rho_s} = \sqrt{k_{z,0}^2 - k_c^2},
\label{eq9}
\end{equation}
where $k_c\equiv 2\sqrt{\pi\rho_s}$.
By applying continuity conditions to the wavefunctions and to their
derivatives at $z=0$, the Fresnel equations for reflectance, $r(k_{z,s})$,
and transmission, $t(k_{z,s})$, are obtained,
\begin{equation}
r(k_{z,s}) = { { {k_{z,0} - k_{z,s}} \over {k_{z,0} + k_{z,s}} } }, \qquad
t(k_{z,s}) = { 2k_{z,0} \over {k_{z,0} + k_{z,s}}}.
\label{t1}
\end{equation}
The measured reflectivity from an ideally flat interface, $R_F$,
is usually displayed as a function of the momentum
transfer $Q_z = k_{z,0}+k_{z,s} \approx 2k_{z,0}$,  and is given by
\begin{equation}
R_F(Q_z) = \left |r(k_{z,s})\right |^2 .
\label{R_F}
\end{equation}
Below a critical momentum transfer,
$Q_c \equiv 2k_c \equiv 4\sqrt{\pi\rho_s}$, $k_{z,s}$ is an imaginary number
and $R_F(Q_z) = 1$; total external reflection occurs.
Notice that whereas the critical momentum
transfer does not depend on the X-ray wavelength, the critical angle for total
reflection does, and it is given by
$\alpha_c \approx \lambda\sqrt{\rho_s/\pi}$.
Typical values for critical angles for X-rays of wavelength 
$\lambda_{CuK_\alpha}=1.5404${\AA},
are listed in Table\ \ref{tb1}.
For  $Q_z \gg Q_c$, $R_F(Q_z)$ can be approximated to a form that is
known as the Born approximation,
\begin{equation}
R_F(Q_z) \sim \left ({ {Q_c}\over 2{Q_z} }\right )^4 .
\label{Fresnel_approx}
\end{equation}
This form of the reflectivity at large $Q_z$'s is also valid for
internal scattering, i.e., reflectivity from liquid into the vapor phase.
However, total reflection does not occur for the internal reflectivity case.
Calculated external and internal reflectivity curves from an ideally 
flat surface, $R_F$, displayed versus momentum transfer
(in units of the critical momentum transfer)
are shown in Fig.\ \ref{fig2}(a).  Both reflectivities converge at large momentum transfer where 
they can be both approximated by Eq.\ \ref{Fresnel_approx}.
The dashed line in the same figure shows the approximation $(Q_c/2Q_z)^4$, which fails in
describing the reflectivity close to the critical momentum transfer.

The photon transmission at a given $k_{z,s}$  is given by
\begin{equation}
T(k_{z,s}) = \left |t(k_{z,s})\right|^2 { {{\rm Re}(k_{z,s})}
\over {{\rm Re}(k_{z,0})}}.
\label{eq12}
\end{equation}
where the ratio on the right hand side accounts for the flux 
through the sample.  In the case of external reflection, and for
values of $k_{z,0}$ that are smaller
than $k_c$, the real part of $k_{z,s}$ is zero, and there is
no transmission, whereas above the critical angle $k_{z,s}$ is real,
and the transmission is given by,
\begin{equation}
T(k_{z,s}) = { {4k_{z,0}k_{z,s}} \over {(k_{z,0} + k_{z,s})^2} } \mbox{  for } k_{z,0} > k_c
\label{Transmission}
\end{equation}
and the conservation of photons is fulfilled in the scattering process,
\begin{equation}
T(k_{z,s}) + R(k_{z,s}) = 1.
\label{eq14}
\end{equation}
In Fig.\ \ref{fig2}(b) the transmission amplitude
$|t(k_{z,s})|$ for external (solid line) and for internal (dashed line)
reflections are shown.  This amplitude modulates non-specular scattering
processes at the interface as will be discussed later in this unit.

The effect of absorption on the reflectivity can be incorporated by
introducing $\beta$ into the generalized potential in Eq.\ \ref{eq6}, so that
$k_{z,s}=\sqrt{k_{z,0}^2-k_c^2+2i\beta}$ is used in the Fresnel equations
Eq.\ \ref{t1}.
Calculated reflectivities from water and liquid mercury
demonstrating that the effect of
absorption is practically insignificant for the former yet has
the strongest influence near the critical angle for the latter, 
are shown in Fig.\ \ref{fig3}.

\subsubsection{Multiple Stepwise and Continuous Interfaces}
On average, the electron density  across a liquid interface
is a continuously varying function, and is a constant
far away on both sides of the interface, as is shown in Fig.\ \ref{fig4}.

The reflectivity for a general function $\rho(z)$ can be then calculated
by one of several methods classified into two major categories:
{\it dynamical} and {\it kinematical}  solutions.
The dynamical solutions are in general more exact and include all
the features of the scattering, in particular the low angle regime, 
close to the critical angle where multiple scattering processes occur.
For a finite number of discrete interfaces exact solutions can be obtained
by use of standard recursive \cite{Parratt54} or matrix 
\cite{Born59} methods.  These methods can be
extended to compute with very high accuracy the scattering from any
continuous potential by {\it slicing} it into a finite but
with sufficient number of interfaces.  On the other hand, the kinematical
approach neglects multiple scattering effects and fails in describing 
the scattering at small angles.
\centerline{\it 2a. The Matrix Method}

In this approach the scattering length density with variation over a
characteristic length $d_t$ is {\it sliced} into a histogram
with  $N$ interfaces.
The matrix method,
is practically equivalent to the Parratt formalism \cite{Born59}.
For each interface, the procedure described previously
for the one interface is  applied.
Consider an arbitrary interface, $n$, separating two regions, of
a {\it sliced} SLD (as in Fig.\ \ref{fig4}), with
$\rho_{n-1}, \mbox{and } \rho_{n}$ at position $z=z_n$ with the following
wavefunctions
\begin{equation}
\nonumber
\begin{array}{ccc}
\qquad \rho_{n-1}     & \vline &  \qquad\rho_{n} \\
R_{n-1,n}{\rm e}^{-ik_{n-1}z}\qquad{\leftarrow}&  \vline &
{\leftarrow}\qquad R_{n,n+1}{\rm e}^{-ik_{n}z} \\
T_{n-1,n}{\rm e}^{ik_{n-1}z}\qquad \qquad{\rightarrow} & \vline &
{\rightarrow}\qquad T_{n,n+1}{\rm e}^{ik_{n}z}      \\
&\mbox{ }z = z_n &
\end{array}
\nonumber
\end{equation}
where $k_n \equiv \sqrt{k^2_{z,0}-4\pi\rho_n}$.  The effect of absorption 
can be taken into account as described earlier.
For simplicity, the subscript $z$  is omitted from the component of the
wavevector so that $k_{z,n} = k_n$.
The solution at each interface in terms of a transfer
matrix, $M_n$, is given by
\[ \left (\begin{array}{c} T_{n-1,n}\\ R_{n-1,n} \end{array} \right ) =
\left ( \begin{array}{ll} {\rm e}^{-i(k_{n-1}-k_{n})z_n} &
r_{n}{\rm e}^{-i(k_{n-1}+k_{n})z_n}  \\
r_{n}{\rm e}^{i(k_{n-1}+k_{n})z_n}
& \quad {\rm e}^{i(k_{n-1}-k_{n})z_n}  \end{array} \right )
 \left (\begin{array}{c} T_{n,n+1}\\ R_{n,n+1} \end{array} \right )\]
where $r_{n} = { {k_{n-1}-k_{n}} \over  {k_{n-1}+k_{n}}} $ is the
Fresnel reflection function through the $z_n$ interface separating the
$\rho_{n-1} \mbox{ and } \rho_n$ SLD's.
The solution to the
scattering problem is given by noting that beyond the last interface,
(i.e., in the bulk), there is a transmitted wave only
for which an arbitrary amplitude  of the form  $(^{1}_{0})$
can be assumed (i.e., the reflectivity is normalized to the incident beam anyway).
The effect of all interfaces is calculated as follows

\[
\left (\begin{array}{c} T_{0,1}\\ R_{0,1} \end{array} \right ) =
\left (\begin{array}{c} M_1 \\  \end{array} \right )
\left (\begin{array}{c} M_2 \\  \end{array} \right )
\quad ....
\left (\begin{array}{c} M_n \\  \end{array} \right )
\quad ....
\left (\begin{array}{c}M_{N+1} \\  \end{array} \right )
\left (\begin{array}{c} 1 \\ 0 \end{array} \right )
\]

with $r_{N+1} = { {k_{N}-k_{s}} \over  {k_N+k_{s}}} $ in the  $M_{N+1}$ matrix
given in terms of the substrate $k_s$.
The reflectivity is then given by the ratio
\begin{equation}
R(Q_z) = \left | { {R_{0,1}}\over{T_{0,1}} }\right |^2
\end{equation}

Applying this procedure to the one {\it box model}
of thickness $d$ with two interfaces yields
\begin{equation}
R(Q_z\equiv 2k_s) = \left | { {r_1 + r_2{\rm e}^{i2k_sd}}
\over {1 + r_1r_2{\rm e}^{i2k_sd}} } \right |^2
\end{equation}
Figure\ \ref{fig5} shows the calculated reflectivities from a flat liquid
interface with two kinds of films (one {\it box}) of the same
thickness $d$ but with
different scattering length densities, $\rho_1$ and $\rho_2$.  The reflectivities
are almost indistinguishable when the normalized SLD's ($\rho_i/\rho_s$) of the films are
complementary to one ($\rho_1/\rho_s + \rho_2/\rho_s = 1$), except for a very minute difference near
the first minimum.  In the  kinematical method described below,
the two potentials shown in Fig.\ \ref{fig5} yield identical
reflectivities.

The matrix method can be used to calculate the exact
solution from a finite number of interfaces, and it is most powerful
when used with computers by slicing any continuous scattering length
density into a histogram.
The criteria for determining the optimum  number of
slices to use is based on the convergence of the calculated reflectivity
at a point where slicing the SLD into more boxes does not change the
calculated reflectivity significantly.

\centerline{\it 2b. The Kinematical Approach}

The kinematical approach for calculating the reflectivity is only
applicable under certain conditions where multiple scattering is not
important.  It usually fails in calculating the reflectivity at very small
angles (or small momentum transfers) near the critical angle.
The kinematical approach, also known as the Born-Approximation,
gives physical insight in the formulation of $R(Q_z)$
by relating the Fresnel normalized reflectivity,
R/R$_F$, to the Fourier transform of {\it spatial changes} in
$\rho(z)$ across the interface\cite{Als-Nielsen89} as discussed below.

As in the dynamical approach, $\rho(z)$ is sliced
so that $k(z)=\sqrt{k_{z,0}^2-k_c^2}$ and the reflectance
across an arbitrary point $z$ is given by
\begin{equation}
 r(z) = { {k(z+\Delta z) - k(z)} \over {k(z+\Delta z) + k(z)}}\approx
 -{ {4\pi} \over {4k(z)^2}} { {{\rm d} \rho} \over {{\rm d}z}} {\rm d}z
 \approx (Q_c/2Q_z)^2 {1\over {\rho_s}}
 { {{\rm d} \rho} \over {{\rm d}z}} {\rm d}z
\end{equation}
In the last step of the derivation, $r(z)$ was multiplied and 
divided by $\rho_s$,
the SLD of the subphase, and the identity $Q^2_c\equiv 16\pi\rho_s$ was 
used. Assuming no multiple scattering, the reflectivity is calculated
by integrating over all reflectances at each point, $z$, 
with a phase factor ${\rm e}^{iQ_zz}$ as follows,
\begin{equation}
R(Q_z) =  R_F(Q_z) \left|\frac{1}
{\rho_s}{\int}(\frac{d\rho}{dz})e^{iQ_z}dz
\right|^2 =  R_F(Q_z)|\Phi(Q_z)|^2
\label{Born_approx}
\end{equation}
where $\Phi(Q_z)$ can be regarded as the {\it generalized
structure factor} of the
interface, analogous to the structure factor of a unit cell
in 3D crystals.
This formula can be also derived by using the  Born Approximation,
as is shown in the following section.

As an example of the use of Eq.\ \ref{Born_approx} we assume that the 
SLD at a liquid interface can be approximated by a sum of
error functions as follows
\begin{equation}
\rho(z) = \rho_{0} + \sum_{j=1}^N{ {(\rho_{j}-\rho_{j-1})} \over 2}
\left [1+{\rm erf}\left ({{z-z_j}\over {\sqrt{2}\sigma_j}}\right )\right ]
\label{erfc}
\end{equation}
where $\rho_0$ is the SLD of the vapor phase and $\rho_N=\rho_s$.
Using Eq.\ \ref{Born_approx} the  reflectivity
is given by
\begin{equation}
R(Q_z) = R_F (Q_z)\left |
\sum_{j}\left ({{\rho_j-\rho_{j-1}}\over {\rho_s}} \right )
\exp^{-\frac{\left (Q_z\sigma_j\right)^2}{2}} \exp^{iQ_zz_j} \right |^2.
\label{Ref_erfc}
\end{equation}
Assuming one interface at $z_1 = 0$ with surface roughness
$\sigma_1=\sigma$, the Fresnel reflectivity, $R_F(Q_z)$,
is simply modified by a Debye-Waller-like factor
\begin{equation}
R(Q_z) = R_F(Q_z) \exp^{-(Q_z\sigma)^2}.
\label{Rf+roughness}
\end{equation}
The effect of surface roughness on the reflectivities from water 
and from liquid mercury surfaces assuming gaussian smearing of the interfaces,
is shown in Figure\ \ref{fig3}.                      
Braslau et al.\cite{Braslau88} have demonstrated that the 
Gaussian smearing of the interface due to capillary waves in simple 
liquids is sufficient in modeling the data, and that more complicated
models cannot be supported by the X-ray data.

Applying Eq.\ \ref{Ref_erfc} to the one box model discussed above (See
Fig.\ \ref{fig5}), and assuming conformal roughness, $\sigma_j=\sigma$, 
the calculated reflectivity in terms of SLD normalized to $\rho_s$ 
is
\begin{equation}
R(Q_z) = R_F(Q_z) \left [ (1-\rho_1)^2 +\rho_1^2 + 2\rho_1(1-\rho_1)\cos(Q_zd)
\right ] \mbox{e}^{(-Q_z\sigma)^2}.
\end{equation}
In this approximation the roles of the normalized SLD of the one box,
$\rho_1$, and that for the {\it complementary} model $\rho_2 = 1-\rho_1$
are equivalent and demonstrates that the reflectivities 
for both models are {\it mathematically} identical.
This is  the simplest of many examples where two or more
distinct SLD models yield identical reflectivities in the Born Approximation.  When using
the kinematical approximation to invert the reflectivity
to SLD there is always a problem of facing a non-unique result. 
Ways to distinguish between such models is discussed in the Data
Analysis section.

In some instances, the scattering length density
can be generated by several step functions that are smeared
with one gaussian ({\it conformal roughness $\sigma_j=\sigma$}),
representing different moieties of the molecules on the surface.
The reflectivity can be calculated
by using a combination of the {\it dynamical} and the {\it kinematical}
approaches\cite{Als-Nielsen89}.
First, the exact reflectivity from the step-like functions ($\sigma=0$) 
is calculated using the matrix method, $R_{dyn}(Q_z)$,
and the effect of surface roughness is  incorporated by
multiplying the calculated reflectivity with a Debye-Waller-like
factor as follows\cite{Als-Nielsen89}
\begin{equation}
R(Q_z) = R_{dyn}(Q_z) \exp^{-(Q_z\sigma)^2}.
\label{R+roughness}
\end{equation}

\subsection{Non-specular scattering}
The geometry for non-specular reflection is shown in Figure\ \ref{fig1}(b).
The scattering from a 2D system is very weak and enhancements
due to multiple scattering processes at the interface is taken advantage of.
As is shown in Figure\ \ref{fig1}(b) the  momentum transfer
$\bf Q $   has a finite component parallel to the liquid surface
($\bbox{Q}_\perp \equiv \bbox{k}_\perp^i-\bbox{k}_\perp^f$)
enabling determination of lateral correlations in the 2D plane.  Exact
calculation of scattering from surfaces is practically impossible
except for special cases, and the Born Approximation (BA)\cite{Schiff68}
is usually applied.  When the incident beam or the scattered beam
are at grazing angles (i.e., near the critical angle),  
multiple scattering effects modify the scattering
and these can be accounted for by a higher order approximation known as
the Distorted Wave Born Approximation (DWBA).
The features due to multiple scattering at grazing angles
provide evidence that the scattering processes indeed occur
at the interface.
\subsubsection{The Born Approximation}
In the BA for a general potential $V(\bbox{r})$ the
scattering length amplitude is calculated as follows\cite{Schiff68},
\begin{equation}
F(\bbox{Q}) =
\frac{1}{4\pi}\int
V(\bbox{r}){\rm e}^{i\bbox{Q\cdot r}}\mbox{d}^3\bbox{r}.
\label{BA}
\end{equation}
where in the present case, $V(\bbox{r}) = 4\pi\rho(\bbox{r})$.
From the scattering length
amplitude  the differential cross section is  calculated
as follows\cite{Schiff68}
\begin{eqnarray}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega}& = & |F(\bbox{Q})|^2 \nonumber\\
& =& {\int}
\left [ \rho(\bbox{0})\rho(\bbox{r}) \right ]
{\rm e}^{i\bbox{Q\cdot r}}\mbox{d}^3\bbox{r}.
\label{BA1}
\end{eqnarray}
where $\left [ \rho(\bbox{0})\rho(\bbox{r}) \right ] \equiv
\int[\rho(\bbox{r}^\prime-\bbox{r})\rho(\bbox{r}^\prime)]
\mbox{d}^3\bbox{r}^\prime$ is the density-density
correlation function.
The measured reflectivity is a convolution of the differential cross-section
with the instrumental resolution,
as discussed below and in the literature\cite{Schiff68,Braslau88,Sinha88}.

The scattering length density, $\rho$, for a liquid-gas interface can be
described as a function of the actual height of the surface,
$z(x,y)$, as follows,
\begin{equation}
\rho(\bbox{\mu},z) = \left\{ \begin{array}{ll}
        \rho_s & \mbox{ for  $z < z(\bbox{\mu})$}  \\
         0  & \mbox{ for $z > z(\bbox{\mu})$}
\end{array}
\right.
\label{z(x,y)}
\end{equation}
where $\bbox{\mu}=(x,y,0)$ is a 2D in-plane vector.
The height of the interface $z$ is also time dependent and temperature
dependent due to capillary waves, and therefore thermal averages of $z$
are used \cite{Buff65,Evans79}.
Inserting the SLD Eq.\ \ref{z(x,y)}, in Eq.\ \ref{BA1} and performing
the integration over the $z$ coordinate yields
\begin{equation}
F(\bbox{Q_\perp},Q_z) =
{\frac{\rho_s}{iQ_z}}
\int{\rm e}^{i[\bbox{Q_\perp\cdot\mu}+Q_zz(\bbox{\mu})]}d^2\bbox{\mu}
\label{BA2}
\end{equation}
where $\bbox{Q}_\perp=(Q_x,Q_y,0)$ is an in-plane scattering vector.
This formula properly predicts the reflectivity from
an ideally flat surface, $z(x,y)=0$ within the kinematical approximation
\begin{equation}
F(\bbox{Q_\perp},Q_z) =
\frac{4\pi^2\rho_s}{iQ_z}\delta^2(\bbox{Q}_\perp)
\end{equation}
with a 2D $\delta$ function that guarantees specular reflectivity only.
The differential cross-section is then given by
\begin{equation}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega} =
 \pi^2\left  ({\frac{Q_c^2}{4Q_z}}\right )^2
\end{equation}
where $Q_c^2 \equiv 16\pi\rho_s$.
This is the general form for the Fresnel reflectivity in terms of the
differential cross-section ${d\sigma}/{d\Omega}$, which is
defined in terms of the flux of the incident beam on the surface.
In reflectivity measurements however, the scattered intensity is
normalized to the intensity of the incident beam and therefore
the flux on the sample is angle dependent and is proportional
to $\sin{\alpha_i}$.
In addition the scattered intensity is integrated over the polar
angles $\alpha_f \mbox{ and } 2\theta$
with $k_0^2\sin\alpha_i\mbox{d}\alpha_i\mbox{d}(2\theta)
= \mbox{d}Q_x\mbox{d}Q_y$,
correcting for the flux and integrating
\begin{equation}
R_F(Q_z) \approx \sigma_{tot}(Q_z) = \int\int \left (\frac{\mbox{d}\sigma}
{\mbox{d}\Omega} \right)
\frac{\mbox{d}Q_x \mbox{d}Q_y}{4\pi^2k_0^2\sin\alpha_i\sin\alpha_f}
= \left (\frac{Q_c}{2Q_z}\right)^4
\end{equation}
as approximated from the exact solution, given in Eq.\ \ref{Fresnel_approx}.

Taking advantage of the geometrical considerations above, the
differential cross-section to the reflectivity measurement can be readily
derived in the more general case of scattering length density that varies 
along $z$ only, (i.e., $\rho(z))$.
In this case, Eqs.\ \ref{BA} can be written as 
\begin{equation}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega}
= {4\pi^2}\delta^2(\bbox{Q}_\perp)
\left |{\int}\rho(z)\mbox{e}^{iQ_zz}\mbox{d}z \right |^2
\label{BA3}
\end{equation}
If we normalize  $\rho(z)$ to the scattering length density
of the substrate, $\rho_s$, and use a standard identity
between the Fourier transform of a function and its derivative, we obtain,
\begin{equation}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega} =
 \pi^2 \left  (\frac{ Q_c^2}{4Q_z} \right )^2\left |
 \int \frac{1}{\rho_s} \frac{\mbox{d}\rho(z)}{\mbox{d}z}
 \mbox{e}^{iQ_zz}\mbox{d}z\right |^2
\end{equation}
which with the geometrical corrections yields 
Eq.\ \ref{Born_approx}.

Thermal averages of the scattering length density under the influence of
capillary waves and the assumption that
the SLD of the gas phase is zero can be approximated as follows
\cite{Buff65,Evans79},
\begin{equation}
[\rho(\bbox{0})\rho(\bbox{r})] \sim \frac{\rho_s}{2} \left
(1 + {\rm erf}[\frac{z}{\sqrt{2}\sigma(\bbox{\mu})}] \right )
\label{sig1}
\end{equation}
where $\sigma(\bbox{\mu})$ is the height-height correlation function.

Inserting Eq.\ \ref{sig1} into Eq.\ \ref{BA1} and integrating over $z$, 
results in the differential cross-section
\begin{equation}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega}
=\frac{2\pi}{Q_z^2}
{\int} \mbox{e}^{i\bbox{Q_\perp\cdot\mu}-Q_z^2\sigma^2(\bbox{\mu})}
\mbox{d}^2\bbox{\mu}.
\end{equation}
and assuming isotropic correlation function in the plane yields
\cite{Sinha88}
\begin{equation}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega}
 = \frac{2\pi}{Q_z^2}{\int} \mu \mbox{J}_0
(Q_\perp\mu)\mbox{e}^{-Q_z^2\sigma^2({\mu})}\mbox{d}{\mu}
\end{equation}
where J$_{0}$ is  a Bessel function of the first kind.  This
expression was used by Sinha et al.
to calculate the diffuse scattering
from rough liquid surfaces with a height-height density correlation
function that diverges logarithmically due to capillary waves
\cite{Sinha88,Sanyal91}.
\subsubsection{Distorted Wave Born-Approximation (DWBA)}
Due to the weak interaction of the electromagnetic field (X-rays) with
matter (electrons), the BA is a sufficient approach
to the scattering from most surfaces.  However, as we have already
encountered with the reflectivity, the BA fails (or is invalid) when
either the incident beam or the scattered beam is near the critical angle,
where multiple scattering processes take place.
The effect of the bulk on the scattering from the surface can be
accounted for by defining the scattering length density
as a superposition of two parts as follows
\begin{equation}
\rho({\bbox r}) = \rho_1(z) +\rho_2({\bbox \mu},z).
\end{equation}
Here $\rho_1(z)$ is a step function that defines
an  ideally sharp interface separating the liquid and gas phases
at $z=0$,
whereas the second term, $\rho_2({\bbox \mu},z)$, is a quasi two-dimensional
function in the sense that it has  a characteristic average
{\it thickness} $d_c$ such that $\lim_{z \rightarrow \pm d_c/2}
\rho_2({\bbox \mu},z)= 0$.
It can be thought of as {\it film}-like
and is a detailed function with features that relate to
molecular or atomic distributions at the interface.
Although the definition of $\rho_2$ may depend on the location of
the interface, ($z=0$) in $\rho_1$,
the resulting calculated scattering must be invariant
for equivalent descriptions of $\rho({\bbox r})$.
In some cases $\rho_2$ can be defined as either a totally external
or totally internal function with respect to the liquid bulk, (i.e.,
$\rho_1$).
In other cases, especially when dealing with liquid surfaces, 
it is more convenient to locate the interface at some intermediate
point coinciding with the center of {\it mass} of
$\rho_2$ with respect to $z$.
The effect of the substrate term $\rho_1(z)$ on the scattering from $\rho_2$
can be treated within the
distorted wave Born approximation (DWBA) by using the exact solution from the
ideally flat interface (Sec. II) to generate the Green function
for a higher order
Born approximation\cite{Schiff68,Rodberg67,Vineyard82,Sinha88}.
The  Green function in the presence of an ideally
flat interface, $\rho_1$,
replaces the {\it free} particle Green function that is commonly used
in the Born approximation.
The scattering amplitude in this case is given by\cite{Schiff68,Rodberg67}
\begin{equation}
F_{DWBA}(\bbox{Q}) = F_F(Q_z) + F_2(\bbox{Q}) =
-i\pi Q_zr_F(Q_z)+ \int \tilde{\chi}^*_{\bbox{k}\prime}(\bbox{r})
\rho_2(\bbox{r}) {\chi}_{\bbox{k}}(\bbox{r}) d\bbox{r}
\label{DWBA}
\end{equation}
where the exact Fresnel amplitude $F_F(Q_z)$ is written in the
form of a scattering amplitude so that Eq.\ \ref{DWBA}
reproduces the Fresnel reflectivity in the absence of $\rho_2$.

The exact solution of the step function $\rho_1$,  ${\chi}_{k}(\bbox{r})$,
is given by
\begin{equation}
{\chi}_{k}(\bbox{r}) = {\rm e}^{i\bbox{k^i_\perp\cdot\mu}}
\left \{
\begin{array}{ll}
\mbox{e}^{ik^i_{z,0}z} + r^i(k^i_{z,s})\mbox{e}^{-ik^i_{z,s}z}
& \mbox{for $z>0$}\\
t^i(k^i_{z,s})\mbox{e}^{ik^i_{z,s}z}& \mbox{for $z>0$}
\end{array}
\right.
\end{equation}
 and the $\tilde{\chi}^*_{k}(\bbox{r}) $
is the time reversed and complex conjugate solution of
an incident beam with $\bbox-\bbox{k_i}$,
\begin{equation}
\tilde{\chi}^*_{k}(\bbox{r}) = {\rm e}^{-i\bbox{k^f_\perp\cdot\mu}}
\left \{
\begin{array}{ll}
t^f(k^f_{z,0})\mbox{e}^{-ik^f_{z,0}z}
& \mbox{for $z>0$}\\
\mbox{e}^{-ik^f_{z,s}z} + r^f(k^f_{z,0})\mbox{e}^{ik^f_{z,0}z}
& \mbox{for $z< 0$}.
\end{array}
\right.
\label{DWBA1}
\end{equation}
In the absence of $\rho_1$, Eq.\ \ref{DWBA}
transforms into the standard Born Approximation for $\rho_2$ 
and this reproduces  Eq.\ \ref{BA}.
The notation for transmission and reflection functions
indicate scattering of the wave from the air onto the subphase
and {\it vice-versa} according to
$k^i_{z,s} = \sqrt{(k^i_{z,0})^2 - k_c^2}$, and
$k^f_{z,0} = \sqrt{(k^f_{z,s})^2 + k_c^2}$ respectively.
In the latter case, total reflectivity does not occur except
for the trivial case $k^f_{z,s}=0$,
and no enhancement due to the evanescent wave is expected.
In this approximation, 
the final momentum transfer in the $Q_z$ direction is a superposition of
momentum transfers from $\rho_2$ (the {\it film}) and 
from the liquid interface, $\rho_1$.
For instance, there could be a wave scattered with $Q_z=0$ with respect
to $\rho_2$ but reflected from the surface with a finite $Q_z$.
Assuming that the scattering from the {\it film} is as strong for
$Q_z$ as for $-Q_z$ (as is the case for an ideal  2D system with
equal scattering along the rod, i.e.,  $\rho_2$ is symmetrical under the
inversion of $z$), we can write $\tilde\rho_2(\bbox{Q}_\perp,Q_z)
\approx \tilde\rho_2(\bbox{Q}_\perp,-Q_z)$. 
It can be shown that the cross-section can be approximated
as follows\cite{Vineyard82,Feidenhas'l89,Sinha88}
\begin{equation}
\frac{\mbox{d}\sigma}{\mbox{d}\Omega} \approx \left \{
\begin{array}{lll}
& \left |
t^i(k^i_{z,s})
\tilde\rho_2(\bbox{Q}_\perp,Q^\prime_z)
t^f(k^f_{z,s})
\right | ^2 & \mbox{  for } z >0 \\
& \left |
t^i(k^i_{z,s})
\tilde\rho_2(\bbox{Q}_\perp,Q^{\prime\prime}_z)
t^f(k^f_{z,0})
\right | ^2 & \mbox{  for }z  < 0
\end{array}
\right. 
\label{DWBA2}
\end{equation}
where $\bbox{Q}_\perp \equiv \bbox{k}_\perp^i-\bbox{k}_\perp^f$
and $Q_z^\prime = k^i_{z,0}- k^f_{z,s}$ and
$Q_z^{\prime\prime} = k^i_{z,s}- k^f_{z,0}$.
Notice that the transmission functions modulate the
scattering from the {\it film} ($\rho_2$), and in particular they
give rise to enhancements as $k^i_{z,s}$  and $k^f_{z,s}$ are scanned
around the critical angle as depicted  in Figure\ \ref{fig2}(b).
Also, it is only by  virtue of
the $z$ symmetry of the scatterer that such enhancements occur for an
exterior film.  From this analysis we notice that there will be
no enhancement due to the transmission function of the final
wave for an interior {\it film}.

To examine the results from the DWBA method,
we consider scattering from a single scatterer
near the surface.  The discussion is restricted to that where the detection
of the scattered beam is performed in the vapor phase only.
The scatterer can be placed either in the vacuum
($z > 0 $) or in the liquid (see Fig.\ \ref{fig6}).
When the particle is placed in the vacuum there are
two major relevant incident waves: a direct one from the source,
labeled $1^i$ in Figure\ \ref{fig6}(a); and a second one
reflected from the surface before scattering from $\rho_2$, labeled $2^i$.
Assuming inversion symmetry
along $z$, both waves scatter into a finite
$\bbox{Q}_\perp$ with similar strengths at $Q_z$ and $-Q_z$, giving rise
to an enhancement near the critical angle if the incident beam
is near the critical angle.  Another multiple
scattering process that gives rise to enhancement at the critical angle
is one in which beam $1^i$ does not undergo a change in
the momentum transfer along ($z \quad Q^{film}_z \approx 0$)
before scattering from the liquid interface.
The effect of these processes gives rise to
enhancements if either the incident beam or the reflected beam are scanned
along the $z$ direction.  Slight modifications of the momentum transfer
along the $z$ direction (such as
$Q^\prime_z = k_z^i + \sqrt{(k_z^f)^2 - k_c^2}$) are neglected in the
discussion above).
The effective amplitude from the scatterer outside the
medium is given by the following terms
\begin{equation}
[{\rm e}^{iQ_zz} + {\rm e}^{iQ^\prime_zz} r(k^f_{z,s}) ]
\approx t(k^i_{z,s})                                               
\end{equation}
where the approximation is valid since at small momentum transfers
the phase factor can be neglected, and $1 + r(k_{z,s}) = t(k_{z,s})$.
At large angles the reflectivity is negligible and the transmission
function approaches $t(k_{z,s}) \approx 1$.
Similar arguments hold for the outgoing wave.
Neglecting the small changes in the momentum transfer due to
dynamical effects, the transmission function modulates the scattering
as is shown in solid line in Fig.\ \ref{fig2}(b).

The scattered wave from a particle that is embedded in the medium is
different due to the asymmetry between external and internal reflection from
the liquid subphase.
The wave scattered from the particle re-scatters from the liquid-gas
interface.   Upon traversing the liquid interface the index of refraction
increases from $n=1-\delta$ ($\delta = (2\pi/k_0^2)\rho$ to $1$ and  no total internal
reflection occurs as discussed earlier; thus there is no evanescent 
wave in the medium.
The transmission function for this wave is given by
${t}(-k^i_{z,s})$, like that of a wave emanating from the liquid
interface into the vapor phase.  In this case, the transmission function
is a real function for all $k^i_{z,s}$ and does not have the
enhancements around the critical angle
(as shown in Figure\ \ref{fig6}(b)), with zero intensity at the horizon.
\subsubsection{Grazing Incidence Diffraction (GID), and Rod Scans}
In some instances ordering of molecules at liquid interfaces occurs.
Langmuir monolayers spread at the  gas-water
interface usually order homogeneously at high enough lateral pressures. 
\cite{Kjaer87,Dutta87,Kjaer94}.
Surface crystallization of n-alkane molecules on molten alkane
has been observed recently\cite{Wu93a,Ocko97}.
In these cases, $\rho_2$ is a periodic function in  $x$ and $y$, 
and  can be expanded as a Fourier series in terms
of the 2D reciprocal lattice vectors ${\bbox\tau}_{\perp}$ as follows
\begin{equation}
\rho_{2}({\bbox \mu},z) = \sum_{\bbox \tau_{\perp}}
F({\bbox \tau_{\perp}},z)
{\rm e}^{i{\bbox \tau}_{\perp}{\bbox\cdot \mu} }
\label{2D_density}
\end{equation}
Inserting Eq.\ \ref{2D_density} in Eq.\ \ref{DWBA2} and integrating yields 
 the cross-section for quasi-2D Bragg reflection at
${\bbox Q}_\perp = \bbox{\tau}_\perp$
\begin{equation}
{\frac{d\sigma}{d\Omega}} \sim
 P(\bbox{Q})\left |t(k^i_{z,s}) \right  |^2
\left < \left | F(\bbox{\tau}_\perp,Q_z) \right |^2 \right >
DW(Q_\perp,Q_z) \left |{t}^f(k^f_{z,s})\right |^2
\delta({\bbox{Q} - \bbox{\tau}_\perp})
\label{2D-Bragg}
\end{equation}
where $P(\bbox{Q})$ is a polarization correction, and
the 2D unit cell structure factor is given as a sum over
the atomic form factors $f_j(Q)$ with appropriate phase
\begin{equation}
F(\tau_\perp,Q_z) = \sum_{j} f_j(Q){\rm e}^{i{\bbox \tau\cdot r_j}+Q_zz_j}.
\end{equation}
The structure factor squared is averaged for multiplicity due to domains
and weighted for orientation relative to the surface normal.
The ordering of monolayers at the air-water interface is usually in the form
of 2D {\it powder} consisting of crystals with random orientation
in the plane.
From Eq.\ \ref{2D-Bragg}
we notice that the conservation of momentum expressed with 
the $\delta$ function allows for observation of the Bragg reflection at
any $Q_z$.  A rod scan can be performed by varying either the incident 
or reflected beam, or both.  The variation of each will produce some
modulation due to both the transmission functions and to the
average molecular structure factor along the $z$-axis.
The Debye-Waller factor , $DW(\bbox{Q}_\perp,Q_z)$,
which is due to the vibration of molecules  about their own equilibrium
position with time dependent molecular
displacement  $\bbox{u}(t)$ is given by
\begin{equation}
DW(\bbox{Q}_\perp,Q_z) \sim {\rm e}^
{-(C_\perp\bbox{Q}^2_\perp<u_\perp^2>+Q_z^2\sigma^2)}
\end{equation}                                     
The term due to capillary waves on the liquid surface
is much more dominant than the contribution from the in-plane
intrinsic fluctuations.  The Debye-Waller factor in this case is an average over a
crystalline size and might not reflect  
surface roughness extracted from reflectivity measurements, where it is averaged over 
the whole sample. 


\section{Experimental Considerations and Data Analysis}
The minute sizes of interfacial samples on the sub-microgram level
combined with the weak interaction of X-rays with matter
result in very weak GID and reflectivity (at large $Q_z$) 
signals that require highly intense incident beams, which are  
available at X-ray synchrotron sources.
A well prepared incident beam for reflectivity experiments
at a synchrotron (for example, the X22B beam-line
at the National Synchrotron Light Source at
Brookhaven National Laboratory) has an intensity of
$10^9 - 10^{10}$ photons/second, whereas for a similar resolution,
an 18 kW rotating anode generator produces $10^4 - 10^5$ photons/second.
Although reflectivity measurements can be carried out with standard
X-ray generators, the measurements are limited to
almost half the angular range accessible at synchrotron sources and
they take hours to complete compared to minutes at the synchrotron.
GID experiments are practically impossible with X-ray generators
since the expected signals (2D-Bragg reflections, for example)
normalized to the incident beam are on the order of $10^{-8}-10^{-10}$.

\subsection{Reflectivity}
X-ray reflectivity and GID measurements of liquid surfaces
are carried out on special
reflectometers that enable manipulation of the incident
as well as the outgoing beam. A prototype liquid surface
reflectometer was introduced for the first time
by Als-Nielsen and Pershan\cite{Als-Nielsen83}.
In order to bring the beam to an angle of incidence $\alpha_i$
with respect to the liquid surface,
the monochromator is tilted by an angle  $\chi$ either about
the axis of the incident beam (indicated by $\chi_1$ in Fig.\ \ref{fig7})
or about the axis normal
to the reciprocal lattice  wave vector of the monochromator,$\bbox{\tau_0}$
($\chi_2$).
Figure\ \ref{fig7} shows the geometry that is used to deflect the beam
from the horizon onto the liquid surface at an angle $\alpha_i$
by tilting the monochromator.
At the Bragg condition,  the surface of the monochromator crystal,
is at an angle $\psi$ with respect to the incoming beam.
Tilting over the incident beam axis is like tracing the Bragg reflection
on the Debye-Scherer cone so that the $\psi$ axis remains fixed,
with a constant wavelength at different tilting angles.
The rotated reciprocal lattice vector and the final
wave-vector in this frame are given by
\begin{eqnarray}
\bbox{\tau_0} &=& \tau_0(-\sin\psi,\cos\psi\cos\chi_1,\cos\psi\sin\chi_1)
\nonumber \\
\bbox{k_f}& =& k_0(\cos\alpha_i\cos\phi,\cos\alpha_i\sin\phi,\sin\alpha_i)
\label{tilting}
\end{eqnarray}
where $\phi$ is the horizontal scattering angle.  
The Bragg conditions for scattering are given by,
\begin{eqnarray}
\bbox{k_i}+\bbox{\tau_0}& =& \bbox{k_f}  ; \qquad
| \bbox{k_f}| = k_0
\label{Bragg1}
\end{eqnarray}
Using Eqs.\ \ref{tilting} and \ref{Bragg1} the following relations for the 
monochromator axes are obtained,
\begin{eqnarray}
\sin\psi &=& \frac{\tau_0}{2 k_0}  \nonumber \\
\sin\chi_1 &=& \frac{k_0}{\tau_0}\cos\psi\sin\alpha_i  \\
\cos\phi & =& \left (1 - \frac{\tau_0^2}{2k_0^2} \right )/\cos\alpha_i
\nonumber
\end{eqnarray}
  And we notice that
the monochromator angle $\psi$ is independent of $\alpha_i$.  However,
the scattering angle $\phi$ has to be modified as $\alpha_i$ is varied.
This means that the whole reflectometer arm has to be rotated.
Similarly, for the configuration where the monochromator is tilted
over the axis normal to $\bbox{\tau_0}$ we get
\begin{eqnarray}
\sin\psi & = &{{\tau_0}\over{2 k_0 \cos\chi_2}} \nonumber \\
\sin\chi_2 &=& \frac{k_0}{\tau_0}\sin\alpha_i\\
\label{Mono2}
\cos\phi &= &\left (1 - \frac{\tau_0^2}{2k_0^2} \right )/\cos\alpha_i
\nonumber.
\end{eqnarray}
From these relations the conditions for a constant wavelength operation
for any angle of incidence, $\alpha_i$, can be calculated and applied
to the reflectometer.
Here, unlike the previous mode, deflection of the beam to different
angles of incidence requires both, the adjustment of $\psi$ as well as $\phi$
in order to maintain a constant wavelength.
If $\psi$ is not corrected in  this mode of
operation, the wavelength varies as $\chi$ is varied.
This mode is sometimes desirable, especially when the
incident beam hitting the monochromator consists of a
continuous distribution of wavelengths around the wavelength at horizontal
scattering, $\chi_2=0$.
Such continuous wavelength distribution exists when operating
with X-ray tubes,
or when the tilting monochromator is facing the {\it white} beam of
a synchrotron.  Although, the variation in the wavelength is negligible
as $\chi_2$ is varied,
without the correction of $\psi$,
the exact wavelength and the momentum transfer can be
computed using the relations in Eq.\ \ref{Mono2}.
In both modes of monochromator tilting, the surface height as well
as the height of the slits are adjusted with vertical translations.

The exact angle of the monochromatic
incident beam on the surface is determined by at least two {\it horizontal}
slits located between the sample and the
source.   One of these slits is usually located as close as possible to the
sample,  and the other as close as possible to the source.
These two slits determine the resolution of the incident beam.
By deflecting the beam from the horizon the shape of the
beam changes and that may change incident beam intensity going through
the slits,
and the use of a monitor right after the slit in front of the sample
is essential for the absolute determination of the reflectivity.
The size of the two slits defining the incident beam, is chosen
in such a way that the foot-print of the beam is much smaller than the
width of the reflecting surface so that total reflectivity occurs.
Figure\ \ref{fig8} shows the reflected beam and the direct beam from
a flat surface of water demonstrating total reflectivity at $Q_z=0.85Q_c$.
In this experiment the detector slit is  wide open at about ten
times the opening of the sample slit.
As is demonstrated, the effect of absorption is negligible for water,
and roughness is significantly reduced  by damping  surface waves.
The damping can be achieved by reducing the height of the water film to
about $\approx 0.3mm$  and placing a passive as well as an
active anti-vibration unit underneath the liquid sample holder, suppressing
mechanical vibrations\cite{Kjaer94}.

\subsection{Non-specular Scattering - GID, Diffuse Scattering
and Rod Scans}
X-ray GID measurements are performed at angles of
incidence below the critical angle $\approx 0.9\alpha_c$.
Operating with the incident beam below the critical angle
enhances the signal from the surface with respect to that of the bulk,
by creating an  evanescent wave in the medium that is exponentially
decaying as,
\begin{equation}
E(z) =  t(k_{z,s})\mbox{e}^{-z/\Lambda} \nonumber \\
\end{equation}
where
\begin{equation}
\frac{1}{\Lambda} = \sqrt{k_c^2-k^2_{z,0}}.
\end{equation}
For water at $k_z\approx 0.9k_c, \quad \Lambda \sim 1000 $ {\AA}.

As illustrated in Figure\ \ref{fig1}(b), the components of the
momentum transfer for GID are given by,
\begin{eqnarray}
Q_z & = & k_0(\sin\alpha_i + \sin\alpha_f) \nonumber  \\
Q_x & = & k_0(\cos\alpha_i-\cos\alpha_f)\cos 2\theta \nonumber \\
Q_y & = & k_0\cos\alpha_f\sin 2\theta
\label{Q_comp}
\end{eqnarray}
In most cases, the 2D order on liquid surfaces is powder like,
and the lateral scans are displayed in terms of
${Q_\perp}$  which is given by,
\begin{equation}
Q_\perp  =  k_0\sqrt{\cos^2\alpha_i+\cos^2\alpha_f - 2\cos\alpha_i
\cos\alpha_f\cos2\theta}
\label{Qperp}
\end{equation}
To determine the in-plane correlations the horizontal resolution of the
diffractometer can be adjusted with a soller collimator that consists
of vertical absorbing foils stacked together between the surface and the
detector.  The area that is probed at each scattering angle $2\theta$ is
proportional to $S_0/\sin 2\theta$, where $S_0$ is the area probed at
$2\theta = \pi/2$.  The probed area must be taken into account in
the analysis of a GID scan that is performed over a wide range of angles. 

Position sensitive detectors (PSD) are commonly used to measure the
intensity along the 2D rods.  It should be kept in mind that the intensity
along the PSD is not a true rod scan of a Bragg reflection at
a nominal $Q_\perp$ because of the variation in $Q_\perp$ as $\alpha_f$
is varied as is seen in Eq.\ \ref{Q_comp}.
\subsection{Data Analysis}
The task of finding the SLD from a reflectivity curve is similar to
that of finding an effective potential for Eq.\ \ref{WaveEq1} from
the modulus of the wave-function.
Direct inversion of the scattering
amplitude to SLD is not possible except for special cases when the
BA is valid\cite{Sacks93}.
If the modulus and the phase are known they can be converted by the method of
Gelfand-Levitan-Marchenko\cite{Sacks93} to SLD (GLM method).  
However, in reflectivity
experiments the intensity of the scattered beam alone is measured, and
phase information is lost.

Direct reconstruction of step-like potentials have been developed
recently by retrieving the phase from the
modulus i.e., reflectivity and then using the GLM method
\cite{Sacks93,Clinton93}.
Model-independent methods which are based on optimization
of a model to reflectivity, without the requirement of any
knowledge of the chemical composition of the SLD at the interface 
were also developed recently\cite{Pedersen92,Zhou95}.  Such
models incorporate a certain degree of objectivity.
These methods are based on the kinematical and the dynamical
approaches for calculating the reflectivity.
One method\cite{Pedersen92} uses indirect Fourier transformation
to calculate the correlation function of  $\mbox{d}\rho/\mbox{d}z$ which
is subsequently used in a square-root deconvolution model
to construct the SLD model.
Zhou and Chen, on the other hand,
developed a {\it groove tracking method} that is based on
an optimization algorithm  to reconstruct the SLD using the {\it dynamical}
approach to calculate the reflectivity at each step\cite{Zhou95}.

The most common procedure to extract
structural information from reflectivity is by using standard
non-linear least squares refinement of an initial SLD model.
The initial model is defined in terms of a P-dimensional set of independent
parameters, $\bf{p}$, using all the available information in guessing
of $\rho(z,\bf{p})$.  The parameters are then refined by calculating the
reflectivity ($R[Q_z^i,{\bbox p}]$) with the tools described earlier
and by minimizing the $\chi^2(p)$ quantity,
\begin{equation}
\chi^2({\bbox p}) = { {1}\over{N-P}}\sum_{i=1} \left [
{ {R_{exp}(Q_z^i) -R(Q_z^i,{\bbox p})}\over {\epsilon(Q^i_z)} }\right ]^2.
\end{equation}
where $\epsilon(Q^i_z)$ is the uncertainty of the measured reflectivity,
$R_{exp}(Q_z^i)$ and $N$ is the number of measured points.
The criteria for a good fit can be found in \cite{Bevington68}.
Uncertainties of a certain  parameter can be obtained by
fixing it at various values and for each value refining the
rest of the parameters until $\chi^2$ is increased by a factor of at 
least $1/(N-P)$.

The direct methods and model-independent procedures
of reconstruction SLD do not guarantee
uniqueness of the potential i.e.,  there can be
multiple SLD profiles
that essentially yield the same reflectivity curve, as discussed
with regard to Figure\ \ref{fig5}, for example.
The uniqueness can be achieved  by  introducing  physical
constraints that are incorporated into the parameters of the model.
Volume, in-plane density of electrons etc., are among such
constraints that can be used, (applying such constraints is 
discussed briefly in the Examples 
section\cite{Vaknin91a,Vaknin91b,Gregory97}).
These constraints reduce the uncertainties and make
the relationship of the SLD to the actual molecular arrangement apparent.
In the {\it dynamical} approach no two potentials yield exactly the same
reflectivity although the differences between two models 
might be too small to be detected in an experiment.

An experimental method to solving such a problem  was
suggested by Sanyal et al.,  using anomalous X-ray reflectivity
methods.   Two reflectivity curves from the same sample are  measured
with two different X-ray energies, one below and one above an absorption
edge of the substrate atoms, thereby varying
the scattering length density of the substrate\cite{Sanyal92}.
Subsequently the two reflectivity curves  can be used to perform a direct
Fourier reconstruction\cite{Sanyal92} or by refinement methods 
to remove ambiguities.
This method is not efficient when dealing with liquids that consist
of light atoms because of the very low energy of the absorption edge 
with respect to standard X-ray energies.
Another way to overcome the problem of uniqueness
is by performing reflectivity experiments on similar
samples with X-rays and with neutrons.
In addition, the SLD, $\rho(z)$ across the interface can be changed
significantly, in neutron scattering experiments, by
chemical exchange of isotopes that change $\rho(z)$, but maintain
the same structure\cite{Vaknin91a}.
The reflectivities (X-ray as well as neutrons) can be fitted to
one structural model that is defined in terms of geometrical
parameters only, calculating the SLD's from
scattering lengths of the  constituents and the geometrical parameters
\cite{Vaknin91a,Vaknin91b}.
\section{Examples}
Since the pioneering work of Als-Nielsen and Pershan\cite{Als-Nielsen83},
X-ray reflectivity and GID became standard tools for the characterization
of liquid surfaces on the atomic length scales.  The techniques have been
exploited in studies of the physical properties of simple liquids
\cite{Braslau88,Sanyal91,Ocko94},
Langmuir monolayers
\cite{Kjaer87,Dutta87,Kjaer89,Kjaer94,Als-Nielsen89,Vaknin91a},
liquid metals\cite{Rice86,Magnussen95,Regan95},
surface crystallization\cite{Wu93a,Wu93b,Wu95,Ocko97},
liquid crystals\cite{Pershan87}, surface properties of
quantum liquids\cite{Lurio92},
protein recognition processes
at liquid surfaces\cite{Vaknin91c,Vaknin93,Loesche93}
and many other applications.
Here, only several examples are briefly described in order to demonstrate the
strengths and the limitations of the techniques.
In presenting the examples, there is no intention of giving a full
theoretical background of the systems.

\subsection{Simple Liquids}
The term simple liquid is usually used for mono-atomic systems governed by
Van der Waals type interactions such as, liquid argon.
Here, the term is extended to include all classical
dielectric (non-metallic) liquids such as water, solvents
(methanol,ethanol,chloroform etc.) and others.
One of the main issues regarding dielectric
liquids  is the
determination of the average density profile across the interface,
$N(z)$.
This density is the result of folding  the {\it intrinsic}
density $N_I(z)$ of the interface due molecular size, viscosity, and
compressibility of the fluid with
density fluctuations due to capillary waves, $\delta N_{CW}(z)$.
The continuous nature of the density
across the interface due to capillary waves was worked out by Buff,
Lovett and Stillinger (BLS)\cite{Buff65} assuming that $N_I(z)$ is an ideal
step like function.
The probability for the displacement is taken be to proportional to
the Boltzmann factor, $\mbox{e}^{-\beta U(z)}$, where $U$ is the 
free energy necessary to  disturb
the surface from equilibrium state (i.e., $z(x,y)=0$), and
$\beta = 1/k_BT$.
The free energy of an incompressible and non-viscous liquid surface
consists of two terms; a surface tension ($\gamma$) term,  that
is proportional to the changes in area from the ideally flat
surface and a gravitational term as follows,
\begin{eqnarray}
U& =  {\int} \left(\gamma\left [ \sqrt{1+ \left |\nabla z \right |^2}-1
\right ] +
\frac{1}{2}m_sg z^2\right )\mbox{d}^2\bbox{\mu} \nonumber             \\
& \approx  \frac{1}{2}{\int} \left( \gamma\left |\nabla z \right |^2  +
m_sg z^2\right )\mbox{d}^2\bbox{\mu}
\end{eqnarray}
where $m_s$ is the mass density of the liquid substrate.
By using standard Gaussian approximation methods Buff et al find,
that $U(z) \sim \frac{z^2}{2\sigma_0^2}$.
After convolution of the probability with a step like function, 
representing the intrinsic density of the liquid surface,
yields the following density function,
\begin{equation}
N(z) = N_s\mbox{erfc}\left (\frac{z}{\sqrt{2}\sigma} \right )
\end{equation}
with a form similar to the one given in Eq.\ \ref{sig1}.
The average surface roughness at temperature $T$, is then given by
\begin{equation}
\sigma_{CW}^2
= \frac{k_BT}{2\pi\gamma}
\mbox{ln} \left ( \frac{L}{a_0} \right )
\label{sigma_CW}
\end{equation}
where $a_0$ is a molecular diameter and $L$ is the size of the surface.
Notice the logarithmic divergence of the fluctuations as the size of the
surface increases, as expected of a 2D
system\cite{Landau80}.  This model was further refined by assuming that the
intrinsic profile has a finite width\cite{Evans79}.
In particular if the width due to the intrinsic profile is also
expressed by a Gaussian then, the effective surface roughness is
given by
\begin{equation}
\sigma^2_{eff} =\sigma_I^2+ \sigma_{CW}^2
\label{sig_eff}
\end{equation}
and the calculated reflectivity is similar to
Eq.\ \ref{Rf+roughness} for an interface that is smeared
like the error function.
\begin{equation}
R_{CW} = R_F(Q_z)\mbox{e}^{-\sigma_{eff}^2Q_z^2}
\label{Ref_rough}
\end{equation}

Fig.\ \ref{fig9} shows the reflectivity from pure water measured
at the synchrotron\cite{VakninUnpub} where it is shown that 
using Eq.\ \ref{Ref_rough} for fitting the reflectivity data
is satisfactory implying that the error function type of
density profile (BLS model) for the liquid interface is sufficient.
In the refinement procedure only one parameter, the
surface roughness $\sigma$, is varied ($\sigma =$ 2.54 {\AA}).
This small roughness value depends on the attenuation of
capillary waves by minimizing the depth of the water to about 0.3mm
by placing a flat glass under the water\cite{Kjaer94}.
The validity of gaussian approximation of $N(z)$ (BLS model) was examined by
various groups and for a variety of systems\cite{Braslau88,Sanyal91,Ocko94}.
Ocko et al. have measured the reflectivity of liquid alkanes, over
a wide range of temperatures verifying that the surface roughness 
is of the form given
in Eqs.\ \ref{sigma_CW}and \ref{sig_eff}\cite{Ocko94}.

Experimentally,  the  reflectivity signal at each $Q_z$
from a rough interface is
convoluted with the resolution of the spectrometer in different
directions.  The effect of the resolution along the $Q_z$ can be calculated
analytically or convoluted numerically by computation.
For simplicity,  we
consider that the resolution functions can be approximated
as a Gaussian with a width of $\Delta Q_z$
along the $Q_z \mbox{can be taken as e}^
{-Q^2_z/\Delta Q_z^2}$ with appropriate normalization factor\cite{Bouwman96}.
The resolution, $\Delta Q_z$, is $Q_z$ dependent as the
angles of incidence and scattering are varied\cite{Ocko94}.
However, if we assume that around a certain  $Q_z$ the resolution is
a constant and we measure $\sigma_{exp}$
the convolution of the {\it true} reflectivity with the resolution
function yields the following relation,
\begin{equation}
\frac{1}{\sigma^2_{exp}} \approx \frac{1}{\sigma_{eff}^2} + {\Delta Q_z^2}
\end{equation}
from which the {\it effective roughness} can be extracted as follows,
\begin{equation}
\sigma_{eff} \approx \frac{\sigma_{exp}} {\sqrt{1 - \sigma_{exp}^2\Delta Q_z^2}}
\end{equation}
Thus, if the resolution is infinitely {\it good} i.e., a $\Delta Q_z =0$
the measured and effective roughness are the same.
However, as the resolution
is relaxed, the measured roughness gets smaller than the {\it effective
roughness}.
The effect of the resolution on the determination of true
surface roughness was discussed rigorously by Braslau et al.\cite{Braslau88}.

Diffuse scattering from liquid surfaces is practically inevitable,
due to the presence of capillary waves.
Calculation of the scattering from disordered interfaces of
various characteristics were treated in\cite{Sinha88}.
In the Born approximation, true specular
scattering from liquid surfaces exist only by virtue of the finite
cutoff-length of the  mean-square height fluctuations.
In other words, the fluctuations due to capillary waves diverge
logarithmically and only due to the finite instrumental resolution
that true specular reflectivity is observed.
The theory for the diffuse scattering from
{\it fractal} surfaces and other rough surfaces was 
developed in\cite{Sinha88}.

\subsection{Langmuir Monolayers}
A Langmuir monolayer (LM) is a monomolecular amphiphilic film spread
at the air-water interface.
Each amphiphilic molecule consist of a polar head
group (hydrophilic moiety)
and a nonpolar tail typically
hydrocarbon (hydrophobic) chains \cite{Gaines66,Swalen87}. Typical examples
are fatty acids, lipids, alcohols and others.
The length of the hydrocarbon chain can be varied chemically,
affecting the hydrophobic character of the
molecule.  Whereas, the head-group
can be ionic, dipolar, or with a certain shape that might attract
specific compounds present in the aqueous solution.
One important motivation to studying LM's is their close relationship 
to biological systems.  Membranes of all living cells and 
organelles within cells consist of a lipid bilayers interpenetrated with
specific proteins,
alcohols, and other organic compounds that combine to give
functional macromolecules that determine transport
of matter and energy through them.  It is well known that biological
functions are structural, and structures can be determined by XR and GID.
In addition, delicate surface chemistry can be carried out at the
head-group interface with molecules from the aqueous solution.
From the physics point of view, the LM belongs to  an important class of
quasi- 2D system with which statistical models that depend on the
dimension of the system can be examined.

Herein,  results from a simple lipid,
dihexadecyl hydrogen phosphate (DHDP),
consisting of a phosphate head group (PO$_4^-$) and two hydrocarbon
chains attached to it, are presented.
Figure\ \ref{fig10}(a) displays the normalized reflectivity
of a DHDP monolayer at the air-water interface at a lateral
pressure of 40 mN/m.
The corresponding electron density profile is shown in the inset
as a solid line.  The profile in the absence of surface roughness
($\sigma=0$) is displayed as a dashed line.  The bulk water
subphase corresponds to $z < 0$,
the phosphate headgroup region is at $ 0 \geq z \geq 3.4${\AA} and
the hydrocarbon tails are at the
 $ 3.4${\AA} $ \geq z \geq 23.1$ {\AA} region.
As a first stage analysis of the reflectivity  a model
SLD with minimum number of {\it boxes}, $i=1,2,3...$, is constructed.  
Each box is characterized by
a thickness $d_i$ and an electron densities $N_{e,i}$,
and one surface roughness , $\sigma$ for all interfaces.  
Refinement of the reflectivity with Eq.\ \ref{R+roughness} 
shows that the two {\it box} model is sufficient. 
In order to improve the analysis   we can take advantage of information
we know of the monolayer i.e.,  the constituents used and the molecular 
area determined from the
lateral-pressure versus molecular area isotherm. 
If the monolayer is homogeneous and not necessarily ordered, 
we can assume an average area per molecule at the interface $A$, and
calculate the electron density  of the tail region as follows
\begin{equation}
\rho_{tail} = \mbox{N}_{e,tail}r_0/(Ad_{tail})  \\
\label{tailSLD}
\end{equation}
where $\mbox{N}_{e,tail}$ is the number of electrons
in the hydrocarbon tail and $d_{tail}$ is the length of the tail
in the monolayer.  The gain in this description is two fold; first,
the number of independent parameters can be reduced, and
constraints on the total number of electrons can be introduced.
However, in this case, the simple relation
$\rho_{head} = \mbox{N}_{e,phosphate}/(Ad_{head})$ is not satisfactory
and in order to get a reasonable fit  {\it additional} electrons are
necessary in the head-group region.
These additional electrons can be associated with water molecules 
that interpenetrate the head group region which is not densely packed.  
The cross section of the phosphate head group is smaller than the
area occupied by the two hydrocarbon tails allowing for water molecules 
to penetrate the head group region.  We therefore introduce  an extra
parameter $\mbox{N}_{H_2O}$, the number of water molecules with ten electrons 
each.  The
electron density of the head group region is given by,
\begin{equation}
\rho_{head}=(\mbox{N}_{e,phosphate}+10\mbox{N}_{H_2O})/(Ad_{head}).
\end{equation}
This approach gives a physical insight into the chemical constituents
at the interface.  In modeling the reflectivity with the above assumptions
we can either apply volume constrains or equivalently examine
the consistency of the model with the literature values of
closely packed moieties.  In this case the following volume constraint can be
applied,
\begin{equation}
V_{headgroup} = Ad_{head} = \mbox{N}_{H_2O}V_{H_2O} + V_{phosphate}
\end{equation}
where $V_{H_2O} \approx 30$ {\AA}$^2$ is known from the density of water.
The value of $V_{phosphate}$ determined from the refinement, 
should be consistent within error with known values 
extracted from crystal structures of salt phosphate\cite{Gregory97}.

Another parameter that can be deduced from the analysis is, the
average tilt angle, $t$, of the tails with respect to the surface from the
relation,
\begin{equation}
d_{tail}/l_{tail} = \cos t
\end{equation}          
where $l_{tail}$ is the full length of the extended alkyl chain evaluated
from the crystal data for alkanes\cite{Gregory97}.
Such a relation is valid under the condition that the
electron density of the tails when tilted is about the same
as that of closely packed hydrocarbon chains in a crystals
$\rho_{tail}\approx 0.32 e/{\AA}^3r_0$ as observed\cite{Kjaer89}.
Such a tilt of the hydrocarbon tails would lead to an average increase in the
molecular area compared to the cross section of the hydrocarbon tails
($A_0$),
\begin{equation}
A_0/A = \cos t.
\end{equation}
Gregory et al. found that at lateral pressure, $\pi = 40 mN/m $ the
average tilt angle is very close to zero ($\approx 7 \pm 7^\circ $)
and extract an  $A_0 \approx 40.7 {\AA}^2$ compared with
a value of $39.8{\AA}^2$ for closely packed crystalline
hydrocarbon chains.  The small discrepancy was attributed to
defects at domain boundaries.

The GID for the same monolayer is shown in Fig.\ \ref{fig10}(b) where a
lowest order Bragg reflection at $1.516 \AA^{-1}$ is observed.  
This reflection corresponds  to the hexagonal ordering of the individual 
hydrocarbon chains\cite{Kjaer87,Kjaer89} with lattice constant 
$d = 4.1144${\AA},  and molecular area per 
chain $A_{chain}= 19.83${\AA}$^2$.
Note that in DHDP the phosphate group is anchored to a pair of hydrocarbon
chains with molecular area $A = 39.66${\AA}$^2$, and it is surprising 
that ordering of the head group with a larger unit cell
(twice that of the hydrocarbon unit cell) is not observed
as is evidenced from  Fig.\ \ref{fig10}(b).
Also shown in the inset of Fig.\ \ref{fig10}(b)
is a rod scan of the Bragg reflection.  To model the rod scan in terms
of tilted
chains the procedure developed in \cite{Kjaer89} is followed.
The structure factor of the chain can be expressed as
\begin{equation}
F_{chain} (\bbox{Q\prime}_\perp,Q\prime_z) =
F(Q_\perp) \frac{\sin (lQ_z^\prime /2)}{(lQ_z^\prime/2)}
\label{Molec-SF}
\end{equation}
where $F(Q^\prime_\perp)$ is the in-plane Fourier transform of the
cross section of the electron density of chain weighted with the
atomic form factors of the constituents. The second term accounts
for the length of the chain and is basically a Fourier transform
of a one dimensional aperture of length $l$.
If the chains are tilted with respect to the surface normal
(in the y-z plane) by an
angle $t$, the $\bbox{Q}^\prime$ should be rotated as follows,
\begin{eqnarray}
Q_x^\prime & = & Q_x \cos t + Q_z \sin t \nonumber \\
Q_y^\prime & = & Q_y \nonumber \\
Q_z^\prime & = & -Q_x\sin t + Q_z \cos t
\end{eqnarray}
For small chain-tilt angles this rotation mainly affects the $Q_z$
dependent part of the chain structure factor,
since the $Q_\perp$ changes.
Applying this transformation to molecular structure factor,
Eq.\ \ref{Molec-SF}, and averaging over all six domains (see more
details in Ref.\ \cite{Kjaer89} with the
appropriate weights to each tilt direction we find
that at 40 mN/m the hydrocarbon chains are practically normal to the
surface consistent with the analysis of the reflectivity.


In recent Brewster Angle Microscopy (BAM) and X-ray studies
of C$_{60}$-propylamine spread at the air-water
interface (see more details on fullerene films\cite{Vaknin96}),  
a broad in-plane GID signal was observed\cite{Fukuto97}.
The GID signal was analyzed
in terms of a 2D radial distribution function that implied
short range positional correlations extending to only few molecular
distances.  It was demonstrated  that the local packing of molecules
on water is hexagonal, forming a 2D amorphous solid.  This is a detailed
study demonstrating how to analyze homogeneously disordered
2D system by combining X-ray scattering techniques and visible light
microscopy.

\subsection{Surface Crystallization of Liquid Alkanes}
Normal alkanes are linear hydrocarbon chains (CH$_2$)$_n$
terminating with CH$_3$ groups similar to fatty acids and lipids
that by contrast posses a hydrophilic head groups at one end.
Recent extensive X-ray studies of pure and mixed liquid alkanes
\cite{Wu93a,Wu93b,Wu95,Ocko97} reveal a rich and remarkable properties near
their melting temperature, $T_f$.
In particular, a single crystal monolayer is formed at the
surface of an isotropic liquid bulk up to $\approx 3^{\circ}C$ above $T_f$
for a range of hydrocarbon number $n$.  The surface freezing phenomena
exists for a wide range of chain lengths $ 16 \geq n \geq 50$.
The molecules in
the ordered layer are hexagonally packed and show three  distinct
ordered phases: two rotator phases, one with the molecules oriented
vertically ($16 \geq n \ge 30$) and the other tilted toward nearest neighbors.
($30 \geq n \geq 44$).  The third phase
($44 \geq n$) orders with the molecules
tilted towards next-nearest neighbors.
In addition to the 2D Bragg reflections observed in the GID studies,
reflectivity curves from the same monolayers were found to be
consistent with a one {\it box} model of densely packed
hydrocarbon chains, and a thickness that corresponds to slightly tilted
chains.
This is an excellent demonstration where no other technique but
the X-ray experiments carried out at a synchrotron
could be applied to get the detailed structure of the monolayers.
Neutron scattering from this system would have yielded
similar information, however the intensities available today
from reactors and spallation sources
are smaller by at least a factor of $10^5$ counts/sec for similar
resolutions and will not allow observation of any GID 
signals above background levels.

\subsection{Liquid Metals}
Liquid metals unlike dielectric liquids consist
of the classical {\it ionic} liquid and quantum free electron gas.
Scattering of conduction electrons at a step like potential (representing
the metal-vacuum interface), give rise to quantum interference effects and
lead to oscillations of the electron density across the
interface\cite{Lang70}.
This effect is similar to the Friedel oscillations in the screened
potential arising from the
scattering of conduction electrons by an isolated charge in a metal.
By virtue of their mobility, the ions in a liquid metal can in turn
rearrange and conform to these oscillations to form layers at the
interface, not necessarily commensurate
with the conduction electron density\cite{Rice86}.
Such theoretical predictions of atomic layering at surfaces of liquid
metals were known for a long time and were only recently
confirmed by X-ray reflectivity studies for liquid
gallium and liquid mercury\cite{Magnussen95,Regan95}.
X-ray reflectivities of these liquids were extended to
$Q_z \sim 3 \mbox{\AA}^{-1}$ showing a single
peak that indicates layering with spacing on the order of atomic
diameters.  The exponential decay for layer penetration into the bulk of Ga
(6.5\AA) was found to be larger than that of Hg ($\sim 3$ \AA).
Figure\ \ref{fig11} shows a peak in the reflectivity of liquid Ga under
{\it in situ} UHV oxygen free surface cleaning\cite{Regan95}.
The normalized reflectivity was fitted to a model
scattering length density shown in Figure\ \ref{fig11}(b) of the
following oscillating and exponentially decaying form\cite{Regan95},
\begin{equation}
\rho(z)/\rho_s = \mbox{erf}[(z-z_0)/\sigma] +
\theta(z)A\sin(2\pi z/d)\mbox{e}^{-z/\xi}
\end{equation}
where $\theta(z)$ is a step function, $d$ is the inter-layer 
spacing, $\xi$ the
exponential decay length, and $A$ an amplitude.  Fits to this model are shown
in Fig.\ \ref{fig11} with $ d=2.56 \AA, \xi =5.8 \AA$.
The layering phenomena in Ga showed a strong temperature dependence.
Although liquid Hg exhibits layering with a different decay length
the reflectivity at small momentum transfers, $Q_z$ are significantly
different than that of liquid Ga indicating fundamental differences in the
surface structures of the two metals.
The layering phenomena suggests in-plane correlations that might be different
than those of the bulk, but had not been observed yet with GID studies.
\acknowledgments
The author would like to thank Prof. P. S. Pershan for
providing a copy of Fig. 11 for this publication.
Ames Laboratory is operated by Iowa State
University for the U.S. Department
of Energy under Contract No. W-7405-Eng-82.
The work at Ames was supported by the Director for Energy
Research, Office of Basic Energy Sciences.

\appendix
\subsection*{APPENDIX A.}
\subsection*{ p-polarized X-ray beam}

A p-polarized X-ray beam has a magnetic field component
that is parallel to the stratified medium
(along the x-axis, see Fig.\ \ref{fig1}), and
straightforward derivation of the wave equation\ \ref{WaveEq1}
yields
\begin{equation}
  {{\rm d}\over {\rm d}z} \left ({{\rm d}B\over \epsilon{\rm d}z}\right ) +
\left [{{ k^2_{z} - V({z})} } \right ]B = 0.
\label{A1}
\end{equation}
By introducing a dilation variable $Z \mbox{ such that},
{\rm d}Z = \epsilon{\rm d}z$
Eq.\ \ref{A1} for $B$ can be transformed to a form  
similar to Eq.\ \ref{eq7}
\begin{equation}
{{\rm d}^2B \over {\rm d} Z^2} + [{ {k^2_{z} - V(z)} \over {\epsilon} }]B = 0.
\label{A4}
\end{equation}
The solution of Eq.\ \ref{A4} for an ideally flat interface in
terms of $r_p(k_{z,s})$ and $t_p(k_{z,s})$ is then simply given by
\begin{equation}
r_p(k_{z,s}) = { { {k_{z,0} - k_{z,s}/\epsilon} \over {k_{z,0} + 
k_{z,s}/\epsilon} } },
 \qquad
t_p(k_z,s) = { 2k_{z,0} \over {k_{z,0} + k_{z,s}/\epsilon}}
\label{A5}
\end{equation}
The critical momentum transfer for total external 
reflectivity of the p-type X-ray
beam is $Q_c = 2k_c = 4\sqrt{\pi\rho_s}$, 
identical to the one derived for the s-type wave.
Also, for $2k_z^B \gg Q_z \gg Q_c$, ($k_z^B$ is defined below), 
$R_F(Q_z)$ can be approximated as
\begin{equation}
R_F(Q_z) \sim \left ({ {Q_c}\over 2{Q_z} }\right )^4 
\left({2 \over {1+\epsilon} }\right)^2.
\end{equation}
The factor on the right hand side is one for all practical liquids,
and thus the Born approximation is basically the same as for 
the s-polarized X-ray beam (Eq.\ \ref{Fresnel_approx}).
The main difference between the s-type and p-type waves 
occurs at larger angles near a Brewster angle that is given by
$\theta_B  = \sin^{-1}(k_z^B/k_0)$.  At this angle, total transmission
of the p-type wave occurs ($r_p(k_{z,s})$ = 0).
Using Eqs.\ \ref{eq9} and \ref{A5}, $k_z^B$ can be derived,
\begin{equation}
{k_z^B\over k_0} = \frac{1} {\sqrt{2 - 4\pi\rho_s/k_{0}^2} }.
\end{equation}
The Brewster angle for X-rays is then given by
$\theta_B = \sin^{-1}(k_z^B/k_0) \approx \pi/4$.
This derivation is valid for solid surfaces, including 
crystals, where the total  transmission effect of the p-polarized
wave at a Bragg reflection is used to produce polarized and 
monochromatic X-ray beam.
  

%   lines and directly read in your .bbl file if you use bibtex.
\parindent=0.0cm
%\bibliographystyle{alpha}
\begin{thebibliography}{alpha}
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\bibitem{Azzam77} Azzam, R.M.A., and Bashara, N.M.,  1977. Ellipsometry and Polarized Light, North Holland: New York.
\bibitem{Bevington68} Bevington, P.R., 1968. {\it Data Reduction and Error Analysis}, McGraw Hill New York.
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\begin{figure}
\caption{The geometry of incident and scattered beam in specular
reflectivity (a) and in a non-specular scattering(b) experiments.}
\label{fig1}
\end{figure}

\begin{figure}
\caption{Calculated reflectivity curves for external
(solid line) and internal (dashed line) scattering from an
ideally flat interface versus momentum transfer given
in units of the critical momentum transfer,
$Q_c = 4(\pi\rho_s)^{1/2}$.
The dotted line is kinematical approximation $(Q_c/2Q_z)^4$.
The lower panel shows the amplitude of the wave in the medium for external
(solid line) and external reflection (dashed line).}
\label{fig2}
\end{figure}
\begin{figure}
\caption{
Calculated reflectivities from H$_2$O and liquid mercury (Hg)
showing the effects of absorption and surface roughness.
The absorption modifies
the reflectivity near the critical momentum transfer for mercury
with insignificant effect on the reflectivity from H$_2$O.
The dashed line shows the calculated reflectivity
from the same interfaces with root mean square surface roughness,
$\sigma = 3 {\AA}$.
}
\label{fig3}
\end{figure}                      
\begin{figure}
\caption{An illustration of a continuous scattering length
density, {\it sliced} into a histogram.
}
\label{fig4}
\end{figure}
\begin{figure}
\caption{Calculated reflectivities
for two films with identical thicknesses but with two distinct
normalized electron densities $\rho_1$ (solid line) and
$1-\rho_1$ (dashed line) and corresponding calculated
reflectivities using the dynamical approach ($\sigma=0$).
The two reflectivities
are almost identical except for a minute difference near the
first minimum (see arrow in figure).
The Born approximation (dotted line) for the two models yields
identical reflectivities.
The inset shows the normalized reflectivities
near the first minimum. As $Q_z$ is increased the three curves
converge.
This is the simplest demonstration of the phase problem, the
non-uniqueness of models where two different potentials give
the same reflectivities.
}                        
\label{fig5}
\end{figure}
\begin{figure}
\caption{(a)  Illustration of wave paths for  exterior (a) and
interior(b) scatterer near a step-like interface.  In both
cases the scattering is enhanced  by the transmission
function when the angle of the incidence  is varied around the critical
angle.
However, due to the asymmetry between external and internal reflectivity
the rod scan of the final beam modulates the scattering
differently as is shown on the right hand side in each case.
}
\label{fig6}
\end{figure}
\begin{figure}
\caption{Monochromator geometry to tilt a Bragg reflected beam
from the horizon on a liquid surface.
Two possible tilting configurations  about the primary beam axis and
about an axis along the surface of the reflecting planes are shown.
}
\label{fig7}
\end{figure}
\begin{figure}
\caption{Superposition of the reflected-beam (circles) below the critical
angle and direct beam (triangles), demonstrating
total reflectivity of X-rays from the surface of water.
Severe surface roughness reduces the intensity and widens the reflected
signal.  A reduction from total reflectivity can also occur if the
slits of the incident beam are too wide, so that the {\it beam-footprint}
is larger than the surface sample.
}
\label{fig8}
\end{figure}
\begin{figure}
\caption{
Experimental reflectivity from the surface of water. The dashed line
is the calculated Fresnel reflectivity from an ideally flat water-interface,
$R_F$.
The Normalized reflectivity versus $Q_z^2$ is fitted to a
the form $R/R_F = \mbox{e}^{-(Q_z\sigma)^2}$, demonstrating the validity
of the capillary-wave model [Buff. et al., 1965].
}
\label{fig9}
\end{figure}
\begin{figure}
\caption{
(a) Normalized X-ray reflectivity from Dihexadecyl-phosphate (DHDP) monolayer
at the air-water interface with
best fit electron density, $N_e$, shown with solid line in the inset.  The
calculated reflectivity from the best model is shown with a solid line.
The dashed line in the inset shows the {\it box} model with no
roughness $\sigma=0$.
(b) A diffraction from the same monolayer showing a prominent 2D Bragg
reflection corresponding to the hexagonal ordering of individual
hydrocarbon chains at $Q^B\perp =1.516$ {\AA}$^{-1}$.
The inset shows a rod scan from the quasi-2D Bragg reflection at
$Q^B_\perp$,  with a calculated model for tilted chains denoted by
solid line (see text for more details).
}
\label{fig10}
\end{figure}
\begin{figure}
\caption{ Upper panel shows measured reflectivity for liquid Ga.
Data marked with X were collected prior to sample cleaning whereas
the other symbols correspond to clean surfaces
(for details see Regan et al., 1995).
Calculated Fresnel reflectivity from liquid Ga surface convoluted
with a surface
roughness due to capillary waves ($\sigma = 0.82${\AA}), and the
atomic form factor for Ga is denoted with a solid line.
The lower panel shows the normalized reflectivity, with a solid line
that was calculated with the best fit by an exponentially decaying sine
model shown in the inset (Courtesy of Regan et al. 1995}
\label{fig11}
\end{figure}

\begin{table}
\caption{Electron number density, SLD,  critical angles and
momentum transfers, and absorption term for water and liquid mercury.}
\begin{tabular}{llllll}
        & $N_e (e/{\AA}^3)$  & $\rho_s ({\AA}^{-2}\times 10^{-5})$
        & $Q_c(\AA^{-1})$              & $\alpha_c$ (deg.) &
        $\beta (\times 10^{-8})$ \\
&&& &for $\lambda=1.5404\AA$& \\
\hline
H$_2$O    & 0.334 & 0.942   & 0.02176 & 0.153 & 1.2   \\
Hg      & 3.265 & 9.208   & 0.06803 & 0.478  & 360.9 \\
\end{tabular}
\label{tb1}
\end{table}
\end{document}

cond-mat0101142.extracted_bib
\begin{thebibliography}{alpha}
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\bibitem{Als-Nielsen89} Als-Nielsen. J., and  Kjaer, K., 1989. X-ray reflectivity and diffraction studies of liquid surfaces and surfactant monolayers.  {\it In} Phase Transitions in Soft Condensed Matter. T. Riste and D. Sherrington, editors.  Plenum Press, New York. 113-138.
\bibitem{Azzam77} Azzam, R.M.A., and Bashara, N.M.,  1977. Ellipsometry and Polarized Light, North Holland: New York.
\bibitem{Bevington68} Bevington, P.R., 1968. {\it Data Reduction and Error Analysis}, McGraw Hill New York.
\bibitem{Binnig83} Binnig, G., and  Rohrer, H., 1983. Scanning tunneling microscopy. {\it Surf. Sci.}  126: 236 -244.
\bibitem{Binnig92} Binnig, G 1992. Force Microscopy. {\it Ultarmicroscopy} 42-44:7-15.
\bibitem{Born59} Born M., and Wolf, E., 1959. Principles of Optics, MacMillan, New York.
\bibitem{Bouwman96} Bouwman, W.G., and Pedersen, J.S., 1996. Resolution function for two-axis specular neutron reflectivity. {\it J. Appl. Crystall.} 28:152-158.
\bibitem{Braslau88} Braslau, A., Pershan, P.S.,  Swislow, G., Ocko, B.M., and Als-Nielsen, J., 1988. Capillary waves on the surface of simple liquids measured by X-ray reflectivity.  {\it Phys. Rev. B}  38:2475-2469.
\bibitem{Buff65} Buff, F.P., Lovett, R.A., and Stillinger, Jr., F.H.,1965. Interfacial density profile for fluids at the critical region. {\it Phys. Rev. Lett.} 15:621-623.
\bibitem{Clinton93}Clinton, W.L., 1993. Phase determination in X-ray and neutron reflectivity using logarithmic dispersion relations. {\it Phys. Rev. B} 48:1-5.
\bibitem{Ducharme90} Ducharme, D., Max, J.-J., Salesse, C., and Leblanc, R.M., 1990. Ellipsometric study of the physical tates of phosphotadylcholines at the air-water interface. {\it J. Phys. Chem.} 94: 1925-1932.
\bibitem{Dutta87} Dutta, P., Peng, J.B., Lin, B.,  Ketterson, J.B., and  Prakash, M. P. Georgopoulos, and S. Ehrlich, 1987.  X-ray diffraction studies of organic monolayers on the surface of water. {\it Phys. Rev. Lett.} 58:2228-2231.
\bibitem{Evans79} Evans, R., 1979. The Nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform, classical fluids. {\it Advances in Physics}  28:143-200.
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\end{thebibliography}