|  Temperature dependence in fast-atom diffraction at surfaces
---|---
|  Peng Pana, Maxime Debiossaca and Philippe Roncina
|  Grazing incidence fast atom diffraction at crystal surfaces (GIFAD or FAD)
has demonstrated coherent diffraction both at effective energies close to one
eV ($\lambda_{\perp}\approx$ 14 pm for He) and at elevated surface
temperatures offering high topological resolution and real time monitoring of
growth processes. This is explained by a favorable Debye-Waller factor
specific to the multiple collision regime of grazing incidence. This paper
presents the first extensive evaluation of the temperature behavior between
177 and 1017 K on a LiF surface. Similarly to diffraction at thermal energies,
an exponential attenuation of the elastic intensity is observed but the
maximum coherence is hardly limited by the attraction forces. It is more
influenced by the surface stiffness and appears very sensitive to surface
defects.

##

††footnotetext: a Address, Université Paris-Saclay, CNRS, Institut des
Sciences Moléculaires d’Orsay (ISMO), 91405 Orsay, France

##  1 Introduction

The characterization of new materials requires a variety of techniques to
analyze their physical and chemical properties. These can be real space
microscopic techniques such as scanning tunneling microscopy and atomic force
microscopy but also reciprocal space techniques using X-rays, neutrons,
electrons, or atoms. Since atoms with kinetic energies below a few eV cannot
penetrate below the surface, thermal energies atom scattering (TEAS), also
known as helium atom scattering (HAS), is a valuable tool to investigate
surfaces and 2D materials 1, 2. It is insensitive to the presence of magnetic
or electric fields and does not induce any direct damage or charging of the
surface. However, its geometry is not compatible with a standard molecular
beam epitaxy (MBE) vessel which requires that no instrument prevents the gas
from the evaporation cells to reach the surface. MBE also requires elevated
surface temperatures in order to control the mobility of the deposited atom or
molecule so that these can reach an optimum location in a reasonable timescale
without being trapped too long in undesirable sites 3. In this context,
grazing incidence fast atom diffraction (GIFAD4 or FAD5) which is the high
energy, grazing incidence counterpart of TEAS offers decisive advantages; it
has shown to be compatible with a harsh UHV environment and with the MBE
geometry. At variance with TEAS, the full diffraction image can be recorded in
seconds so that it can be used as an real time diagnostic of the structure of
the terminal layer. More unexpected, diffraction can be recorded on a surface
at elevated temperatures. This is illustrated in Fig.1 taken from Ref 6 and
recorded inside a MBE vessel with a GaAs surface around 850 K, whereas TEAS is
mainly performed at low temperatures. This interesting aspect of GIFAD to
operate at elevated surface temperatures is poorly documented 7, 8, 9 and no
systematic experimental investigation has been reported.

Fig. 1: Schematic view of a GIFAD setup, the well-aligned row of atoms acts as
a grating for the projectile atomic wave. A detector located almost a meter
downstream records the diffraction image. The one here, taken from Ref.6 was
recorded directly in a MBE vessel on a GaAs(001) surface at $\sim$ 850 K. The
bright spots corresponding to elastic diffraction are located on the Laue
circle of energy conservation.

This paper presents experimental investigations on temperature dependence
under a wide variety of experimental conditions of energy, angle of incidence
and temperatures for helium atoms impinging on a LiF surface. The paper is
organized as follows, the experimental setup is described in Sec.2 with the
protocol used to transform raw data into transverse and polar scattering
profiles from which the coherence ratio is defined. In Sec.3, the strategy
adopted to performed stable temperature variations is presented before an
introduction to theoretical aspects of elastic and inelastic diffraction in
Sec.4. The results are presented and discussed in Sec.5.

##  2 Grazing incidence fast atom diffraction (GIFAD)

The grazing incidence fast atom diffraction at crystal surfaces uses atoms in
the keV energy range as probed with incidence angles $\theta_{i}$ around 1
deg. so that the full diffraction pattern can be recorded in one take on a
position-sensitive detector 10, 11, 12, 13 as sketched in Fig.1. A commercial
ion source delivers ions at the desired energy, they pass inside a charge
exchange cell filled with the same gas, where a significant fraction is
neutralized by resonant electron capture, see e.g. Ref.14. After this cell,
the ion fraction is deflected away and the spatial extent and angular
divergence of the neutral beam is controlled by two co-linear diaphragms
adjustable between 20 and 100 $\mu$m, separated by a distance close to half a
meter before entering into the UHV chamber with the target. If the projectile
encounters a large enough terrace, it undergoes quasi-specular reflection and
the projectiles are scattered within a cone with an opening angle of
$\theta_{i}$. Since keV atoms are easily detected and imaged by micro-channel
plates, GIFAD was able to record a few images per second 15, 16 with an old
ion source.

GIFAD offers a high topological resolution of a few $pm$ on atomic structure,
e.g. surface rumpling 17 or charge transfer 18, simple semi-quantitative
interpretation 14 and, when compared with exact quantum scattering code 4, 19,
20, 21, a parameter free accuracy 6, 22. The temperature of the surface
affects both its nuclear and electronic systems. We use the well-documented
system of helium on LiF(001), where the large band-gap prevents electronic
contributions allowing interpretations of inelastic effects only in terms of
thermal motion of the surface atoms.

Fig. 2: a) quasi polar transform of the raw diffraction image in panel d). The
polar scattering profile b) corresponds to a full projection of a) onto the
vertical axis. It is fitted by the sum of a narrow Gaussian and a broad log-
normal profile used to evaluate the DWF=$I_{e}/I_{tot}$ with
$I_{tot}=I_{e}+I_{ine}$. Panel c) corresponds to the intensity in a narrow
horizontal band centered at the specular angle. The contrast measured on the
Laue circle (c) is 73% much larger than the DWF=13%.  Fig. 3: Three
diffraction images recorded with 500 eV helium impinging with
$\theta_{in}$=0.75 deg. on LiF [100] at temperatures of 177 K a), 687 K b) and
1017 K c). The images are normalized to the maximum intensity corresponding
here to the elastic specular spot. The rainbow color palettes are identical
with a threshold at 3% of the maximum intensity.

A definition of the $(x,y,z)$ axis is displayed in Fig.1 together with a
typical raw diffraction image on GaAs at elevated temperatures. Another raw
diffraction image is plotted in Fig.2d) for a LiF crystal and a helium beam
oriented along the [110] direction. These images correspond to a direct
mapping of the final velocity or wave vector ($k_{fy},k_{fz}$) of the
scattered projectile perpendicular to the crystal axis. Bright elastic
diffraction spots are clearly visible and located on a single circle
corresponding to energy conservation of the motion in the ($y,z$) plane
perpendicular to the crystal axis probed :
$k_{fy}^{2}+k_{fz}^{2}=k_{\perp}^{2}=cst$. A polar-like transformation 23
brings this Laue circle into a straight line displayed in (Fig.2a)). The
intensity in this narrow stripe of maximum intensity is reported on Fig.2c)
and shows well-resolved diffraction peaks equally separated by multiples of
the Bragg angle $\theta_{B}=\arctan(G_{y}/\hslash k_{//})$, where $\hslash
k_{//}=\hslash k\cos{\theta_{in}}$ is the projectile momentum parallel to the
crystal axis. $G_{y}=2\pi/a_{y}$ is the reciprocal lattice vector associated
with the distance $a_{y}$ between atomic rows perpendicular to the probed
crystal axis, taken here as the $x$ direction. To derive the structure factor,
only the elastic intensities should be considered, however, when elastic
intensity is significant, the elastic and inelastic relative intensities on
the Laue circle $I_{m}$ were found identical 24 so that Fig.2c) can be
directly exploited.

The projection of Fig.2a) on the vertical axis produces the polar scattering
profile in Fig.2b) showing a narrow elastic peak on top of a broader inelastic
scattering profile. The relative weight of the elastic peak can be estimated
using a simple fit where the elastic component is represented by a narrow
Gaussian peak and the inelastic one by a broader, slightly asymmetric log-
normal profile
$f(\theta)=\frac{1}{\text{w}\theta\sqrt{2\pi}}\exp(-\frac{(\ln{\theta}-\ln{\theta_{s}})^{2}}{2\text{w}^{2}})$
25, 26, 27. Assuming that the image contains only the gently scattered
projectile that did not encounter major surface defects, this ratio
DWF=$I_{e}/I_{tot}$ is believed to be a direct measurement of the Debye-Waller
factor. It is different from the standard definition used in TEAS
DWF=$I_{e}/I_{0}$ where $I_{0}$ would be the intensity of the direct beam
which, in practice, is never measured directly with the same detector. In
GIFAD, the direct beam is always measured, either directly or through a
calibrated attenuation grid, because both the exact beam position and line-
profile are mandatory for accurate analysis. It reveals that for the present
cleaved LiF sample, the intensity scattered by the surface drops rapidly from
close to unity an incidence angle above to 2-3 deg down-to only a few percent
below 0.5 deg. This reduced reflectively is in part due to the overlap with
the sample surface but is probably mainly due to the distribution of terrace
height resulting from the cleavage.

Fig. 4: For the three images in Fig3, panel a): lateral profiles on the Laue
circle, the lines are here to outline the exponential decay of the peak tail.
Panel b): polar scattering profiles.

Fig.3 shows three diffraction images recorded at temperatures of 177K, 687K,
and 1017K measured with a type-N thermocouple mechanically pressed on the
backside of the Omicron sample plate. The transition from a spotty pattern to
a much more diffuse one is clearly visible and is discussed in detail along
the vertical and horizontal axis corresponding respectively to the polar
profile and lateral deflection. On Fig.4 a) the intensity on the Laue line is
displayed in the log scale. Within experimental uncertainty due to slightly
different beam conditions, the narrow elastic peaks do not change shape but
the inelastic contribution increases both in intensity and in width as
outlined by the full lines.More precisely, in this example, both the
extrapolated inelastic peak intensity and the exponential decay range increase
quasi-linearly with the temperature.

Fig.4b) reports the polar scattering profiles of Fig.3. The broad inelastic
profile is also clearly identified with a relative height and a width growing
with temperature so that the relative elastic intensity decay rapidly.

At this stage, it is useful to compare with the first investigation of the
temperature dependence performed when elastic diffraction was not yet
demonstrated and where all peak shapes seemed to depend on temperature 18, 28,
6, probably because of a limited surface coherence. In this context, the
Debye-Waller factor was tentatively attributed to the ratio of the narrow peak
relative to the total intensity observed on the Laue circle7, 29, 30, e.g. in
Fig.2b) or Fig.4a), rather than from the projected polar profile in Fig.2c) or
Fig.4b). This point of view, obviously underestimates the overall inelastic
intensity and strongly depends on the primary beam profile, and is not used
anymore.

##  3 $\phi$-scan, $\theta$-scan, E-scan and T-scan

A typical GIFAD experiment begins with the introduction or preparation of a
new sample surface and subsequent annealing. The first measurements consist of
a rapid azimuthal scan, $\phi$-scan, where the target surface is rotated in-
plane (around the $z$ axis in Fig.1) in order to identify its crystal axis. It
does not require that diffraction is observed because a simple analysis of the
width of scattered lateral profiles; $\langle k_{fy}^{2}\rangle^{1/2}$ in
Fig.2a, 2c, 2d, 4a), as a function of the target azimuthal angle $\phi$ is
enough to identify the principal crystal axis. The procedure called
triangulation 31, 32, 14 is used where the angle of incidence $\theta$
($\equiv k_{iz}$) is kept constant. In practice, measuring the relative width
$\langle k_{fy}^{2}\rangle^{1/2}/k_{iz}$ compensates for a minor tilt between
the surface normal and the rotation axis 32, 33. After proper alignment on the
desired axis, the surface is prepared by various methods until a good
diffraction pattern is observed. Here, we used LiF single crystal previously
irradiated by $\gamma$-rays 34 giving a pronounced yellow to orange color, and
cleaved in the air just before introduction into the vacuum. Subsequent
heating at 400 ∘C during a few hours is usually enough to recombine the color
centers and to record clean diffraction images with well-resolved elastic
spots. The size of the diffraction spots provides indications of the surface
coherence length. The lateral angular resolution corresponding to 1/10 of the
Bragg angle indicating a transverse coherence length
$\delta_{y}=(k\delta\theta)^{-1}\sim 10\leavevmode\nobreak\ a$ with $a$ the
lattice unit. The equivalent vertical width suggests a longitudinal coherence
length $\delta_{z}/\theta_{in}$ almost hundred times larger for $\theta=0.57$
deg so that $1/\theta=100$ (for circular diaphragms $\delta k_{z}=\delta
k_{y}$ so that $\delta_{z}=\delta_{y}$). Thus coherent specular reflection
requires a crystal without defect on a surface $\delta_{S}$ given by
$\delta_{S}=\delta_{y}^{2}/\theta_{in}\leavevmode\nobreak\ $ $10^{4}$ $a^{2}$
equivalent to a tiny square hundred by hundred lattice units or
$\leavevmode\nobreak\ 10^{5}$ Å2.

In principle, a $T$-scan would consist of a simple variation of the target
temperature leaving all other parameters unchanged. Unfortunately, this is not
compatible with the extreme sensitivity of grazing incidence. In GIFAD, the
target surface is easily positioned within 10 to 20 $\mu$m from the beam: when
it is not inserted enough, the primary beam is still present on the image,
whereas when it is inserted too much, the direct beam impacts the edge of the
$\approx 1$ mm thick crystal and even the scattered beam disappears from the
detector. However, thermal expansion of the target crystal and of the
manipulator induce much larger displacements as well as minor angular tilts
producing major effects in GIFAD (see e.g. the $\theta^{3}$ dependence in
Fig.7). For this reason, we were not able to record the three images displayed
in Fig.3 one after the other. Instead, we had to wait for a stable temperature
before realigning the target and restoring a comparable incidence angle. In
the following, we rationalize this approach by performing several
$\theta$-scan or $E$-scan at different temperatures. From these variations, we
interpolate between measured angular values, for instance with a B-spline, to
restore a temperature variation. It should be noted that at elevated
temperatures, typically above 800 K, particles emitted from the heating
filament produce an additional noise appears on our detector that may ruin a
temperature variation whereas, taking time to find a stable detector bias and
data-acquisition-mode can reduce the noise level allowing a more stable
$\theta$-scan or $E$-scan.

##  4 elastic and inelastic diffraction

For ideal crystalline surfaces with atoms at the equilibrium position, it has
been shown that the rapid movement parallel to the crystal axis ($x$) is
decoupled from the slow one in the ($y,z$) perpendicular plane 20, 35, 36, 37,
21, 38. Therefore, elastic diffraction of fast atoms along a crystal axis is
equivalent to that of a much slower particle with an energy
$E_{\perp}=E\sin^{2}\theta$ evolving in the averaged potential
$V_{2D}(y,z)=\int_{x}V_{3D}(x,y,z)$. Experimentally, this axial channeling
approximation (ASCA) results in diffraction taking place only in the ($y,z$)
perpendicular plane 36, 39, 14 (see also 28, 35 for the limitations). From a
spatial point of view, the elastic diffraction can be seen as the coherent
part of the scattered waves and this later can only build up at a location
corresponding to the equilibrium position, the one of the center of the
vibrational wavefunction. In contrast, inelastic diffraction corresponds to a
situation where momentum and/or energy has been exchanged with the surface
i.e. with the vibration of the surface atoms or phonons, breaking the exact
translation symmetry of the ideal surface and requiring a priori to abandon
the ASCA 8, 9.

In TEAS, x-ray or neutron diffraction, this situation is described by the
Debye-Waller factor DWF=$I_{e}/I_{0}=e^{-\sigma_{\phi}^{2}}$ where
$\sigma_{\phi}^{2}$ is the variance of the phase distribution induced the
thermal displacement $\sigma_{z}$ of surface atoms. In a Debye model of
solids, each atom is described by a local harmonic oscillator with frequency
$\omega_{D}$ and the thermal amplitude is Gaussian $\sigma_{z}(T)$ resulting
in a phase coherence $e^{-\sigma_{\phi}^{2}}=e^{-q^{2}\sigma_{z}^{2}}$ where
$q$ is the projectile momentum. It describes how the elastic diffraction
intensity progressively vanishes when the amplitude $\sigma_{z}$ of the
thermal disorder becomes comparable to the projectile wavelength
$\lambda=h/q$. From a momentum point of view, the DWF can be written as
$e^{-E_{r}/\hslash\omega}$ where $E_{r}=q^{2}/2m$ is the binary recoil energy,
which can be interpreted as the Lamb-Dicke probability of recoil-less emission
that the harmonic oscillator exchanges the momentum $2q$ without changing its
energy level. This DWF does not apply directly to GIFAD, even when considering
$\lambda_{\perp}=\lambda/\sin\theta_{in}\gg\lambda$, because the momentum
exchange of $2k\sin\theta_{in}$ needed for specular reflection of the primary
beam is acquired via several successive fast gentle collisions with surface
atoms. With a rigid lattice model and an exponential repulsive mean planar
potential of stiffness $\Gamma$, $V(z)=V_{0}e^{-\Gamma z}$, the trajectory is
analytic and the binary momentum exchanges can be calculated to obtain the
classical energy loss $\Delta E_{Cl}$ in eq.1 with $m_{p}$ the projectile mass
and $a$ the lattice unit26, 24. The effective number $N=\frac{6}{\Gamma
a\theta}$ of binary collision is defined such that $N$ times the individual
deflection $d\theta$ is the specular deflection $2\theta_{in}$ and that $N$
times the individual binary recoil energy $E_{r}$ matches the analytic total
energy loss $\Delta E_{Cl}=N\leavevmode\nobreak\ E_{r}$. The DWF for GIFAD
writes:

$DWF=\exp\left({-\frac{3\Delta
E_{Cl}}{\hslash\omega_{D}}\coth{\frac{T_{D}}{2T}}}\right),\leavevmode\nobreak\
\Delta E_{Cl}=\frac{2\leavevmode\nobreak\ m_{p}}{3\leavevmode\nobreak\
m}E\Gamma a\theta_{i}^{3}\leavevmode\nobreak\ \equiv NE_{r}$ (1)

For the He-LiF system, the product $\Gamma a$ is close to 14 22, 27 so that
$N\approx 2/\theta$ can be large explaining why elastic diffraction could be
observed with $E_{\perp}$ close to one eV 39, 14 whereas TEAS is usually
limited below 100 meV. Alternately, with this reduced decoherence, GIFAD can
explore the higher temperatures that are needed for MBE.

This naive, purely repulsive description was improved by taking into account
an attractive part of the potential, for instance, van der Waals contributions
40, 41, 42, 43, responsible for the physisorption well of depth $D$. In
elastic diffraction, the effect of such attraction is the presence of bound-
state resonances 44, 45 and the increase of the rainbow angle at low energy
22. These are naturally accounted for using a quantum approach 45, 22 or
modeled in semi-classical 46 or optical method such as the hard corrugated
wall by the Beeby correction indicating that the effective impact energy
$E_{\perp}$ increases to $E_{\perp}+D$ 40, 47, 48 or considering a modified
angle of incidence $\theta_{\text{eff}}=\sqrt{\theta_{in}^{2}+D/E}$. The Beeby
correction also decreases significantly the DWF in TEAS 49. In GIFAD, we also
found that the mere presence of a tiny well significantly modifies the
stiffness of the potential by bringing the turning point much closer to the
surface plane 27. This can be expressed quantitatively using a Morse potential
$V_{M}(z)=D(e^{-\Gamma(z-z_{0})}-2e^{-\Gamma/2(z-z_{0})})$ and looking for the
turning point $z_{t}$ where $V_{M}(z_{t})=E_{\perp}$ the effective stiffness
is :

$\Gamma_{\text{eff}}(E_{\perp})=\Gamma\left[1+\left(1+\frac{E_{\perp}}{D}\right)^{-1/2}\right]$
(2)

Fig. 5: a), b) and c) are raw diffraction images of 500 eV helium incident at
1.4∘ on LiF at 180 K oriented along the [110], [100], and random direction,
respectively (in part, taken from 27). The resulting polar scattering profiles
are almost identical, with a narrow elastic peak at $\theta_{out}=\theta_{in}$
on top of a broader peak fitted by a log-normal profile 25 (green curves).

This increased stiffness was already identified in TEAS 50 but it has much
less consequence since at normal incidence the projectile hits a single
surface atom, only the time scale $\tau\approx\Gamma/v_{\perp}$ depends on
$\Gamma$ not the magnitude of the exchanged momentum $2q$ and therefore not
the coherence ratio. It is the reverse in grazing scattering, for identical
values of $E_{\perp}$ the time scale $\tau$ for bouncing from the surface are
identical in TEAS or GIFAD, but the time needed for a single quasi binary
collision is now $\tau^{\prime}\approx v_{//}/a=\tau/N$ independent on
$\Gamma_{\text{eff}}$, while its magnitude $\approx 2q/N$ depends directly on
the effective stiffness $\Gamma_{\text{eff}}$. In summary, for GIFAD, the
stiffness $\Gamma_{\text{eff}}$ governs the momentum transferred in each
collision, a stiffer interaction potential needs fewer collisions for specular
reflection and each of them becomes more violent leading to an overall
reduction of the DWF.

At the atomic level, the temperature is modeled by the spatial extend of a
surface atom of mass $m$, which, in a Debye harmonic model is Gaussian profile
with
$\sigma_{z}^{2}(T)=\frac{3\hslash}{2m\omega}\coth{\frac{T_{D}}{2T}}\sim\frac{3\hslash}{2m\omega}\frac{2T}{T_{D}}$,
where $T_{D}$ is the Debye temperature such that $\hslash\omega=k_{B}T_{D}$
with $k_{B}$ the Boltzmann constant.

##  5 results

###  5.1 Orientation of the surface

When investigating the polar scattering profile 27, it was shown that the
shape of the polar scattering profile does not depend significantly on the
orientation of the surface. This is illustrated in Fig.5 and we have checked
that this similarity remains valid at different temperatures. Since both the
data acquisition and the data analysis are simpler for random orientation of
the surface where only one specular order ($m=0$) is present, most of the
temperature variations presented here correspond to this crystal orientation.
The other data were recorded along the [100] direction where the corrugation
amplitude is reduced generating fewer diffraction orders as visible when
comparing Fig. 3 with Fig.2.

Fig. 6: The relative intensities $I_{m}$ of ( )$m=0$, ( )$m=\pm 1$ and
($\triangle$)$m=\pm 2$ extracted from the elastic intensities recorded in
three $\theta$-scan performed at temperatures of  177 K,  687 K,  1017 K, fall
on top of each other.  Fig. 7: The DWF=$\frac{I_{e}}{I_{t}}$ from 3 LiF
samples are reported as a function of $E\theta^{3}$. The dotted lines are fits
to the data while the solid lines are numerical prediction of eq.1 with (red
A=.94,B=.19) and without corrections (black A=1,B=.17) due to the attraction
towards the surface

###  5.2 Bragg angle and form factor

The main focus of this paper is the temperature dependence of the elastic
ratio which is presumed to depend mainly on the thermal movement of the
surface atoms. When analyzing the intensity on the Laue circle as displayed in
Fig.2c), we can extract the value of the Bragg angle as well as the relative
intensities $I_{m}$ associated with each diffraction order $m$. Both could
change with the temperature due to thermal expansion and possible
reconstruction of the surface. This could be investigated in detail but we
have only checked that the surface equilibrium positions and GIFAD form factor
do not change significantly. The evolution of the Bragg angle is compatible
with the thermal expansion coefficient measured by TEAS 51 and the Fig.6 shows
that in spite of important variation of the scattering profiles visible in
Fig. 3 and 4, the relative intensities $I_{m}$ measured along the [100]
direction do not change significantly with temperature. This aspect is not
discussed further because, in practice, it is by-passed when analyzing the
random direction where the reciprocal lattice vector is absent.

Fig. 8: Panels a) c), report the DWF as a function of the reduced parameter
$E\theta^{3}$ for two different LiF samples. The dotted lines are B-spline
interpolation used to derive the temperature dependence at fixed values of
$E\theta^{3}$ in b) d) and reproduced by a global formula. At very low angles
of incidence, corresponding to $E\theta^{3}\leqslant$ 1 meV, where a diffuse
scattering background is present an error bar estimated to 5% is plotted.

###  5.3 Sample quality

All our samples have been prepared by cleaving at air LiF crystals previously
irradiated by $\gamma$ rays 34, however, the Debye-Waller factor varies from
sample to sample and also depend on the actual part of the surface illuminated
by the atomic beam. As a worst case, we experienced a variation up to almost a
factor two, after changing the target orientation and position when switching
from Random to [100] direction. Exploring a few target positions is usually
enough to optimize the DWF. We also observed a slow degradation of the
diffraction images with time even at a few $10^{-10}$ mb pressure. When taking
a sample that was left in a vacuum for a few weeks, the measured DWF is
systematically lower, even after a short thermal treatment. This is
illustrated in Fig.7 where three sets of data recorded on different samples
are reported. In spite of a scattering of data points in each set, the data
follow different curves and therefore different DWF, but the trends are
similar. In the final analysis, we have kept only two samples with large
enough temperature variations and we provide the results as separate sets.

###  5.4 Energy and angular variations

As detailed in section 4, the effective DWF adapted to GIFAD is expected to
scale with $\exp(-E\theta_{in}^{3})$ where $E$ is the primary beam energy and
$\theta_{in}$ the angle of incidence. This was suggested from the spatial
approach 7 considering a that the scattering by a row of $N$ atoms should have
a mean thermal amplitude $\langle\sigma_{z}^{2}\rangle=\sigma_{z}^{2}/N$ or
from momentum transfer along the trajectory 26, 24 because the classical
recoil energy loss is also expected to scale with $E\theta_{in}^{3}$. This
dependence was first observed at room temperature by reporting DWF measured on
a large set of energy and incidence angles 27. The Fig.8a) and 8c) display
similar dependence’s recorded during E-scan and $\theta$-scan at different
temperatures. All curves indicate a pronounced exponential decay illustrated
by the straight lines resulting from independent fits
$DWF=A_{i}\,\exp(-B_{i}\,E\theta^{3}\,T/300)$ where both $A_{i}$ and $B_{i}$
were left free and without any weighting of the data.

###  5.5 Temperature dependence

According to the announced strategy, we do not use the fits of Fig.8a) to
evaluate the temperature dependence but we use the B-spline interpolation of
the data, plotted as dotted lines Fig.8a) to produce the temperature
dependence displayed in Fig.8b) at fixed values of $E\theta^{3}$ between 0.5
meV and 6 meV. Here again, the exponential character is readily visible. In
contrast with Fig.8a), we now try to apply a unique formula to describe all
the data. The adjustment was performed by changing step-wise the parameters
$A$ and $B$ until a decent visual impression is obtained. In practice, we
could have used the data from the fits through the $E\theta^{3}$ variation
Fig.8a,c) but this would induce a possible bias forcing the final dependence.

In TEAS, the Beeby correction to the DWF alone is significant and limits the
maximum coherence, it is often used to estimate the $D$ value (see e.g. 40,
52, 2). This is not the case in GIFAD, a numerical evaluation, reported as
full lines in Fig.7 indicates that these corrections have a weak influence
both on the magnitude and the exponential range.

Fig. 9: a) and b) inelastic scattering width w as a function of
$E_{\perp}=E\theta^{2}$. The increase at low $E_{\perp}$ is mainly due to
$\Gamma_{\text{eff}}(E_{\perp})$. The w measured at $E_{\perp}$=500 meV are
reported in panel c) with the mean thermal amplitude $\sigma_{z}^{2}(T)$. The
lines in a) and c) correspond to Eq.3 with different angles in the evaluation
of $N$. (see text) The green curve marked as Quantum takes the zero point
energy into account.

###  5.6 The inelastic scattering width

The polar inelastic scattering profile is fitted log-normal form 25 and the
relative width w are reported in Fig.9a) and 9b) as a function of the
perpendicular energy $E_{\perp}$, a quantity that governs the distance of
closest approach to the surface. The fact that this width was found 27 to
depend mainly on $E_{\perp}=E\theta^{2}$ indicates that it is sensitive to the
magnitude of the most violent inelastic collisions along the trajectory rather
than to the integral effect of such collision which would be closer to
$E\theta^{3}$. Assuming that the inelastic collision is well-modeled by
classical mechanics, the thermal motion $\sigma_{z}(T)$ of the surface atom
induces, for each collision a log-normal scattering profile having a width
$\text{dw}=\Gamma\sigma_{z}$ 26, 24 or, equivalently a contribution to the
angular straggling $d\sigma_{\theta}=\Gamma\sigma_{z}d\theta$ where $d\theta$
is the elastic deflection angle in this collision already estimated as
$d\theta=(\theta_{in}+\theta_{out})/N$. Adding the $N$ individual variances
$d\sigma_{\theta}^{2}$ or using the mean log-normal width
$\text{w}=\text{dw}/\sqrt{N}$ the classical scattering width is predicted:

$\text{w}_{Cl}=\Gamma\sigma_{z}/\sqrt{N},\text{with }N=6/\Gamma a\theta_{in}$
(3)

The comparison with experiment is tricky, the Eq.3 reproduced the evolution
during an $E$-scan but not during a $\theta$-scan while the experiment gives
similar results during $E$-scan and $\theta$-scan 27. In this respect, the
following discussion is only qualitative. More precisely, during an $E$-scan
where $\theta_{i}$ is fixed $N$ would stay constant so that, neglecting for
here the Beeby correction, $\text{w}_{Cl}\propto\Gamma\sigma_{z}$ would remain
constant. However, as already observed at room temperature 27 and in Fig.9a)
and 9b) a sharp increase of w is observed at low $E_{\perp}$. The agreement
was established by taking into account the attractive forces i.e. by replacing
$\Gamma$ by $\Gamma_{\text{eff}}$ from Eq.2. The increase at low energy could
be then be attributed to the enhanced stiffness at low energy 27. The full,
dashed and dotted lines indicate how the absolute values of $\text{w}_{Cl}$
depend on the angle of incidence of the hypothetical equivalent $E$-scan. The
9c) reports the value of w measured at $E_{\perp}$ = 500 meV where is becomes
stable. The lines using the same Eq.3 now indicate that the evolution of the
plateau values in Fig.9a) and 9b) is compatible with the expected variation of
$\sigma_{z}$ even if the low-temperature zero point motion (green curve) is
not visible. The physical parameter $D$=8.5 meV 44 and $T_{D}$=550 K 17
correspond to well-accepted values in the literature and $\Gamma$=3.5 Å-1 was
produced in a quantum calculation 45. It is close to the asymptotic value
$\Gamma=2\sqrt{2W}=3.55\pm 0.15$ Å-1 where $W=12.2\pm 0.5$ eV is the work-
function of LiF. Once again, the semi-quantitative agreement should be
balanced by the fact that the model does not predict the observed similar
behavior of $\text{w}(E_{\perp})$ during an $E$-scan and a $\theta$-scan.

##  6 Summary and Conclusion

Using a definition based on the analysis of the polar scattering profile to
isolate the elastic and inelastic components, the DWF can be evaluated for
each diffraction image. At comparable energy and incidence angle, the DWF is
found independent of the crystal axis probed. Due to the extreme sensitivity
to mechanical deformations associated with temperature variations, the
$T$-scans were performed indirectly via interpolation of $\theta$-scan and
$E$-scan at different temperatures. At each temperature, the DWF specific to
the multiple collision regime of GIFAD is shown to depend primarily on
$E\theta^{3}$, differing from $E\theta^{2}$ in TEAS where a single collision
regime prevails. Within the present accuracy, a simple exponential decay is
observed but, different LiF samples produce slightly different decay
parameters and maximum coherence suggesting an important contribution of the
defect density and terrace size distribution. The effect of the attractive
forces towards the surface have been investigated in TEAS. It produces an
increased impact energy, known as the Beeby correction, and an increased
stiffness of the surface means planar potential energy surface due to a closer
distance of approach towards the surface 50. They also have the same
consequences regarding the elastic diffracted intensities in GIFAD or TEAS but
very different consequences in the inelastic behavior of GIFAD and TEAS. In
TEAS, the Beeby correction is known to limit the maximum possible DWF 40 while
in GIFAD, the Eq.1 and Fig.7c) indicate a limited influence on the maximum
DWF. As to the effective stiffness $\Gamma_{\text{eff}}$50, it does not
directly affect the DWF in TEAS, whereas, it enters the DWF factor in GIFAD
because each binary collision becomes more violent as evidenced by the sharp
increase of the inelastic scattering width w at low values of $E\theta^{2}$ in
Fig.9a),b). The effect of $\Gamma_{\text{eff}}$ on the DWF, though larger than
the Beeby correction is also reduced because it is, in part, balanced by the
reduced number of collisions needed for specular reflection.

With the LiF samples used here, the Fig.8b and d) indicate that GIFAD is not
able to offer an internal temperature with an accuracy better than 20-50
${}^{\circ}C$. Combining consistently the width w and the DWF could improve
the accuracy but the sensitivity to sample quality appears as a severe
limitation. In contrast, if the main focus is to optimize the growth
parameters to improve the surface quality in terms of coherence length, i.e.
mean distance $L_{C}$ between defects, GIFAD offers a unique handle with a
very broad range of operation. First, a simple $\phi$-scan 32 can identify
crystallographic axis even without diffraction offering a first estimate of
$L_{C}$ via the peak to background ratio of the $\phi$-scan 14. When
diffraction becomes visible, the presence of elastic diffraction, and its
associated elastic peak width, readily gives insights on $L_{C}$. Then,
optimizing the DWF could give real time access to very large defect-free
surfaces and the diagnostic is performed simultaneously on an illuminated
surface $S\approx\oslash^{2}/\theta$ on the order of 1 mm2 for a diaphragm
size of $\oslash$=100 $\mu$m. This diagnostic is complementary to the elastic
diffracted intensity which indicates in real time the detailed topology of the
terminal layer. We have shown here that the width w of the scattering profile
can be understood qualitatively in terms of a classical model using an
effective stiffness $\Gamma_{\text{eff}}(E_{\perp})$ and a thermal amplitude
$\sigma_{z}(T)$. This suggests that classical scattering simulation in grazing
incidence, in general 53 and in the context of GIFAD 8, 9 should produce a
fair estimate of the inelastic profile. However, In fine, a quantum inelastic
treatment as developed in TEAS 54 and recent attempts to encompass both
elastic and inelastic aspects under grazing incidence 55 should help
connecting to the real world of surface phonons and their possible specific
coupling to the multiple collision regime.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

The image in Fig.1 was recorded inside a MBE vessel in collaboration with P.
Atkinson, M. Eddrief, V. Etgens, F. Finocchi from Institut des NanoSciences de
Paris and H. Khemliche, A. Momeni A.G. Borisov, A. Zugarramurdi from ISMO. We
are grateful to Hynd Remita from the Institut de Chimie Physique for the
irradiation of the LiF samples and Andrew Mayne for his help in reading the
drafts. This work has been funded by the Agence Nationale de la Recherche
(contract ANR-2011-EMMA-003-01) and Chinese Scholarship Council (CSC) Grant
reference number 201806180025.

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