THE JOURNAL OF CHEMICAL PHYSICS 138, 124705 (2013)

Knudsen layer formation in laser induced thermal desorption
Akihiko Ikeda,a) Masuaki Matsumoto, Shohei Ogura, Tatsuo Okano,
and Katsuyuki Fukutanib)

Institute of Industrial Science, The University of Tokyo, 4-6-1, Komaba, Meguro, Tokyo 153-8505, Japan(Received 6 January 2013; accepted 6 March 2013; published online 27 March 2013)
Laser induced thermal desorption of Xe atoms into vacuum from a metal surface following the nanosecond pulsed laser heating was investigated by the time-of-flight (TOF) measurement. The desorption flow was studied at a wide range of desorption flux by varying the initially prepared Xecoverage ! (1 ML = 4.5 × 1018 atoms/m2 ). At ! = 0.3 ML, the TOF of Xe was well represented
by a Maxwell-Boltzmann velocity distribution, which is in good agreement with thermal desorption
followed by collision-free flow. At ! > 0.3 ML, the peak positions of the TOF spectra were shifted
towards the smaller values and became constant at large !, which were well fitted with a shifted
Maxwell-Boltzmann velocity distribution with a temperature TD and a stream velocity u. With TD
fixed at 165 K, u was found to increase from 80 to 125 m/s with increasing ! from 1.2 to 4 ML.
At ! > 4 ML, the value of u becomes constant at 125 m/s. The converging feature of u was foundto be consistent with analytical predictions and simulated results based on the Knudsen layer formation theory. We found that the Knudsen layer formation in laser desorption is completed at Knudsen
number Kn <0.39. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4795827]
I. INTRODUCTION

Laser induced desorption of atoms and molecules fromsolid surfaces is a vital phenomenon to investigate fundamentals such as surface electronic structures and dynamics.1–5 Besides, laser desorption is an essential technique for pulsed
laser deposition used in thin film growth6–12 and mass spectrometry of protein employing matrix-assisted laser desorption ionization.13 When desorption flux is small, the velocity distribution of desorbed atoms is directly governed by
the desorption mechanism. When the desorption flux is large
enough, on the other hand, the post-desorption collision between desorbed particles may become significant and modify
the velocity distribution in the vicinity of the surface after thedesorption.
Manifestations of the collision effect in laser induced
desorption have been reported both by experiments and simulations as the modifications of the angular and velocity distribution of desorbing atoms and molecules.14–17 Cowin et al.
investigated the angular dependence of the translational temperature of D2 desorbed from tungsten surfaces under a pulsed
laser irradiation.14 The translational temperature of D2 desorbed in the surface normal direction was higher than those in
oblique directions. They attributed the variation of the translational temperature to the collision effect. Noorbatcha et al.
used the direct Monte Carlo simulation of desorbing atoms toinvestigate the collision effect. They showed that even in the
sub-monolayer regime the collision noticeably modifies the final angular, velocity, and rotational-energy distributions.15–17
However, there has not been any model that can quantitativelya) Present address: Institute for Solid State Physics, The University ofTokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan. E-mail:ikeda@issp.u-tokyo.ac.jpb) E-mail: fukutani@iis.u-tokyo.ac.jp

0021-9606/2013/138(12)/124705/6/$30.00

estimate the degree of modification by the post-desorption
collision in laser desorption so far.
Knudsen layer formation theory has been developed in
rarefied gas dynamics to model the steady flow of the strong
evaporation from the surface.18–20 As shown in Fig. 1, theKnudsen layer(a)Surface

u

(b)

TK

TS
0

vz

0

vz

FIG. 1. (a) A schematic sketch of the Knudsen layer formed above the surface following the laser induced thermal desorption. The desorption flux isassumed to be large enough for the post-desorption collisions to take place
and to further form the Knudsen layer. The arrows denote the velocity vectors of desorbed atoms at the surface and at the end of the Knudsen layer.
(b) Schematic plots of the velocity distribution of the desorbed atoms in the z(surface normal) direction at the surface and at the end of the Knudsen layer.
TS , TK , and u denote the surface temperature, the translational temperature of
the desorbed atoms at the end of the Knudsen layer, and the stream velocity,respectively.

138, 124705-1© 2013 American Institute of Physics

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initial velocity distribution of thermally desorbed species atthe surface is well described by a “half-range” MaxwellBoltzmann velocity distribution.18 In the Knudsen layer theory, as a result of intensive post-desorption collisions, thehalf-range velocity distribution at the surface is thermallyequilibrated to a full-range Maxwell-Boltzmann velocity distribution with a stream velocity at some distance from thesurface. This thermalization layer is defined as the Knudsen
layer. The theory analytically predicts for monoatomic gasthat the ratio of the translational temperature at the end ofKnudsen layer TK to the surface temperature TS and the Mach
number of the desorption flow at the end of the Knudsen layer
become 0.65 and 1.0, respectively.21, 22
Kelly and Dreyfus discussed that the Knudsen layer
formation theory may be applicable to the pulsed desorption
flow with a large desorption flux. However, it is not straightforward because the pulsed desorption involves complex time
evolution of the density and velocity distributions of desorbed
atoms.23 Sibold and Urbassek have shown by means of theMonte Carlo simulation of the Boltzmann equation that thepulsed desorption flow at an intense flux is well characterized
by the above values predicted by the Knudsen layer formation
theory. Although previous experimental studies recognized
the collision effect in laser desorption,24–28 the formation of
the Knudsen layer in laser desorption has been discussed onlyby theory and simulations.21, 22, 29 So far, any experimental
confirmation of Knudsen layer formation in laser desorption
has not been presented. For an experimental verificationof the theory, a systematic observation of the translationaltemperature and stream velocity of the desorption flow as afunction of the desorption flux is strongly required.
In the present paper, we investigated the laser induced
thermal desorption (LITD) of Xe from an Au(001) surface bymeans of the time-of-flight (TOF) measurement as a functionof the wide range of desorption flux by varying the initial Xe
coverage !. We found that at ! close to 0 ML, the TOF waswell analyzed by a Maxwell-Boltzmann velocity distribution.
Hence, the desorption at ! close to 0 ML is rationalized by thethermal desorption followed by the collision-free flow. At !
close to monolayer, we observed that the peak positions of the
TOF spectra shift towards smaller values. Assuring that thedesorption is only thermally activated, we regard this modification of the TOF as the manifestation of the collision effect.
At larger !, the peak positions of the TOF spectra become
constant. We deduced the Mach number M of the desorption
flow to be 0.96 at large ! under the assumption that TK /TS= 0.65. The obtained value of M and the saturating behaviorof u at ! > 4 ML both well coincide with the previously reported values by the Knudsen layer theory and simulations.
The facts suggest the formation of the Knudsen layer in LITD
at large !. Furthermore, we tentatively estimated the Knudsen number of the initial desorption flow.

ultra high vacuum (UHV) chamber with a base pressure of 2
× 10−8 Pa. The sample surface was cleaned by several cycles
of Ar+ sputtering and annealing at 700 K in the UHV by theelectron bombardment. The cleanliness of the specimen was
confirmed by the observation of a clear reconstructed (5×20)
pattern30 by low-energy electron diffraction and no contamination in the Auger electron spectrum. The sample temperature was monitored with a Chromel-Alumel type thermocouple directly spot-welded to the side of the Au disk. The disk
was cooled down and kept at 23 K by a closed-cycle He compressor type refrigerator during all LITD experiments.
The Xe gas was dosed onto the Au surface by backfillingthe chamber. The Xe coverage ! was monitored by the LITD
yield. The LITD yield showed a linear dependence on the Xe
exposure in the present experimental condition. The saturation coverage at the sample temperature of 80 K was defined
to be 1 ML = 4.5 × 1018 atoms/m2 as in Ref. 31. ! was varied
by varying the dosage of Xe gas onto the Au surface. We notethat the desorption flux is defined as ! divided by the mean
desorption period τ . On the basis of the result employing the
LITD simulation, τ was shown to be constant at about 4 ns
at ! concerned in the present study.32 Hence, the desorption
flux can be varied by varying !.
The TOF measurement of the LITD of Xe from an Au
surface was carried out in the following manner. The experimental setup is schematically shown in Fig. 2. An ArF
excimer laser (Lambda Physik) was used as a pulsed laser
source. The wavelength was 193 nm and the time duration was 8 ns. The incidence angle was 25◦ from the surface normal direction. The irradiated area on the sample disk
was ∼0.1 cm2 . The laser pulse was guided on to the surface through an aperture and a quartz window. The laser
power absorbed by the sample was fixed at 80 mJ/cm2 inall experiments taking the reflectivity into consideration. At
this laser power absorbed by the sample, the LITD of Xe
is the dominant desorption process as shown in Ref. 32.
The reflected light escapes the UHV chamber through another quartz window. A quadrupole mass spectrometer (QMS:

Substrate on a cold headBS
Lens

Aparture

Laser
PD

QMS

TriggerC. Amp.

Oscilloscope

II. EXPERIMENTThe Au specimen and the physisorbed Xe layers wereprepared in the following manner. An Au disk with 001 orientation was cut out from a single-crystal rod and was chemically and mechanically polished. The Au disk was put into an

FIG. 2. A schematic drawing of the experimental setup for the time-of-flightmeasurement of Xe following the laser induced thermal desorption from Au
surfaces. C. Amp., PD, and BS denote fast current amplifier, photo diode, andbeam splitter, respectively.

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source described as,33

300
Exp.
Shifted M-B----- M-B

200
100

Θ = 10 ML

0Intensity (Arb. units)

10050

Θ = 6.7 ML

020
Θ = 1.7 ML

1003

Θ = 0.3 ML

0
0.0

0.5
1.0
Time of Flight (ms)FIG. 3. Time-of-flight (TOF) spectra of Xe from Au surfaces following
pulsed laser irradiations. The Xe coverage ! are 10, 6.7, 1.7, and 0.3 MLfrom top to bottom, respectively. The dashed curves and the solid curves are
the Maxwell-Boltzmann (M-B) velocity distribution and the shifted M-B velocity distribution fitted to the experimental results, respectively. The verticaldashed line and the arrows indicate the peak position of each spectrum.

Balzers QMA125) was placed in the surface normal direction
at a distance of 10 cm from the Au specimen. The QMS wastuned for a high-sensitive detection of Xe with a low mass selectivity. The ion current of the QMS was amplified with a fastcurrent amplifier (Keithley 427) and sent to an oscilloscope(Tektronixs: TDS620B). The desorption of Xe from theAu(001) surface was induced with only one laser pulse in each
LITD experiment. The respective TOF was recorded using theoscilloscope with a single acquisition mode.

III. RESULTS

We obtained a series of TOF spectra of desorbed Xefrom Au surfaces following pulsed laser irradiation as shownin Fig. 3. The TOF spectrum of Xe from the Au surface at! = 0.3 ML is shown at the bottom of Fig. 3. The vertical dashed line at 0.42 ms shows the peak position of thespectrum. The spectrum was well analyzed with a MaxwellBoltzmann velocity distribution in a flux from a thermal

"
!
mv 2
dv,
J (v)dv = Av exp −
2kTD
3

(1)

where m, k, and v are the mass of a Xe atom, the Boltzmann
constant, and the velocity of Xe atoms, respectively, and A
and TD are fitting parameters. In the analysis, we convert
Eq. (1) to a TOF function for the density sensitive detector,
which results in the form f(t)dt = at−4 exp ( − bt−2 )dt. By fitting Eq. (1) to the TOF of Xe at ! = 0.3 ML, we obtained TD
of 255 K as illustrated by the dashed curve at the bottom of
Fig. 3.
At the top and middle of Fig. 3, the TOF spectra of desorbed Xe at ! = 10 – 1.7 ML are shown. We note that thepeak positions of those TOF spectra, which are indicated by
the arrows, are shifted towards smaller values with increasing! compared with that of ! = 0.3 ML. It was found that thepeak positions of the TOF spectra become constant at about
0.34 ms at ! ≥ 5.0 ML.
IV. DISCUSSION
A. Desorption mechanism

We first discuss that the observed desorption proceeds
only via thermal activation. In the previous study,32 we investigated the same system by varying the photon fluence at! = 1 ML. At the small fluence region where the laser poweris below 20 mJ/cm2 , we observed a desorption of Xe only at a
photon energy of 6.4 eV and no desorption at 2.3 eV. We assigned this photodesorption as a non-thermal desorption via atransient Xe− formation. At a large fluence region where laser
power exceeds 60 mJ/cm2 we observed, on the other hand,that thermal desorption is the dominant desorption process
regardless of the photon energy. These observations are confirmed by comparing the experimentally observed TD and Xe
desorption yield with the results of LITD simulations. Therefore, we regard that the initial desorption is only thermally
activated in the present experiment.

B. Analysis of TOF spectra

In order to analyze the TOF spectra at ! > 0.3 ML, we
introduce a shifted Maxwell-Boltzmann velocity distribution
which is characterized by u. Generally, the velocity distribution in a desorption flux shows an angular distribution andis expressed by the elliptical distribution which involves anangular dependent translational temperature as described inRef. 34. In the present experiment, the detector was fixed in
the surface normal direction. Therefore, the translational temperature only in this direction is detected in the present experiment. In this case, the shifted Maxwell-Boltzmann velocity distribution in a flux from a thermal source is expressed
as,21, 29$#m(v − u)2
dv.
J (v)dv = Av exp −
2kTD
3

(2)

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J. Chem. Phys. 138, 124705 (2013)

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124705-4

u (m/s)

150
100
50
0
0

5
Θ (ML)

10

FIG. 4. The stream velocity u of the desorbed Xe atoms from Au surfaces
following the pulsed laser irradiations as a function of the Xe coverage !.
The data are obtained by analyzing the time-of-flight spectrum with theshifted Maxwell-Boltzmann velocity distribution. The solid line is a guidefor eyes. u becomes constant at above 4 ML which is denoted by the arrow.

Under the condition of u = 0, Eq. (2) is identical to Eq. (1).
In the analysis, Eq. (2) was also converted to a TOF function
for the density sensitive detector.
The TOF spectra at ! > 0.3 ML were well analyzed using Eq. (2). However, we found it difficult to obtain a unique
set of TD and u. The spectra were well reproduced with anarbitrary value of TD from 165 to 310 K by adjusting u from
125 to 0 m/s. We note that, with a fixed value of TD between
165 and 310 K, the obtained value of u monotonically increases with increasing ! from 0.3 to 4 ML and becomes
constant at ! > 4 ML. For example, Fig. 4 shows u as afunction of ! obtained by using Eq. (2) with TD fixed at 165
K, of which the fitting curves are shown as solid curves inFig. 3.
C. Moderate desorption at ! ! 0 ML

At ! & 0, the observed feature of the TOF can be rationalized with a model employing thermal desorption followedby collision-free flow. At the instance of thermal desorption,
the desorbed Xe gas is in thermal equilibrium with the surface at the temperature TS . Here, it is reasonable to consider
the half-range Maxwell-Boltzmann velocity distribution fordesorbed Xe expressed as21!
"
mv2
f (v)dv = A exp −
dv,
2kTS
(3)

For a qualitative analysis, we estimated the surface temperature during the present LITD experiment using the firstorder desorption rate equation, assuming that the desorption
activation energy is 240 meV31 and that the desorption period
is 4 ns.35, 36 The calculated results show that the required surface temperature is 260 K, which is in good agreement withthe obtained value of TS = 255 K at ! = 0.3 ML. The agreement indicates that the model of the thermal desorption followed by the collision-free flow well describes the observed
TOF at ! & 0, in good accordance with the results by Cowin14
and Wedler.35
D. Intensive desorption at ! > 4 ML

At ! > 4 ML, we discuss that the desorption flow is
described as an intense flow, where the post-desorption collision modifies the velocity distribution.15, 16 Several theoretical studies have speculated that the so-called Knudsen
layer is formed in the vicinity of the surface as a result ofcollisions.21, 22, 29, 37 As schematically drawn in Fig. 1, in the
Knudsen layer model, the initial velocity distribution is inthermal equilibrium with the surface at TS except that there
is no distribution at vz < 0 as in Eq. (3). As a result of a significant number of post-desorption collisions, the half-range
velocity distribution at z = 0 becomes thermally equilibrated
at some distance from the surface to a full-range MaxwellBoltzmann velocity distribution with a stream velocity
u.18, 19, 21 The model also requires that, at the end of the Knudsen layer, the temperature of the flow becomes identical (TK )in all degrees of freedom. Hence, at the end of the Knudsen
layer, the velocity distribution may be described as18, 19, 21%&m{vx2 + vy2 + (vz − uK )2 }f (v)dv = A exp −
dv,
(4)
2kTK
where vi , uK , and TK are the velocity component in the Cartesian coordinate, stream velocity, and the temperature at theend of the Knudsen layer, respectively. Under the condition
that the flight distance is sufficiently longer than the thicknessof the Knudsen layer besides the conditions described above,
Eq. (4) is converted, as a velocity distribution from a thermal source, to the identical form to Eq. (2). Thus, we see thatTD = TK and u = uK at ! > 4 ML. Equation (2) was shownto well describe the experimental results of the TOFs at !
> 4 ML. This fact, along with the convergence feature of u,
indicates that the experimental result at ! > 4 ML may berationalized with the Knudsen layer formation model.

vz > 0,where A and v are a normalization factor and the velocity vector of Xe atoms in the Cartesian coordinate, respectively. We
note that in Eq. (3) the velocity component in the z direction
has a distribution only at vz > 0,21 assuming the z axis to be
the surface normal direction. Given the desorption period ismuch shorter than the flight time and the irradiation diameter is much smaller than the flight distance, Eq. (3) can be, asa velocity distribution in a flux from a thermal source, transformed to the same form as Eq. (1). Thus, we notice that TD= TS at ! & 0.14, 35

E. Mach number of the flow

We further examine the results at ! > 4 ML by quantitatively comparing the obtained results of uK with previous theoretical reports on Knudsen layer formation in thesteady strong evaporation18, 19 and the pulsed desorption29
from plane surfaces. We compare uK in terms of the Mach
number M at the end of the Knudsen layer. M is defined as
'
m
uK
= uK
,
(5)
M=
c
γ kTK

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TABLE I. Mach number M obtained in the present study and previously
reported values by simulation and theory on the Knudsen layer.
MethodM

Condition

Experimenta
Simulation29
Simulation38
Theory18, 19, 21

0.96
1.0
1.0
0.99

Pulsed flow
Pulsed flow
Steady flow
Steady flowa

TABLE II. Mean gas density n̄, mean free path λ, and Knudsen number Kn
in the vicinity of the surface at the moment of laser desorption as a functionof !.
! (ML)n̄ (1024 atoms/m3 )λ (nm)

Kn

3.3
45
111

2114
159
63

5.2
0.39
0.16

0.3
4.0
10

Present study.

where c and γ are the local velocity of sound and the heat
capacity ratio of the gas, respectively, the latter of which is
5/3 for monoatomic gas as in the present case.
Ytrehus18 and Cercignani19 formulated M and TK /TS byfinding a solution to the Boltzmann equation under the Knudsen layer formation model assuming the conservation of theparticle number, momentum, and the energy flux between thesurface and the end of the Knudsen layer. Sibold and Urbassek
estimated M and TK /TS using the Monte Carlo simulationof the Boltzmann equation in one-dimension for both pulsed
flow29 and steady flow38 conditions. Those reports have statedthat at the end of the Knudsen layer, M should always be close
to unity and that TK /TS is at around 0.65.
As discussed in Sec. IV B, it was difficult to unambiguously determine the value of TD . Here, we assume the relationof TK /TS = 0.65 following the theoretical studies and evaluate
the Mach number. With the value of TS = 255 K, we obtainthe value of TK to be 165 K. With TK = 165 K and Eq. (5),we obtain the value of M to be 0.96. As can be seen in
Table I, the previously reported values of M show a quantitative agreement with the obtained value of M in the present
study. This indicates that the observed trend of u at ! > 4 ML
is consistent with the Knudsen layer formation theory.21
With adiabatic expansion, M should well exceed
unity.37, 39, 40 We note that we observed the saturation of M ataround unity at ! > 4 ML in the present experimental conditions. This indicates that the adiabatic expansion is practicallyabsent, although the slight increase of u at ! > 4 ML may be
due to the adiabatic expansion.
F. Knudsen number of the flow

In order to generalize the present result, we tentatively
consider mean gas density n̄, mean free path λ, and Knudsen
number Kn in the vicinity of the surface at the moment ofdesorption as a function of !. Kn is defined by29
Kn =

λ
1
=√
,
lz
2n̄σ lz˜ and a are the relative coverageas introduced in Ref. 29. !
˜ = !/!S with the monolayer saturation coverage !S and!
the ratio of the area occupied by an atom at !S to σ described
as a = 1/(σ !S ), respectively.
˜ and a. ThereNow, we see that Kn depends simply on !
fore, we can discuss the required condition for the formationof Knudsen layer in LITD for general systems in terms of Kn.
We fixed τ = 4 ns. On the basis of the result employing the
LITD simulation, τ showed little dependence on ! in the region concerned in the present study.
In Table II, we summarized the obtained n̄, λ, and Kn
as a function of !. Kn can also be understood as an inverseof mean collision times per each atoms. In the present study,
we observed the formation of the Knudsen layer at ! > 4
ML, which corresponds to Kn < 0.39 and to more than ∼2.6
collisions per desorbing atom. The manifestation of collisioneffects is observed at ! > 0.3 ML, which corresponds to
Kn > 5.2. The results coincide with the previous theoretical work by Sibold and Urbassek29 that the Knudsen layer
is formed at ! = 2.5 ML and that the collision effect alreadyappears while the Knudsen layer is not formed at ! = 0.25
ML, respectively. The obtained values of Kn also agree wellwith the previous experimental observation of the collision effect in LITD of D2 /W at ! = 1 ML reported by Cowin et al.14
They also coincide with the theoretically estimated collision
number of 2.9 per atom at ! = 1 ML reported by Noorbatcha
et al.15, 16
Finally, we note the observed feature at ! & 1 ML, where
the Knudsen layer is not formed. Thus, the velocity distributions are not necessarily well fitted with Eq. (2). Nevertheless,they were very well analyzed with Eq. (2) in all ! at ! & 1
ML. The result is in good agreement with the simulated results by Sibold and Urbassek,29 although any analytical formulation for the flow at ! & 1 ML has not been presented sofar. Hence, further theoretical and experimental works are required for elucidating the mechanism for the modification ofvelocity distribution at ! & 1 ML due to the post-desorption
collision as pointed out in Ref. 29.

(6)where lz and σ are the representative length of the system andthe van der Waals collision cross section of Xe (1.0 × 10−19m2 ), respectively.
Here, we simply regard that lz is the mean thickness ofthe gas cloud above the surface. It, then, reads that lz = v̄z τ ,where v̄z is the mean thermal velocity of desorbed Xe in the z
direction√ and τ is the mean desorption period. We further takev̄z = 2kTS /πm and estimate that n̄ = !/ lz . By substitut√˜
ing n̄ in Eq. (6), we obtain a simple relation Kn = a/( 2!)

V. CONCLUSION

In conclusion, we observed the TOFs of Xe from aAu(001) surface by LITD at a wide range of !. At ! & 0.3
ML, the TOF is well analyzed by the Maxwell-Boltzmann velocity distribution at 255 K indicating thermal desorption followed by the collision-free flow. At ! > 0.3 ML, the TOF
is well analyzed by the shifted Maxwell-Boltzmann velocity
distribution, which is regarded as the manifestations of the
post-desorption collision effects. At ! > 4 ML, the value of

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u under the assumption that TK /TS = 0.65 became constantat around 125 m/s, which corresponds to M = 0.96, being ingood agreement with the prediction by the Knudsen layer formation theory. In the LITD experiment, we found that the collision effect already appears at Kn & 5.2 and that the Knudsen
layer is formed at Kn < 0.39, which corresponds to a mean
collision number of greater than 2.6 per atom for generalsystems.

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ACKNOWLEDGMENTS

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This work was supported by Grant-in Aid for Scientific
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