- Email: [email protected]
Quantum statistical theory of adsorption and desorption of a gas at a solid surface
Surface Science D North-Holland
100 (1980) Publishing
178-198 Company
QUANTUM STATISTICAL THEORY OF ADSORPTION OF A GAS AT A SOLID SURFACE *
AND DESORFTION
H.J. KRJSJZER Theoretical Physics Institute and Department Alberta, Canada T6G 2JI Received
4 August
1979; accepted
of Physics, University of Alberta, Edmonton,
for publication
12 October
1979
The nonequilibrium initial value problem of phonon-mediated adsorption and desorption of a gas at a solid surface is formulated explicitly in a quantumstatistical theory and adsorption, flash desorption and isothermal desorption times are calculated using a non-local separable surface potential. We find that the flash desorption time is a function of both initial and final temperatures Tg and T, and can, for fixed Tg, be approximated by Frenkel’s formula over a limited range of flash temperatures T,. We fit our theory to flash desorption data for He desorbing from Constantan and find excellent agreement. Predictions are given for the H/NaCl system. Isothermal and flash desorption times are correlated, and experiments are suggested to demonstrate the differences.
1. Introduction The kinetics of adsorption and desorption of a gas at a solid surface is primarily controlled by the molecular interactions between the particles making up the gas and solid phases. Information on the time evolution of a gas in front of a solid surface can be gotten via basically three different experiments. In adsorption experiments a clean solid surface is suddenly exposed to a gas and the build-up of the adsorbate, i.e. of those gas particles that get bound to the solid surface, is followed as a function of time and characterized by an adsorption time t,. Though simple in principle, adsorption experiments are quite difficult in practice [I], so much so that most studies of the kinetics of the gas-solid system have been performed in desorption experiments [ 1,2] of which there are basically two variants. In a flash desorption experiment one typically proceeds as follows: a pure gas of known composition is adsorbed at a surface, previously cleaned at a high temperature. After this controlled adsorption process is completed and an equilibrium adsorbate has been built up at the gas temperature Ts equal to the initial solid temperature, the latter is raised suddenly by a controlled amount [3] to a new solid temperature * Work supported
by the Natural
Sciences
and Engineering
178
Research
Council
of Canada.
H.J. Kreuzer /Quantum
statistics of ad- and desorption
179
T,. As a result, the surplus adsorbate is desorbed in a flash desorption time Id(f) = tdcfj(Tg, T,) which is obviously a function of the initial temperature Tg and the flash temperature T, of the solid [4]. In an isothermal desorption experiment, on the other hand, one typically follows the change in the adsorbate after a sudden isothermal change in one of the external control variables, e.g. after a sudden substantial reduction of the gas pressure either by fast pumping or by chopping a molecular beam impinging onto the surface. In the latter case desorption kinetics can be deduced from the decay time of the desorption signal and can be characterized by an isothermal desorption time td(i)(Ts) which is obviously a function of the temperature Ts of the solid and the gas (isothermal conditions!). Though the kinetics of adsorption and desorption are controlled by the same molecular interactions between the particles making up the gas and solid phases no matter what the experimental procedure is, it is nevertheless necessary to analyse the various experiments by setting up the appropriate initial value problems of nonequilibrium statistical mechanics to calculate and correlate the various characteristic times measured. We do this here for adsorption, flash desorption and isothermal desorption for the simplest gas-solid system, namely one in which localized physisorption dominates. Furthermore we restrict ourselves to pressures such that the gas can be treated as ideal. Also we consider only systems with low surface coverage such that interactions between particles in the adsorbate can be neglected. Many experiments are, indeed, performed under such conditions. Numerical results, comparisons with experiments and suggestions for further experiments will be given for the He/constantan system for which flash desorption experiments have been performed [S]. Estimates for the H/NaCl system will also be given. All our calculations will be performed within one and the same quantum-statistical model as defined by the Hamiltonian (11). In the next section we will study the adsorption of a gas at a virgin surface. We will see that the adsorption proceeds in two steps with greatly different time scales. The first stage is one of fast transients in which the gas particles take notice of the solid surface by readjusting their wavefunctions to the presence of the surface potential. These transients, which we will calculate in an exactly soluble quantum-statistical model, typically evolve in times of the order lo-l3 s and will prepare the gas-solid system in the appropriate state from which the second phase starts. The latter, kinetic regime is controlled by energy-dissipating mechanisms and will lead to the build-up of the gas adsorbate on the surface, i.e. to the filling of the surface bound states. We will calculate the characteristic relaxation time for phonon-mediated adsorption. In sections 3 and 4 we will then calculate the desorption times for flash desorption and isothermal desorption experiments, respectively. We will see that these times are close for low temperatures but differ significantly at moderate and high temperature (“low”, “moderate” and “high” with respect to a typical bound state energy!) so much so that they could easily be distinguished in desorption experiments for He desorbing from constantan. We will also show that the parametriza-
180
H.J. Kreuzer
tion of desorption td = tad exP(Q/knT)
/ Quantum
statistics
of ad- and desorption
times in terms of Frenkel’s formula a
0)
is of limited value. For systems with weak bound states like He/constantan or He/W tEfis of order lo-’ s whereas for systems with several deep bound states t8 is typically lo-i2 s and less, and, in addition, (-ln ta increases with Q, as also found empirically [6], with Q in magnitude generally larger than the energy of the deepest bound state. In section 5 we will present a semi-quantitative discussion of gas-solid systems for which the surface potential develops deep bound states, i.e. of energies large in magnitude compared to the Debye energy of the solid. These arguments will show the direction in which a quantum-statistical theory of chemisorption can be developed.
2. Adsorption
of a gas at a virgin surface
In this section we want to set up and solve the initial value problem for the time evolution of an ideal gas in front of a virgin solid surface. One must expect that this time evolution will in general proceed in two steps with greatly different characteristic times, namely an initial phase of fast transients followed by a much slower phase controlled by energy-dissipating mechanisms. It is only the time scale of the second phase that is generally measured in adsorption experiments, the transients being much too fast to be resolved experimentally. To get a feeling of how such transients might look like and to see what can be learnt from them we have recently set up an exactly soluble one-dimensional quantum statistical model [7], the main results of which we want to summarize briefly. One assumes that for times t < 0 a gas is prepared at a temperature T = T, and is enclosed in a finite box with perfect mathematical walls represented by an external potential that is zero inside the box and infinite outside of it [8]. At time t = 0 the (attractive) wall potential is switched on. The Hamiltonian of the system then reads in second quantization
where ak are annihilation operators of gas particles of mass m in momentum states with energies ek = Azk2/2m in a box of length L. vkk’ are the (momentum space) matrix elements of the wall potential. To specify the wall potential we must note that experimental information about it can be obtained from scattering experiments in the form of phase shift over a limited range of energies and bound state energies. It must be stressed that these data alone do not specify vkk’ uniquely. A first principles calculation of the gassolid interaction, on the other hand, will construct a potential that is in general energy-dependent and non-local in coordinate space. Most frequently the wall po-
k
H.J. Kreuzer / Quantum statistics of ad- and desorption
181
tential is then approximated by a local, energy-independent potential like the Morse potential. However, there is no compelling reason for such a choice because we could equally well approximate the wall potential by a sum of separable terms in such a way that experimental phase shifts and bound state energies are reproduced [9]. For a given local potential such separable approximations are well defined [lo]. They have the great advantage of allowing a complete calculation of single particle wave functions and thus the diagonalization of the static Hamiltonian H,,. If the wall potential develops only one bound state, as it seems to be in the He-W and He-constantan systems [5,11], we can approximate V,,* by just one separable term (this is called the unitary pole approximation [lo], i.e. we write vkk’
= fltik’
.
(3)
In what follows we will choose the potential
form factor
uk = k/(k’ + r2)
(4)
(i.e. the unitary pole approximation for a Hulthen potential [12]), where X = 7-r is the range of the wall potential. In coordinate space this form factor is exp(-yx). So much for the choice of the static wall potential. Returning to our initial value problem we observe that the physical property of interest is the time evolution of the local gas density n(x, t) after the potential v,&’ is switched on. It follows from the time evolution of the statistical operator pt = exp(-%t)
POexdflstO
and is given by (fi = 1) n(x, t) = Tr]Gt(x)
G(x)
where the time evolution $(x, t) = exp(iHJ)
PA= Tr[rLt(x, t) 44, t>PO],
(9
has been shifted from pt to the field operators
+(x) exp(-rHStt)
= (2/L)lj2 q
sin kx ak(t) ,
(6)
and where p e = Z;r exp(-OH,,); Z, = Tr exp(-OH,,) is the statistical operator of the gas at times t < 0 in equilibrium at a temperature T = (k&)-l. We have evaluated eq. (5) exactly for the Hamiltonian (2). Rather than presenting the analytical work involved [7] we show in fig. 1 the density n(x, r)/n plotted over the resealed space-time manifold. Here n is the asymptotic gas density a large distance away from the surface and r = 2 V&i measures the time in units of -h/2 V. where Vo = gX is the strength of the wall potential. The parameters chosen are those for He adsorbing on constantan. The surface potential is such that it develops a bound state for He particles at a energy E,/k, = -2.5 K [5]. The system starts at t = 0 with a density profile n(x, t = 0) = 1 - exp(-x22mkT/h2),
(7)
where we have assumed that the density n is so low that classical Maxwell-Boltz-
182
H.J. Kreuzer / Quantum statistics of ad- and desorption
Fig. 1. Perspective view of the local density evolution n(x, 7)/n, eq. (9, of a gas in a static wall potential, plotted over the resealed space-time manifold. Parameters appropriate for the He/ constantan system at T= 4 K; Y= 0.06,6 = 72, see eq. (18).
mann statistics can be used, though the calculation can be carried out as easily for quantum statistics. Due to the attractive potential mass is drawn towards the wall where it builds up an enhancement ridge within the range of the potential. This surplus mass is taken from a depletion layer further away from the wall which in turn generates secondary small ridges and valleys, barely visible in fig. 1, running as density waves away from the wall, Le. more or less diagonally across the (x, r) plane. The oscillations on the main enhancement ridge are caused by the fact that a suddenly switched-on potential; at first overshoots the redistribution of mass for a few trials before it settkes at t = 00 into a stationary distribution (“over&ability” in the sense of Edd~gton). The basic time scale in this problem so far is &/ZYe = 2.5 X IO-r4 s. However, the oscillations along the bound state ridge have a somewhat larger period of h/2 I,!?, 1% 1O-l3 s.
H.J. Kreuzer / Quantum statistics of ad- and desorption
183
x/(x + 2.5X) Fig. 2. Initial (7 = 0) and final (T = -1 density distribution of the separable potential (4) in coordinate space.
from
fig. 1.
V(x) is the form factor
In fig. 2 we compare the initial and final density profiles, also indicating the POtential form factor in coordinate space. It is clear that a substantial increase in the mass density within the range of the potential has occurred. This is caused by the readjustment of the wave functions in the (external) wall potential, with the shape of the enhancement peak resembling closely the square of the bound state wave function. This large increase in the probability of finding gas particles closest to the wall will be of great significance for the effective coupling of the gas to the phonon system of the solid, as we will show below. Quite a different enhancement mechanism has been discussed recently by Knowles and Suhl [ 131. In their treatment it comes about as a result of the exchange of virtual phonons between gas particles and the solid (surface polarons). We have so far calculated the transients during the initial time evolution of a gas exposed to a virgin solid surface. One might be tempted to think that the large time limit of this evolution coincides with the equilibrium distribution of the gas in front of the attractive wall. This, however, is not the case, as one can see from the following consideration: we have to recall that initially for times t < 0 the gas’was in equilibrium and described by the ideal gas distribution pe = exp(-fiHe)/Ze. Switching on the wall potential at t = 0, will lead to a single-particle evolution due to independent elastic scattering events at the wall. Thus no energy can be exchanged between the particles and between particles and wall, which, in particular, prevents the filling of bound states. All that happens is a readjustment of the single-particle
184
H.J. Kreuzer
/ Quantum
statistics
of ad- and desorption
wavefunctions, enhancing the density in the range of the wall attraction. This purely quantum mechanical effect is, of course, absent in a clasical treatment, which, on the contrary, in the absence of any inelastic processes, leads to a decrease of the density within the range of an attractive potential that is suddenly switched on. This result is due to the fact that classical particles will accelerate in the region of the negative potential, and, therefore, spend less time there. Only in the equilibrium situation of a classical system, particles will have dropped into the potential well by inelastic scattering and oscillate near the bottom, enhancing the density in this region. For the purely elastic time evolution we find, thus, an obvious contrast between the density enhancement found in the quantum mechanical treatment and the depletion found classically, despite the fact that we are dealing in both models with a macroscopic system in the limit of classical Maxwell-Boltzmann statistics. However, we have to observe that the wall potential only extends over atomic distances and that the low quantum number wavefunctions give the dominant contribution to the local density. This is strictly a quantum effect which cannot be simulated in a classical model as one knows from the harmonic oscillator in which only high quantum number states approach the classical probability distribution. Let us stress once more that after the transients, considered so far, have died out the system will find itself with the wavefunctions of the gas paryicles adjusted to the static wall potential but still with no particles trapped into bound states. To do the latter, energy-dissipating mechanisms must be included in the theory which we do by coupling the gas particles to the phonons of the solid by extending the Hamiltonian (2) to read
where b, are annihilation
operators for phonons
gp = 2~(~12%%~,)“2
>
of momentum
p and where (9)
withM, the mass of an atom in the solid of which there are N, in total. To calculate the adsorption time, i.e. the time scale characterizing the slow energy-dissipating phase during the evolution of a gas in front of a virgin surface, we want to perform a time dependent perturbation theory [ 141 on the interaction part of eq. (8). To do this properly we have te rewrite eq. (8) in terms of the annihilalation operators of H,,. We get that by writing for eq. (6) Jl(x, t) =q
(10)
&c(X) a!&) >
where Qk’s are the bound state and continuum eigenfunctions of Hst which, for a separable potential, can be calculated exactly [7,9]. The Hamiltonian (8) then reads Ao,b;bp
H =c
+ E,&,
P
t L-’
c
k,k’
w;wko ,.j&(bjt, P
+ c
k>O
WI&
+ bp)
abk’
>
(11)
N.J. Kreuzer /Quantum
statistics of ad- and desorption
185
with
where the &(q)‘s are the Fourier transforms of @k(x). We have simplified the Hamiltonian to a system for which the wall potential develops only one bound state of energy E,. Gas particles are put into or taken out of that bound state by the action of the operators CL$and oo, respectively. If we then want to know how that bound state fills we have to calculate its occupation no(t) = Tr[c&r)
oe@)
exp(-PHo)l/TrEexp(-PHo)l .
(12)
To second order in g this gives %(MO
= tr,
- r&l
2~lMr
,
(13)
with
5 ;
I?,, :=
(1
exp(-See)
6)
X
w + (2-i/2
WD-EO
3
ei12
- @)a
!
z!WD (1s
dw exp(-6 w) 0
1
1+exp[&(w
(14)
+ eO)] - 11 ’
and exp(-St-0)W,
v, f0)
do exp[6,(w + eO)] --1 ’
cl
where
X Y exp(&‘)
erfc(&S)
+ 4 + (v2 -
[
e0)
i
(
+
)IeAf2
26
exp(6ee) erfc [(~e~)‘/~l
-4(U2-Eo)t/(i . n We also introduced 2lrA
I
the equilibrium
bound state occupation 07)
186
H.J. Kreuzer /Quantum
and the dimensionless
statistics of ad- and desorption
parameters
v = yA/(4m V,,)“2 , 6 = 2 Vo/kBT ,
Gj, = AC+)/2 vo )
where oD is the Debye frequency of the solid. rem is of adsorption due to spontaneous emission (the factor (14)) and stimulated emission (the factor {exp[&,(w a phonon upon impact of a gas particle at the wall. rabs of a phonon by a gas particle close to the wall. Before we analyze F,, and Tabs in detail let us note
co = IEol/2vo ,
(18)
the contribution to the rate 1 in the last bracket in eq. ee)] - 1) -’ in eq. (14)) of is caused by the absorption that eq. (13) is to be inter-
He/Constantan: E,Jke = -25 K x = 2.5 A fm,/ kg = 384 K
10-l-
12 -
la3 B
2 2 lo4 -
lc? -
lo4 -
16’
0
‘/
20
I 40
I 60
I 80
I 100
I 120
L
I
140
160
180
-h'D/keT
Fig. 3. Adsorption tem .
time
ta and pre-exponential
factor
t,“, eq. (22), for the He/constantan
sys-
H.J. Kreuzer / Quantum statistics of ad- and desorption
187
preted as the lowest order term in an expression no(t) = Z0 [ 1 - exp(-t/t,)]
,
(19)
where c, = @/2V0)(rem
- ~&J’
,
(20)
is the time of adsorption. This comment also clarifies the necessity of our, so far, ad hoc normalization of no(t) with the equilibrium bound state occupation n,. Several obvious features of rem and rabs should be pointed out. If we keep the solid at zero temperature, T, = 0, or 6, = m, no phonons are present and only the first term in eq. (14) accounting for spontaneous emission of (non-equilibrium) phonons, survives with stimulated emissions and absorption of (nonexistent thermal) phonons absent. Similarly at high T and T,, one finds from eqs. (14) and (15) a complete compensation of stimulated emission and absorption processes. Eqs. (14) and (15) suggest that we write
r em=r:rnexp(-hd,
rabs = r:,,, exp(-b)
,
(21)
and thus get from eq. (20) t,= -
’
r:m
r.
$- exp(beo)
abs
0
= tt exp(Ge0) .
(22)
In fig. 3 we plot t, and t,” as a function of Aq,/k,T. We see that ti, the pre-factor in Frenkel’s formulas (1) and (22), is indeed only weakly dependent on T. It-is amazing to see that in the He-constantan system at T = 20 K, the time of adsorption t, - lo-’ s is larger than the atomic time h/2V0 - IO-l4 s by about seven orders of magnitude, slowing down to about 3 ms at T = 3 K. Unfortunately there seem to be no measurements of these adsorption times in the He-constantan system. A connection with sticking coefficients has been made in ref. [7].
3. Flash desorption We now want to formulate the initial value problem appropriate for a flash desorption experiment within the framework of the quantum statistical theory outlined so far [ 151. We want to consider in this section the three-dimensional version of our model of localized physisorption by assuming that the localized surface potential is given in coordinate space by
V(r, r’) = g
V(r) V(i)
where we choose
)
(23)
188
H.J. Kreuzer
/ Quantum
statistics
of ad- and desorption
where rc’ is the range of the potential normal to the surface (along the z-direction) and yF1 is the range along the surface with p being the (two-dimensional) component of r in the surface. In three dimensions the Hamiltonian (11) reads H=H,,
+Hs+Hdyn
(25) where u enumerates tors and
the various phonon
J&7, Q’) = gW244)1’2
branches,
k2, [2Y$ + i(G -
ePO are their polarization
vec-
kdl
x w> W’) A@) 4$&q .
(26)
Recal that a flash desorption experiment starts from an initial state in which gas and solid are in equilibrium with each other at a temperature Tg. At time t = 0 the temperature of the solid is suddenly raised to a value T, > T, and the resulting nonequilibrium time evolution, in which surplus adsorbate particles are desorbed, is measured, and, assuming an exponential decay, characterized by a desorption time t,(T,, T,). To formulate the corresponding initial value problem we can assume that the initial equilibrium occupation for t < 0 is adequately described by the static part of the Hamiltonian (25). The macroscopic time evolution in the system is then started at time [ 161 t = 0 by raising the temperature of the phonon bath to T,. This produces a non-equilibrium state of the gas and the solid which can only be equilibrated by the dynamic part of H. Therefore we switch on Hdyn at time t = 0 by writing H = Hs + H,, + e(t) Hdyn >
(27)
where 0(t) = 0 for t < 0 and O(t) = 1 for t > 0. The physical quantity to be calculated is the time dependent occupation of the bound state which is given by eq. (12) where the time dependent statistical operator is D(t) = exp(-iHt/t?)
p(0) exp(-iHt/Fz) = exp(-tit/A)
@,
exp(-iHt/h)
,
(28)
and Ps = exp [-P&H,,
-
&&I /Tr{exp[-P,& - ruir,)l) ,
is the initial equilibrium
(29)
statistical operator for the gas. (30)
is the gas particle
number
operator
and n is the chemical potential.
Similarly we
H.J. Kreuzer /Quantum
statistics of ad- and desorption
189
identify P, = exp(-P&,)/Tr
]exp(-P&)1
,
(31)
as the initial equilibrium statistical operator for the phonons at temperature T, = l/(k&) which is higher than the gas temperature Tg = l/(k&J. The inherent assumption here is that the phonon system reaches its equilibrium at T, much faster than the desorption process itself. Note that the initial equilibrium operator in eq. (28) can always be written in product from p(O) = fi& because H, and H,, and fis commute. We identify the flash desorption time from a second order time-dependent perturbation calculation which yields for a gas-solid system with one surface bound state [15] no(t),‘no(0)=
1 + tp (32)
where 1 nq(0) =
exp [P,&
(33) - ~>1
’
is the initial equilibrium distribution function for the gas particles with the + (-) sign applying if the gas particles obey Fermi-Dirac (Bose-Einstein) statistics. The energies Eq are the eigenvalues of H,, with 4 = 0 referring to the bound state and 4 # 0 enumerating the continuum states. Similarly (33) gives the phonon distribution at inverse temperature 0,. Within the framework of perturbation theory used to derive eq. (32) the linear time dependence is interpreted (Fermi’s Golden Rule) as the lowest order term in an expression n O(t) ‘= not=)
+ b O(O) - fi O(=)l
exp(-t/td(f))
(34)
.
We must choose such an ansatz (rather than an expression no(t)= no(O) exp [-cd]) to insure that the adsorbate develops from the initial occupation no(O) at temperature Tg to the final equilibrium occupation no(-) at temperature T,. With eq. (34) we can identify the flash desorption time td(f) from eq. (32) as
1 J-
312
-1
(WD-EO)/v2
exp ]@, - 6s) E01
du
0
)
exp[6S(v2z4 + ee)] - 1 _._~_____ x { [ 1 + I(u>l@v2)]2 + [J(u))/(4v~)]2}-’
0
.
(35)
190
H.J. Kreuzer / Quantum statistics of ad- and desorption
Here we introduced
k!
vo=
the quantities
u = y$zl(4mV,)“2
)
6 g,s = 2 Vok&,,s
>
f() = IEo 1/2Vo )
71 ’ r = Y~YI ,
Gj, = &D/2
vo )
(36)
where E. is the bound state energy. We also defined L,,‘=4
X(1 +X2)-3 [ 1 t (2x2
J 0
t fo/v2)1’2] -3(r2x2
t E~/v’)-~‘~ & ,
J;lr X(U - r2x2 - l)(l
Z(u)=4
+x2)-“(u
- 2x2
t
1)-2 dx
CJ 0 m
s
x(1 tx2)-3[1
+ (r2x2 - u)~‘~]-’
dx
d/r J(u) = 4
s 0
, 1
+ x~)-~(U ~ r2x2 + 1)-2 dx .
X(U - r2x2)“‘(1
The parameter v is connected
(37)
with the bound state energy (-co) via
3 = 1
2 s x( 1 +x2)-3 [ 1 + (r2x2 + eO/v2)] -2 dx . 0
(38)
Let us note that in the one-dimensional version of our model (which results if we take the limit r = y~/y~ + 0 and identify y = 73 the flash desorption time is given by _1 _
2V,
--T~TM,
*w
x
3
m
v (I-)
3 &/‘[I
-(~~2expl(s,-63Col]-1
sL;D-Eo
0
exp[(6, - S,)(w t eO)] - 1 I_. ‘w w + (2-1’2 - #2)2 expP,(w + CO)] - 1
dw
.
(39)
We can test our theory of flash desorption against an experiment performed by Cohen and King [5] for He desorbing from constantan. This system is particularly suited for our theory because its low heat of adsorption of the order of 30 K indicates that the He-constantan surface potential develops only one bound state. The experiments of Cohen and King [5] have been conducted starting from an initial temperature Tg - 2 K. Their experimental flash desorption times for He desorption from constantan are given as, see eq. (l), *d = (2.0 f 0.5) X 10e7 s exp[(31
f 1) K/T,]
,
(40)
H.J. Kreuzer / Quantum statistics of ad- and desorption
191
He/Constantan Edke .3 _
= -25K
XI _ Xl, =X = 2.5 -ho,/ kg = 304K
B,
b
7
10
30
50
70
90
110
130
fiwdksTs Fig. 4. Flash desorption times td(f) versus flash temperature Ts for He desorbing from Constantan, mass ratio m/M, = 0.065. Initial temperature Tg = 2 K. Dashed curve and point: measurements by Cohen and King (51 with errors indicated by cross hatched area. Solid curve calculated from eq. (35) (with r = 1 three-dimensional) or from eq. (39) (with r = 0 one-dimensional) agree to within a few percent.
for 4 K < T,< 18 K with an additional points at T, = 26 K where td = 1.4 X lo-’ s, which seems doubtful to the authors [5]. To fit our theory to these experiments we observe that our theory has in principle three independent parameters, namely E,, yi and yl. We must, however, expect that E. s’ Q and that 77’ =S + z%0.2-0.3 run. One finds that the slope of the linear part of log tdcfj versus TJ-'is mainly controlled by yI for reasonable values of r = yu/yI. In fig. 4 we give the best fit of our theory to the experiments of Cohen and King [5] for both the 3D and the 1D versions. Several points, demonstrated in detail in ref. [ 1.51, are noteworthy: (i) The bound state energy IE,I/k, = 25 K is about 25% less than the slope Q/k, = (31 + 1) K, the slope generally increasing with increasing E,. (ii) Changing ri’ from 0.1-0.3 nm increases &j(f) by a factor 25. The best value rll =
H.J. Kreuzer / Quantum statistics of ad- and desorption
192
0.25 mn agrees with related calculations [ 171. (iii) Flash desorption times in the 1D and the 3D versions of the theory (with all parameters the same) are comparable. This is in contrast to speculations in the literature [ 171 (not supported by explicit calculations) that pre-exponential times tz of the order of lo-’ s cannot be obtained in 1D theories with reasonable potential parameters [ 181. (iv) Frenkel’s formula (1) can only be a valid parametrization over a limited range of flash temperatures, for the system under study for T, = T, = 2 K up to about T, - 15 K. For higher T, the desorption times are shorter. It is interesting to note that the experimental point at T, = 25 K which Cohen and King [S] seem to doubt (primarily, it seems, because it does not fit Frenkel’s formula) lies exactly on the theoretical curve!
4. Isothermal
desorption
Next we formulate the initial value problem for isothermal desorption after a sudden substantial reduction of the gas pressure 1191. The latter instant is t = 0. The initial state for t < 0 is now one in which the gas and the solid are in thermal equilibrium at temperature T,. The final state is one in which all gas particles have been pumped out of the system. In eq. (34) we must therefore take n&t = -) = 0 and find for the resulting isothermal desorption time in the 1D model
0
w
+
(2 ‘I2 - &r/2)2 exp[6,(e0
+ w)] - 1 ’
(41)
Note that the term exp[(6, -- 6J(eo + w)]/{exp[?i,(w + eO)] - 1) in eq. (39) which corresponds to readsorption of gas particles accompanied by spontaneous and stimulated emission of phonons is absent in eq. (4 l), the remaining term simply representing the absorption of phonons by gas particles desorbing from the solid surface. Almost all calculations of desorption kinetics in the literature deal with isothermal desorption [ 17,181, though most of them are rather phenomenological in nature [20]. Also note that at very low temperatures, i.e. for k,T < IEo 1, the adsorption time t, in eq. (20) coincides with the isothermal desorption time d(i) in eq. (41). In fig. 5 we plot the isothermal desorption time for He desorbing from Constantan. It is obvious that only in the low-temperature region, i.e. for T, 2 4 K, can td(t) be approximated by Frenkel’s formula Id(i) = TV exp(Q/kBTs), with f$(t) and Q given in fig. 5. Note that the energy constant Q is larger than the binding energy (-E,) by some 20% and that the prefactor t8~i, is of order 1Oe7 s for this system. To compare isothermal desorption with flash desorption times we first look at the lat-
H.J. Kreuzer / Quantum statistics of ad- and desorption
A0
10
193
TsIKl
A
2
10-l -
10-Z-
10-3-
3
R .=_ 5
IF4
-
10-5 -
Fig. 5. Isothermal desorption time td(i), eq. (41), for He desorbing from constantan. It coincides to within a few percent with the equal temperature flash desorption time Id(f) (Ts, 7’s), eq. (42).
ter for an infinitesimal t&(Ts,TJ=
TlimT t-’ d S*
temperature
step, i.e. we take
Tg , T,)
g
(42)
H.J. Kreuzer / Quantum statistics of ad- and desorption
194
80
B J
40
T&l I5
25
td(fi
Fe = 25 K. Ts)
‘0.
10.
IO
20
30
hwdke‘rs
Fig. 6. Various desorption times for He desorbing from constantan.
Note that in this limit no real time evolution takes place as nothing is done to the system. Still this limit can be useful for comparison with isothermal desorption times. For He desorbing from Constantan it turns out that td(i)(Ts) is less than td(f)(Ts, T,) by less than 1% for T, 2 4 K, increasing to some 30% at T, = 40 K. The latter temperature region is explored in some detail in fig. 6 where we plotted the isothermal desorption time fd(i)(Ts), the equal temperature limit td(f)(Ts, T,) of the flash desorption time and also the full t,j(f)(Ts, T,) for RGJ&BT~ = 15 fixed. Note also the limit
H.J. Kreuzer / Quantwn statisticsof ad- and desorption
195
It would be interesting to see whether these differences between isothermal and flash desorption times can be verified experimentally.
5. Resorption from deep bound states We have so far been dealing with localized physisorption in gas-solid S~Sin which the surface potential develops only one bound state; gas particles trapped in that bound state define the adsorbate. We have also tacitly assumed that the energy Ee of the bound state is, in magnitude, less than the Debye energy of the solid, i.e. that l&o]<&on, an assumption well justified e.g. in the He/constantan system. Under such c~cumstances the bound state occupation can be changed by the emission or adsorption of one phonon implying that relaxation times can be calculated in second order time dependent perturbation theory. The situation changes drastically if the surface potential of a particular gas-solid system deveIops a deep bound state of energy Fa (or several of them) i.e. one for which ]Es I > fiioD besides one (or several) with IE, [ <&.Q,. A gas particle trapped in such a deep bound state can leave it, i.e. desorb, via basically two mechanisms. Firstly, it can absorb two pbonons (of total energy larger than ]Es i) simultaneou~y; this process is very unlikely at low temperatures where the number of phono~s, particularly those with energy&w S k,T, is very low. The desorption from a deep bound state can, secondly, proceed in a two-step cascade E2 +E, + continuum in which two phonons are emitted in succession, For either process higher, i.e. at least fourth, order time dependent perturbation theory must be invoked, an afternative to this approach being a kinetic one in which microscopically calculated relaxation times are used in phenomenological rate equations. It seem plausible that an adsorption cascade, as just outlined, is one possible mechanism of chemisorption: a gas particle must be physisorbed into a weak bound state before it can undergo a chemical reaction and get trapped into a very deep bound state. Work along these lines is in progress [24,25]. We want to conclude this paper by estimating the isothermal desorption time for atomic hydrogen desorbing from NaCi. At low coverage and at low temperatures hydrogen has essentially zero mobility along the surface and will desorb in atomic form [21]. The bound states of hydrogen on an NaCl(OOI~surface have recently been measured by Iannotta and Valbusa [22] to be E&u = -351 K, E2/kB = -247 K, Es = -166 K, and Ed,fkB= -1lOK. The Debye temperature of NaCl is FiqJkB = 280 K for T > 50 K dropping to 275 K for T - 35 K below which it rises to 3 10 K [23 J. Hydrogen can therefore desorb from the higher bound states E,, Es and E4 by absorbing a single phonon, Neglecting the lowest bound state Er for the moment, we can estimate the desorption time by simply adding the properly normalized rates from n = 3 individual bound states within reach of a single phonon, i.e. a bound state of energy Ei is produced by a separable potential of the form (3) and (41 of the same range X = y-r, but of variable strength V, = g&. This yields from terns
H.J. Kreuzer / Quantum statistics of ad- and desorption
196
eq. (41)
X
1 (43)
exp[&‘,(ci - W)] - 1 ’
where fi=
iE,l/2Vf,
Vi = yfil(4m Vi)1’2 .
Gb = FlWL)/2Vi ,
In this approximation we estimate the isothermal drogen desorbing from NaCl(001) to be td(i) =f&i) exp(Q/kd’)
- 3.05 X lo-”
s
desorption
exp(265 K/7’) ,
time for atomic hy-
for lS
cd(i) - 1 .O3 X lo-’
s
exp(25 1 K/T) ,
for4
10K. (45)
In the higher temperature range td(i) is of the order of milli- to microseconds. Note also that as the temperature is lowered the preexponential factor t:(i) increases and the energy constant Q decreases in such a way that -1n f&i) -
Q,
(46)
is satisfied very well, as found experimentally on other systems with deep bound states [6]. To see how severe an approximation the neglect of the deepest bound state actually is, one can look at a system with the Debye energy raised artificially so that Fiio,, > IE, 1. Taking fiuD/kB = 360 K we have calculated the isothermal desorption time for all four bound states of the H/NaCl system and found td(i) - 3.2 X lo-r” td(i) - lo-’
S
S
exp(384 K/Z’) ,
exp(365 K/T) ,
for 15
(47)
for4
(48)
The pre-exponential factors are very close to the ones estimated in eqs. (44) and (45) whereas the energy constants Q are somewhat larger. Again the empirical rule (46) is very well satisfied. A detailed calculation of desorption times for the H/NaCl system, including multi-phonon processes, will be given elsewhere [24,25].
References [l]
A discussion of adsorption experiments Ehrlich, Advan. Catalysis 14 (1963) 25.5.
and some
experimental
results
are given by G.
H.J. Kreuzer f Quantum statistics of ad- and desorption [2] For a recent review see, e.g. D. Menzel, mer (Springer, New York, 1975).
in: Topics
in Applied
Physics,
197
Vol. 4, Ed. R. Go-
[3] If the temperature of the solid is raised in a sequence of small steps between each of which the system has enough time to reach its new equilibrium, the procedure is known as temperature-programmed desorption. [4] A temperature-programmed desorption experiment ideally measures the limit limTs+Tg Id(f) (Tg, Ts). [5] S.A. Cohen and J.G. King, Phys. Rev. Letters 31 (1973) 703. (61 J. Lapujoulade, Nuovo Cimento Suppl. 5 (1967) 433; A.K. Mazumdar and H.W. Wassmuth, Surface Sci. 34 (1972) 617. [7] H.J. Kreuzer and G.M. Obermair, J. Phys., to be published; Similar models have been studied in a different context by H.J. Kreuzer and R. Teshima, Physica 87A (1977) 453; and H.J. Kreuzer, Nuovo Cimento 45B (1978) 169. [8] Transient phenomena in the time evolution of a system depend quite sensitively on the initial conditions, i.e. on the preparation of the initial perturbation of the system to which the transients are the response. Our model assumes that the gas fills and adjusts to the container before a dynamical interaction with the walls takes place, i.e. we assume that fi/kT Q fi/IEol, where E, is an energy characterizing the solid-gas interaction, e.g. the energy of the weakest bound state of the surface potential. [9] See, e.g., N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions (Oxford Univ. Press, London, 1965); K.M. Watson and J. Nuttall, Topics in Several Particle Dynamics )Holden-Day, San Franco, 1967). [lo] For a recent review see, e.g., J.S. Levinger, in: Springer Tracts in Modern Physics, Vol. 71, Ed. G. HBhler (Springer, Berlin, 1974) in particular pp. 119-166; also Z.W. Gortel, H.J. Kreuzer and R. Teshima. Can. J. Phys. 58 (1980) 376. [ 1 l] F.O. Goodman and H.Y. Wachman, Dynamics of Gas-Surface Scattering (Academic Press, New York, 1976). [12] A.G. Sitenko, V.F. Kharchenko and N.M. Petrov, Phys. Letters 28B (1968) 308. (13) T.R. Knowles and H. Suhl, Phys. Rev. Letters 39 (1977) 1417. [ 141 This type of approach was advocated by J.E. Lennard-Jones and C. Strachan, Proc. Roy. Sot. (London) A150 (1935) 442;and J.E. Lennard-Jones and A.F. Devonshire, Proc. Roy. Sot. (London) Al56 (1936) 6,29; Al58 (1937) 242,253. [ 151 Z.W. Gortel, H.J. Kreuzer and D. Spaner, J. Chem. Phys. 72 (1980) 234. [ 161 More precisely we put t = 0 at that instant when the rapid heating of the solid from Tg to TS is completed. This implies that the heating time is much less than the desorption time, a necessary condition for performing a flash desorption experiment. [17] B. Bendow and S.C. Ying, Phys. Rev. B7 (1973) 622,637. [18] Our conclusions are supported by recent model calculations by F.O. Goodman and I. Romero, J. Chem. Phys. 69 (1978) 1086. [19] Z.W. Gortel and H.J. Kreuzer, Chem. Phys. Letters 67 (1979) 197. [20] F.O. Goodman, Surface Sci. 24 (1971) 667 and 60 (1976) 45; G. Armand, in: 6th Rarefied Gas Dynamics Symp., Cambridge, MA, 1968; P.J. Pagni and J.C. Keck, J. Chem. Phys. 58 (1973) 1162; P.J. Pagni, J. Chem. Phys. 58 (1973) 2920; L. Trilling, in: 8th Rarefied Gas Dynamics Symp. (Academic Press, New York, 1974). [21] E.g., hydrogen adsorbed on tungsten only shows diffusion for T > 180 K as observed by R. Gomer, R. Wortman and R. Lundy, J. Chem. Phys. 26 (1957) 1147. For a recent discussion, see K.J. Vette, T.W. Orent, D.K. Hoffman and R.S. Hansen, J. Chem. Phys. 60 (1974) 4854;and D.K. Hoffman, J. Chem. Phys. 65 (1976) 95. [22] S. Iannotta and U. Valbusa, Surface Sci. 100 (1980) 28.
198
H.J. Kreuzer / Quantum statistics of ad- and desorption
[23] E.W. Kellermann, Proc. Roy. Sot. (London) Al79 (1941) 17. For a recent discussion see ,4.A. Maradudin, E.W. Montroll, G.H. Weiss and I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, in: Solid State Physics, Suppl. 3, Eds. H. Ehrenreich, F. Seitz and D. TurnbuU (Academic Press, New York, 1971). [ 241 A complete fourth order calculation including all one-phonon and two-phonon processes has just been completed by Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B, to be published. [25] Desorption via one-phonon cascades has been calculated by Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B, submitted. For the H/NaCI system, we find tscij = 7.2 X 10-l 2 s, somewhat shorter than the estimate in eq. (47).
100 (1980) Publishing
178-198 Company
QUANTUM STATISTICAL THEORY OF ADSORPTION OF A GAS AT A SOLID SURFACE *
AND DESORFTION
H.J. KRJSJZER Theoretical Physics Institute and Department Alberta, Canada T6G 2JI Received
4 August
1979; accepted
of Physics, University of Alberta, Edmonton,
for publication
12 October
1979
The nonequilibrium initial value problem of phonon-mediated adsorption and desorption of a gas at a solid surface is formulated explicitly in a quantumstatistical theory and adsorption, flash desorption and isothermal desorption times are calculated using a non-local separable surface potential. We find that the flash desorption time is a function of both initial and final temperatures Tg and T, and can, for fixed Tg, be approximated by Frenkel’s formula over a limited range of flash temperatures T,. We fit our theory to flash desorption data for He desorbing from Constantan and find excellent agreement. Predictions are given for the H/NaCl system. Isothermal and flash desorption times are correlated, and experiments are suggested to demonstrate the differences.
1. Introduction The kinetics of adsorption and desorption of a gas at a solid surface is primarily controlled by the molecular interactions between the particles making up the gas and solid phases. Information on the time evolution of a gas in front of a solid surface can be gotten via basically three different experiments. In adsorption experiments a clean solid surface is suddenly exposed to a gas and the build-up of the adsorbate, i.e. of those gas particles that get bound to the solid surface, is followed as a function of time and characterized by an adsorption time t,. Though simple in principle, adsorption experiments are quite difficult in practice [I], so much so that most studies of the kinetics of the gas-solid system have been performed in desorption experiments [ 1,2] of which there are basically two variants. In a flash desorption experiment one typically proceeds as follows: a pure gas of known composition is adsorbed at a surface, previously cleaned at a high temperature. After this controlled adsorption process is completed and an equilibrium adsorbate has been built up at the gas temperature Ts equal to the initial solid temperature, the latter is raised suddenly by a controlled amount [3] to a new solid temperature * Work supported
by the Natural
Sciences
and Engineering
178
Research
Council
of Canada.
H.J. Kreuzer /Quantum
statistics of ad- and desorption
179
T,. As a result, the surplus adsorbate is desorbed in a flash desorption time Id(f) = tdcfj(Tg, T,) which is obviously a function of the initial temperature Tg and the flash temperature T, of the solid [4]. In an isothermal desorption experiment, on the other hand, one typically follows the change in the adsorbate after a sudden isothermal change in one of the external control variables, e.g. after a sudden substantial reduction of the gas pressure either by fast pumping or by chopping a molecular beam impinging onto the surface. In the latter case desorption kinetics can be deduced from the decay time of the desorption signal and can be characterized by an isothermal desorption time td(i)(Ts) which is obviously a function of the temperature Ts of the solid and the gas (isothermal conditions!). Though the kinetics of adsorption and desorption are controlled by the same molecular interactions between the particles making up the gas and solid phases no matter what the experimental procedure is, it is nevertheless necessary to analyse the various experiments by setting up the appropriate initial value problems of nonequilibrium statistical mechanics to calculate and correlate the various characteristic times measured. We do this here for adsorption, flash desorption and isothermal desorption for the simplest gas-solid system, namely one in which localized physisorption dominates. Furthermore we restrict ourselves to pressures such that the gas can be treated as ideal. Also we consider only systems with low surface coverage such that interactions between particles in the adsorbate can be neglected. Many experiments are, indeed, performed under such conditions. Numerical results, comparisons with experiments and suggestions for further experiments will be given for the He/constantan system for which flash desorption experiments have been performed [S]. Estimates for the H/NaCl system will also be given. All our calculations will be performed within one and the same quantum-statistical model as defined by the Hamiltonian (11). In the next section we will study the adsorption of a gas at a virgin surface. We will see that the adsorption proceeds in two steps with greatly different time scales. The first stage is one of fast transients in which the gas particles take notice of the solid surface by readjusting their wavefunctions to the presence of the surface potential. These transients, which we will calculate in an exactly soluble quantum-statistical model, typically evolve in times of the order lo-l3 s and will prepare the gas-solid system in the appropriate state from which the second phase starts. The latter, kinetic regime is controlled by energy-dissipating mechanisms and will lead to the build-up of the gas adsorbate on the surface, i.e. to the filling of the surface bound states. We will calculate the characteristic relaxation time for phonon-mediated adsorption. In sections 3 and 4 we will then calculate the desorption times for flash desorption and isothermal desorption experiments, respectively. We will see that these times are close for low temperatures but differ significantly at moderate and high temperature (“low”, “moderate” and “high” with respect to a typical bound state energy!) so much so that they could easily be distinguished in desorption experiments for He desorbing from constantan. We will also show that the parametriza-
180
H.J. Kreuzer
tion of desorption td = tad exP(Q/knT)
/ Quantum
statistics
of ad- and desorption
times in terms of Frenkel’s formula a
0)
is of limited value. For systems with weak bound states like He/constantan or He/W tEfis of order lo-’ s whereas for systems with several deep bound states t8 is typically lo-i2 s and less, and, in addition, (-ln ta increases with Q, as also found empirically [6], with Q in magnitude generally larger than the energy of the deepest bound state. In section 5 we will present a semi-quantitative discussion of gas-solid systems for which the surface potential develops deep bound states, i.e. of energies large in magnitude compared to the Debye energy of the solid. These arguments will show the direction in which a quantum-statistical theory of chemisorption can be developed.
2. Adsorption
of a gas at a virgin surface
In this section we want to set up and solve the initial value problem for the time evolution of an ideal gas in front of a virgin solid surface. One must expect that this time evolution will in general proceed in two steps with greatly different characteristic times, namely an initial phase of fast transients followed by a much slower phase controlled by energy-dissipating mechanisms. It is only the time scale of the second phase that is generally measured in adsorption experiments, the transients being much too fast to be resolved experimentally. To get a feeling of how such transients might look like and to see what can be learnt from them we have recently set up an exactly soluble one-dimensional quantum statistical model [7], the main results of which we want to summarize briefly. One assumes that for times t < 0 a gas is prepared at a temperature T = T, and is enclosed in a finite box with perfect mathematical walls represented by an external potential that is zero inside the box and infinite outside of it [8]. At time t = 0 the (attractive) wall potential is switched on. The Hamiltonian of the system then reads in second quantization
where ak are annihilation operators of gas particles of mass m in momentum states with energies ek = Azk2/2m in a box of length L. vkk’ are the (momentum space) matrix elements of the wall potential. To specify the wall potential we must note that experimental information about it can be obtained from scattering experiments in the form of phase shift over a limited range of energies and bound state energies. It must be stressed that these data alone do not specify vkk’ uniquely. A first principles calculation of the gassolid interaction, on the other hand, will construct a potential that is in general energy-dependent and non-local in coordinate space. Most frequently the wall po-
k
H.J. Kreuzer / Quantum statistics of ad- and desorption
181
tential is then approximated by a local, energy-independent potential like the Morse potential. However, there is no compelling reason for such a choice because we could equally well approximate the wall potential by a sum of separable terms in such a way that experimental phase shifts and bound state energies are reproduced [9]. For a given local potential such separable approximations are well defined [lo]. They have the great advantage of allowing a complete calculation of single particle wave functions and thus the diagonalization of the static Hamiltonian H,,. If the wall potential develops only one bound state, as it seems to be in the He-W and He-constantan systems [5,11], we can approximate V,,* by just one separable term (this is called the unitary pole approximation [lo], i.e. we write vkk’
= fltik’
.
(3)
In what follows we will choose the potential
form factor
uk = k/(k’ + r2)
(4)
(i.e. the unitary pole approximation for a Hulthen potential [12]), where X = 7-r is the range of the wall potential. In coordinate space this form factor is exp(-yx). So much for the choice of the static wall potential. Returning to our initial value problem we observe that the physical property of interest is the time evolution of the local gas density n(x, t) after the potential v,&’ is switched on. It follows from the time evolution of the statistical operator pt = exp(-%t)
POexdflstO
and is given by (fi = 1) n(x, t) = Tr]Gt(x)
G(x)
where the time evolution $(x, t) = exp(iHJ)
PA= Tr[rLt(x, t) 44, t>PO],
(9
has been shifted from pt to the field operators
+(x) exp(-rHStt)
= (2/L)lj2 q
sin kx ak(t) ,
(6)
and where p e = Z;r exp(-OH,,); Z, = Tr exp(-OH,,) is the statistical operator of the gas at times t < 0 in equilibrium at a temperature T = (k&)-l. We have evaluated eq. (5) exactly for the Hamiltonian (2). Rather than presenting the analytical work involved [7] we show in fig. 1 the density n(x, r)/n plotted over the resealed space-time manifold. Here n is the asymptotic gas density a large distance away from the surface and r = 2 V&i measures the time in units of -h/2 V. where Vo = gX is the strength of the wall potential. The parameters chosen are those for He adsorbing on constantan. The surface potential is such that it develops a bound state for He particles at a energy E,/k, = -2.5 K [5]. The system starts at t = 0 with a density profile n(x, t = 0) = 1 - exp(-x22mkT/h2),
(7)
where we have assumed that the density n is so low that classical Maxwell-Boltz-
182
H.J. Kreuzer / Quantum statistics of ad- and desorption
Fig. 1. Perspective view of the local density evolution n(x, 7)/n, eq. (9, of a gas in a static wall potential, plotted over the resealed space-time manifold. Parameters appropriate for the He/ constantan system at T= 4 K; Y= 0.06,6 = 72, see eq. (18).
mann statistics can be used, though the calculation can be carried out as easily for quantum statistics. Due to the attractive potential mass is drawn towards the wall where it builds up an enhancement ridge within the range of the potential. This surplus mass is taken from a depletion layer further away from the wall which in turn generates secondary small ridges and valleys, barely visible in fig. 1, running as density waves away from the wall, Le. more or less diagonally across the (x, r) plane. The oscillations on the main enhancement ridge are caused by the fact that a suddenly switched-on potential; at first overshoots the redistribution of mass for a few trials before it settkes at t = 00 into a stationary distribution (“over&ability” in the sense of Edd~gton). The basic time scale in this problem so far is &/ZYe = 2.5 X IO-r4 s. However, the oscillations along the bound state ridge have a somewhat larger period of h/2 I,!?, 1% 1O-l3 s.
H.J. Kreuzer / Quantum statistics of ad- and desorption
183
x/(x + 2.5X) Fig. 2. Initial (7 = 0) and final (T = -1 density distribution of the separable potential (4) in coordinate space.
from
fig. 1.
V(x) is the form factor
In fig. 2 we compare the initial and final density profiles, also indicating the POtential form factor in coordinate space. It is clear that a substantial increase in the mass density within the range of the potential has occurred. This is caused by the readjustment of the wave functions in the (external) wall potential, with the shape of the enhancement peak resembling closely the square of the bound state wave function. This large increase in the probability of finding gas particles closest to the wall will be of great significance for the effective coupling of the gas to the phonon system of the solid, as we will show below. Quite a different enhancement mechanism has been discussed recently by Knowles and Suhl [ 131. In their treatment it comes about as a result of the exchange of virtual phonons between gas particles and the solid (surface polarons). We have so far calculated the transients during the initial time evolution of a gas exposed to a virgin solid surface. One might be tempted to think that the large time limit of this evolution coincides with the equilibrium distribution of the gas in front of the attractive wall. This, however, is not the case, as one can see from the following consideration: we have to recall that initially for times t < 0 the gas’was in equilibrium and described by the ideal gas distribution pe = exp(-fiHe)/Ze. Switching on the wall potential at t = 0, will lead to a single-particle evolution due to independent elastic scattering events at the wall. Thus no energy can be exchanged between the particles and between particles and wall, which, in particular, prevents the filling of bound states. All that happens is a readjustment of the single-particle
184
H.J. Kreuzer
/ Quantum
statistics
of ad- and desorption
wavefunctions, enhancing the density in the range of the wall attraction. This purely quantum mechanical effect is, of course, absent in a clasical treatment, which, on the contrary, in the absence of any inelastic processes, leads to a decrease of the density within the range of an attractive potential that is suddenly switched on. This result is due to the fact that classical particles will accelerate in the region of the negative potential, and, therefore, spend less time there. Only in the equilibrium situation of a classical system, particles will have dropped into the potential well by inelastic scattering and oscillate near the bottom, enhancing the density in this region. For the purely elastic time evolution we find, thus, an obvious contrast between the density enhancement found in the quantum mechanical treatment and the depletion found classically, despite the fact that we are dealing in both models with a macroscopic system in the limit of classical Maxwell-Boltzmann statistics. However, we have to observe that the wall potential only extends over atomic distances and that the low quantum number wavefunctions give the dominant contribution to the local density. This is strictly a quantum effect which cannot be simulated in a classical model as one knows from the harmonic oscillator in which only high quantum number states approach the classical probability distribution. Let us stress once more that after the transients, considered so far, have died out the system will find itself with the wavefunctions of the gas paryicles adjusted to the static wall potential but still with no particles trapped into bound states. To do the latter, energy-dissipating mechanisms must be included in the theory which we do by coupling the gas particles to the phonons of the solid by extending the Hamiltonian (2) to read
where b, are annihilation
operators for phonons
gp = 2~(~12%%~,)“2
>
of momentum
p and where (9)
withM, the mass of an atom in the solid of which there are N, in total. To calculate the adsorption time, i.e. the time scale characterizing the slow energy-dissipating phase during the evolution of a gas in front of a virgin surface, we want to perform a time dependent perturbation theory [ 141 on the interaction part of eq. (8). To do this properly we have te rewrite eq. (8) in terms of the annihilalation operators of H,,. We get that by writing for eq. (6) Jl(x, t) =q
(10)
&c(X) a!&) >
where Qk’s are the bound state and continuum eigenfunctions of Hst which, for a separable potential, can be calculated exactly [7,9]. The Hamiltonian (8) then reads Ao,b;bp
H =c
+ E,&,
P
t L-’
c
k,k’
w;wko ,.j&(bjt, P
+ c
k>O
WI&
+ bp)
abk’
>
(11)
N.J. Kreuzer /Quantum
statistics of ad- and desorption
185
with
where the &(q)‘s are the Fourier transforms of @k(x). We have simplified the Hamiltonian to a system for which the wall potential develops only one bound state of energy E,. Gas particles are put into or taken out of that bound state by the action of the operators CL$and oo, respectively. If we then want to know how that bound state fills we have to calculate its occupation no(t) = Tr[c&r)
oe@)
exp(-PHo)l/TrEexp(-PHo)l .
(12)
To second order in g this gives %(MO
= tr,
- r&l
2~lMr
,
(13)
with
5 ;
I?,, :=
(1
exp(-See)
6)
X
w + (2-i/2
WD-EO
3
ei12
- @)a
!
z!WD (1s
dw exp(-6 w) 0
1
1+exp[&(w
(14)
+ eO)] - 11 ’
and exp(-St-0)W,
v, f0)
do exp[6,(w + eO)] --1 ’
cl
where
X Y exp(&‘)
erfc(&S)
+ 4 + (v2 -
[
e0)
i
(
+
)IeAf2
26
exp(6ee) erfc [(~e~)‘/~l
-4(U2-Eo)t/(i . n We also introduced 2lrA
I
the equilibrium
bound state occupation 07)
186
H.J. Kreuzer /Quantum
and the dimensionless
statistics of ad- and desorption
parameters
v = yA/(4m V,,)“2 , 6 = 2 Vo/kBT ,
Gj, = AC+)/2 vo )
where oD is the Debye frequency of the solid. rem is of adsorption due to spontaneous emission (the factor (14)) and stimulated emission (the factor {exp[&,(w a phonon upon impact of a gas particle at the wall. rabs of a phonon by a gas particle close to the wall. Before we analyze F,, and Tabs in detail let us note
co = IEol/2vo ,
(18)
the contribution to the rate 1 in the last bracket in eq. ee)] - 1) -’ in eq. (14)) of is caused by the absorption that eq. (13) is to be inter-
He/Constantan: E,Jke = -25 K x = 2.5 A fm,/ kg = 384 K
10-l-
12 -
la3 B
2 2 lo4 -
lc? -
lo4 -
16’
0
‘/
20
I 40
I 60
I 80
I 100
I 120
L
I
140
160
180
-h'D/keT
Fig. 3. Adsorption tem .
time
ta and pre-exponential
factor
t,“, eq. (22), for the He/constantan
sys-
H.J. Kreuzer / Quantum statistics of ad- and desorption
187
preted as the lowest order term in an expression no(t) = Z0 [ 1 - exp(-t/t,)]
,
(19)
where c, = @/2V0)(rem
- ~&J’
,
(20)
is the time of adsorption. This comment also clarifies the necessity of our, so far, ad hoc normalization of no(t) with the equilibrium bound state occupation n,. Several obvious features of rem and rabs should be pointed out. If we keep the solid at zero temperature, T, = 0, or 6, = m, no phonons are present and only the first term in eq. (14) accounting for spontaneous emission of (non-equilibrium) phonons, survives with stimulated emissions and absorption of (nonexistent thermal) phonons absent. Similarly at high T and T,, one finds from eqs. (14) and (15) a complete compensation of stimulated emission and absorption processes. Eqs. (14) and (15) suggest that we write
r em=r:rnexp(-hd,
rabs = r:,,, exp(-b)
,
(21)
and thus get from eq. (20) t,= -
’
r:m
r.
$- exp(beo)
abs
0
= tt exp(Ge0) .
(22)
In fig. 3 we plot t, and t,” as a function of Aq,/k,T. We see that ti, the pre-factor in Frenkel’s formulas (1) and (22), is indeed only weakly dependent on T. It-is amazing to see that in the He-constantan system at T = 20 K, the time of adsorption t, - lo-’ s is larger than the atomic time h/2V0 - IO-l4 s by about seven orders of magnitude, slowing down to about 3 ms at T = 3 K. Unfortunately there seem to be no measurements of these adsorption times in the He-constantan system. A connection with sticking coefficients has been made in ref. [7].
3. Flash desorption We now want to formulate the initial value problem appropriate for a flash desorption experiment within the framework of the quantum statistical theory outlined so far [ 151. We want to consider in this section the three-dimensional version of our model of localized physisorption by assuming that the localized surface potential is given in coordinate space by
V(r, r’) = g
V(r) V(i)
where we choose
)
(23)
188
H.J. Kreuzer
/ Quantum
statistics
of ad- and desorption
where rc’ is the range of the potential normal to the surface (along the z-direction) and yF1 is the range along the surface with p being the (two-dimensional) component of r in the surface. In three dimensions the Hamiltonian (11) reads H=H,,
+Hs+Hdyn
(25) where u enumerates tors and
the various phonon
J&7, Q’) = gW244)1’2
branches,
k2, [2Y$ + i(G -
ePO are their polarization
vec-
kdl
x w> W’) A@) 4$&q .
(26)
Recal that a flash desorption experiment starts from an initial state in which gas and solid are in equilibrium with each other at a temperature Tg. At time t = 0 the temperature of the solid is suddenly raised to a value T, > T, and the resulting nonequilibrium time evolution, in which surplus adsorbate particles are desorbed, is measured, and, assuming an exponential decay, characterized by a desorption time t,(T,, T,). To formulate the corresponding initial value problem we can assume that the initial equilibrium occupation for t < 0 is adequately described by the static part of the Hamiltonian (25). The macroscopic time evolution in the system is then started at time [ 161 t = 0 by raising the temperature of the phonon bath to T,. This produces a non-equilibrium state of the gas and the solid which can only be equilibrated by the dynamic part of H. Therefore we switch on Hdyn at time t = 0 by writing H = Hs + H,, + e(t) Hdyn >
(27)
where 0(t) = 0 for t < 0 and O(t) = 1 for t > 0. The physical quantity to be calculated is the time dependent occupation of the bound state which is given by eq. (12) where the time dependent statistical operator is D(t) = exp(-iHt/t?)
p(0) exp(-iHt/Fz) = exp(-tit/A)
@,
exp(-iHt/h)
,
(28)
and Ps = exp [-P&H,,
-
&&I /Tr{exp[-P,& - ruir,)l) ,
is the initial equilibrium
(29)
statistical operator for the gas. (30)
is the gas particle
number
operator
and n is the chemical potential.
Similarly we
H.J. Kreuzer /Quantum
statistics of ad- and desorption
189
identify P, = exp(-P&,)/Tr
]exp(-P&)1
,
(31)
as the initial equilibrium statistical operator for the phonons at temperature T, = l/(k&) which is higher than the gas temperature Tg = l/(k&J. The inherent assumption here is that the phonon system reaches its equilibrium at T, much faster than the desorption process itself. Note that the initial equilibrium operator in eq. (28) can always be written in product from p(O) = fi& because H, and H,, and fis commute. We identify the flash desorption time from a second order time-dependent perturbation calculation which yields for a gas-solid system with one surface bound state [15] no(t),‘no(0)=
1 + tp (32)
where 1 nq(0) =
exp [P,&
(33) - ~>1
’
is the initial equilibrium distribution function for the gas particles with the + (-) sign applying if the gas particles obey Fermi-Dirac (Bose-Einstein) statistics. The energies Eq are the eigenvalues of H,, with 4 = 0 referring to the bound state and 4 # 0 enumerating the continuum states. Similarly (33) gives the phonon distribution at inverse temperature 0,. Within the framework of perturbation theory used to derive eq. (32) the linear time dependence is interpreted (Fermi’s Golden Rule) as the lowest order term in an expression n O(t) ‘= not=)
+ b O(O) - fi O(=)l
exp(-t/td(f))
(34)
.
We must choose such an ansatz (rather than an expression no(t)= no(O) exp [-cd]) to insure that the adsorbate develops from the initial occupation no(O) at temperature Tg to the final equilibrium occupation no(-) at temperature T,. With eq. (34) we can identify the flash desorption time td(f) from eq. (32) as
1 J-
312
-1
(WD-EO)/v2
exp ]@, - 6s) E01
du
0
)
exp[6S(v2z4 + ee)] - 1 _._~_____ x { [ 1 + I(u>l@v2)]2 + [J(u))/(4v~)]2}-’
0
.
(35)
190
H.J. Kreuzer / Quantum statistics of ad- and desorption
Here we introduced
k!
vo=
the quantities
u = y$zl(4mV,)“2
)
6 g,s = 2 Vok&,,s
>
f() = IEo 1/2Vo )
71 ’ r = Y~YI ,
Gj, = &D/2
vo )
(36)
where E. is the bound state energy. We also defined L,,‘=4
X(1 +X2)-3 [ 1 t (2x2
J 0
t fo/v2)1’2] -3(r2x2
t E~/v’)-~‘~ & ,
J;lr X(U - r2x2 - l)(l
Z(u)=4
+x2)-“(u
- 2x2
t
1)-2 dx
CJ 0 m
s
x(1 tx2)-3[1
+ (r2x2 - u)~‘~]-’
dx
d/r J(u) = 4
s 0
, 1
+ x~)-~(U ~ r2x2 + 1)-2 dx .
X(U - r2x2)“‘(1
The parameter v is connected
(37)
with the bound state energy (-co) via
3 = 1
2 s x( 1 +x2)-3 [ 1 + (r2x2 + eO/v2)] -2 dx . 0
(38)
Let us note that in the one-dimensional version of our model (which results if we take the limit r = y~/y~ + 0 and identify y = 73 the flash desorption time is given by _1 _
2V,
--T~TM,
*w
x
3
m
v (I-)
3 &/‘[I
-(~~2expl(s,-63Col]-1
sL;D-Eo
0
exp[(6, - S,)(w t eO)] - 1 I_. ‘w w + (2-1’2 - #2)2 expP,(w + CO)] - 1
dw
.
(39)
We can test our theory of flash desorption against an experiment performed by Cohen and King [5] for He desorbing from constantan. This system is particularly suited for our theory because its low heat of adsorption of the order of 30 K indicates that the He-constantan surface potential develops only one bound state. The experiments of Cohen and King [5] have been conducted starting from an initial temperature Tg - 2 K. Their experimental flash desorption times for He desorption from constantan are given as, see eq. (l), *d = (2.0 f 0.5) X 10e7 s exp[(31
f 1) K/T,]
,
(40)
H.J. Kreuzer / Quantum statistics of ad- and desorption
191
He/Constantan Edke .3 _
= -25K
XI _ Xl, =X = 2.5 -ho,/ kg = 304K
B,
b
7
10
30
50
70
90
110
130
fiwdksTs Fig. 4. Flash desorption times td(f) versus flash temperature Ts for He desorbing from Constantan, mass ratio m/M, = 0.065. Initial temperature Tg = 2 K. Dashed curve and point: measurements by Cohen and King (51 with errors indicated by cross hatched area. Solid curve calculated from eq. (35) (with r = 1 three-dimensional) or from eq. (39) (with r = 0 one-dimensional) agree to within a few percent.
for 4 K < T,< 18 K with an additional points at T, = 26 K where td = 1.4 X lo-’ s, which seems doubtful to the authors [5]. To fit our theory to these experiments we observe that our theory has in principle three independent parameters, namely E,, yi and yl. We must, however, expect that E. s’ Q and that 77’ =S + z%0.2-0.3 run. One finds that the slope of the linear part of log tdcfj versus TJ-'is mainly controlled by yI for reasonable values of r = yu/yI. In fig. 4 we give the best fit of our theory to the experiments of Cohen and King [5] for both the 3D and the 1D versions. Several points, demonstrated in detail in ref. [ 1.51, are noteworthy: (i) The bound state energy IE,I/k, = 25 K is about 25% less than the slope Q/k, = (31 + 1) K, the slope generally increasing with increasing E,. (ii) Changing ri’ from 0.1-0.3 nm increases &j(f) by a factor 25. The best value rll =
H.J. Kreuzer / Quantum statistics of ad- and desorption
192
0.25 mn agrees with related calculations [ 171. (iii) Flash desorption times in the 1D and the 3D versions of the theory (with all parameters the same) are comparable. This is in contrast to speculations in the literature [ 171 (not supported by explicit calculations) that pre-exponential times tz of the order of lo-’ s cannot be obtained in 1D theories with reasonable potential parameters [ 181. (iv) Frenkel’s formula (1) can only be a valid parametrization over a limited range of flash temperatures, for the system under study for T, = T, = 2 K up to about T, - 15 K. For higher T, the desorption times are shorter. It is interesting to note that the experimental point at T, = 25 K which Cohen and King [S] seem to doubt (primarily, it seems, because it does not fit Frenkel’s formula) lies exactly on the theoretical curve!
4. Isothermal
desorption
Next we formulate the initial value problem for isothermal desorption after a sudden substantial reduction of the gas pressure 1191. The latter instant is t = 0. The initial state for t < 0 is now one in which the gas and the solid are in thermal equilibrium at temperature T,. The final state is one in which all gas particles have been pumped out of the system. In eq. (34) we must therefore take n&t = -) = 0 and find for the resulting isothermal desorption time in the 1D model
0
w
+
(2 ‘I2 - &r/2)2 exp[6,(e0
+ w)] - 1 ’
(41)
Note that the term exp[(6, -- 6J(eo + w)]/{exp[?i,(w + eO)] - 1) in eq. (39) which corresponds to readsorption of gas particles accompanied by spontaneous and stimulated emission of phonons is absent in eq. (4 l), the remaining term simply representing the absorption of phonons by gas particles desorbing from the solid surface. Almost all calculations of desorption kinetics in the literature deal with isothermal desorption [ 17,181, though most of them are rather phenomenological in nature [20]. Also note that at very low temperatures, i.e. for k,T < IEo 1, the adsorption time t, in eq. (20) coincides with the isothermal desorption time d(i) in eq. (41). In fig. 5 we plot the isothermal desorption time for He desorbing from Constantan. It is obvious that only in the low-temperature region, i.e. for T, 2 4 K, can td(t) be approximated by Frenkel’s formula Id(i) = TV exp(Q/kBTs), with f$(t) and Q given in fig. 5. Note that the energy constant Q is larger than the binding energy (-E,) by some 20% and that the prefactor t8~i, is of order 1Oe7 s for this system. To compare isothermal desorption with flash desorption times we first look at the lat-
H.J. Kreuzer / Quantum statistics of ad- and desorption
A0
10
193
TsIKl
A
2
10-l -
10-Z-
10-3-
3
R .=_ 5
IF4
-
10-5 -
Fig. 5. Isothermal desorption time td(i), eq. (41), for He desorbing from constantan. It coincides to within a few percent with the equal temperature flash desorption time Id(f) (Ts, 7’s), eq. (42).
ter for an infinitesimal t&(Ts,TJ=
TlimT t-’ d S*
temperature
step, i.e. we take
Tg , T,)
g
(42)
H.J. Kreuzer / Quantum statistics of ad- and desorption
194
80
B J
40
T&l I5
25
td(fi
Fe = 25 K. Ts)
‘0.
10.
IO
20
30
hwdke‘rs
Fig. 6. Various desorption times for He desorbing from constantan.
Note that in this limit no real time evolution takes place as nothing is done to the system. Still this limit can be useful for comparison with isothermal desorption times. For He desorbing from Constantan it turns out that td(i)(Ts) is less than td(f)(Ts, T,) by less than 1% for T, 2 4 K, increasing to some 30% at T, = 40 K. The latter temperature region is explored in some detail in fig. 6 where we plotted the isothermal desorption time fd(i)(Ts), the equal temperature limit td(f)(Ts, T,) of the flash desorption time and also the full t,j(f)(Ts, T,) for RGJ&BT~ = 15 fixed. Note also the limit
H.J. Kreuzer / Quantwn statisticsof ad- and desorption
195
It would be interesting to see whether these differences between isothermal and flash desorption times can be verified experimentally.
5. Resorption from deep bound states We have so far been dealing with localized physisorption in gas-solid S~Sin which the surface potential develops only one bound state; gas particles trapped in that bound state define the adsorbate. We have also tacitly assumed that the energy Ee of the bound state is, in magnitude, less than the Debye energy of the solid, i.e. that l&o]<&on, an assumption well justified e.g. in the He/constantan system. Under such c~cumstances the bound state occupation can be changed by the emission or adsorption of one phonon implying that relaxation times can be calculated in second order time dependent perturbation theory. The situation changes drastically if the surface potential of a particular gas-solid system deveIops a deep bound state of energy Fa (or several of them) i.e. one for which ]Es I > fiioD besides one (or several) with IE, [ <&.Q,. A gas particle trapped in such a deep bound state can leave it, i.e. desorb, via basically two mechanisms. Firstly, it can absorb two pbonons (of total energy larger than ]Es i) simultaneou~y; this process is very unlikely at low temperatures where the number of phono~s, particularly those with energy&w S k,T, is very low. The desorption from a deep bound state can, secondly, proceed in a two-step cascade E2 +E, + continuum in which two phonons are emitted in succession, For either process higher, i.e. at least fourth, order time dependent perturbation theory must be invoked, an afternative to this approach being a kinetic one in which microscopically calculated relaxation times are used in phenomenological rate equations. It seem plausible that an adsorption cascade, as just outlined, is one possible mechanism of chemisorption: a gas particle must be physisorbed into a weak bound state before it can undergo a chemical reaction and get trapped into a very deep bound state. Work along these lines is in progress [24,25]. We want to conclude this paper by estimating the isothermal desorption time for atomic hydrogen desorbing from NaCi. At low coverage and at low temperatures hydrogen has essentially zero mobility along the surface and will desorb in atomic form [21]. The bound states of hydrogen on an NaCl(OOI~surface have recently been measured by Iannotta and Valbusa [22] to be E&u = -351 K, E2/kB = -247 K, Es = -166 K, and Ed,fkB= -1lOK. The Debye temperature of NaCl is FiqJkB = 280 K for T > 50 K dropping to 275 K for T - 35 K below which it rises to 3 10 K [23 J. Hydrogen can therefore desorb from the higher bound states E,, Es and E4 by absorbing a single phonon, Neglecting the lowest bound state Er for the moment, we can estimate the desorption time by simply adding the properly normalized rates from n = 3 individual bound states within reach of a single phonon, i.e. a bound state of energy Ei is produced by a separable potential of the form (3) and (41 of the same range X = y-r, but of variable strength V, = g&. This yields from terns
H.J. Kreuzer / Quantum statistics of ad- and desorption
196
eq. (41)
X
1 (43)
exp[&‘,(ci - W)] - 1 ’
where fi=
iE,l/2Vf,
Vi = yfil(4m Vi)1’2 .
Gb = FlWL)/2Vi ,
In this approximation we estimate the isothermal drogen desorbing from NaCl(001) to be td(i) =f&i) exp(Q/kd’)
- 3.05 X lo-”
s
desorption
exp(265 K/7’) ,
time for atomic hy-
for lS
cd(i) - 1 .O3 X lo-’
s
exp(25 1 K/T) ,
for4
10K. (45)
In the higher temperature range td(i) is of the order of milli- to microseconds. Note also that as the temperature is lowered the preexponential factor t:(i) increases and the energy constant Q decreases in such a way that -1n f&i) -
Q,
(46)
is satisfied very well, as found experimentally on other systems with deep bound states [6]. To see how severe an approximation the neglect of the deepest bound state actually is, one can look at a system with the Debye energy raised artificially so that Fiio,, > IE, 1. Taking fiuD/kB = 360 K we have calculated the isothermal desorption time for all four bound states of the H/NaCl system and found td(i) - 3.2 X lo-r” td(i) - lo-’
S
S
exp(384 K/Z’) ,
exp(365 K/T) ,
for 15
(47)
for4
(48)
The pre-exponential factors are very close to the ones estimated in eqs. (44) and (45) whereas the energy constants Q are somewhat larger. Again the empirical rule (46) is very well satisfied. A detailed calculation of desorption times for the H/NaCl system, including multi-phonon processes, will be given elsewhere [24,25].
References [l]
A discussion of adsorption experiments Ehrlich, Advan. Catalysis 14 (1963) 25.5.
and some
experimental
results
are given by G.
H.J. Kreuzer f Quantum statistics of ad- and desorption [2] For a recent review see, e.g. D. Menzel, mer (Springer, New York, 1975).
in: Topics
in Applied
Physics,
197
Vol. 4, Ed. R. Go-
[3] If the temperature of the solid is raised in a sequence of small steps between each of which the system has enough time to reach its new equilibrium, the procedure is known as temperature-programmed desorption. [4] A temperature-programmed desorption experiment ideally measures the limit limTs+Tg Id(f) (Tg, Ts). [5] S.A. Cohen and J.G. King, Phys. Rev. Letters 31 (1973) 703. (61 J. Lapujoulade, Nuovo Cimento Suppl. 5 (1967) 433; A.K. Mazumdar and H.W. Wassmuth, Surface Sci. 34 (1972) 617. [7] H.J. Kreuzer and G.M. Obermair, J. Phys., to be published; Similar models have been studied in a different context by H.J. Kreuzer and R. Teshima, Physica 87A (1977) 453; and H.J. Kreuzer, Nuovo Cimento 45B (1978) 169. [8] Transient phenomena in the time evolution of a system depend quite sensitively on the initial conditions, i.e. on the preparation of the initial perturbation of the system to which the transients are the response. Our model assumes that the gas fills and adjusts to the container before a dynamical interaction with the walls takes place, i.e. we assume that fi/kT Q fi/IEol, where E, is an energy characterizing the solid-gas interaction, e.g. the energy of the weakest bound state of the surface potential. [9] See, e.g., N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions (Oxford Univ. Press, London, 1965); K.M. Watson and J. Nuttall, Topics in Several Particle Dynamics )Holden-Day, San Franco, 1967). [lo] For a recent review see, e.g., J.S. Levinger, in: Springer Tracts in Modern Physics, Vol. 71, Ed. G. HBhler (Springer, Berlin, 1974) in particular pp. 119-166; also Z.W. Gortel, H.J. Kreuzer and R. Teshima. Can. J. Phys. 58 (1980) 376. [ 1 l] F.O. Goodman and H.Y. Wachman, Dynamics of Gas-Surface Scattering (Academic Press, New York, 1976). [12] A.G. Sitenko, V.F. Kharchenko and N.M. Petrov, Phys. Letters 28B (1968) 308. (13) T.R. Knowles and H. Suhl, Phys. Rev. Letters 39 (1977) 1417. [ 141 This type of approach was advocated by J.E. Lennard-Jones and C. Strachan, Proc. Roy. Sot. (London) A150 (1935) 442;and J.E. Lennard-Jones and A.F. Devonshire, Proc. Roy. Sot. (London) Al56 (1936) 6,29; Al58 (1937) 242,253. [ 151 Z.W. Gortel, H.J. Kreuzer and D. Spaner, J. Chem. Phys. 72 (1980) 234. [ 161 More precisely we put t = 0 at that instant when the rapid heating of the solid from Tg to TS is completed. This implies that the heating time is much less than the desorption time, a necessary condition for performing a flash desorption experiment. [17] B. Bendow and S.C. Ying, Phys. Rev. B7 (1973) 622,637. [18] Our conclusions are supported by recent model calculations by F.O. Goodman and I. Romero, J. Chem. Phys. 69 (1978) 1086. [19] Z.W. Gortel and H.J. Kreuzer, Chem. Phys. Letters 67 (1979) 197. [20] F.O. Goodman, Surface Sci. 24 (1971) 667 and 60 (1976) 45; G. Armand, in: 6th Rarefied Gas Dynamics Symp., Cambridge, MA, 1968; P.J. Pagni and J.C. Keck, J. Chem. Phys. 58 (1973) 1162; P.J. Pagni, J. Chem. Phys. 58 (1973) 2920; L. Trilling, in: 8th Rarefied Gas Dynamics Symp. (Academic Press, New York, 1974). [21] E.g., hydrogen adsorbed on tungsten only shows diffusion for T > 180 K as observed by R. Gomer, R. Wortman and R. Lundy, J. Chem. Phys. 26 (1957) 1147. For a recent discussion, see K.J. Vette, T.W. Orent, D.K. Hoffman and R.S. Hansen, J. Chem. Phys. 60 (1974) 4854;and D.K. Hoffman, J. Chem. Phys. 65 (1976) 95. [22] S. Iannotta and U. Valbusa, Surface Sci. 100 (1980) 28.
198
H.J. Kreuzer / Quantum statistics of ad- and desorption
[23] E.W. Kellermann, Proc. Roy. Sot. (London) Al79 (1941) 17. For a recent discussion see ,4.A. Maradudin, E.W. Montroll, G.H. Weiss and I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, in: Solid State Physics, Suppl. 3, Eds. H. Ehrenreich, F. Seitz and D. TurnbuU (Academic Press, New York, 1971). [ 241 A complete fourth order calculation including all one-phonon and two-phonon processes has just been completed by Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B, to be published. [25] Desorption via one-phonon cascades has been calculated by Z.W. Gortel, H.J. Kreuzer and R. Teshima, Phys. Rev. B, submitted. For the H/NaCI system, we find tscij = 7.2 X 10-l 2 s, somewhat shorter than the estimate in eq. (47).