A simple stochastic description of desorption rates

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A simple stochastic description of desorption rates

G. Iche, Ph. Nozières

To cite this version:

G. Iche, Ph. Nozières. A simple stochastic description of desorption rates. Journal de Physique, 1976,

37 (11), pp.1313-1323. �10.1051/jphys:0197600370110131300�. �jpa-00208528�

A SIMPLE STOCHASTIC DESCRIPTION OF DESORPTION RATES

G.

ICHE,

^Ph.

NOZIÈRES

Institut

Laue-Langevin,156 ^{X, Centre}

^{de Tri}

38042 Grenoble

Cedex,

^France

(Reçu

^{le 11 mars}

1976, accepti

^le

3 juin 1976)

Résumé. ²⁰¹⁴ La théorie

cinétique

des vitesses de

désorption

d’un adatome en surface est reprise

sous un angle stochastique, permettant ^de

généraliser

^les résultats de Kramers (fondés ^{sur une}

équation

^{de Fokker} Planck). ^{Pour un} puits ^de

potentiel profond,

1a vitesse de

désorption

^{est reliée}

à la

probability

^de piégeage moyenne d’un ^atome thermique incident. Celle-ci

dépend

^{à son tour de} l’énergie moyenne cédée ^au substrat par la particule

pendant

^une oscillation. En général, le transfert

d’énergie

^aux phonons ^est si efficace

qu’il

justifie ^a posteriori le modèle absolu de Eyring. ^On peut

généraliser

le modèle à des situations plus

compliquées (adparticule

^avec degré de liberté interne,

barrières activées, etc...). Enfin, ^un formalisme

dépendant

^du temps, ^valable pour des

puits

peu

profonds,

^est

esquissé

^{2014 pour des}

puits profonds,

^ce formalisme

général

^{se réduit au modèle}

simple

du

depart.

Abstract. ²⁰¹⁴ The kinetic

theory of desorption

^{rates for an adatom on a} solid surface is reconsidered from a stochastic point of view. The results of Kramers, obtained within the framework of a Fokker- Planck equation, ^are thereby

generalized.

^{For a} deep potential well, ^the

desorption

^{rate is related}

to the average

sticking

probability ^{of an} incoming ^thermal

particle.

The latter in turn

depends

only

on the average energy loss 03B4 of the

particle

^to the substrate

during

^a

single

oscillation.

It is shown that in most cases the loss to lattice vibrations is so strong ^{as to}

justify

^a posteriori

the absolute rate theory ^of

Eyring.

^The theory is extended to more

complicated

situations

(adparticle

with internal

degrees

^of freedom, ^activated barrier, ...). Finally, ^a

fully time-dependent

formulation,

valid for shallow wells, ^is sketched; it is shown how, ^for

deep

wells, it reduces to the above

simple

formulation, ^{based on} detailed balance

only.

Classification

Physics ^Abstracts

7.854

1. Introduction. ^- The

theory

of chemical reaction rates is a well-established field ^- and as such it has reached a level of considerable

sophistication

^over

the past

forty

years. As a

result, simple physical

^ideas

often tend to be hidden.

In the present paper, ^we consider the

problem

^from

a somewhat unusual

point

^of

^view,

stochastic in nature, ^which

clearly points

^{to some} of the essential

physical

facts. Our purpose is ^to

lay

^{down a formula-}

tion,

^and to stress

qualitative

^{results : we thus deli-}

berately

^choose

oversimplified

^models.

We consider

only

situations in which the

reacting species

^{are in close contact with a heat}

bath,

^whose

many

degrees

of freedom can

easily

^{absorb or}

provide

energy. The

simplest example

^{of such a} reaction is the

desorption

^{of a}

single

^{atom A}

trapped

^{on a solid}

surface :

there,

^the

underlying

^substrate

provides

^the

thermal

agitation

^that

ultimately

will lead the atom A into free _space. The chemical reaction _may be viewed

as a brownian motion in a

potential

^well.

(In

^section

4,

we consider

briefly

^the

generalization

^{to more}

compli-

cated cases,

using

the standard concept ^{of a reaction}

path.)

The

application

of stochastic theories to chemical kinetics is not new. In

1958,

^for

instance,

^Montroll

and Shuler

[1]

treated in that _way the

desorption of a particle trapped

^{inside a} truncated harmonic well.

They

^{relied on a master}

equation

^{in order to describe}

transitions between the successive

quantized

^levels.

The present

approach

^is similar in

spirit

^{to this}

pioneering work,

but rather different in its formula- tion. We consider

only

^a

purely

^classical

regime.

Also,

^we

emphasize

the role of inelastic _processes

near the

top

^{of the}

potential

^{barrier - a feature}

which was

apparently

^not

^included

^{in the work of}

Montroll and

Shuler.

As usual in stochastic

theories,

^{all that we} say is standard

practice

^for

professional

statisticians : ^our

problem

^of

desorption

^is

isomorphous

^{to the}

general

Markov chain with

absorbing

^barriers

(as ^treated,

for

instance,

in the standard text of Feller

[2]).

^Granted

that the stochastic method is in no way

original,

we found it easier to construct from scratch our own

probabilistic formulation,

rather than

translating

the usual mathematical

language

^{into our}

problem.

In section

2,

^{we set} up the

general problem

^of

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370110131300

desorption,

^{and we} discuss the

corresponding

^absolute

rate uo ^as obtained from the well-known

Eyring theory. Using

^a detailed balance argument, ^{we show} that the actual

desorption

^{rate u} differs from _uo, the ratio

uluo being

^{related to the} average

sticking probability

^{that an} incident thermal atom remains

trapped

^when

hitting

^the surface. That u should

depart

from _uo was noted back in 1940

by

^Kramers

[3]

^who

treated brownian motion ⁱⁿ the framework of a

Fokker-Planck

equation.

^The

problem

^{was revived}

recently by

^{Suhl and coworkers}

[4].

The various

regimes

^of Kramers appear

naturally

^{in our}

language.

Such a stochastic

description

^{is based on the}

assumption

^{that the} process is markovian ^{on an}

appropriate

time scale t

(here

^the

period

of oscillation of the adatom in the surface

potential well).

^One may take the

opposite view,

^{and consider a} non-stochastic

substrate,

^which

responds

^{to the adatom}

according

to the usual deterministic laws of mechanics. The calculation of the

sticking probability

^{is then a}

purely

^classical

problem

^{- as} described for instance in recent articles

of Beeby et

^al.

[5].

Several

possible

extensions of our

simple

^{model are}

sketched in section 3 : influence of the three-dimen- sional structure

(motion

of the adatom

parallel

^{to the}

surface),

^{of the internal}

degrees

of freedom of the adsorbed

particle.

^{We also} consider reactions between two

potential wells,

instead of one well vs outer space.

In section

4,

^{we return to the case of}

desorption,

and we show that our

simple

detailed balance _argu- ment is

only approximate : actually,

^{it assumes that}

one has two very different time scales : a fast scale

(a

few oscillation

periods)

^{for the}

adparticle

^{to fall}

in the

well,

^{and a} slow scale

(the desorption time)

^for

the

trapped particle

^to

finally

escape.

Trapping

^and

desorption

^can be identified

only

^{if their two scales}

are well

separated,

^a condition which is met if the

potential

^well

depth

^{V is much}

larger

^{than the} tempe-

rature T.

In the most

general

^{case, we} show how the calcula- tion of the reaction rate may be reduced to a

simple

mathematical

form, namely

the search for

eigen-

values of the

appropriate

stochastic operator. ^The results of section 2 are reached in the

limit V >

^T.

2.

Desorption

^{rate vs}

sticking probability.

^{- We}

consider

point particles,

^{with mass} m,

trapped

^{in a}

potential

^well

V(x)

^{near a} solid surface. In this

section,

we consider

only

^motion

along

^the

x-direction

^normal

to the surface : the

problem

^is one-dimensional. The

potential profile

is sketched in

figure

^{1. We assume}

that there is no activation barrier to be overcome

before

entering

^{the well}

(the

influence of such ^a barrier is discussed in section

3).

^The

origin

^of energy is chosen at the vacuum

level;

^the

depth

^of the well is V.

Were it not for the friction on the

substrate,

^a

particle

with _energy 8 in the _range

(- V, 0)

would oscillate with a

frequency w(E), easily

found from m and

V(x).

Implicit

^{in this}

description

^{is the}

assumption

^{that we}

FIG. 1. - The profile ^{of a} one-dimensional, unactivated surface

potential ^barrier.

can separate ^{a static} interaction with the substrate

(yielding

^the

potential

^well

V(x))

^{from a}

dissipative

friction mechanism. To the extent that it is the same

interaction that

provides

^the

binding

^{and the}

^friction,

such a

separation

is far from obvious. It can be achie- ved for

heavy adparticles

^{that move}

slowly

^{as com-}

pared

^to

typical

^substrate

frequencies (R6sibois

^and

Lebowitz

[6]).

We limit ourselves to a

purely

^classical

regime,

^valid

if hw « T. We can then build a wave

packet

the size of which is small

compared

^to the width of the well.

The temperature ^is

imposed by

^the substrate. In thermal

equilibrium,

^{each state has an}

occupation.

The net number of

trapped particles

^is

where p

⁼

1/hw(e)

^{is the}

density

^{of states.}

The

integral

ⁱⁿ

(1)

^is controlled

by

^a

region

^of

width - T near the bottom of the well. For

deep

^wells

( V > T),

^{we can}

replace W(B) by

^{its value Wm near the}

bottom : we thus find

In

principle, (2)

^holds

only

if the adsorbed atoms are

in thermal

equilibrium

^{with a} gas

phase having

^the

same

(u, T).

Assume now that there is no gas

phase.

The adsorbed atoms are in thermal

equilibrium

with the substrate except ^{for a thin}

layer

^{near the} top ^of the well. For

a

deep well, (2)

remains valid. Let J be the net flow of

particles

^that

quit

^{the well} per unit time. The

desorption

^{rate u} is defined as

In the standard

approach

^of

Eyring [7],

^{one assumes}

that the thermal

equilibrium

distributions

fo(e)

^extends

to all

energies, including

^{those atoms above vacuum}

level

(E

>

0)

^{which move} away from the surface. The net flow is found from

elementary

^kinetic

theory :

where dx ⁼ v dt is the distance travelled

during

time dt. On

comparing (2)

^and

(3),

^{we find the so-}

called absolute rate

of desorption

(4)

has the usual Arrhenius form. Note that

(4) only

involves

equilibrium quantities (1).

As shown

by

^Kramers

[3]

the result

(4)

^{cannot hold}

in all cases. Let us describe the

dissipation

^{on the}

surface as a friction force

proportional

^{to the}

velocity

^v

If n >>

m, the mean free

path

^{of the}

adparticle

^{is short}

as

compared

^{to the well}

^size,

^{and the}

desorption

may be viewed ^{as a}

plain

^diffusion process in the _presence of an

applied

^force

(- grad V).

^The net current

density J(x)

^{is a} result of conduction and diffusion

For a

deep

well under

stationary conditions,

^Kramers

shows that

(5)

^{can be solved}

exactly, yielding

^{a reaction}

rate

(xm

is the well

minimum).

In the

opposite limit I

^ill, the motion of the

particle

^{is a}

damped

oscillation. In the

limit q - 0,

Kramers shows that the

desorption

may be viewed as

diffusion in energy space; the

corresponding

^{rate u is}

proportional to q :

^{it vanishes}

when q - ^0,

for lack of

a mechanism which could

replenish

^the upper ^states

(c

>

0)

^{once the first}

particles

^are gone.

More

generally,

^{let us consider an adsorbed}

layer

ⁱⁿ

thermal

equilibriwn

^{with a} vapor

phase.

^{Consider a} gas

particle

^with energy ^s that hits the surface : let

P(s)

be the

probability

that it remains stuck.

By

^{stuck we}

mean that the

particle

^falls

deep enough

into the well

(losing

^an energy

bigger

^than

T),

^{in such a} way ^{as to} remain

trapped

near the bottom for a

long

^time

(see

^{section 5 for more}

details).

^{The net flow of}

particles

that hit the surface is

given by (3).

^Of

^these,

^{a fraction P}

(1) ^In (4), the factor _wm comes from _no, and it has nothing ^{to do}

with kinetics. The argument according ^to which it is the frequency

at which the particle ^tries to escape is meaningless.

is

trapped :

hence in thermal

equilibrium

^{a net}

ingoing flux

Jin

^{must be balanced}

by

^the

desorption flow, Jout

⁼ no ^u.

We thus obtain an exact

expression

^{for the}

desorption

rate u

Eyring’s

^rate uo is corrected

by

^the average

sticking probability

^P

(2).

^Note that the result

(8) depends only

^on detailed balance

arguments.

^{It is}

completely general, equally

valid in the

overdamped

case

(where,

combined with

(6),

^it

yields

easy ^{access to}

P ),

^{and in}

the

underdamped limit q

^m.

We now limit ourselves to the low friction _case,

" n ^{to. The}

trajectory

^{is then an}

oscillatory

^round

trip

with random energy. We focus our attention ^on the

particle

when it is reflected on the external side. Let si 0 be the energy ^at such a

turning point :

^the

brownian motion _may be viewed as

hopping

^from

si ^to ei+1

in

^{the course of one round}

trip. Eventually,

the

particles

will reach a

positive

energy en ^at the

edge

of the barrier

(beyond

which friction is

negligible) :

the

particle

is then desorbed.

Conversely,

^an

incoming particle

^with energy ^e > 0 will

hop

^into

turning energies

81, E2, ...,

falling

down into the bottom unless is

again

^{hits a}

positive

En.

Let

W(s, 8’)

de’ be the

probability

^{that the}

particle hops

^from energy ^{e to} the _range

(e’,

^{s’ +}

de)

^{in the}

course of one round

trip

^{between two}

turning points.

W(s, s) provides

^the

only physical ingredient

^{of our}

formalism. We assume that the _process is markovian from one round

trip

^{to the next : there are no} statistical correlations between two successive oscillations.

(Such

a markovian

approximation

is less restrictive than the Fokker-Planck

description

^used

by Kramers,

^{as it}

applies only

^on the finite scale of one round

trip).

We do not make _any

assumption

^{on the}

shape

^of

w(Bg g’).

Consider the

sticking probability P(8)

^{of an incident}

particle

^with energy ^e > 0. After one round

trip,

^it

has energy

8’,

^{with a}

probability

distribution

W(e, c’).

If s’ >

0,

^the

particle

escapes and it is not

trapped ;

if e’

0,

^{it starts over}

again

^{with a}

sticking probability

now

equal

^to

P(e’).

^{The function}

P(s)

^thus

obeys

^the

simple integral equation

(9)

^must be solved with the

boundary

^{condition P - 1}

when s -+ - oo. The reaction rate u is thus

entirely

(1) ^The relation between desorption ^{rate and} sticking probability

is well known-for instance, ^{in the case of solid} evaporation.

determined

by

^the energy loss

probability

^{for a}

single

round

trip, W(s, s’).

^{Such a}

point

of view has been used in the past ^{in other}

problems,

^for

instance,

^the capture ^of

electrons ^by

^{traps in} semi-conductors ^- the so-called Giant trap

theory of

^cascade processes.

The

pioneering

^{work of M. Lax}

[8]

^{in this} respect ^is based on an

equation

^{similar to}

(9).

In order to

proceed ^further,

^we examine in some

detail the behaviour of W.

Obviously,

it is normalised to 1 :

For

simplicity,

^{we assume that W}

depends only

^on

the difference

(e’

^-

E)

^{in the}

important

range

(of

width

T)

^{near the}

top

^{of the barrier}

(3).

^{It then}

displays

the behaviour sketched in

figure

^{2 : an} average loss per oscillation

6,

and fluctuations around this value with an

amplitude

^{4. W is}

subject

^{to a further cons-}

traint : it must

respect

detailed balance in thermal

equilibrium.

The condition of

equal

^{flow between}

the energy ranges

(s, e

⁺

de)

^and

(c’,

^{e’ +}

de) implies

FIG. 2. ^- A sketch of the _energy transfer probability W(8,8) ^for

a single oscillation in the potential ^well.

We may

distinguish

^two

regimes : (i) 6 «

^{T : we can}

expand

^{the factor}

and

consequently (11) implies

(12)

^is

nothing

^but Einstein’s relation in energy space,

relating

energy diffusion

( N A 2’)

^to average energy loss 6 .

(ii) 6 »

^{T : then}

(11) implies

W = 0 if e’ - e > 0 :

we can

only

^lose energy

(except

^{for a small tail of}

width

T).

(3) ^{Such a} simplification may be questioned in the real case of a

barrier which is rounded off near the top : ^{then the} oscillation period becomes longer ^and longer ^{when 8} approaches ^0. If however we assume that dissipation ^occurs only ^{in a} region of finite width a near

the surface, the energy transfers to the heat bath do not vary appre-

ciably ^{when 8} goes through ^{zero : in that} case, W depends only

on (E’ ^- e).

We now return to the

integral ^equation (9) governing

the

sticking probability P(e),

^{and we} consider first

case

(ii), i.e., 6 >>

^{T. If W were}

strictly

^{zero when}

e’ > s, the

question

^of

trapping

would be answered after the

first

^round

trip (since

^the

particle

^cannot

regain energy).

The solution of

(9)

^{is then}

In

practice,

^{the small}

positive

^{tail in}

W(8’ - s)

^will

round off P near

-s = 0,

^{over a} range - T : this is ^a small effect which leaves

P(s)

very close to 1 in that

region.

^Hence

P ~

1. In the limit

6 » T,

^{the absolute}

rate

uo is therefore

^correct,

irrespective

of all the details of individual _energy losses.

We consider next the

opposite

^limit

^{6 «}

^{T. The}

behaviour of

P(s)

^{is then sketched in}

figure

^3.

P(8)

vanishes

if 8 »

^L1

(in

^{which case the}

particle

^leaves

after the first

oscillation).

^For

negative

s, it _goes to 1 when s - - T : the maximum _energy the

particle

can

regain

from the thermostat is T

(after n

^oscilla-

tions the

particle

^{loses an energy}

^nb,

^{while it can at}

best

regain In

^{L1 from}

diffusion ;

the maximum energy it can accumulate is

d 2/b

⁼

T).

^{What we want}

is the _average

P

defined in

(7) (shaded

^{area of}

Fig. 3a).

FIG. 3. ^- The behaviour of the sticking coefficient P(c) ^{as a func-}

tion of _E, in the limiting ^{cases 3 « T.}

Actually, (9)

^{can be solved}

exactly

^{in that}

^limit,

^as

shown in

appendix

A. One finds that

where 6 is the _average energy

loss,

^{defined as the first}

moment of W

The actual

desorption

^{rate u} is thus reduced ^as

compared

^to uo.

As an

example,

^{let us consider} Kramers’ model where

dissipation

^occurs

through

^a friction force

- mnv. Then

(the integral

^{runs over one}

period

^of

oscillation).

^The

value of 6

depends

^{on the exact}

potential profile ;

it is of order

where V is the well

depth

^and

Wõ 1

the time it takes to cross a

dissipation layer.

^{We recover} the low friction result of

Kramers,

^which

applies

^if

As a

conclusion,

^{let us stress a few}

important points.

First of

all,

any information

regarding dissipative

mechanisms _appears

only

ⁱⁿ

P.

The other factors in u are

equilibriwn quantities,

^{that have}

nothing

^{to do}

with friction.

Second,

^{and more}

important, P

^is

entirely

controlled

by

^what

happens

^{in a narrow}

layer

of width T _near

the top of

the barrier : when

trying

to escape from the

potential well,

it is the last T that counts, ^the

highest

^{hurdle to overcome. Below that}

layer,

the distribution is in thermal

equilibrium, independent

of friction.

In

principle,

these results

depend

^{on our markovian}

approximation,

^{which as we shall see} in section 3 is often doubtful.

Actually,

^{such a} restriction does not hold if 6 »

T,

^{in which case after a}

single

oscillation an

incoming

^thermal

particle

^is

already

well below the

top layer

^{of width T :} the chance that

subsequent

^oscilla-

tions will make for this lost _energy are

negligible (the

heat bath can at best

provide

^an energy

T).

^{Thus we}

may

safely

^conclude

that, irrespective

of statistical correlations between successive

oscillations,

^the trap-

ping probability P

^{is 1} when d > ^{T : the} absolute rate

theory

^holds.

3. Generalization to more

complicated problems.

^-

Up

to now, ^{we were} concerned

only

^{with a} very

simplified problem : desorption

^{of a}

point particle, trapped

^{in a} one-dimensional

well,

in the absence of any activation barrier. We now consider how such a

crude model could be

extended,

^{at least in}

principle.

Let us first assume that the

adparticle

has additional

degrees

^of

freedom, beyond

the motion in the x-

direction normal to the surface. These

degrees

^of

freedom _may be for instance motion

along the y

and z axis

parallel

^to the surface

(in

the three-dimen- sional

problem). They

may also be internal motion of a molecular

adparticle,

whether rotational or

vibrational. Let n denote the state of these extra

degrees

of freedom. The net energy of the

particle

^is

It is clear that

only Ex

is relevant as

regards desorption

and

trapping :

^the

particle

^will escape if it hits the

edge

of the barrier with a

positive

Ex. The extension of our

approach

^{is then}

straightforward.

^Let

Pn(ex,)

^{be the}

probability

^{that a}

particle

with normal energy Ex,

internal state n, ^be

ultimately trapped

^{in the}

potential

well.

Similarly,

^let

Wnn,(Ex, E;x

^{be the}

probability

^that

in one round

trip

^the

particle

goes from

(n, E,x

^to

(n’, Ex)

^x

From W,

^we infer the

equation obeyed by

^the

sticking probability

(18)

^{is to be solved}

(for

^instance

by iteration),

^{with the}

boundary

conditions

Pn(Ex)

^{- 1 if} Bx --> - oo. Once

we know

Pn(Ex),

^we

proceed

^{to the} average

sticking

coefficient of

incoming

^thermal

particles

We are now in a

position

^to

apply

^our

simple

^detailed

balance

argument.

The number of

particles trapped

in the well is

where

Zm

is the internal

partition

function for

particles

near the bottom

(note

the difference with

(2)

^{due to}

the additional

degrees

^of

freedom). Similarly,

^{the flux}

of

particles arriving

^{from the gas}

phase

^{onto the}

surface is no

longer (3)

^{but rather}

where

Zo

is the internal

partition

^{function near the} top of the well

(not necessarily equal

^to

Zm).

^On

equating

the in and out

fluxes,

^{we find the}

desorption

^rate

The absolute ^rate is corrected

by

^the

internal ^degrees

of freedom ^- but the relation

u/uo

⁼

P

is unaffected.

The main effect of the internal

degrees

^{of freedom}

is to blur the

probability

^{W :} the normal _energy

loss, (ex - 8§),

may ^now arise from a transfer between internal _{energy u}

and ex itself,

^a

purely

^elastic process

(which

^{adds to} the former inelastic

exchange

^{with the}

heat

bath).

In any case, whatever the _average loss 6 in normal _{energy was,} additional

degrees of freedom

will make it

bigger.

The two

limiting

^{cases are}

again

trivial. Let

bn

^{be the}

energy loss

averaged

^{over all final channels n’ :}

If

dn > T,

^the

trapping probability

^{P is}

1, irrespective

of the

details,

^and

Eyring’s

^{absolute rate} is correct,

a situation even more

likely

^{than was}

anticipated

ⁱⁿ

section 3. If

by

chance elastic _processes allowed

dn T,

^a

simple

extension of

Appendix

^{A would}

show that

We now return to a one-dimensional

point particle,

and we consider another

complication, arising

^when

the

potential profile V(x)

^{is not} monotonic. In the activated

desorption

barrier of

figure 4,

^the

particle

must climb a hill of

height Va (the

activation

energy)

before

falling

into the well of

depth Vm. Clearly,

^the

reverse

desorption

process should

depend

^{on the net}

desorption

^barrier

height

FIG. 4. ^- An activated desorption ^barrier.

More

generally,

^{we can consider reactions between}

two

equilibrium positions

^{1 and 2}

(one

^{of them meta-}

stable),

^{as shown in}

figure

^5.

FIG. 5. ^- A reaction path ^{between two} equilibrium positions ¹

and 2.

Such a

picture

is often used to represent ^reactions inside a molecule

(for

instance in an adsorbed

layer).

The one-dimensional model

(4)

is then the reaction (4) In such molecular reactions, the one-dimensional reaction

path ^is only ^{a first} approximation. Eventually, ^{one must also} worry about motion in the directions orthogonal ^{to that} path (up the

flanks of the hill at the saddle point ⁱⁿ V). Such motion can be viewed as additional internal degrees of freedom superimposed ^on

the motion in the x-direction along the reaction path.

path,

^the steepest ^dscent

path

^{that overcomes the}

saddle

point separating

^two

valleys

ⁱⁿ the multidi- mensional _energy surface.

We take the

origin

^{of energy at the} top ^{of the}

barrier,

and we denote

by Vi

⁼

Va - Vmi,

^the

depth

^{of the}

well on either side. As

before,

^{we assume}

deep wells,

i.e.

Vi >>

^{T. In thermal}

equilibrium,

the number of

particles

^on each side is thus

where _Wi is the oscillation

frequency

^at the bottom of each well

(see (2)).

^{We now} empty ^{well 2. Let}

J1-+2

be the flow of

particles coming

^{from 1 that fall to the}

bottom of 2 : the reaction rate is defined as

A similar definition holds for the reverse rate U2-+ 1.

In the absolute rate

theory,

one assumes thermal

equilibrium everywhere, including

^{near the} top ^{of the} barrier. Then the kinetic argument ^of

Eyring yields

at once

Of _course, detailed balance is verified

Such a result is

subject

^{to the same} limitations as

in section 2 : it assumes

complete trapping

^of

particles

that _pass

through

the barrier at x ⁼ xa. If in

figure

⁵

they

^cross the barrier from left to

right, they

^are

trapped

ⁱⁿ

^{2 ;}

^if

they

go from

right

^to

left, they

^{fall in 1.}

Assuming

^{this to be true,}

Eyring’s

^result

(22)

^{is exact.}

In order to calculate

departures

from the absolute rates, ^{we must} describe more

precisely

^this

problem

of

trapping.

^{In the}

large

^{friction case}

(il

>

w),

^the

motion is

diffusive,

and it is easier to calculate u

directly,

^{as done}

by

Kramers. One recovers result

(6),

except ^{that the}

integral

in the denominator extends from _xl to X2

(anyhow,

it is controlled

by

^the

vicinity

of

xa). Here,

^{we consider}

only

^{the weak}

damping

case

(n w),

where the motion is

oscillatory.

Let

Pi(e)

^{be the}

probability

^{that a}

particle ^entering

well i

(1 or 2)

^{will be}

trapped

^{in that} well without ever

crossing again

the barrier at xa

(it

remains in

well

ⁱ until it falls to the

bottom). Conversely,

^let

Q(e)

^de

be the

probability

per unit time that a

particle starting

from the bottom of well i crosses the barrier ^at _xa with an energy in the _range

(s,

^{B +}

ds)

^and

for

^the

first

time.

Clearly, Pi

^and

Øi

^{are related}

by

^detailed

balance;

it is

easily

^{shown that}

Now,

^{when a}

particle

^with energy ^{8 enters} well

1,

it must end _up

somewhere,

^with

probability Ql(g)

to be in well

1, [1 - Ql(e)]

^to be in well 2. This

time,

there is no restriction on

Q, :

^the

particle

may ^return

as much as it wants to the 2 side before

eventually falling

^{into 1.}

Similarly,

^{we denote as}

Q2(e)

^the pro-

bability

^{that a}

particle crossing

the barrier toward 2 will

eventually

^end up in 2. With all these

definitions,

the net rates from one well to the other are

easily found,

e.g.

(we identify

^the

first

passage

through

^{x =}

xa). Using (22)

^and

(23),

^{we see that}

(24)

^{extends our} former result

(7).

^{It is shown in}

appendix

^{B that} a12 ⁼ a21, ^so that detailed balance is

preserved.

In order to

proceed

^fur

ther,

^we introduce the condi- tional

probability W1 (8, E)

^that

during

^{one round}

trip

in well

1,

^the

particle

goes from _energy 8 to 8’. Simi-

larly, W2(E, 8’)

^{refers to one round}

trip

^{in well 2.}

Upon isolating

^the

first

^round

trip

ⁱⁿ

1,

^{we obtain}

easily

^the

equations obeyed by Pl

^and

Q 1

(equations

^for

P2

^and

Q2

^{are obtained}

by

^{the inter-}

change

¹ ->

2).

^Once

again,

^the

important

parameters

are

61

^and

62,

the average energy losses

during

^one

oscillation in well 1 or well 2. The same two

limiting

cases are

simple.

(i) If ð1, ð2 > T, trapping

^is

complete

^{in the} energy range of

interest,

^{8 - T :}

According

^to

(24),

the absolute ^rate

theory

^{is correct,}

a conclusion which is insensitive to our various _appro- ximations

(markovian evolution, etc...) :

^a

particle

that crosses the barrier cannot

regain

^the energy lost in one oscillation.

(ii) If ð1, ð2 T,

^{we know how to calculate}

Pi(e).

Moreover, ^{we show in}

appendix

^{B that in the}

range

of

interest,

⁰ 8 %

T,

^the

probabilities Ql

^and

Q2

are

essentially

^constant

((26)

^{was indeed to be}

expected :

^since

statistically trapping

^occurs

only

^after many

oscillations,

^{it does}

not matter either where or in which direction one crosses the barrier : hence

Q,

^{= 1 -}

Q2, prorated by

the _energy losses

61

^and

62)- Using (13)

^and

(26),

we find the correction to

Eyring’s

^rate

As an

application,

^{let us consider}

desorption

^with

an activation

barrier,

^{as shown in}

figure

^{4. «1» is}

the

well,

^{0 2 » the} open space. The absolute rate,

given by (22)

involves the total

height V

⁼

(Ya - V oJ

of the barrier to be climbed. In order to find the

correcting

^factor

(24),

^{we note that a}

particle entering region

^{2 never returns. Hence}

The correction to the absolute rate thus retains the

same form

(7)

^{as in the case of no} activation.

4. Time

dependent

stochastic formulation. ^- All the

foregoing analysis

^{is based on an}

implicit

assump- tion that we can

unambiguously

separate ^two pro-

cesses :

(i) trapping

^{in the}

potential well, (ii)

^subse- quent

desorption.

^{The two} processes ^are treated as

statistically independent.

^{We now examine to what}

extent such an

assumption

^{is valid.}

Let us consider an

incoming

gas

particle. Strictly speaking,

^{it is never}

trapped !

^{Even if it} falls down to the bottom of the

well,

^{it will}

eventually evaporate again.

Thus absolute

trapping

^is

meaningless.

^What

we must consider instead is the

probability P(8, t)

that the

incoming particle

^{be still}

trapped after

^a

time t. As a function

of t,

^{we then} expect ^two stages :

(i)

^At

first,

^the

particle

remains close to the top ^of the

barrier,

^{within a}

layer

^{of width T.} Then there is still a fair chance that it can

pick

up

enough

energy from the heat bath to escape

right

away. The corres-

ponding

time scale t, is

typically

^{a few}

periods

^of

oscillation. If one waits

too long,

^the

systematic

^loss

after n round

trips, n3,

^overcomes any diffusion

effect ~ ilL1] :

^{with an}

overwhelming probability,

the

particle,

if it has not

escaped before,

^{falls toward}

the bottom of the well.

(ii)

^Then

begins

^the

desorption period :

^{the fallen}

particle

^tries

again

^and

again

^{to climb} up the

potential

well ^- and it will

eventually succeed,

^{after a}

time t2

of order

1 /u.

The

corresponding

behaviour of

P(8, t)

^{is sketched}

in

figure

^{6. At t =}

0,

^{we start from some} energy ⁸ close to the top : ^the

particle

^{is in and}

P(8, 0)

^{= 1.}

At each round

trip,

^the

particle

^{has some}

probability

to escape, and

P(8, t)

^decreases

quickly.

^{When the}

particle

^{has lost more than an} energy -

T,

^the escape