Atom trapping on surfacesAshok T. Modak and Patrick J. Pagni Citation: The Journal of Chemical Physics 59, 2019 (1973); doi: 10.1063/1.1680288 View online: http://dx.doi.org/10.1063/1.1680288 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/59/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Calculations of trapping and desorption in heavy atom collisions with surfaces J. Chem. Phys. 130, 064703 (2009); 10.1063/1.3073831 Quantum studies of H atom trapping on a graphite surface J. Chem. Phys. 122, 014709 (2005); 10.1063/1.1827601 Residence Time Analysis of Rare-Gas Atoms Trapped at a Solid Surface J. Vac. Sci. Technol. 10, 411 (1973); 10.1116/1.1317081 Trapping and Energy Transfer in Atomic Collisions with a Crystal Surface. II. Impurities J. Chem. Phys. 39, 1317 (1963); 10.1063/1.1734433 Trapping and Energy Transfer in Atomic Collisions with a Crystal Surface J. Chem. Phys. 38, 523 (1963); 10.1063/1.1733690

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

[HE JOURNAL OF CHEMICAL PHYSICS VOLUME 59, NUMBER 4 15 AUGUST 1973

Atom trapping on surfaces

Ashok T. Modak and Patrick J. Pagni University of California, Berkeley, California 94720

(Received 20 November 1972)

The trapping of gas atoms on collision with a solid surface is described using the soft cube model. Fraction trapped is defined as that fraction of an incident flux which has a normal energy less than the gas surface potential well depth, after a single collision with the surface. Sticking coefficient is synonymous with fraction trapped. The functional dependence of the sticking coefficient on the gas and surface atom masses, temperatures and the interaction potential parameters is obtained. Sticking coefficients are predicted over a wide temperature range for xenon, argon, and neon on gold, silver, and platinum, and also for cadmium, gallium, titanium, cesium, silver, and rubidium on tungsten. Good agreement is obtained with experimental results for xenon on tungsten, platinum, and silver and for silver on tungsten. Argon and krypton on tungsten, platinum, and silver obtain reasonable agreement. Potential param­eters for xenon, krypton, and argon on tungsten, platinum, and silver are derived from comparison with sticking coefficient data. These parameters are analogous to the gas-gas potential parameters obtained from gas phase transport properties, and though more approximate, may eventually prove to be as useful.

I. INTRODUCTION

The collision of a gas atom with a solid surface is a fundamental inelastic process. The functional de­pendence of the energy exchange on the collision parameters has been obtained previously. 1 All in­cident atoms with energy after collision less than the gas surface interaction potential well depth are trapped on the surface. Fraction trapped is defined as the fraction of an incident flux which is trapped after a single collision with the surface. Sticking coefficient and capture coefficient are synonymous with fraction trapped. Fraction trapped is obtained here as a function of gas and surface masses and temperatures, and gas-surface potential parame­ters. Good agreement is obtained with experimen­tal results for xenon on tungsten, platinum and sil­ver and for silver on tungsten. Argon and krypton on tungsten, platinum, and silver obtain reason­able agreement.

Previous claSSical mechanical analyses include the work of Goodman, 2 who obtained the fraction trapped on a stationary lattice at 0 OK, and of Lo­gan and Keck3 and Logan4 who used the soft cube theory with an exponential repulSion to obtain an implicit expression for the fraction trapped. Com.,. parison with Logan's results provides an empirical relation between exponential and harmonic poten­tial parameters. Other classical theories have been reviewed by Weinberg and Merrill5 who have obtained an expression for the sticking coefficient as a function of thermal accomodation coefficients.

The purpose of this paper is to provide an ac­curate description of the temperature and atomic species dependence of the sticking coefficient. The results of the analysis are physically consistent. Predicted fraction trapped decreases with increas-

2019

ing nondimensional well depth, tending to unity for deep wells. Fraction trapped is predicted for com­parison with future experiments and compared with available experimental data. Gas surface potential parameters obtained here from sticking coefficient data have been used to predict adsorption and de­sorption and desorption lifetimes. 6 This classical theory applies when the gas and surface tempera­tures are larger than the surface Debye tempera­ture, and when the deBroglie wavelength of the in­cident atom is small compared to the lattice spac­ing. The model predicts a resonant behavior. At resonance collisions are adiabatic. While inde­pendent evidence 7

,8 exists for this phenomenon, more experimental verification is required to de­termine the extent to which resonance survives when anharmonicities and a distribution of surface oscillator strengths are taken into account.

The following section briefly describes the har­monic soft cube model with its assumptions and principal results. The fraction trapped is obtained in Sec. m. Com pari suns are made with previous

DYNAMIC HARMONIC REPULSION

_-0 ------~7SrATIC ATTRACTION

/ ,,~ ___ -'--_'-- ...- '" 0

T = y-z

FIG. 1. Collision model. Gas surface potential well formed by superposition of dynamic repulsive and static attractive potentials. An incident atom at energy E is trapped in the well with energy E' after collision.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

2020 A. T. MODAK AND P. J. PAGNI

calculations of Goodman2 and Logan4 and with ex­perimental results of Stoll, Smith, and Merrill9

in Sec. IV. Additional predictions are presented. Section V summarizes our conclusions.

II. GAS-SURF ACE INTERACTION MODEL

A. Gas-Surface Collision

A detailed description of the harmonic soft-cube model has been previously presented. 1 The inter­action between an incident gas atom and the sur­face is described classically as a two-body col­lision represented by a potential consisting of an oscillating, harmonic inner half and a stationary outer half of height Ii. The gas and surface atom displacements from their respective equilibrium positions are y(t) and z(t) as shown in Fig. 1. At time t = 0 the gas atom impacts at the end of the gas-surface" spring". For the collision shown, the gas atom approaches the solid surface from the free state with a kinetic energy E =mgu2/2kTs non­dimensionalized on the surface temperature. The energy after colliSion, E I, is less than the well depth, Ii = D /kTs , and the atom is trapped on the surface.

The assumptions impliCit in the model are: (1) the tangential component of the gas atom velocity is conserved; (2) a single "spring" of stiffness ks, represents lattice effects; and (3) the surface os­cillators have an equilibrium distribution of veloci­ty amplitudes. There is experimental evidence showing that at least in the region of the maximum of the scattering pattern the first assumption is satisfactory. The second assumption is valid for high energy collisions where the collision dura­tion is short compared to the period of the surface oscillator. 5

The collision duration, T c ' is defined as the time between collision initiation and the return of the gas-surface spring to its equilibrium exten­sion with the gas atom moving away from the sur­face atom, and is given implicitly by the smallest root of y(T cl = z(Tcl which may be written as

COS(W1 Tc + <PI) = (- 10'21 wUI 0'11 w~) COS(W2Tc + <P2)' (1 )

The normal mode frequencies of the coupled mo­tion are

W 1,2 = {[ v2 + f,L + 11 ± [(v2 + f,L + 1)2 _ 4v2] II 2 Y 12/ 4v,

(2) where f,L=m g /11I s is the inertia parameter, v=wslwg= [(k s /k g)(Jn g/m s)l1/2 is the frequency parameter. 0'1,2 and <P1,2 are the amplitudes and phases of the normal modes. Knowing T c' one can find the final energy and the energy transfer A(E), defined as

(3)

where Y(Tc) is obtained from a normal mode solu­tion of the equations of motion as

Y (Te) = [tvs WI /(1 - t)] Sinw1 Te

- [(1 + Ve - w~ v2)/(1 - t)wf v2] COSW1 Te

- rVs w2/(1- t)] sinw2Te

+ [(1 + Vc - wf v 2)/(1- t) wf v2] cosw2 Te • (4)

An interesting feature of this model is the res­onance which occurs at

For these values of W1,2, Te is such that Y(Tcl= 1 and the energy transfer in the collision goes to zero. On the basis of Eq. (2), resonance values of the frequency parameter are given by

vr(f,L, n) = «2/X){1 +[1- >..(1 + f,Lll1/2}- (1 + f,L))1/2,

where

B. Energy Transition Kernel

Pagni and Keck1 obtained the one-way equilib­rium transition rate R(E, Ve , Vs)dE dVedVs as the flux in phase space from the state E to the state

(5)

(6)

E'. Dividing this rate by the Boltzmann factor one obtains the energy transition kernel

G(E, Ve' V.)dEdVedVs= (Ehf,L)(1 +Vc)

x exp [ - (E/ f,L)(V~ + V ~)] dE dVe dVs (7)

which gives the probability of an incident atom with energy E striking a surface atom of velocity Vc at the position Vs'

Transition kernel contours are shown in Fig. 2 for an incident energy, E = 3 and inertia parame­ter f,L = O. 72. For E 2: 2, the transition kernel has a strong maximum in initial condition (Ve , V s )

space at

Vs=O, Ve=H(1+2f,L/E)1/2_1]-O. (8)

This peaked behavior will be utilized in evaluating the fraction trapped in the next section.

III. STICKING COEFFICIENTS

A. Analysis

Consider a flux of gas atoms with an energy dis­tribution B(E) incident on a surface. The fraction of all collisions with initial conditions in' the range E to E +dE, Vc +dVe , and Vs to Vs +dVs is given by B(E)G(E, Ve, Vs)dEdVedVs ' Since all incident atoms with an energy, after colliSion, less than the well depth are trapped, the sticking coefficient is

ft=r J. 5B(E)G(E, E')dE'dE/f'[B(E)G(E, E')dE'dE.

6 0 6 0 (9)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

ATOM TRAPPINGON SURFACES 2021

Vs

1.2 E = 3 fJ-=O.72

G=O.03

resonance region in (/1, lJ) space, 2. 5s lJ S 3. 5, where previous linear approximations6 are valid.

The cross term in the expansion is eliminated by the transformation

Ve= V~ cosO - V~sinO,

V.= V~ sinO + V~cosO ,

(13)

where 0 = t cos-1( (g - d)/2/]. Substituting Eqs. (13) in Eq. (11) gives E' /E as

1.2 Vc (~sin20+~cos20+/cososino)

-1.2

FIG. 2. Probability of initial conditions, G(€, Ve, Vs ),

in initial condition space for incident energy, €= 3 and inertia parameter, /-1= O. 72. The maximum occurs at Vs=O and Ve=O.1.

The numerator of Eq. (9) can be integrated over all the initial conditions which correspond to final en­ergies in the range 0 to 1)

r~ JV s(6,E) JVe(6,e, v s) / It= J. B(E) v (0 e) V (0 e V) G(E, Ve , Vs)dVedVsdE

v s t C' , S

~~ B(E) (s':_: f~e:_: G(E, Ve, Vs)dVedVsdE. (10)

The strong maxima observed in Fig. 2 permits a Taylor series for E'/E(Ve, V.) about (ire, V.) to be truncated at the second order terms,

E' /E = 6./E + 1 =a +bV.+c(Ve - VJ + (d/2) V~ - - 2 +1 V s(Ve - Vel + (g /2)(Ve - Ve), (11)

where the coefficients a, b, c, d, I, and g are de­fined as

a - y' 21_ b = 2y (aY / a V s) I ve ' 0 , - Ve,O,

c=2y(ay/aVe)lv 0, e'

d=2[y(a2y/aV~)+(aY/aV.)2]ve'0 (12)

1= 2[ y(a2y/aV~) + (ay/aVe)2]ve,0

g = 2 [y (a 2y /av. aVe) + (ay /a v.)(aY /a Ve)Jve,O

Contours of the coefficients a, b, c, d,f, andg in (/1, lJ) space are shown in Figs. 3(a) to 3(f), respec­tively. These figures may also be used to esti­mate moments of the energy transfer and rate con­stants1 given in terms of the coefficients in Eq. (11). At resonance values of the frequency parameter the final and initial energy are equal for all initial conditions Ve and Vs so that a is unity while b, c, d, I, andg are zero. Figure 3 defines the near

( , b sinO +c cosO )2

X Ve + d sin20 +g cos20 +1 sin20

+ (~ cos20 + ~ sin20 - I cosO sino)

( , b cosO - c sinO )2

x Vs+ dcos20+gsin20-/sin20

E' (b sinO +c COSO)2 = E -a+ 2(dsin20+gcos20+/sin20)

(b cosO - c sinO)2 (14)

+ 2(d cos20 +g sin20 -I sin20)

The coefficient of one of the transformed variables on the left-hand side is at least three orders of magnitude larger than the coefficient of the other over the full parameter range, 10-2< /1< 1.3 and 1 < lJ < 5. Since one of the second order terms is small, Eq. (14) becomes either a pair of parallel lines oriented about the V~ axiS, V~= ~1' ~2 where

~ = -(bsinO+ccosO) 1,2 dSin20+gcos20+/sin20

±{(d sin20 +g2COS20 +/sin20)

x [~ _ a + (b sinO +C COSO)2 E 2(d sinzO +g cos~O +1 sin20)

(b cosO - c sinO)2 lll/2 + 2(d cos20 +g sin20 -I sin20)J \ (15a)

or a pair of parallel lines oriented about the V~ axis, V~ = ~ 1, ~2 where

~ = -: (b cosO -c sinO) 1,2 d cos 20 +g sin20 -I sin20

±{(a cos20 +g2Sin2o - ISin20)

[(' (b sinO +c COSO)2

x E - a + 2(d sinzO +g cos20 +/sin20)

(bcosO-csinO)2 J}1/2 + 2(d cos20 +g sin20 -I sin20) (15b)

For (' = 0, the radical in Eqs. (15) is very nearly zero so that only (' = 1) is of interest. The numera tor of Eq. (10) is then well approximated by the

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

2022 A. T. MODAK AND P. J. PAGNI

::i.

0:: W I-

/.0

w 0.8 :::E ~ 0:: ~

c... 0.6 ~

I-0::

~ 0.4 0.99

O. 2 L::--.£.1.-::--"""'----L_L....,-.l...-_---..JIL.-_---LIL.-~ 1.5 1.75 2.0 2.25 2.5 2.75

::i.

~

I-0::

1.0

~ 0.4

3.0 FREQUENCY PARAMETER, V

,7/

1.2 r---,--~rr---.------,---,------,-..____,

~

I-0::

0.2 L-.._----L--"_.....J..:l~_.=.._...LJ~_---L_~ 1.5 1.75 2.0 2.25 2.5 2.75 3.0

FREQUENCY PARAMETER, 7/

1.2 .------.-----~.------.------~----~

1.2.----~~--~--~-~......,..----,---~

~ 0.4

FIG. 3. Contours of constant Taylor series coefficients a, b, c, d, f, and g as defined by Eq. (11) in (11, v) space. The coefficients of the first six terms are displayed as follows: (a) a, zeroth order coefficient; (b) b, first order coef. ficient with respect to Vs; (c) c, first order coefficient with respect to Vc; (d) d, second order coefficient with respect to Vs; (e) f, second order coefficient with respect to Vs and Vc; (f) g, second order coefficient with respect to Vc' At resonance values of v, a=l, and b=c=d=f=g=O.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

ATOM TRAPPING ON SURFACES 2023

volume bound by G(E:, Va' V.) and O(E, Va V.) in (G, Va' V.) space.

Applying the rotation transformation to Eq. (7) one obtains

G (E, V~, V~ )dE dV~ dV~ = (E!7T /l)(1 + V~ cosO - V~ sinO)

or

xr(E//l)(V~2+ V~2)]dEdV~dV~ • (16)

Changing variables in Eq. (10) to V~, V~ allows integration of the numerator between the limits ~1 < V~ < ~2 and - 00 < V~ < 00; or ~ 1 < V~ ~ 2 and - 00

< V~<oo, depending on whether Eq. (15a) or (15b) applies, to obtain the sticking coefficients

(17a)

ft = i~ B(E){~ {erf[(~ y/2 ~2 ] -erf [(~y/2 ~lJ} -t sinO (; ) 1/2 [exp (- ; ~~) - exp (- ; ~~) ] }d{i ~B(E)dErl,

where the lower limit on Vc in the denominator was approximated as - 00.

B. Limiting Cases

Three limiting forms of the incident energy dis­tribution are considered: (a) a monoenergetic in­cident beam at energy Eo,

where N is the number density of the incident beam as it strikes the surface; and (b) a collimated beam from an oven at Te normalized on the flux per solid angle to the surface,

(17b)

B(E) dE = (N!7T) (kT./21)1Tmg)1/2[172E/(1 +170)]

xexp[-17(E-O)]dE, (19)

where 17 -= T .ITe and N is the number density in the oven; and (c) a one~dimensional flux to a surface immersed in a gas with a Boltzmann distribution about the temperature T e'

B(E) dE = N(kT ./2171Tmg)1/217 exp [ -17(E - 0)] dE, (20)

where N is the gas phase number density.

The fraction trapped for the limiting case of a monoenergetic incident beam at Eo is obtained by substituting Eq. (18) in Eq. (17)

ft=~{erf[(~r/2 ~2J-erf[(~y/2 ~lJ}+tcOSO(1T~Oy/2[exp·(- ~ ~~) -exp(-~~~)J (21a)

or, depending on whether Eq. (15a) or (15b) applies;

where ~1 and ~2 are given by Eqs. (15) with E'/E = olEo.

(22)

as expected, since the energy transfer required for trapping is large and the probability of such transfer is small. Now consider Eo- O. For large () the first terms in Eqs. (15) are small com­pared to the radical so that ~ 1 = - ~2' Since the

error function is odd, Eqs. (21) give

lim ft=Herf[(EoI/l)1/2 x] +erf[(EoI/l)1/2 X]}= 1, 60""6- co

(23) where X = ~ 1 -= - ~2' For small 0, expansions of the error functions and the exponentials in Eqs. (21) yields

(24a) or

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

2024 A. T. MODAK AND P. J. PAGNI

lim It = (Eohr J.L)I/ 2(~2 - ~ lH1 - t sinO (~2 + ~dl = 0 eo~ 6~ 0

(24b) which corresponds to the condition of no atoms be-

or

ing trapped due to the absence of a well.

The fraction trapped for a collimated incident beam is

(25a)

while the fraction trapped for a Boltzmann incident beam is

or

It = 111~ exp[ -17(E - O)l{tierf[ (~r 2~2J -erf[(; t2~IJ} -t sinO(:E t 2 [exp (- ;~~)- exp (- ~ ~~)J }dE. (26b)

All integrations are performed numerically. Equations (25) and (26) are equal to within ten percent for 170 > 10. ~1 and ~2 are given by Eqs. (15) with E' = O. The results of these calculations are shown in Figs. 4 and 5.

f­Z W

U

LL

~ 01 o u

(3 = 50 7) = 1.0

--I-' = 0.2 -1"=1.0

( kT 112

BIEJdE~N ~;~) ry"pl-ryle-&J}dE /_, 2ryrrmg

/ \ I \ I \ I \ I \ I \ I I

I I I I

I' I' I' II II II II

" " II

" " i I

\ I \ I \ I \ I \ I

I I

II II II 1/

" II U ,

I -­/

FIG. 4. Sticking coefficient, ft, versus the frequency parameter I' for 0 = 50, p. = 0.2 and 1. 0, for a monoener­getic beam at EO 0 = 6 and a Boltzmann beam with 1) = 1. O. The effect of resonance is displayed by it - 0 at resonance values of I'r(/.l, n).

I or--,---,--,-------r=::::::=::=:t=='f'===j

'" U

f-en 0.3

TJ = 0.1

WELL DEPTH 8

FIG. 5. Sticking coefficient, ft, versus the nondimen­sional well depth, 6, for a Boltzmann incident beam parametrized in 1), with 1)=0.01, 0.1, 1 and "'atp.=0.2 and 1'=1.90.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

ATOM TRAPPING ON SURFACES 2025

Fraction trapped is plotted versus frequency parameter in Fig. 4 for a monoenergetic incident beam at Eo= 0 and a Boltzmann beam with 1] unity. The fraction trapped for the Boltzmann beam ap­proaches the monoenergetic limit as 1]- 00. At resonance the energy transferred is zero and the fraction trapped approaches zero. Anharmonicities of real systems tend to round the local minima shown here. The fraction trapped tends to unity for v:s 2 for deep wells.

The dependence of the fraction trapped on the well depth, 0, parameterized in 1] for a Boltzmann incident beam is shown in Fig. 5. The fraction trapped shows a strong dependence on both Tg and Ts. In the limit 1]-0, it-a, since the more ener­getic gas atoms are less likely to be trapped. At any fixed 1], in the limit 0- 00 , it-1 and in the limit 0- 0, it- 0.

Potential parameters for gas-surface systems may be obtained by comparing data with the frac­tion trapped contours in ("M, v) space shown in Fig. 6 for a typical well depth, 0 = 5 and temperature parameter 1] = 1. Approaching from either side of resonance, the fraction trapped monotomically de­creases to zero at VrUl).

IV. COMPARISONS

The comparison between Logan's results4 for a Boltzmann beam at the surface temperature and Eq. (26) is shown in Fig. 7. The harmonic potential parameters used provide the best agreement with Logan's exponential potential results. This analysis is compared in Fig. 8 with Goodman's results2 for a static lattice with a Boltzmann incident beam. The predicted values of it given by Eq. (26) and those of static lattice are equal within 10% in each case.

It is interesting that agreement over the wide temperature range shown in Figs. 7 and 8 was ob­tained with the single set of potential parameters listed in Table I for each gas-surface system. Sat­isfactory agreement with Goodman and Logan leads to the conclusion, albeit tentative, that the fraction trapped may be relatively insensitive to the as­sumed functional form of the repulsive potential.

A. Interaction Potentials

To correlate alternative interaction potential functions the harmonic parameters used in this analysis are matched with other potential parame­ters. Four functional form!? for the interaction po­tential are considered. The harmonic repulsive potential used here is

(27)

where kg is the gas-surface spring stiffness. The exponential repulsive potential used by Logan4 is

1.75 2.0 2.25 2.5 2.75 3.0 FREQUENCY PARAMETER, V

FIG. 6. Contours of constant sticking coefficients in (/l, v) space given by Eq. (25) for 0=5 and 7]=1. Poten­tial parameters may be estimated from Fig. 6 using experimental sticking coefficients.

V(r)=Dexp(-r/K), (28)

where K is the range of interaction. The Morse po­tential is

V(r)=D[1-2exp(-yr)]2; (29)

and the Lennard-Jones potential is

V(r) = D - 2D [r* /(r + 2-1 / 6r *)]6 +D [r* /r + 2-1/ 6r *)]12,

(30) where r= ° is placed at the inside top of the well and D is the well depth. Expanding in a series about r= ° with V(O) = D and comparing first order coefficients yields the matching condidtions

kg =D /2K2 = 81'2 D = 363D /(r*)2, (31)

comparing second order coeffiCients yields

kg=D/K2=12y2D=575D/(r*)2. (32)

A best fit might be expected to lie between these values. Comparing the harmonic and exponential parameters which gave the agreement shown in Fig. 7 results in the matching expression

kg=0.2D1.17/K2, (33)

where kg is in dynes cm-1, D is in OK and K is in A. The harmonic frequency parameter is then

(34)

and it is assumed that the well depth remains con­stant.

B. Experimental Comparisons

One method for obtaining it consists in measur­ing the steady state scattering pattern obtained

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

2026 A. T. MODAK AND P. J. PAGNI

1.0 ,------,--.-------,.----,---,------,

+" 4-

0.9

0.8

- 0.7 f­z W

~ 0.6 LL w o u '-" 0.5 z

" u f- 0.4 en

0.3

0.2

~ .. Xe- W 7/ = 2.02

\ \ \

Kr-W \ 7/=2.2 \

\

\ , Ar-W " 7/ = 2.25 ,

,

"

7) = 10 -- HARMONIC SOFT CUBE --- EXPONENTIAL SOFT CUBE

\ \

"

, ~

:,.

" " "-"-

"

FIG. 7. Sticking coefficient, ft, versus T., for a Boltzmann incident beam with 1) = 1. 0 of xenon, krypton and argon on tungsten. The dashed lines are soft cube calculations with a repulsive exponential potential. The heavy lines are the results of Eq. (26).

when a collimated beam strikes a clean, well de­fined surface in an evacuated chamber.l0 The pat­tern is resolved into a diffuse component and a specular component. The latter is assumed to be rotationally symmetric about the angle of maxi­mum scattering. The diffuse component is fitted to a cosine law such that the sum of the symmetric specular lobe and the fitted diffuse component pro­vides a mass balance with the incident beam. The ratio of the number flux in the diffuse component to the number flux in the incident beam is the fraction accommodated and approximately gives the fraction trapped. These are difficult experi­ments. The best data available are those of Stoll, Smith, and Merrill9 of accuracy ± 15%. Using modulated beam techniques with the fraction trapped given by one minus the ratio of the instantaneous flux from the surface to the incident flux, Cho and Hendricksll report an accuracy of ± 1 %.

To compare analytical results with experiment, Jl, lJ, 11, and Ii are required. The temperature parameter is

(35)

where 01 is the angle of incidence with the sur­face normal. Gas-surface interaction potential

parameters, lJ and Ii, are obtained by comparison with independent experimental results analogous to obtaining gas-gas potential parameters from gas phase transport properties. Harmonic gas-sur­face potential parameters for cadmium, gallium, titanium, ceSium, silver, and rubidium on a tung­sten substrate have been determined6 by compari­son between predicted and experimental adsorp­tion lifetimes. For rare gases on tungsten and gold, Logan' S3,4 exponential potential parameters may be converted to lJ via Eq. (34). For rare gases on platinum and silver, lJ is determined by iterative comparison of Eq. (25) with the data of Weinberg and Merrill5 and Stoll, Smith, and Mer­rill. 9

These comparisons are shown in ·Figs. 9 and 10. Fraction trapped versus Ii for a collimated beam at 11= 1 and 10 for rare gases on platinum and silver is shown in Figs. 11 and 12, respective­ly. At intermediate values of 11 the fraction trapped may be obtained by linear extrapolation between 11 = 1 and 11 = 10. Similar comparison of Eq. (25) with the data of Stoll, Smith and Merrill for rare gases on tungsten is shown in Fig. 13. It is im­portant to note that the parameters used for the comparison in Fig. 13 are those obtained by Logan4

from independent comparisons with accomodation coefficient experiments. The agreement obtained using these parameters is encouraging. Figure 14 shows fraction trapped versus Ii for a collimated beam with 11 = 1 and 10 for rare gases on tungsten.

While the over-all agreement in Figs. 9, 10, and 13 is good, some discrepancies arise particu-

f­Z W U 0.6 LL. LL. W o u 0.4 o z

"" u

tI; 0.2 ---

o~ __ ~ __ ~~~~~~~~ o ~o ~o Tg OK

FIG. 8. Sticking coefficient, ft, versus gas tempera­ture, TK, for a Boltzmann incident beam of xenon, kryp­ton and argon on tungsten at 0 0 K. The dashed lines represent static lattice results. The heavy lines are the predictions of this model for a lattice - 0 oK.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

A'

1.0 .----,-1---.---1---,-�- . r . d aN' 'OVd

..:: 08 r-,... z w

-- Xe

--- Kr

_.- Ar . '" 1000

GO~r- ~

~ r---~--------;--------:-8 0 1::.

0.4r- _--.... ------ -== ~ f-.---- 0

:x:: 0 _-----.-u "' ___ --------,... 02f----(f) r- <;>---._.----v~

1--'-'---'--t-

o I I I I 3~070--~40~0~--~50~0~--6~0~0--~7~O~O--~8~OO

SURFACE TEMPERATURE. TS OK

FIG. 9. Sticking coefficient, Ito versus the surface temperature, T., for xenon, krypton, and argon at Tg=3000K and Tg=10000K, onplatinum. The corre­sponding experimental results of Stoll, Smith, and Merrill for an incident beam at 61 = 45° to the surface normal are indicated.

larly for argon on tungsten at high temperatures. These are easily explained. Firstly, the experi­mental points represent the fraction of the incident atoms which fully accommodate with the surface and then desorb while the calculated values rep­resent the fraction of the incident atoms which have a final energy less than the well depth after the first collision and are initially trapped. These quantities are not identical and only order of magni­tude agreement may be expected. Since some of the atoms counted as trapped by Eq. (25) are de­sorbed after only a few collisions, they are not fully accommodated to the surface and will appear

0.8.-----,1,-----,-1---,-1--.--1---,

,...- 0.6r­

Tg=3000K _ Xe-Ag

--_ Kr-Ag --_ Ar-Ag

-

~~ F.---A~-----------------------9 u l.i.. l.i.. OAr­w

-o u

~ --------------'-'" f------ --- - • z 0.2r- _

"_.--_._'-,... (f)

~--.-------.

o ~ __ ~I_~I ____ ~L-I---~I--~. ~O 400 500 WO roo

SURFACE TEMPERATURE. TsoK 800

FIG. 10. Sticking coefficient, Ito versus T. for xenon, krypton, and argon on silver at Tg = 300 ° K. Experi­mental results of Stoll, Smith, and Merrill are indicated.

ON SURFACES 2027

09,-----r----,-----,-----,-----,------,

0.8

0.7

~ - 0.6

f-z w

~ 0.5 l.i.. w 0 U

t9 OA z :x:: u f- 0.3 (f)

0.2

0.1

0 0

-- Xe oPt Kr - Pt

-'- Ar -Pt

-- --7)= 10_-----

IC!_--------------

/

I /

.......­/'

/ I

I

2

/ /

/

/

---

/' /'

/

..- /'

...­..-...-

...­..-......... ""

--­......... ''''''''''

4 6 8 WEL L DEPTH 8

7) = I ........ "-­........ -

-' :.....-.--.

10 12

FIG. 11. Sticking coeffiCient, Ie. versus the well depth, 6, from a collimated beam given by Eq. (25) with 1)= 1 and 10 for krypton, xenon, and argon on platinum. Linear interpolation between 1) = 1 and 10 is possible.

in the experiment as nearly reflected atoms. Thus the experimental fraction accommodated should be less than the theoretical fraction trapped for the same physical system. This effect will be most significant for systems with a small fraction trapped such as the high temperature results for krypton and argon and least Significant for the deep welled xenon-tungsten system. Also the presence of the well as a barrier and possible adatom nonequilib­rium1

,6 alter the distribution of normal energies. Therefore an additional approximation is intro­duced by assuming a cosine distribution for the dif­fuse component.

Secondly, some discrepancy arises from the as­sumption of rotational symmetry of the specular component of the reflected beam. This assumption is inconsistent with the soft cube model assumption of conservation of tangential momentum. The good agreement of the soft cube model predictions with experimental scattering patterns previously re­ported3 lends credence to the conservation of tan­gential momentum. There is no known mechanism for transfer of tangential momentum save surface roughness since the dynamic mechanism affects only the normal momentum. Since dynamic energy

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

2028 A. T. MODAK A

0.9.-----,---,---------,---,----,----,

0.8

0.7

- 0.6 I­Z W

u 0.5 u.. LL W o u '-" OA z YO U

I- OJ en

/

/ /

Xe -Ag

Kr -Ag Ar -Ag

.-' .-'

/ --- 10 _ ----.----

1 _--­-----0.2 ---.... ,.... .1----

/' ---.---// --'--

// -- -----/ --/-

0.1

o o 12

FIG. 12. Sticking coefficient, ft, versus ('ifrom a col­limated beam with 1) = 1 and 10 for xenon, krypton, and argon on silver.

transfer mechanisms dominate the in-plane scat­tering, while the physically quite distinct static mechanisms effect the out-of-plane scattering, the assumption of symmetry is difficult to justify.

Perhaps data reduction can be facilitated by re-

1.0,--,----,1--,---,--1--.-.-1-,----,1--,---,

Tg =300 0 K _ Xe-W --- .... Kr-W -

~0.8- .. ___ Ar-W _

I­Z

i::-

w O.6f- _

~ ~~---------;------------~ ... -----u OAI- ... __ ------ ---.--.--- ..

1----

o III I 300 500 700 900 1 100 1300

SURFACE TEMPERATURE, Ts OK

FIG. 13_ Sticking coefficient, ft, versus Ts for xenon, krypton, and argon on tungsten at Tg = 300 0 K_ Experi­mental results of Stoll, Smith, and Merrill are indicated.

, NI

0.9.------,---,-------,---,---,-----,

0.8

07

~ - 0.6

I-z w

~ 05 u.. w o u '-" OA z YO U

i= 0.3 en

0.2

0.1

-- Xe-W --- Kr-W --- Ar-W

10 _ --.--------

o IL-_~I---~-~---L--L-~~ o 2 12

FIG. 14. Sticking coefficient, ft. versus {) for a colli­mated beam with 1) = 1 and 10 for xenon, krypton, and argon on tungsten.

placing rotational symmetry with an ellipsodial distribution. The larger of the minor axes of the ellipsoid is defined by the width of the in-plane scat­tering pattern while the new, smaller, minor axis is defined by the width of the out-of-plane scatter­ing pattern. This width could be approximately measured as the width of the in-plane scattering pattern at low temperatures where roughness scat­tering dominates dynamic scattering.

The assumption of rotational symmetry becomes more tenuous as T s increases, increasing the in­plane width of the specular lobe while the out-of­plane width remains approximately constant, lead­ing to an underestimate of the experimental fully accommodated fraction of the incident atoms. This effect is negligible in the case of xenon on tung­sten, where due to the high values of the sticking coefficient the reflected component is small. The error may, however, be significant for krypton and argon on high temperature surfaces. It would be useful to determine ft for the systems described here from modulated beam experiments.

In Fig. 15 the sticking coefficient predicted by Eq. (25) is plotted versus well depth for rare gases on gold using v given by Eq. (34) for 1) = 1 and 1) = 10. The predictions of Figs. 15 and 16 should prove use­ful for comparison with future experiments.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

ATOM TRAPPING ON SURFACES 2029

0.9

0.8

0.7

+-' 4-

- 0.6 f-z w

~ 0.5 u... u... w 0 u

<.':) 0.4 z 0<: U

f- 0.3 Ul

0.2

0.1

0 0

/' /'

/'

/

/

/ /

I

/'

/

,/--­"

---

------__ 1.9..--_----...--

" /' /'

/' /

/ /

/' /

/

...--. ...-- -

--

;---­----

--/"

-- ------~------ Xe-Au

8 . 8

Kr-Au Ne -Au

10 12

FIG. 15. Sticking coefficient, ft, versus 0 for a colli­mated beam with 7J = 1 and 10 for xenon, krypton, and neon on gold.

Figure 16 plots the sticking coefficient versus well depth given by Eq. (26) for a Boltzmann inci­dent beam at T S' 7j = 1, for cadmium, gallium, ti­tanium, cesium, silver, and rubidium on a tung­sten substrate. The frequency parameters, v, and well depths, D, for these systems6 are listed in Table I. Strong metallic-adsorption processes, where charge transfer between the adsorbate and the substrate can become important, 12 have well depths of the order of 60 kcal/mole. Under these conditions, initial collisions are highly energetic and the repulsive interaction is less adiabatic than is implied by an harmonic potential. The assump­tion of such a repulsive potential therefore leads to a slight underestimate of the sticking coefficient. The magnitude of this effect may be seen from com­parison of the predicted value of 0.96 for silver on tungsten at 1175 OK (5 = 30) with the experimental value of unity measured by Cho and Hendricks. 11

v_ CONCLUDING REMARKS

This analysis derives the fraction initially trapped using classical energy transition probabilities. Fraction initially trapped is that fraction of an in­cident flux with an energy less tl-jan the well depth after a Single collision. The rather peaked nature of the transition probability permits simple evalua­tion of the sticking coefficients for any incident

distribution. In a manner analogous to the gas­gas interaction problem we have derived potential parameters for rare gases on tungsten, platinum, Silver, and gold, and correlated harmonic poten­tial parameters with parameters of alternative po­tentials. Sticking coefficients are computed over a wide range of gas and surface temperatures.

Comparison of these predictions with experiments of Stoll, Smith, and Merrill shows good agreement for xenon on tungsten. Sticking coefficients of rare gases on tungsten also agree with the calculations of Goodman2 and Logan. 4 Comparison with experi­ments using argon and krypton on tungsten, xenon, argon and krypton on platinum and xenon, argon and krypton on silver gives qualitative agreement. Since experiments measure the fraction equilibrated with the surface while this analysis computes the fraction initially trapped some discrepancy may be expected. Good agreement is obtained with the ex­perimental sticking coefficient for silver on tung­sten. 11 Sticking coefficients for xenon, krypton and argon on silver and cadmium, gallium, titani-

1.0

0.9

0.8

0.7

4!"'

f-O.6 Ti-W z w u

~05 w 0 u

z

Ga-W

140

FIG. 16. Sticking coefficient, ft. versus well depth, ii, for a Boltzmann incident beam with 7J = 1, for cadmium, gal­lium, titanium, cesium, silver, and rubidium on tungsten. Experimental result for silver on tungsten at 0 = 30 is indicated by •.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

2030 A. T. MODAK AND P. J. PAGNI

TABLE 1. Potential parameters for an harmonic repulsive potential derived from comparison with experimental results of Stoll, Smith, and Merrillf and previously reported potential parameters. a.-c

Frequency Inertia Well depth Range of Adsorbate Substrate parameter parameter OK interaction, A

Rb W 1. 61b 0.465 32000b

Ag W 1. 72b 0.585 35500b

Xe Au 1. 75d 0.666 6000a 0.40a Xe Ag 1. 90f 1. 217 2000a 0.21a Xe W 2.02- 0.716 4500a 0.16a

Kr Pt 2.05 f 0.430 1050c

Xe Pt 2.10 f 0.673 3750c

Ar Au 2.10d 0.203 1200a 0.34a

Cs W 2.14b 0.722 33600b

Kr W 2.20- 0.456 2250a 0.14a Ar W 2.24- 0.218 950a 0.13a

Ar Pt 2.30 f 0.205 1600a 0.16a

Kr Ag 2.30 f 0.777 700c

Ne Au 2.39d 0.102 300a 0.24a Ti W 2.45b 0.260 63000b

Ar Ag 2.55f 0.370 420c 0.18a

Ga W 2.66b 0.379 39400b

Cd W 2.67b 0.611 21700b

aReference 4. ~eference 6. cR. Sau and R. P. Merrill, 8th Intl. Symposium on Rarefield Gas Dynamics,

Stanford, California, July, 1972. dCalculated from Eq. (34). -Comparison with Ref. 4. fComparison with Ref. 9.

um, ceSium, and rubidium on tungsten are pre­dicted for comparison with future experiments.

The current range of application of the model is limited to atoms exchanging translational energy nor­mal to the surface since tangential momentum is assumed to be conserved. Rotational and vibration­al excitations are not considered so that molecule­surface interactions are not rigorously described by the model. The analysis is based on classical collision mechanics and hence inapplicable to light slow atoms with de Broglie wavelengths of the or­der of the lattice spacing or for gas or surface temperature below the Debye temperature. This model also neglects surface roughness and lattice interactions. 3 It is assumed, as in the Einstein model that the surface atoms oscillate at a Single frequency. Resonance effects are predicted which would be diminished but not eliminated by including a more realistic continuous distribution for the sur­face atom frequencies and an anharmonic potential. There exists independent evidence for the resonant behavior predicted here in the work of Armand7

and Trilling. 8

Currently this model is being extended to de­scribe the energy distribution of atoms leaving the

surface, f+(E, e) for a given incident distribution f.{E, e) in the rarefied limit. In the future the ad­sorbed atoms could be treated as a two-dimensional gas to allow calculation of adatom recombination and dissociation rates using the diffusion theory 13

or the variational theory14 of reaction rates with the surface acting as an inert background. Such rates along with the adsorption and desorption rates al­ready obtained1

,6 could facilitate interpretation of global heterogeneous reaction rates.

ACKNOWLEDGMENTS

The authors are grateful to Professors J. C. Keck and R. P. Merrill for many useful discussions.

This research was supported by a Faculty Re­search Grant from the College of Engineering, Uni­versity of California, Berkeley. Numerical com­putations were performed at the University of Cali­fornia, Berkeley Computer Center.

'P. J. Pagni and J. C. Keck, 1. Chern. Phys. 58, 1162 (1973). 2F. O. Goodman, Adv. App!. Meeh. Supp!. 3, Vo!' 2,366 (1969). JR. M. Logan and J. C. Keck, J. Chern. Phys. 49, 860 (1968).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24

ATOM TRAPPING ON SURFACES 2031

4R. M. Logan, Surf. Sci. 15, 387 (1969).

sW. H. Weinberg and R. P. Merrill, J. Vac. Sci. Techno!. 8, 718 (1971).

6p. J. Pagni, J. Chern. Phys. 58, 2940 (1973).

'G. Armand, Le Vide 139, 18 (1969). 8L. Trilling, "Physical Adsorption and Desorption of Molecules

on Solid Surfaces," presented at the 8th IntI. Symposium on

Rarefied Gas Dynamics, Stanford, CA, July 1972.

9 A. G. Stoll, D. L. Smith, and R. P. Merrill, J. Chern. Phys.

54, 163 (1971).

laD. L. Smith and R. P. Merrill, i. Chern. Phys. 52, 5861 (1970).

IIA. Y. Cho and C. D. Hendricks, J. Appl. Phys. 40, 3339 (1969).

12G. A. Somorjai, Principles of Surface Chemistry (Prentice Hall, Englewood Cliffs, NJ, 1972), p. 256.

13J. C. Keck and G. F. Carrier, J. Chern. Phys. 43,2284 (1965).

14J. C. Keck, Adv. Chern. Phys. 13, 85 (1966).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to

IP: 128.114.34.22 On: Sat, 22 Nov 2014 16:27:24