The sticking coefficient of hydrogen atoms on a metal: Exact results for a model Hamiltonian

Surface Soence 161 (1985) L573-L577 North-Holland, Amsterdam

L573

SURFACE SCIENCE LETTERS THE STICKING COEFFICIENT OF HYDROGEN ATOMS ON A METAL: EXACT RESULTS FOR A MODEL HAMILTON|AN G.P. B R I V I O Isntuto dt Ftstca dell'Umcerstti~ and Gruppo Naztonale dl Struttura della Materla del CNR, Vm Celorla 16, 1-20133 Mdan, Italy

and T B. G R I M L E Y Donnan Laboratories, Umverstty of Liverpool, P 0 Box 147, Lwerpool L69 3BX, UK

Recewed 2 April 1985, accepted for pubhcatlon 3 June 1985

Previouslycomputed values of the sticking coeffioent of H atoms on a model sohd with energy loss to electron-hole pa~rs, are converted to exact (umtary) values for a model Hamlltonmn which allows sucking, and elastic scanenng only This ts done without further computation because, for the phys,cal process treated, namely the "getterlng" of a gas m a closed vessel, the decay of the ,mtlal state ,s gaven essentially exactly by the quas~-parttcle approxamat~on used in the original computations The only (tnvml) calculaUon needed, stems from the necessity to define carefully the sticking coefficient m terms of the decay rate, and the colhslon number We also comment briefly on certain aspects of the application of text-book scattering theory, to the getterlng process

Brivio a n d G r i m l e y [1] (BG) have calculated a p p r o x i m a t e values of the s u c k i n g c o e f f t o e n t of a reactive atom (hydrogen) o n a metal substrate, with energy loss to the electron system, using an approach based o n the B o r n - O p p e n h e i m e r [2] system of coupled equations. The physical process they treated is the following. A gas a t o m is c o n t a i n e d m a vessel of length b, one wall of which is the metal, the others inert. F o r time t < 0, the n o n - a d i a b a t i c c o u p h n g terms m the B o r n - O p p e n h e i m e r equations are switched off, a n d the system is in the electronically a d m b a t i c state [0,a) describing the gas a t o m in a s t a n d i n g wave state In) o n the g r o u n d - s t a t e adiabatic potential energy curve E o ( R ) w~th energy E o , = % the gas atom's kanetlc energy. At t = 0, the n o n - a d i a b a t i c c o u p h n g ,s switched o n causing the initial state to decay, a n d other adiabatic states [m,fl) to b e c o m e populated. [ m , f l ) describes the n u c l e a r m o t i o n of the gas atom in the state [fl) o n the excited a d m b a t i c p o t e n t m l energy curve E r a ( R ) , with energy E,,a. [fl) can be a b o u n d state of the c h e m l s o r p t i o n well, 0 0 3 9 - 6 0 2 8 / 8 5 / $ 0 3 . 3 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics P u b h s h m g Division)

G P Brwto, T B Gnmley / Stroking coeffictent of H on metal

L574

in which case ]rn,fl) Is a stuck state, or an u n b o u n d state describing inelastic scattering. In the quasi-particle approximation, F0~(Eo~ ), the negative imaginary part of the self-energy of the initial state, determines its hfetime 1-= 1/[2Fo~ (E0~)]. T o lowest order m the non-adiabatic coupling operator IV, the sticking contribution t o / ' 0 4 is F 0 ~ s ( E 0 ~ ) = ~ r Y ",

I(~.OIWIm,B)I28(Eo=-E,,~).

~

(1)

m /3-bound

If Vco. is the n u m b e r of collisions per unit u m e the gas a t o m makes with the metal, a formula for the sucking coefficient s is, s = Rst,ck/~,co . ,

Rs,,c k = rs-' = 2F0,~.s(E0,,).

(2)

B G calculated the sticking coefficient for a simple microscopic model m flus way, and observed that s < 1 is not guaranteed, and that reasonable changes in the parameterlzatlon of the theoretical model used for H atoms interacting with a metal would lead to s > 1. We now wish to point out that: (0 There is a model H a m i l t o n i a n for which (1) is an exact formula, not simply the leading term. This H a m f l t o m a n is,

H=Ho+ V,

Ho=lO,a)Eoa(a,Ol+ Y" .,

v = Y ".

~_.

~..

[m,fl)E,.~(fl,m[,

i

fl = b o u n d

( 3 ~'

([O,a)(a,OlWlrn,fl)(fl,m[+lm,fl)

m .8 = b o u n d

(fl,m [ W IO,a)(a,O J) (n) The quasi-particle a p p r o x i m a u o n , which reqmres

C0o(E0.) <<

(4)

can be m a d e as accurate as we like by m a k m g the dimension b of the vessel containing the gas a t o m large enough. ( i l l ) Eq. (2) ~s only an acceptable definition of s when s << 1. T h e correct definition is

s = R,,,~k/(Rs,,~k + ~coll),

(5)

when there is no inelastic scattering, Le., for the H a m i l t o n i a n (3), and s = R ~t,ck/( R ~t,~k + R,,~, + V~ol,)

(6)

m general, where R,,~ is the rate for inelastic scattering. Observations (i)-(iii) mean that, from the results p u b h s h e d by BG, we can easily obUan exact (i.e. unitary) values of the sucking coefficient of H atoms on a model metal ff (3) describes the dynamics.

G P Bnwo, T B Gnmley / Stwkmg coeffwtent of H on metal

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N o w (1) is self-evident: the states [m,fl) are not coupled together by W so H remains, like H 0, diagonal with respect to these states. Consequently B G ' s diagonalizatlon of the Green function is exact for th~s Hamiltoman. Regarding (u), it is a unique feature of the physical process treated by BG (the "gettering" of a gas in a closed vessel) that, because the operator W for inelastic events is localized at the metal surface, we can, by making the d i m e n s m n b of the vessel large enough, make the decay rate of the initial state as small as we please. Ttus follows because, for large enough b, there is a n o r m a h z a t l o n factor b-1/2 in l a). This makes (a,OI W lm,fl) - b-1/2 if ]fl) ~s a stuck state, and therefore F0~.s - b - 7. If I/3) is an u n b o u n d state, it too has the factor b-b:z, but when we calculate the self-energy contribution for inelastic scattering, we have to include the density of final states which is proportional to b. This gives a result - b-~ again. Of course, increasing b to validate the quasi-partmle approximation, decreases Vcoll because Vco. - b-~ so that s, calculated from (2), (5) or (6) is independent of b. In BG's calculations Fo~..,(Eo~)/Eo~ < 4 x 10-3 for b = 60 AU, so the quasi-particle approximation is in fact valid. Turning now to the defmitlon (5) of s for use with the Hamiltonian (3), we note first that it coincides with (2) when s --+ 0. But the reason why (5) is exact, and (2) is not, can be seen as follows. At t = 0 the initial gas a t o m state is normalized to unity, ( a l a ) = 1, and there is one a t o m in the vessel making elastic collisions with the metal at the rate Vcol!= ( % / 2 M b 2 ) 1 / 2 . For t > O, the state [a> decays due to sticking and (a(t) la(t)) = exp(-t/rs) This decay means that the rate for elastw colhstons is reduced,

co. (t)= ,,co,, <,,(t) I This is the crucial point. The sticking rate is d ( a ( t ) l a ( t ) ) / d t nian (3), and therefore the exact definition of s is

for the Hamllto-

s = StlClong r a t e / ( s t i c k i n g rate + elastic scattering rate)

= Rst,ck(t)/[Rst,ck(t)

+ Pcoll(t)]

= Rst,ck/(Rsuck + VcoU),

where, as the last form shows, we can calculate at t = 0 because the exponentml time dependence cancels. The definition (6) for theoretical models which allow inelastic scattering, is established by a trivial extension of the above argument, and need not detain us here. Once Vcon has been identified as the elastic scattenng rate (not the rate for a// processes) for the "gettering" of a gas, (5) and (6) are obviously correct. Indeed with v~o. replaced by Re~, (6) has been used before [3], and of course it gives unitary results (s < 1) even when the rates are calculated by approximate

G P Brtvlo. T B Grtmley / Sttckmg coefflctent of H on metal

L576 $

0t,

i ,,/ ",, 20

4,0

60

80 E(meV)

Ftg 1 Exact results for the stlckang coeffictent s (full hne) and SBG (broken hne) of H atoms on a model metal for the Harmltoman (3), and the non-adiabatic couphng IzVof ref [l], as functtons of the kinetic energy of the mcormng gas atom

methods. But this is not our reason for using (5) here. We want to use it because BG's results for Rst,c k are exact for a Hamiltonian of the form (3), and therefore we can give exact values of the stickdng coefficient for a specified theoretical model, namely the Hamiltonian (3) with the matrix elements of W defined by BG. We recall that BG found that the r a t t o Rsttck/Vcoll, which we shall now call SBG, was not small at all gas a t o m energies (SBG -- 0.4 at % - 21 meV), and it is important therefore to use the correct formula (5), not (3). This is easy to do because (5) is s = sBo/(sBo

+ 1).

(7)

The results are shown in fig. 1; s ts the full line, SBG is the broken hne, and when s is greater than about 0.1, the difference between s and SBG becomes apparent. Fmally, we c o m m e n t briefly on certain scattering theory aspects of our work. M a n y students of Surface Science are aware that there is a difficulty in defining s in scattenng theory. Text-book accounts of scattering (ref. [4] for example) assume that the scattering potential is meffectlve in the remote past, and in the distant future, because the particle is far from the scatterer, m these hrmts. For a reflected particle this is true, and text-book scattering theory can be applied using a suitable Hamiltonian (but n o t the H a m i l t o m a n (3), see below) to calculate the reflection coefficient r, hence by unitarity (r + s = 1) we obtain s. O n the other hand, for a stuck particle, the elastic part of the a t o m - s o l i d potential is n o t ineffective in the distant future, it is always effecttve, and there would be no sticking otherwise. Consequently text-book scattering theory does not apply, and for example, the S-matrix element ( f , It + ) between a stuck state and the exact scattering state It + ) cannot be used at once to obtain the r a t e ,Rsuck. It was precisely to avoid this dtfftculty that BG calculated the decay of a prepared state and we now emphasise that,

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G P Brtoio, T B Grtmley / Sttckmg coefftctent of H on metal

for the g e t t e r m g H a m i l t o m a n (3), where the final o u t c o m e of the c o l h s i o n process ~s that the gas a t o m is stuck, t e x t - b o o k scattering theory gives (correctly) r = 0 as m a y be verified b y calculating ( a , 0 1 0 , a + ) . T h e scalar p r o d u c t s ~fl, m 10,a + ) are n o n - z e r o a n d these elements o f the S - m a t r i x tell us h o w the gas a t o m is finally d i s t r i b u t e d over the stuck states when the colhsJon is over. This coincides with the usual t e x t - b o o k m e a n i n g of the S - m a t r i x On the o t h e r hand, b y direct c a l c u l a t i o n from (3) using the defimt~on I = W-~ W G W of the T = matrix, we find that, on the energy shell, all elements of the T - m a t r i x are zero except those in the ( I r n , f l ) } subspace. These, a n d o t h e r results, serve to r e m i n d us that gettering ~s a special sort of collision process. O n e o f us (G.P.B.) is grateful to Professor J.R. M a n s o n d~scussion o n g a s - s u r f a c e scattering.

for a useful

References [1] [2] [3] [4]

G P Bnvlo and T B Gnmley, Surface Scl 131 (1983) 475 M Born and R Oppenheimer, Ann. Physlk 84 (1927) 457 G Doyen, Phys. Rev. B22 (1980) 22 M L Goldberger and K M Watson, Cqlhslon Theory (Wdey, New York, 1964) ch 3