Collision Frequency of Adsorbed Particles

Transcription

1 Bulg. J. Phys. 40 (2013) Collision Fequency of Adsobed Paticles N.S. Peev Geogi Nadjakov Institute of Solid State Physics, Bulgaian Academy of Sciences, 72 Tzaigadsko Chaussee Blvd., 1784 Sofia, Bulgaia Abstact. The fequency of collision of adsobed paticles is of inteest fo two-dimensional nuclei fomation in cystal gowth and fo gowth of twodimensional nano-paticles. The inteaction of two paticles and thei coalescence is impossible without collision. The density of paticles (dimmes, timmes etc.) is popotional to fequency of collisions. It has been assumed that the cystal suface is isotopic. The adsobed paticles, consisting of i-atoms, having adius i, migate by velocity v i and fee path length ove the substate. The pobability fo collision ω(x) between two kinds of paticles a i and a k has been detemined. An expession fo the numbe of collisions pe unit suface aea pe unit time has been deived. The numbe of collisions pe unit length of a staight gowth step pe unit time has also been evaluated. PACS codes: h, Aj 1 Intoduction The paticles migation on cystal suface has been investigated since long time by Field Ion Micoscopy using the feeze-and-look technique [1-4]. At the moment two kinds of atoms migations ove the cystal suface have been obseved hopping motion and exchange motion. Exchange tanspot seems to be moe active because fewe bonds ae boken. This tanspot concens only the homo diffusion (fo example Pt on Pt). In case of heteo diffusion the hopping mechanism is moe pobable (fo example Pd on Pt). In geneally, which kind of mechanism takes place, depends on the inteaction between ad-atoms and substate. One supposes that the jumps of the paticles ae not coelated each jump is not influenced by the pevious. Expeiments ae caied out in absence of any gadients (as well as tempeatue and concentation). In a sequence of obsevations, sepaated by shot time intevals, duing that the paticle aangement on the suface does not change substantially, the positions of paticles on the suface has been identified and thei tajectoy detemined [4]. Obsevations Talk given at the Second Bulgaian National Congess in Physics, Sofia, Septembe c 2013 Heon Pess Ltd.

2 Collision Fequency of Adsobed Paticles show that the length of the jumps and thei diections ae quite abitay the lengths ae within the ange (10-60 Å). The detemination of the fequency of collisions of adsobed on the cystal suface atoms is the aim of the pesent wok. The knowledge of the fequency is valuable fo the two-dimensional nano-paticles fomation and as well as fo the cystal gowth nucleation. 2 Pobability fo Collision of Adsobed Paticles The cystal suface is assumed to be isotopic all diections of motion ae equally pobable. Two-dimensional paticles a i monatomic high, having cicumfeence shape with adius i, mass m i, velocity of motion v i and fee path length ae taken into consideation iis the numbe of atoms in the paticle. Two paticles has been consideed A k and B i, whee A k a k and B i a i. Paticle A k is assumed to be immobile. B i moves along the suface with velocity v i. The collision of both paticles occus if the tajectoy of B i lies within the fame of angle α the angle of collision (Figue 1). The pobability fo collisionω is povided by the atioω = α/2π [5]. ki = k + i B i, i α A k, k Figue 1: Angle of collision α. ω 1 () = 1 ( π acsin ki ) fo (0, ) + ki (1) ω 2 () = 1 [ π accos (+ki ) 2 +L 2 ] i 2 ki fo (, ) 2 ( + ki ) whee = L 2 i +2 ki ki. The pobability fo collision is a smooth and uninteupted function of [5], shown in Figue 2, whee x = / ki, C = / ki, x = / ki = C andx (0,C). 215

3 N.S. Peev Pobability fo collision, ω 0,5 0,4 0,3 0,2 0,1 0, x * 10 Distance between the paticles, x Figue 2: Pobability fo collisionω(x): C = 10,x = Fequency of Collision A ing aound the paticle A k has been consideed it is situated between two cicumfeences with adii (+ ki ) and (+ ki +d) espectively. The numbe dn i of paticlesa i within the ing is dn i = n s i0ds = 2πn s i0( + ki )d, whee n s i0 is the density of paticles a i. All the paticles dn i have the same angle of collision α and pobability fo collision ω. That pat of paticles dn i, which will collide with A k within the time τ i, is ωdn i. The entie numbe N i of these paticles is N i = 0 ωdn i =2πn s i0 [ 0 (+ ki )ω 1 d+ (+ ki )ω 2 d ] =2n s i0 ki (2) τ i is the time fo coveing of the fee pat length : τ i = /υ i. The fequency of collision is defined by numbe of collisions pe unit time. How fequently paticle A k will be attacked pe unit time by the paticles a i is given by the expession ν Ai = N i τ i = 2n s i0 ki v i, (3) whee A k is an abitay paticle of a k : A k a k. Each paticle a k will get the same numbe of collisions pe unit time by the paticles a i. The numbe of collisions pe unit aea of substate suface pe unit time is as follows: R ki = n s k0ν Ai = 2n s i0n s k0 ki v i, (4) 216

4 Collision Fequency of Adsobed Paticles whee n S k0 is the density of paticles a k. All the collisions occuing pe unit suface aea pe unit time is given by the sum ove all possible collisions R 0 = 2 m m i=1 k=i+1 n s i0n s k0 ki v i, (5) wheemis the numbe of atoms in the lagest paticlea m of the system. 4 Fequency of Collisions with Staight Gowth Step Adsobed single atoms have the geatest migation velocity with espect to all othe clustes on the suface. One may suppose that the fomation of clustes (dimmes, timmes and etc.) occus mainly by collisions with single atoms. Equation 4 allows detemining the numbe of collisions with single atoms (i = 1) of cluste a k pe unit length of its peiphey pe unit time. The length of the peiphey of the cluste is as follows: l k = 2π k. The fequency of collisions pe unit lengthr 0 Ai of the paticlea k with the ad-atomsa 1 is R 0 A1 = ν A1 l k = 1 π ns 10v 1 k1 k. (6) + d α / 2 α A i Figue 3: Fequency of collisions with a staight gowth step monatomic gowth step is pesented by the black paticles. In the case of a vey lage A k paticle ( k and k ) the peiphey becomes a staight line (Figue 3) and taking into account the equality ( k1 / k ) 1, one obtains the numbe of collisions of the staight gowth step with single ad-atoms pe unit length pe unit time this quantity affects the gowth ate of cystals and epitaxial layes R 0 1 = ns 10 υ 1 π. (7) 217

5 N.S. Peev 5 Discussion and Conclusions The migation of single ad-atoms has been consideed at the moment how the motion of clustes (dimmes, timmes etc.) occus is not sufficiently studied. The density of adsobed single atoms is in the ange ofn S cm 2 [6] this is the numbe of single atoms, simultaneously sojouned on unit aea of the suface. Thei migation velocity is about v cm/s [6]. By Eq. 4 one obtains that the fequency of collisions between the ad-atoms is aboutν cm 2 /s, whee cm. By coalescence of collided adatoms a dimme may aise. If one assumes that the pobability fo coalescence of both collided atoms is about δ and the dimmes life time is about θ d 1 ns, then the equilibium density of the simultaneously sojouned on the suface dimmes will be aboutn S 20 = ν 11θ d δ cm 2. Following Eq. 7 the numbe of collisions of the ad-atoms with a staight gowth step is in the ange of R (cm.s) 1. By such a collision the ad-atom falls into a new situation it becomes an atom in step edge position. In this position the binding enegy with the substate is geate than that in a dimme o in an ad-atom position. The enegy diffeence between the stepedge position and the dimme one is: E = E s.e. E d = 2φ 2, whee φ 2 is the binding enegy with the next neaest neighbos. In case of Lenad-Jones inteaction φ 2 φ 1 potential dops vey abupt. If the life time of an atom of dimme is about θ d 1 ns (as aleady assumed), then the life time in step edge position must be appoximately the same θ d θ s.e.. Then the numbe of atoms in step edge position, available simultaneously along a unit length is as follows: N = θ s.e. R cm 1. The aveaged distance between the atoms is about ρ = N 1 25 Å. As initially assumed, the suface migation is isotopic all diections of motion ae equally pobable. It means that the flow of migating atoms in all diections will be the same and all steps, available on the suface, will have the same fequency of collisions with the ad-atoms pe unit length. The ad-atoms will be aely supplied to the islands step, shote than ρ. Its motion ove the substate will be hindeed. Vice vesa, steps longe than ρ will spead ove the suface substantially faste. Refeences [1] H. Shiakawa and H. Komiyama (1999) Jounal of Nanopaticle Reseach [2] G.L. Kellogg (1993) Applied Suface Science [3] G.L. Kellogg (1992) Suf. Sci [4] T.T. Tsong (1972) Phys. Rev. B [5] N.S. Peev (2013) Comptes endus de l Académie Bulgae des Sciences [6] H. Bune, J. Winttelin, R.J. Behm (1992) Phys. Rev. Lett