Jump processes in surface diffusion - Bilkent University - Faculty of ...

Surface Science Reports 62 (2007) 39–61 

www.elsevier.com/locate/surfrep 

Jump processes in surface diffusion 

Grazyna Antczak a,b,∗ , Gert Ehrlich a 

a Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA 

b Institute of Experimental Physics, University of Wroclaw, Wroclaw, Poland 

Received 2 December 2006; accepted 4 December 2006 

Abstract 

The traditional view of the surface diffusion of metal atoms on metal surfaces was that atoms carry on a random walk between nearest-neighbor 

surface sites. Through field ion microscopic observations and molecular dynamics simulations this picture has been changed completely. Diffusion 

by an adatom exchanging with an atom of the substrate has been identified on fcc(110), and subsequently also on fcc(100) planes. At elevated 

temperatures, multiple events have been found by simulations in which an atom enters the lattice, and a lattice atom at some distance from the 

entry point pops out. Much at the same time the contribution of long jumps, spanning more than a nearest-neighbour distance, has been examined; 

their rates have been measured, and such transitions have been found to contribute significantly, at least on tungsten surfaces. As higher diffusion 

temperatures become accessible, additional jump processes can be expected to be revealed. 

c○ 2007 Elsevier B.V. All rights reserved. 

Keywords: Atom jumps; Surface diffusion; Field ion microscopy; Molecular dynamics 

Contents 

1. Introduction............................................................................................................................................................................ 39 

1.1. Surface diffusivities....................................................................................................................................................... 40 

2. Atom exchange ....................................................................................................................................................................... 40 

2.1. On fcc(110) planes ........................................................................................................................................................ 40 

2.2. On (100) planes ............................................................................................................................................................ 46 

2.3. Via multiple events........................................................................................................................................................ 48 

3. Long and rebound jumps.......................................................................................................................................................... 50 

3.1. Theoretical work........................................................................................................................................................... 50 

3.2. Experimental studies ..................................................................................................................................................... 51 

4. Summary ............................................................................................................................................................................... 58 

Acknowledgements ................................................................................................................................................................. 59 

References ............................................................................................................................................................................. 59 

1. Introduction 

Surface diffusion plays a significant role in crystal and film 

growth, in evaporation and condensation, in surface chemical 

∗ Corresponding author at: Department of Materials Science and Engineering, 

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Tel.: 

+1 2173336752. 

E-mail address: antczak@mrl.uiuc.edu (G. Antczak). 

reactions and catalysis, in sintering as well as in other surface 

processes. As such there has been considerable interest in how 

diffusion occurs; during the last few decades much novel and 

significant material has been discovered. Here we will briefly 

review what has been learned about how diffusion of single 

metal atoms takes place on metal surfaces, what sort of atomic 

jumps occur and how information about these processes has 

been obtained. 

0167-5729/$ - see front matter c○ 2007 Elsevier B.V. All rights reserved. 

doi:10.1016/j.surfrep.2006.12.001

40 G. Antczak, G. Ehrlich / Surface Science Reports 62 (2007) 39–61 

Fig. 1. Hard-sphere models of bcc (a) (110), (b) (211), and (c) (321) planes. 

1.1. Surface diffusivities 

Diffusion is characterized by the material parameter D, the 

diffusivity in the (one-dimensional) diffusion equation: 

J = −D ∂c 

∂x 

connecting the adatom flux J to the concentration gradient 

∂c/∂x. The diffusivity is given by the usual Arrhenius relation 

( 

D = D o exp − E ) 

D 

, (2) 

kT 

where the prefactor D o , usually considered a constant, and 

the activation energy E D are obtained from the temperature 

dependence. Langmuir [1], in 1933, already viewed the atoms 

in surface diffusion as hopping between elementary spaces at a 

separation a, and was able to show that for an individual adatom 

the diffusivity was given in terms of the lifetime τ at a particular 

site by 

D = a 2 /2τ. (3) 

In transition-state theory the one-dimensional diffusivity is 

written more elaborately as: 

( ) ( 

D = υl 2 SD 

exp exp − E ) 

D 

, (4) 

k kT 

where υ gives the effective vibrational frequency of the adatom, 

l its jump length, and S D as well as E D the entropy and 

activation energy for diffusion. We can, however, introduce the 

abbreviation: 

( ) 

SD 

υ o = υ exp , (5) 

k 

so that the diffusivity appears as in Eq. (2), but with 

D o = υ o l 2 . (6) 

On surfaces it would be difficult to measure atom fluxes 

and gradients. The mass flux of material over a surface can 

of course be detected, but it is not clear how to disentangle 

interactions between the atoms. The diffusivity is therefore 

obtained alternatively from the Einstein relation, which ties the 

diffusivity to the mean-square displacement 〈x 2 〉, 

〈x 2 〉 = 2Dt, (7) 

(1) 

Fig. 2. Models of fcc (a) (111) and (b) (100) planes. 

where t is the time interval of the measurements. 

The elementary formalism is now in hand, and using it much 

data has been gathered about diffusion kinetics [2]. But what 

are the jump processes participating in surface diffusion? If 

atoms always jump between nearest-neighbour sites, then since 

a typical vibrational frequency at the surface is ∼10 12 s −1 , and 

the spacing is ∼3 Å, the prefactor D o should, if we neglect 

entropy contributions, be of magnitude ∼10 −3 cm 2 /s. It turns 

out that this is close to the value of the prefactor for many 

diffusion systems [2], starting with one of the first studied, W 

on W(110), for which a value of 2.6×10 −3 cm 2 /s was found [3, 

4]. This agreement suggested that this simple view of diffusion 

processes had considerable merit. However, the story has turned 

out to be much more interesting than that. 

2. Atom exchange 

2.1. On fcc(110) planes 

Early studies of individual metal atoms diffusing on metals 

were with tungsten atoms on (110), (211), and (321) planes 

of bcc tungsten, models of which are shown in Fig. 1. Also 

studied somewhat later were rhodium atoms on rhodium, an fcc 

metal [4]. Examined were (111) and (100), as well as (110), 

(311) and (331) planes, shown in Figs. 2 and 3. What is clear 

is that the (321) and (211) planes on tungsten and (311), (331) 

and (110) planes on rhodium are channelled, and that an adatom 

moving along one of these channels might be expected to stay in 

such a channel and execute strictly one-dimensional diffusion. 

That was in fact found to be the case as shown in Fig. 4 

for W(211), making predictions of the direction in diffusion 

seemingly straightforward.

G. Antczak, G. Ehrlich / Surface Science Reports 62 (2007) 39–61 41 

Fig. 3. Models of fcc (a) (110), (b) (311), and (c) (331) planes. 

Fig. 5. Mechanism for cross-channel diffusion of adatom on Pt(110) proposed 

by Bassett and Webber [5]. (a) Adatom in channel. (b) Vacancy forms in 

channel wall. (c) Adatom has moved into vacancy, lattice atom is now in 

channel site. 

Fig. 4. Location of sites on W(211) at which Rh adatom has been detected after 

diffusion at 197 K. Motion is strictly one-dimensional. Shown at the bottom is 

a count of the number sighted at each site. 

This simple picture was suddenly destroyed, however, by the 

work of Bassett and Webber [5] in 1978, who studied another 

fcc metal, platinum. On (331) and (311) planes diffusion 

occurred, as expected, along the channels of the surfaces. On 

the (110) plane, however, diffusion was two-dimensional! The 

activation energy for in-channel motion was 0.84 ± 0.1 eV, for 

cross-channel movement it was smaller, only 0.78 ± 0.1 eV. 

Similar results were found with the diffusion of iridium atoms 

on these surfaces. What had been established here was that 

diffusion on channelled surfaces could occur in two dimensions, 

depending on the particular system. Predicting the direction in 

which diffusion occurred was not just a matter of looking at 

surface geometry. 

The question still remained how cross-channel motion took 

place. Bassett and Webber [5] had two ideas. One was that 

large fluctuations occurred in the rows of atoms constituting 

the channel walls, creating holes through which adatoms could 

easily jump. A more likely possibility was that the movement 

of a channel atom created a hole that could be filled either 

by the adatom or a surface atom, as shown in Fig. 5. They 

concluded that “Further experimental and theoretical studies are 

required to clarify the nature of inter-channel diffusion”. Even 

without an understanding of the mechanism of cross-channel 

motion, the phenomenon of cross-channel diffusion had been 

established. 

What actually happened in cross-channel atom movement 

was soon uncovered in experiments by Wrigley and Ehrlich [6]. 

They took advantage of the atom probe [7], to measure the 

chemical identity of atoms on Ir(110)-(1 × 1). In this work 

the system was calibrated by first depositing an iridium and 

separately a tungsten atom on the surface kept at ∼50 K. The 

mass of the individual atoms was then obtained by measuring 

the time of flight to a detector ∼1 m away by field desorbing 

the atoms; the results are shown in Fig. 6. With this information 

in hand, a tungsten atom was placed on the clean surface, 

which was heated until cross-channel diffusion took place. The 

atom observed by field ion microscopy after diffusing into 

the adjacent channel was then field evaporated, and as also 

shown in Fig. 6 proved to be iridium, an atom from the lattice. 

Thereafter the top layer of the surface was field evaporated and 

analyzed, as in Fig. 7, and usually a tungsten atom was detected 

in the surface layer, as expected if atom exchange had occurred. 

This was not the case when the surface was probed after crosschannel 

motion had not been seen. These experiments were the 

first direct proof of an interchange between a substrate and an 

adsorbed atom during cross-channel diffusion. 

A number of theoretical investigations were stimulated 

by the results on Pt(110). Halicioglu [8,9] carried out 

calculations for diffusion on Pt(110) with Lennard-Jones twobody 

potentials and found that in cross-channel diffusion an 

adatom interacts with an atom in the channel wall to form 

a dumbbell, as in Fig. 8(b). This could decompose by the 

adatom incorporating into the surface and the surface atom 

going into the adjacent channel, or else by returning to the 

original channel, as indicated in Fig. 8(c). 

Just slightly later, DeLorenzi et al. [10,11] did molecular 

dynamics calculations, again using Lennard-Jones potentials, 

on fcc surfaces. Most interesting were the results for (110) 

planes, shown in Fig. 9, on which both in-channel and


Fig. 6. Distribution of atomic weights of individual atoms field evaporated 

from Ir(110) surface [6]. Top: after deposition of Ir atom. Bottom: after 

deposition of W atom. Center: After deposition of W atom, followed by heating 

and observed cross-channel diffusion event; material desorbed is iridium. 

Fig. 7. Atomic weight distribution of material field evaporated from first 

Ir(110) layer [6]. Left: after cross-channel diffusion has occurred, a W atom 

is detected in the first substrate layer. Right: when no cross-channel motion was 

detected, iridium is desorbed. 

cross-channel motion was discovered, the latter with a 

lower activation energy. In cross-channel diffusion they again 

envisioned a dumbbell intermediate, as shown in Fig. 8(b), 

in which a pair of atoms sat across the ridgeline. When 

this decomposes, the adatom can move back to its original 

channel, or else it can move into the channel wall, placing 

a lattice atom in the adjacent channel. Additional simulations 

of diffusion across channels on a Lennard-Jones fcc crystal 

were done by Mruzik and Pound [12]. On the (110) plane, 

cross-channel motion again occurred by exchange with a lattice 

atom, but movement on the (113) plane was along the channels. 

Garofalini and Halicioglu [13] did similar estimates on the 

Pt(110) plane at both low and high temperatures. At lower 

temperatures both iridium and gold atoms diffused along the 

channels, but at higher temperatures exchange took place for 

iridium, and a platinum lattice atom appeared in the next 

channel. Gold at this temperature also formed a dumbbell 

with an atom from the channel wall, but returned to continue 

diffusion in its original channel, so diffusion was really onedimensional. 

More experimental work was reported in 1982 by 

Wrigley [14], who examined the temperature dependence of 

cross-channel motion, as shown in Fig. 10, with an activation 

energy of 0.74 ± 0.09 eV and a prefactor 1.4(×16 ±1 ) × 

10 −6 cm 2 /s. Data were obtained at only five temperatures, 

and keeping in mind the low prefactor, must be viewed with 

some doubt. What was not clear at this point was the reason 

for the magnitude of the prefactor. Was it associated with the 

complicated mechanism, or were the measurements not detailed 

enough? 

At essentially the same time, Tung and Graham [15] looked 

at self-diffusion on various surfaces of nickel. The behaviour 

of the (110) plane varied depending on whether it had been 

cleaned thermally, or had been subjected to field evaporation 

in hydrogen gas. Diffusion measurements were made after the 

hydrogen treatment and some field evaporation, as well as after 

thermal cleaning only, and gave the results shown in Fig. 11. For 

both treatments self-diffusion was two-dimensional on Ni(110); 

however, diffusion after thermal treatment led to very low 

diffusion prefactors ∼10 −7 –10 −9 cm 2 /s, with an activation 

energy of 0.23 ± 0.04 eV for in-channel movement and 0.32 ± 

0.05 eV across the channels. After hydrogen treatment, the 

characteristics for in-channel motion were a diffusion barrier 

of 0.30 ± 0.06 eV and a prefactor of 10 1 cm 2 /s; for crosschannel 

diffusion the barrier was lower, 0.25 ± 0.06 eV, with 

a prefactor of 10 −1 cm 2 /s. In the temperature range 80–90 

K, cross-channel motion predominated. On Ni(331) and (113), 

however, movement was always one-dimensional. Tung [16] 

also briefly studied self-diffusion on Al(110), determining 

the temperature for the beginning of atom movement. Twodimensional 

diffusion was observed on the (110) plane, at a 

temperature of 154 K for both directions, leading to an estimate 

of 0.43 eV for the activation energy. It should be noted that 

the diffusion characteristics, both for nickel and aluminum 

(110) are uncertain, and the data have been reanalyzed [17]. 

Nevertheless, these observations firmly established crosschannel 

motion on these surfaces. 

Another five years later, in 1986, Kellogg [18] measured 

the rates of in- and cross-channel self-diffusion on Pt(110) 

over a range of temperatures. From an Arrhenius plot, shown 

in Fig. 12, he found an activation of 0.72 ± 0.07 eV with a 

prefactor of 6 × 10 −4 cm 2 /s for diffusion along the channels,


Fig. 8. Schematic of atom exchange process in self-diffusion on fcc(110) surface: (a) Atom in the equilibrium position. (b) At the saddle point, atom pair sits as a 

dumbbell across [1¯10] row of substrate atoms. (c 1 )–(c 4 ) After diffusion, atoms distributed uniformly over allowed sites. 

and 0.69 ± 0.07 eV with a prefactor of 3 × 10 −4 cm 2 /s for 

motion across the channels. It appeared that the prefactors for 

in- and cross-channel motion were similar and normal. 

Chen and Tsong [19] again looked at self-diffusion on 

Ir(110) in 1991, and as shown in Fig. 13, observed crosschannel 

diffusion with an activation energy of 0.71 ± 0.02 eV 

and a prefactor 6 × 10 −2.0±1.8 cm 2 /s; the values for inchannel 

motion were 0.80 ± 0.04 eV and a prefactor of 

4 × 10 −3.0±0.8 cm 2 /s, so cross-channel jumps occurred more 

rapidly. Earlier work by Wrigley [14] was reasonably close in 

the diffusion barrier, but the present prefactor value is clearly 

preferable. Above and beyond the usual measurements, Chen 

and Tsong also looked at the distribution of displacements to 

get additional information about the diffusion process. These 

observations are shown in Fig. 14, with x indicating in-channel 

diffusion and y cross-channel jumping. What is especially 

interesting here is the fact that 80% of the jumps were in the 

〈112〉 direction, shown in Fig. 8, with the remainder along 

〈100〉. According to all the simulations of such self-diffusion 

done at that time, a cross-channel transition should have the 

same probability in four directions, whereas in the experiments, 

transitions in the direction of the moving adatom were favoured. 

These results were not accidental, however. 

Kellogg [20] in the same year, looked at the diffusion of 

platinum on the (110) plane of nickel. For these measurements 

nickel was not cleaned in hydrogen; instead a combination 

of thermal cleaning, sputtering and evaporation was found to 

be essential for a good image of the (110) plane. Although


Fig. 11. Arrhenius plot for in- and cross-channel diffusion of Ni on Ni(110) 

plane [15]. Left: Substrate after thermal treatment. Right: Nickel substrate had 

been hydrogen fired and field evaporated before experiments. 

Fig. 9. Adatom trajectories in diffusion on fcc(110) plane at 0.4T m [11]. 

Transitions across the [1¯10] channels are apparent. 

Fig. 12. Self-diffusion of Pt on Pt(110) surface [18]. In-channel diffusion: 

E D = 0.72 ± 0.07 eV, D o = 6 × 10 −4 cm 2 /s. Cross-channel diffusion: 

E D = 0.69 ± 0.07 eV, D o = 3 × 10 −4 cm 2 /s. 

Fig. 10. Early Arrhenius plot for the cross-channel diffusion of iridium atoms 

on Ir(110) [14]. 

Kellogg did not rely on an atom probe, he was able to 

discriminate between Ni and Pt atoms through differences in the 

voltages necessary for field evaporation—the platinum atoms 

were removed at significantly higher voltages; they also had 

a much larger image spot. When a Pt atom was deposited 

on the cold surface, the image looked as in Fig. 15(a). After 

heating to 112 K, the image in Fig. 15(b) yielded a much 

smaller spot size, indicative of Ni. Upon removing the adatom, 

in Fig. 15(c), and field evaporating the topmost layer, as is 

shown in Fig. 15(d), a platinum atom was retrieved from 

the surface layer. Clearly exchange had occurred between a 

Fig. 13. Self-diffusion of Ir on Ir(110) plane [19]. Both in-channel and crosschannel 

motion is observed. 

platinum adatom and a nickel atom from the substrate, and the 

atom entering the lattice was platinum not nickel. Assuming 

a prefactor of 1 × 10 −3 cm 2 /s, a barrier of 0.28 eV was 

determined for the displacement of platinum. The distribution 

of displacements was also observed in these experiments and it 

was found that in 62.5% the replacement atom appeared in the 

〈112〉 direction, never along 〈001〉; the remaining experiments


Fig. 14. Distribution of displacements in diffusion of Ir on Ir(110) at different 

temperatures [19]. Transitions in direction of moving atom are favoured, 

contrary to theoretical predictions. 

Fig. 16. Exchange between Re adatom and Ir(110) substrate illustrated in field 

ion images [21]. In (b), Re atom has been deposited on the clean Ir(110) 

surface in (a), giving a round image spot, a schematic of which is in (e). After 

one minute at 255 K, oblong atom, Ir, is found in (c), also illustrated in (f). 

Substituted Re atom is revealed on partial field evaporation, in (d). 

Fig. 15. Neon field ion images showing exchange of Pt adatom with Ni from 

Ni(110) surface [20]. (a) Single Pt adatom on Ni(110) plane. (b) After one 

minute at 112 K, adatom has changed to Ni, judged from lower desorption 

field. (c) Adatom has been removed by field evaporation. (d) After removal of 

one layer of nickel, Pt adatom is again apparent. 

ended with an atom in the original row. Kellogg speculated that 

this was due to the atom from the channel wall maintaining its 

cohesion with the adatom, but this conclusion was not reached 

in previous molecular dynamics simulations. The mechanism 

of cross-channel movement is still wreathed in uncertainty. 

A year later, Chen et al. [21] carried out field ion microscopic 

studies of rhenium atom diffusion on the Ir(110) surface. They 

were able to distinguish between Re and Ir atoms, as rhenium 

gave a circular image spot and iridium yielded an elongated 

image. A rhenium atom was deposited on the surface and then 

heated to ∼256 K. This moved the atom spot one space over 

and elongated the image, suggesting that an atom replacement 

had occurred. The top layer of the substrate was then field 

evaporated, as shown in Fig. 16, revealing a bright rhenium 

atom, and demonstrating that exchange with a lattice atom had 

indeed taken place. Finally, it should be noted that in 2002, 

Pedemonte et al. [22] were able to detect some cross-channel 

movement in helium scattering experiments on Ag(110) at 

temperatures above 750 K, but did not derive energetics for 

such movement. 

At the beginning of the nineties there started quite a 

large number of efforts to calculate diffusivities of metal 

atoms on metals. Such estimates, usually using semi-empirical 

interactions, were done for the (110) planes of Al [17,23– 

25], Ni [17,26–28], Cu [17,27–33], Pd [17,27,28,34], Ag [17, 

28,30,32,34,35], Ir [28,36,37], Pt [17,27,28,38,39], Au [17,27, 

28,32], and Pb [40]. In all of these the barrier to diffusion 

along the channels was lower than the barrier for cross-channel 

transitions, despite the fact that for Al, Ni, Ir and Pt the opposite 

had been demonstrated in experiments. In short, diffusion by 

atom exchange on fcc(110) is well established, if not yet well 

understood. What is still not clear is why a particular system 

undergoes exchange while another one does not, and why a 

particular direction is favoured in diffusion.


Fig. 19. Map of sites on Pt(100) at which Pt adatom was found after diffusion at 

175 K [46]. A c(2 × 2) net is formed, indicative of jumps by atom replacement. 

Fig. 17. Atom migration on bcc (100) surface, shown in molecular dynamics 

simulations modelled by Price potential [42,43]. Straight arrows j indicate 

nearest-neighbour jumps, curved arrows e show adatom exchange with 

substrate [41]. 

2.2. On (100) planes 

While all this work studying atom exchange on fcc(110) 

planes was going on, DeLorenzi and Jacucci [41] had been 

continuing their efforts to examine atomic jumps in surface 

diffusion. Their attention turned to bcc materials, which meant 

they had to resort to a different type of interaction for 

simulations. They relied on a metallic potential devised by 

Price [42,43] to describe sodium, and used this in examining 

diffusion on various planes of a bcc lattice. Of specific interest 

here is what was observed on the (100) plane. In the usual 

course of events, we expect diffusion to take place by an atom 

jumping from its normal binding site, at the centre of four 

surface atoms, to a neighbouring binding site in a straight line 

at right angles to the border of the unit cell. What frequently 

happened in the simulations, however, was different and is 

shown in Fig. 17, where short straight arrows indicate normal 

jumps, between nearest neighbours, and curved ones reveal an 

atom-exchange process. 

In this exchange an adatom moves into the outer lattice 

layer, dislodging a lattice atom which ends up on the surface 

in a position diagonal to the starting point, as suggested in 

the schematic in Fig. 18. In their own words, DeLorenzi 

and Jacucci found that “In addition to conventional nearest 

neighbour jumps between surface sites, the adatom undergoes 

migration events reminiscent of exchange processes of the 

intersticialcy in the bulk. In these events, atom A belonging to 

the surface layer is replaced by atom B originally constituting 

the adatom. As a result, atom A ends up as an adatom located 

at a site displaced from the one originally occupied by the 

point defect”. “This is the first observation, in simulations or 

real experiments, of the occurrence of exchange events between 

adatom and a substitutional atom as a quantitatively important 

process contributing to atomic diffusion on isotropic crystal 

surfaces.” A new exchange event had been discovered; one 

question remaining was the generality of this process. Would 

it occur in practical systems? 

Regrettably, nothing was done to test the work of DeLorenzi 

and Jacucci, until five years later two studies appeared: Kellogg 

and Feibelman [44] studied Pt(100) and Chen and Tsong [45] 

simultaneously looked at Ir(100). They both examined the 

displacements carried out on the surface, and found that atoms 

moved diagonally and not just to nearest neighbors, creating 

a c(2 × 2) map of binding sites, as shown in Fig. 19. From 

the mean-square displacement at 175 K, the activation energy 

for diffusion on Pt(100) was estimated as 0.47 eV, assuming 

the usual prefactor of 10 −3 cm 2 /s. Chen and Tsong did better 

in studying iridium. From an Arrhenius plot of the meansquare 

displacement per unit time, shown in Fig. 20, they found 

a barrier of 0.84 ± 0.05 eV and a diffusivity prefactor of 

6.3(×11 ±1 ) × 10 −2 cm 2 /s. Just like Kellogg and Feibelman, 

they observed diagonal atom movement for iridium. On these 

two surfaces, diffusion occurred by atom exchange, as had been 

found earlier by DeLorenzi and Jacucci [41]. 

Fig. 18. Schematic showing exchange between adatom and substrate atom on (100) plane: (a) Adatom at equilibrium site. (b) Adatom and lattice atom in transition 

state. (c) Adatom incorporated into lattice; atom from lattice has turned into adatom.


Fig. 20. Arrhenius plot for diffusion of Ir adatom on Ir(100) observed in field 

ion microscope [45]. 

Fig. 23. Interaction of Re adatom with Ir(100) surface [48]. Re atom is placed 

on (100) plane in (b) and is then heated to ∼230 K, yielding an oblong image 

spot in (c), which rotates on heating to ∼262 K. This sequence is shown 

schematically in (e)–(g). 

Fig. 21. Dependence of Pt atom diffusivity on Pt(100) upon 1/T [46]. 

Fig. 22. Binding sites for Pd on Pt(100) after diffusion at 265 K, indicating 

ordinary hopping [47]. 

Further studies of self-diffusion on Pt(100) were carried out 

in 1991 by Kellogg [46], who again found a diagonal map 

of displacements for platinum atoms, indicating an exchange 

process. The diffusion barrier obtained from the Arrhenius plot 

in Fig. 21 was 0.47 ± 0.01 eV, with a prefactor 1.3(×10 ±1 ) × 

10 −3 cm 2 /s. Subsequent work [47] showed that palladium 

atoms carried out the usual atomic jumps, as indicated in 

Fig. 22, while nickel and platinum underwent atomic exchange 

reactions in diffusion. 

In the next year, Tsong and Chen [48] put a rhenium atom 

on Ir(100). When heated to a temperature of ∼230 K, the 

Re displaced an Ir atom from the surface layer to form a 

dimer above the vacancy, as in Fig. 23. Above 280 K the 

dimer decomposed and the rhenium entered the lattice, with an 

iridium atom left to continue surface diffusion, another example 

of atom exchange. 

Kellogg [49] also studied the behavior of platinum atoms on 

Ni(100). He was able to distinguish between the two in terms 

of the higher voltage required to field evaporate platinum than 

nickel atoms. After depositing platinum on the surface at 77 

K, and then heating to 250 K, the platinum disappeared. On 

field evaporating the surface layer platinum appeared again, as 

in Fig. 24, revealing that an exchange process had taken place. 

Additional confirmation for iridium diffusion on Ir(100) by 

atom exchange was provided by the work of Friedl et al. [50] 

in 1992. They plotted the sites visited in diffusion and found 

a c(2 × 2) net, as expected if exchange took place between 

an adatom and a lattice atom. However, they proposed an 

alternative explanation: the surface could reconstruct and create 

a c(2 × 2) sub-lattice for diffusing atoms. This explanation 

did not survive the test of time, however, since reconstruction 

should occur independent of the type of diffusing atom. Some 

years later, Fu and Tsong [51] again looked at self-diffusion on 

Ir(100) and again observed a c(2 × 2) net. 

In the meantime, quite a number of calculations were 

made for surface diffusion on a variety of fcc(100) planes, 

with a mixture of results. More then ten estimates have been 

carried out for Al(100). Most of them considered exchange


Fig. 24. Replacement of Pt adatom on Ni(100) by Ni atom [49]. (a) Pt atom 

deposited on Ni(100) at 77 K. (b) After heating to 250 K for one minute, Pt 

adatom disappears. (c) After partial field evaporation. (d) On complete field 

evaporation of one layer of nickel, Pt atom reappears. 

as a possible mechanism [17,23,25,52–59], and roughly half 

favored atom exchange during diffusion. For Ni(100) [17,59– 

71], out of a total of nineteen estimates, only eight considered 

exchange as an option, and fewer than half of these indicated 

an exchange of atoms as primary in the diffusion process. 

Here it is important to note the observation of Perkins and 

DePristo [65] as well as of Chang and Wei [60] that the 

activation energy for exchange depends strongly on the size of 

the cell used in the calculations, while the hopping energy is 

not sensitive to this factor. Much more work has been done to 

understand diffusion on Cu(100) [17,29,53,58–68,70–90,110– 

112]. Half the investigations considered exchange as an option, 

but only three studies gave an indication of atom exchange 

taking place. For Pd(100) [17,27,59–62,64,66,69–71,92,93], 

three estimates favored hopping as the principal mechanism, 

but after increasing the size of the cell in the calculations 

three favoured exchange [17,60,65]. Only one out of six studies 

considered exchange in diffusion on Ag(100) [17,27,59–62,64– 

71,76,91,94–97,109–112] possibly indicating a rate of atom 

exchange comparable to atom hopping; the remainder gave 

hopping as the mode of diffusion. Not too much calculational 

work has been done to evaluate diffusion characteristics on 

Ir(100) [37,92], Pt(100) [17,92,98], and Au(100) [17,66,99], 

but all of them indicate that the preferred path for diffusion was 

by atom exchange. 

Although the outcome of some of the theoretical estimates 

is not that certain, the evidence is firm that exchange between 

an adatom and a lattice atom occurs in diffusion on (100) 

planes of Ir [45], Pt [44], and Ni [49], and probably also on 

Au [17,66,99]. There have been attempts to rationalize the 

conditions under which exchange will dominate in diffusion. 

For Al(100), Feibelman [100] pointed out that the transition 

state for atom exchange was stabilized by the covalent nature 

of aluminum. Kellogg et al. [47] correlated exchange diffusion 

with the relaxation of surface atoms around the binding site of 

the adatom. Yu and Scheffler [94] argued that “tensile surface 

stress plays the key role for the exchange diffusion on fcc(100) 

surfaces”, and is important not only for Au(100), but also for 

Fig. 25. Multiple atom exchange process in molecular dynamics simulation of 

adatom on Cu(100) at 900 K [75]. Adatom (black) moves into substrate, causing 

eventual emergence of a substrate atom at some distance from the original entry. 

Al(100) and 5d metals. However, Feibelman and Stumpf [92] 

have done detailed density functional calculations for the (100) 

surfaces of Rh, Ir, Pd, and Pt, and found no clear relation 

between surface stress and the diffusion barrier. Instead, they 

concluded that exchange diffusion was favored, as proposed 

by Kellogg et al. [47], when the relaxation of the substrate 

around an adatom was largest. Right now, however, it appears 

that predicting from experimental information which systems 

will undergo atom exchange in surface diffusion is an uncertain 

matter. 

2.3. Via multiple events 

The results for exchange processes in diffusion described 

so far have been obtained at reasonably low temperatures, 

and have revealed a single event. Experiments at higher 

temperatures, to explore the possibility of multiple processes, 

are difficult and have not yet been explored. However, these 

conditions are accessible to molecular dynamics simulations, 

and have been probed starting in 1993 with the work of Black 

and Tian [75], who studied copper on Cu(100) relying on 

embedded atom potentials [101]. At 900 K, a high temperature 

for copper, they found that an atom adsorbed on the surface 

entered into the surface layer, straining the adjacent surface 

atoms as indicated in Fig. 25. The strain caused a surface atom 

to leave, popping out, not adjacent to the original entry point, 

but farther away. 

This peculiar event was again found in the work of 

Cohen [76], who did similar simulations on Ag, Al, Au, Cu, Pd, 

Pt, and Ni. On the (100) plane of aluminum she found that an 

atom entered the surface and travelled two sites and then over 

one before emerging again. The barrier for this novel diffusion 

was much higher than for ordinary hopping, and for all the 

metals above except for nickel would only become important 

above half the melting point. For nickel, the temperature for this 

diffusion process was expected to be even higher. An Arrhenius 

plot of the different diffusion processes is given in Fig. 26, 

and shows clearly that at elevated temperatures the new type 

of diffusion event makes significant contributions.


Fig. 26. Arrhenius plot for three different mechanisms of diffusion for Ag 

on Ag(100) derived in molecular dynamics simulations [76]. New refers to 

multiple displacement exchange process, which has a higher activation energy 

and therefore only contributes at elevated temperatures. 

Fig. 28. Arrhenius plot for frequency of jumps in self-diffusion on 

Cu(100) [82]. Filled squares—single jumps; open squares—double jumps. 

Filled circles—simple exchange; filled and open triangles—double and triple 

exchange. Rhombic squares—quadruple exchange. 

Table 1 

Jump characteristics of Cu on Cu(100) [82] 

Jump Migration barrier (eV) D o (cm 2 /s) 

Single 0.43 ± 0.02 3.4×10 −3±0.2 

Double 0.71 ± 0.05 38 × 10 −3±0.5 

Simple exchange 0.70 ± 0.04 42 × 10 −3±0.3 

Double exchange 0.70 ± 0.06 45 × 10 −3±0.6 

Triple exchange 0.82 ± 0.08 104 × 

10 −3±1.0 

Quadruple exchange 0.75 ± 0.12 86 × 10 −3±0.9 

Fig. 27. Molecular dynamics simulation of self-diffusion on Cu(100) at 950 

K [82]. Adatom squeezes into the surface layer, and eventually a surface atom 

pops out at a considerable distance from the first entry. 

A more detailed examination of high temperature surface 

processes was carried out by Evangelakis and Papanicolaou 

[82] for copper atoms on the Cu(100) plane. They used 

the quite reliable RGL [102,103] potential to carry out their 

molecular dynamics simulations, which were done at 700 K and 

higher. At more elevated temperatures they observed double 

jumps, as well as exchange processes. Most interesting, however, 

were events such as shown in Fig. 27: an adatom enters 

the surface layer, disturbing a row of surface atoms, the last 

of which exits the surface to become an adatom. From observations 

of the jump frequency at different temperatures they 

were able to construct the Arrhenius plot in Fig. 28, yielding 

the diffusion characteristics of the different processes, shown 

in Table 1. It is important to recognize that the atom exchange 

events all have much the same kinetics, independent of the 

length of the process, which may be related to the extent of 

Fig. 29. Schematics showing atom movement in correlated jump-exchange 

(je), exchange-jump (ej), and finally jump-exchange-jump (jej) [104]. Dark 

grey indicates atom initially placed in channel; light grey shows atom from 

surface row taking part in exchange. 

the strain field; these transitions also occur over barriers much 

higher than for ordinary jumps. 

Also to be noted is the work of Ferrando on Ag(110) [104], 

who at temperatures above 600 K observed not only exchange 

but also correlated exchange-jump movements, as sketched in 

Fig. 29. He also observed a drastic change in the frequency of 

events when correlation between jumps and exchange occurred. 

Correlated jump and exchange processes were also observed on 

Cu(110) [32].


Fig. 30. Temperature dependence of length of correlated jumps in diffusion of 

adatom on Lennard-Jones fcc(100) surface [107]. 

Fig. 32. Adatom trajectories on bcc(110) plane modelled with Price 

potential [42,43] during 200 ps at ∼0.4T m [41]. (a) Single jumps. (b) Double 

jumps. (c) Complicated trajectories. 

3. Long and rebound jumps 

3.1. Theoretical work 

Fig. 31. Adatom trajectories on Lennard-Jones (100) surface at T = 0.34T m , 

revealing non-nearest neighbour transitions [11]. 

The study by Evangelakis and Papanicolaou [82] has been 

the most detailed work on multiple exchange events, but 

these processes were rediscovered several years later by Xiao 

et al. [105,106]. Using EAM potentials they found multiple 

atom exchange events for strained Cu(100) and Pt(100), which 

were designated as crowdions. 

Although as yet there are no experiments to demonstrate 

such large scale exchange processes, there is little doubt that 

they will be found at elevated temperatures. However, detailed 

theoretical studies of such complicated processes will have 

to wait until procedures for investigating simple exchanges 

become reliable. One thing is certain: much more extended 

cell sizes will be needed for such calculations. From the 

experimental point of view, STM should be the most suitable 

tool to uncover such transitions. 

The view that in diffusion over a surface, atoms jump at 

random between nearest-neighbor sites, remained widespread 

until the end of the seventies. At that time a number of 

simulations appeared which suggested a more complicated 

picture of atomic events. Tully et al. [107] in 1979 carried out 

ghost particle simulations for a (100) surface of a Lennard- 

Jones crystal at different temperatures below the melting point 

T m . They discovered that, as shown in Fig. 30, the average 

jump length more than doubled as the temperature increased 

from 0.2T m to 0.6T m . More extensive molecular dynamics were 

carried out at much the same time by DeLorenzi et al. [10,11], 

again on an fcc Lennard-Jones crystal. Most interesting was 

the fact that on the (100) plane at ∼0.3T m long jumps were 

observed, between sites as much as three to four spacings apart. 

This is clear from the atom trajectories in Fig. 31. 

Corrections to diffusion rates predicted by transition-state 

theory for Rh(100) stemming from transitions to sites farther 

away than a nearest-neighbour distance were done by Voter 

and Doll [108], again with Lennard-Jones interactions. Only 

in the vicinity of 1000 K were jumps to other than nearestneighbour 

positions found. In the same year, 1985, DeLorenzi 

and Jacucci [41] published molecular dynamics simulation 

on various bcc surfaces modelled with a metallic potential 

developed by Price [42,43] for sodium. On the most densely 

packed surface, the (110), at a temperature of 0.4T m , they 

found not only jumps between nearest-neighbour sites, as in 

Fig. 32(a), but also double and more elaborate transitions, 

shown in Fig. 32(b) and (c). Evangelakis and Papanicolaou [82] 

also saw that double jumps started to be active on the (100) 

plane of copper, an fcc metal, at temperatures above 750 K. 

Above 800 K, more complicated processes, such as quadruple 

exchange began playing a role as well.


the length of atomic jumps can be obtained, as the probability 

that after a time t an atom will be at a displacement x from the 

origin was given by Wrigley et al. [114] as 

Fig. 33. Schematic of single, double and triple transitions in one-dimensional 

atom motion. 

In 1996, Ferrando [104] looked at self-diffusion on Ag(110) 

and found that single jumps represented ∼90% of the total, the 

rest were double or more complicated transitions. Investigations 

of long jumps were done by Montalenti and Ferrando [32] who 

looked at self-diffusion on the (110) planes of gold, silver and 

copper. Although the activation energy for movement on gold 

and silver is almost the same, and for copper slightly lower, 

their behaviour with respect to long jumps differed greatly. At 

450 K, long jumps were absent on gold, there were 3% long 

jumps on silver, and 6% on copper. For copper, the fraction 

of long jumps increased to 15% at 600 K, but never got to 

this value on the Ag(110) surface. Ferrón et al. [113] saw long 

jumps as well as rebound jumps in self-diffusion on Cu(111). 

They claimed that at 500 K, 95% of jumps were correlated; at 

100 K this decreased to 50%. 

What is quite clear from these simulations is that at elevated 

temperatures, larger than 0.2T m , long jumps should participate 

significantly in surface diffusion. The question still remained, 

however, how to detect the transitions in an experiment. 

3.2. Experimental studies 

Information about the characteristics of surface diffusion 

has usually been obtained from measurements of the meansquare 

displacement. However, these measurements provide 

no simple connection to the types of transitions carried out 

by atoms diffusing over the surface. Usually an increase in 

the diffusivity is expected due to long transitions, since the 

diffusivity in Eq. (4) depends on the jump length squared, but 

so far this expectation has not been realized in experiments. 

More information is definitely needed. From measurements of 

the distribution of displacements in one dimension insight into 

p x (t) = exp[−2(α + β + γ )t] 

∞∑ 

∞∑ 

× I k (2γ t) 

k=−∞ 

j=−∞ 

I j (2βt)I x−2 j−3k (2αt). (8) 

Here it is assumed that single jumps at the rate α, double jumps 

at the rate β, and triple jumps at the rate γ take place on 

the surface, as is illustrated in Fig. 33; I m (u) is the modified 

Bessel function of the first kind, of order m and argument u. 

The jump rates clearly affect the probability of finding a set 

of atom displacements, and that is more immediately evident 

from the plots in Fig. 34, where single and double transitions 

participate in diffusion. All that needs to be done is to carry out 

an adequate number of observations of atomic displacements, 

and from these deduce the probability p x (t). To derive the 

individual jump rates, Eq. (8) is not all that useful, however, 

as it holds for diffusion on an infinite plane. For experiments in 

the field ion microscope the individual planes are quite small, 

and may extend over only ∼20 spacings, so that Monte Carlo 

simulations must be employed to extract rates. 

The first attempt to derive jump rates from the measured 

distribution of displacements was made in 1989 by Wang 

et al. [115] in one-dimensional diffusion on W(211). Tested 

were Re, Mo, Ir, and Rh atoms. As shown for rhenium in 

Fig. 35, the best fit to the distribution measured at 300 K 

was obtained assuming only single jumps. Even though long 

jumps were not found, for the first time the picture of diffusion 

as occurring by random jumps between nearest-neighbor sites 

had been demonstrated. Much the same also held for the other 

atoms studied, although for iridium and rhodium there were 

negligibly small contributions from double jumps. 

The next step was taken by Senft [116–118], who surmised 

that energy transfer between the moving adatom and the 

lattice would affect the participation of long jumps in surface 

diffusion. She therefore elected to study palladium as well as 

nickel, with low values of the diffusion barriers amounting to 

0.314±0.006 eV for Pd and 0.46±0.06 eV for Ni, which imply 

rather poor energy transfer. The displacement distribution was 

tested for self-diffusion of tungsten on W(211) at 307 K, and as 

Fig. 34. Effect of double jumps on the distribution of atomic displacements in diffusion with a mean-square displacement of two.


Fig. 35. Distribution of Re adatom displacements at 300 K on W(211), obtained from field ion observations [115]. Best fit to experiments obtained with an entirely 

negligible contribution from long jumps. 

Fig. 36. Displacement distribution for W adatom on W(211) at 307 K [117]. 

Best fit to field ion experiments derived with negligible contribution of β double 

or longer jumps. 

Fig. 37. Distribution of Pd atom displacements on W(211) at 133 K [118]. Best 

fit with double/single jumps equal to 0.20, and triple/single of 0.13. 

shown in Fig. 36 was found to be due entirely to single jumps 

between nearest-neighbor sites, as expected at the time. 

The same was found for the diffusion of palladium at 114 

and 122 K. However, at 133 K, the distribution in Fig. 37 for 

the first time gave a clear indication of significant contributions 

from long jumps; the ratio of double to single jumps was 0.20, 

Fig. 38. Distribution of Ni adatom displacements on W(211) plane at 179 

K [117]. Best fit of observations with double/single jumps equal to 0.058. 

and even triple jumps were detected, at a ratio of 0.13 for triple 

to single transitions. In the diffusion of nickel atoms on W(211), 

Senft [117] found a distribution shown in Fig. 38, best fit with 

a ratio of doubles to singles of 0.058 at T = 179 K. What 

was surprising about these findings is not just the detection 

of long jumps, but long jumps at quite a low temperature, < 

0.1T m , and at a rate very temperature sensitive. For palladium, 

a diminution of the temperature by 11 K sufficed to eliminate 

all long transitions. 

With long jumps now firmly established in surface diffusion, 

Linderoth et al. [119] decided to explore their rates in selfdiffusion 

on the reconstructed Pt(110)-(1 × 2) plane, shown 

in Fig. 39. Jumps were observed with the scanning tunneling 

microscope, which yielded the distribution at 375 K in Fig. 40, 

with a ratio of “double” to single jumps of 0.095. They also 

made measurements over a temperature range of 60 K to come 

up with the Arrhenius plot in Fig. 41, which gave a barrier 

of 0.81 ± 0.01 eV for single jumps and a somewhat higher 

value, 0.89 ± 0.06 eV, identified as coming from doubles. 

This identification turned out to be premature, however. A year 

passed and Montalenti and Ferrando [120] did simulations of 

diffusion on Au(110)-(1×2), relying on RGL interactions [102, 

103]. They discovered two prevalent jumps, illustrated in


Fig. 39. Hard-sphere model of (1×2) reconstructed fcc(110) surface, in which 

every second 〈111〉 row has been removed. 

Fig. 41. Arrhenius plot for single and double jump rates presumed to occur in 

self-diffusion on Pt(110)-(1 × 2) [119]. 

Fig. 40. Distribution of atomic displacements in self-diffusion on Pt(110)- 

(1 × 2) plane [119]. Best fit obtained with ratio of “double” to single jumps 

of 0.095. 

Fig. 42: transitions along the bottom of the diffusion channel, 

and in addition transitions in which the atom jumps to the 

(111) sidewalls and continued diffusion there. The number of 

these metastable transitions exceeded that of long jumps in the 

channel. Another year later, Lorensen et al. [121] published 

density functional estimates for diffusion on Pt(110)-(1 × 2), 

which confirmed what had previously been found for gold— 

metastable jumps were the likely explanation for the findings 

of Linderoth et al. [119]. 

Up to this point, long jumps had been identified 

experimentally only on the channeled, one-dimensional surface 

of the W(211) plane, and only for rapidly diffusing atoms. Oh 

et al. [122] in 2002 undertook tests to see if they also occurred 

on W(110), in the two-dimensional diffusion of palladium. Of 

course this is a rather more complicated system, and jumps may 

take place in quite a number of ways, as indicated in Fig. 43. 

The Arrhenius plot in Fig. 44 did not show any evidence of 

long jumps. However, an Arrhenius plot is not a good indicator 

of the jump processes in diffusion. For this we have to rely 

on the distribution of displacements, shown in Fig. 45. At a 

Fig. 42. Trajectories of Au adatom on reconstructed Au(110) surface with 

〈110〉 rows missing, obtained in molecular dynamics simulations at 450 

K [120]. Left column: in-channel jumps single, double, and triple. Right 

column: metastable transitions single, double, and triple. 

temperature of 210 K this distribution gave a significant number 

of double β jumps along 〈111〉 as well as vertical δ y transitions. 

After correction for effects due to transient temperatures during 

sample heating and cooling, the ratio of β/α proved to be 

0.12 ± 0.06, and 0.11 ± 0.10 was found for δ y /α, the first 

experimental demonstration of long jumps in two-dimensional 

diffusion. Worth noting here is the difference between vertical


Fig. 43. Schematic of possible atom jumps on W(110) plane. 

Fig. 45. Displacement distribution of Pd adatom on W(110) plane at 210 

K [122]. Double/single jump rate β/α is 0.12, δ y /α is 0.11. 

Fig. 44. Dependence of the diffusivity of single Pd adatom on W(110) upon 

the reciprocal temperature [122]. Data corrected for edge effects as well as for 

migration during transients. 

and horizontal jumps, which is huge. It has been possible to 

derive a significant energy difference between the two kinds of 

jumps. 

Oh et al. [123] also examined the self-diffusion of tungsten 

atoms on W(110), and surprisingly found behaviour similar to 

that of palladium. At 365 K, the distribution yielded β/α = 

0.22, δ x /α = 0.36, and δ y /α = 0.43. Half of the transitions 

now were long jumps. Of particular interest are the vertical δ y 

and horizontal δ x transitions. It is not yet clear how they take 

place, but they can be envisioned as starting as jumps in the 

〈111〉 direction, which are then deviated either toward the x- or 

y-axis. 

Although long jumps had now been found for a variety 

of atoms in both one- and two-dimensional diffusion, nothing 

was known about the rates of these transitions. This matter 

was tackled by Antczak [124]. In 2004 she examined in detail 

the distribution of displacements of tungsten atoms on W(110) 

Fig. 46. Arrhenius plot for diffusivities of W atom on W(110) along 〈100〉 and 

〈110〉 direction [124]. Best fit is obtained with a straight line. 

over a range of temperatures and with very good statistics of 

1200 observations. From an Arrhenius plot of the diffusivities, 

in Fig. 46, she obtained straight lines, indicating a barrier 

of 0.92 ± 0.02 eV for diffusion along 〈100〉 and much the 

same barrier of 0.93 ± 0.01 eV along 〈110〉. Again there 

were no indications in the diffusivity of anything to suggest 

contributions from long jumps. However, the distribution of 

displacements at elevated temperatures clearly showed such 

transitions, as is evident from Fig. 47. It must be noted that at 

these high temperatures significant displacements occur during 

the temperature rise before the diffusion interval and during the 

fall at the end. The distribution of displacements during these 

temperature transients therefore has to be determined and is 

also shown in Fig. 47. The final rates were obtained using the 

relation 

r = Rt − r ot o 

t e 

(9)


Table 2 

Jump rate parameters for W on W(110) [124] 

Rate Activation energy (eV) Frequency factor υ o (s −1 ) 

α (low temperature) 0.94 ± 0.03 5.92(×2.5 ±1 ) × 10 12 

β 1.24 ± 0.13 8.06(×8.1 ±1 ) × 10 15 

δ x 1.28 ± 0.13 3.78(×8.6 ±1 ) × 10 16 

δ y 1.37 ± 0.13 1.00(×4.4 ±1 ) × 10 18 

Fig. 49. Arrhenius plots for δ x and δ y jumps of W adatom on W(110) [124]. 

Fig. 47. Distribution of W displacements on W(110) at 364 K [124]. Best fit 

obtained with significant contributions from long jumps. Inset gives distribution 

during temperature transients. Rates not corrected for effects from transients. 

Fig. 48. Arrhenius plots for rates of single α jumps and double β jumps of 

W adatom on W(110) [124]. Single jump rate is fitted at low temperatures. At 

elevated temperatures, rate drops significantly below this straight line. 

where R is the rate measured in the experiment for an interval 

t, r o and t o give these quantities determined for the transients, 

and t e is the effective time interval. 

In Fig. 48 are shown the Arrhenius plots for the rates of 

tungsten single jumps α and double jumps β. What is most 

interesting here is that above ∼340 K, the rate α drops below 

the straight line extrapolated from low temperatures. This is 

not the case for the other rates β, nor for δ x or δ y shown 

in Fig. 49. Clearly the rate of single transitions is influenced 

by the occurrence of the other jumps. The kinetics of these 

transitions are listed in Table 2. The barriers to the long jumps 

are all considerably higher than for single transitions; so are 

the frequency prefactors, 5.92 × 10 12 s −1 for single jumps and 

1.00 × 10 18 s −1 for vertical transitions. 

How can all of this be understood? The important thing to 

recognize is that the various rates are not independent. If the 

sum of all the rates is displayed on an Arrhenius plot, as in 

Fig. 50, a straight line is obtained, not a line curved concave 

upward. This can be understood in terms of the schematic in 

Fig. 51. An atom starts a transition toward a nearest-neighbor 

site, but may continue beyond this. Every time one of these 

other transitions β, δ x , or δ y occurs, the number of single jumps 

is diminished. At higher temperatures, where the higher barriers 

to these transitions can be overcome, the rate α will therefore 

become smaller. 

How can we account for the high barriers and prefactors 

of the long transitions? For the range of temperatures in the 

experiments, we can write the rate r i of transition i as 

r i = R × p i , (10) 

where R is the basic jump rate, given by 

( 

R = υ o exp − E ) 

o 

, (11) 

kT


Fig. 50. Arrhenius plot of the sum of all jump rates for W adatoms on W(110), 

giving a straight line fit [124]. 

Fig. 52. Arrhenius plot for Ir adatom diffusivities on W(110) plane along 〈100〉 

and 〈110〉 [125]. 

Fig. 51. Schematic of jump rates on W(110) plane [124]. Basic jump is α 

transition along 〈111〉. At elevated temperatures, this can proceed beyond 

nearest neighbour end point, giving β, δ x , or δ y . 

and p i is the probability of a particular transition i. This 

probability is given by a normalized expression similar to 

Eq. (11), so that the activation energy for the given rate r i 

is just the sum of the barriers, and the prefactor the product 

of the separate prefactors, which makes the results obtained 

understandable. 

The diffusion of tungsten on W(110) is not the only system 

considered so far. Antczak [125] also examined the behaviour 

of iridium atoms on W(110). Arrhenius plots in Fig. 52 for 

diffusion along 〈100〉 and 〈110〉 are again quite straight, without 

any indication of multiple jump processes. Nevertheless, longer 

jumps occur frequently, as is clear from the distribution of 

displacements during experiments at 366 K, shown in Fig. 53 

together with zero-time experiments, to catch displacements 

during temperature transients. For the long jumps were found 

β/α = 0.15, δ x /α = 0.38, and δ y /α = 0.32. Just as previously 

with tungsten atoms above 350 K, the rate of single α transitions 

falls below the straight extrapolation from low temperatures; 

the other jump rates β, δ x , and δ y fit on normal Arrhenius 

plots. The explanation is the same as for tungsten atoms—the 

Fig. 53. Distribution of Ir adatom displacements on W(110) surface at 

366 K [125]. Both normal experiments, and experiments during transient 

temperatures are shown. Values for ratios of jump rates shown obtained after 

corrections for transient effects. 

jumps are not independent and arise by conversion from one 

elementary process. 

What is important here is that the long transitions of both 

iridium and tungsten replace the single jumps on W(110) 

gradually and at quite a low temperature, less than a tenth 

of the melting point. There they start playing a leading role 

in diffusion, and at higher temperatures single transitions 

disappear completely. Surprising is the difference in the 

activation energies of vertical and horizontal jumps on W(110), 

which as shown in Fig. 54 is huge for palladium atoms, much 

lower for tungsten, and disappears for iridium atoms. It turns 

out that this difference has a linear dependence on mass, but it 

is so far based on only three points.


Fig. 54. Differences in the activation energies for vertical and horizontal jumps 

of Pd, W, and Ir adatoms on W(110). 

Fig. 56. One-dimensional distribution of W adatom displacements at 325 K 

on W(211) [126]. Zero-time measurements to detect transient contributions (in 

inset) have been used to give listed rates. 

Fig. 55. Arrhenius plot for self-diffusion of W adatom on W(211) plane [126]. 

On tungsten, long jumps are not limited to two-dimensional 

diffusion. Senft [116–118] already showed this for Pd and 

Ni on W(211), but with tungsten she came to the conclusion 

that long jumps did not occur. More recently Antczak [126] 

has probed one-dimensional self-diffusion on W(211), but at 

higher temperatures than Senft, and longer transitions were 

discovered for this system as well. A normal Arrhenius plot 

for tungsten, with an activation energy of 0.81 ± 0.02 eV and 

a prefactor of 3.41(×2.40 ±1 ) × 10 −3 cm 2 /s was found, as 

shown in Fig. 55. At an elevated temperature of 325 K the 

distribution of displacements in Fig. 56 now revealed a ratio 

β/α of double to single jumps of 0.66; transitions during the 

temperature transients are given in the same figure. The plot of 

the single jump rate α in Fig. 57 drops sharply at temperatures 

of 300 K and above. At 325 K the rate of single jumps has gone 

Fig. 57. Arrhenius plot for α single jumps of W on W(211) [126]. At ∼320 K, 

rate α has decreased so much it is no longer discernable. 

to zero and only long jumps contribute to diffusion; the rate of β 

double jumps behaves normally with temperature, as is evident 

from Fig. 58. 

Long jumps are clearly demonstrated in this onedimensional 

system, but there is also a big surprise. As 

indicated in Fig. 59, the sum of the two rates measured does 

not plot as a linear Arrhenius graph, as had been found on 

W(110); it appears that here not all types of jumps have been 

detected. What is missing are rebound transitions, illustrated in 

Fig. 60. An atom starts on a jump; at the adjacent site it can 

settle down, creating a single jump. It can also continue on to 

the next site to form a double transition, or else it can rebound 

to return to the starting point. No displacements arise from


Fig. 60. Schematic of different types of jump processes for an adatom on 

W(211) [126]. β R denotes rebound transition. 

Fig. 58. Normal Arrhenius plot for β double jumps of W adatom on 

W(211) [126]. 

Fig. 61. Rate of rebound jumps, obtained as the difference between the 

sum of single plus double jumps and total jump rate extrapolated from low 

temperatures [129]. 

Fig. 59. Arrhenius plot for the sum of all jump rates measured for W adatom 

on W(211). For temperatures ≥310 K, rate falls below linear plot [126]. 

such a rebound transition, and it is therefore not detected in 

the normal diffusion measurements, but rebounds were found in 

molecular dynamics simulations in 1989 by DeLorenzi [127], 

and a few years later by Sanders and DePristo [128] and Ferrón 

et al. [113]. 

A way to measure such rebounds has recently been 

discovered. Antczak [129] has pointed out that the sum of the 

measured jump rates has to be subtracted from the straightline 

Arrhenius plot obtained by extrapolating the sum from 

low temperatures. The rebound rate so obtained is shown 

in Fig. 61; it has an activation energy of 1.03 ± 0.06 eV 

and a frequency prefactor of 1.40(×10.3 ±1 ) × 10 16 s −1 , and 

thus lies midway between single and double jumps. What 

is surprising is that rebounds were not detected in twodimensional 

diffusion on W(110) at all, suggesting it may be an 

effect tied to the channelled structure of W(211) that is involved 

in these transitions. However, they have been seen previously in 

simulations of self-diffusion on the Cu(111) plane [113]. 

It should be noted that long jumps are not limited to the 

movement of single atoms. As is apparent from Fig. 62, 

long jumps were observed by Wang [130,131] for clusters of 

iridium atoms on Ir(111), and for the large organic molecules 

decacyclene and hexa-tert-butyl-decacyclene by Schunack 

et al. [132]. However, information about the kinetics of these 

processes is limited and crying for more work. 

Finally, it should be said that theoretical predictions about 

long jumps have been very important in pointing to their 

contributions in diffusion. However, even the limited number 

of experiments done so far have already revealed that these 

transitions are rather more varied and more significant than 

predicted, especially at low temperatures. 

4. Summary 

This survey should make it clear that a whole variety of 

different jumps contributes to surface diffusion on metals. Atom 

exchange processes are of course very much affected by the 

chemistry of the interacting partners, but have so far only 

been observed in experiments with fcc metals. On channelled 

surfaces the direction favoured in diffusion is still puzzling.


Fig. 62. Centre-of-mass displacements for Ir 19 cluster during 10 s intervals at 

690 K [131]. Observed number of displacements shown bold, best fit in outline 

numerals just below. α, β, as well as γ transitions contribute significantly. 

Very interesting are the multiple exchange processes seen in 

simulations at elevated temperatures, but these have not yet 

been found in experiments. 

The situation may well be different for the rate of long 

jumps. These have so far only been detected in experiments on 

tungsten surfaces, on which it has been possible to determine 

jump rates. There it is clear that in self-diffusion these jumps 

already occur at low temperatures less than 1/10 the melting 

point; at more elevated temperatures they supplant single atom 

jumps as carriers of the diffusion process. So far these long 

transitions have not been observed on other metals, even though 

the expectation is that long jumps will prove to be quite 

general, independent of the substrate, and important at elevated 

temperatures, where new types of jumps may also be found. 

Surface diffusion has now been explored on the atomic level 

for forty years, but new effects are still being discovered, and 

that suggests important work is ahead. 

Acknowledgements 

This work was done while supported by the Petroleum 

Research Fund, administered by the ACS under Grant ACS 

PRF 44958-AC5. We are also indebted to Mary Kay Newman 

for her help with the literature. 

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