890:
1000:
1072:
912:
1096:
936:
154:
1103:
1079:
925:
837:) and self-diffusion (e.g. Pt adatoms on Pt). It is still unclear from a theoretical point of view why the atomic exchange mechanism is more predominant in some systems than in others. Current theory points towards multiple possibilities, including tensile surface stresses, surface relaxation about the adatom, and increased stability of the intermediate due to the fact that both atoms involved maintain high levels of
1206:. By visualizing the displacement of atoms or clusters over time, it is possible to extract useful information regarding the manner in which the relevant species diffuse-both mechanistic and rate-related information. In order to study surface diffusion on the atomistic scale it is unfortunately necessary to perform studies on rigorously clean surfaces and in
1172:: the rate of cluster diffusion has a strong dependence on the size of the cluster, with larger cluster size generally corresponding to slower diffusion. This is not, however, a universal trend and it has been shown in some systems that the diffusion rate takes on a periodic tendency wherein some larger clusters diffuse faster than those smaller than them.
63:, this motion is typically a thermally promoted process with rates increasing with increasing temperature. Many systems display diffusion behavior that deviates from the conventional model of nearest-neighbor jumps. Tunneling diffusion is a particularly interesting example of an unconventional mechanism wherein hydrogen has been shown to diffuse on clean
776:
726:, diffusion in this regime is now also dependent on the formation energy of mobile adparticles. The exact nature of the diffusion environment therefore plays a role in dictating the diffusion rate, since the formation energy of an adparticle is different for each type of surface feature as is described in the
987:
can occur in the case of channeled surfaces. Typically in-channel diffusion dominates due to the lower energy barrier for diffusion of this process. In certain cases cross-channel has been shown to occur, taking place in a manner similar to that shown in figure 8. The intermediate "dumbbell" position
904:
Figure 6. Surface diffusion jump mechanisms. Diagram of various jumps that may take place on a square lattice such as the fcc (100) plane. 1) Pink atom shown making jumps of various length to locations 2-5; 6) Green atom makes diagonal jump to location 7; 8) Grey atom makes rebound jump (atom ends up
928:
Figure 8. Cross-channel diffusion involving an adatom (grey) on a channeled surface (such as fcc (110), blue plus highlighted green atom). 1) Initial configuration; 2) "Dumbbell" intermediate configuration. Final displacement may include 3, 4, 5, or even a return to the initial configuration. Not to
697:
describes the process at higher level of coverage where the effects of attraction or repulsion between adatoms becomes important. These interactions serve to alter the mobility of adatoms. In a crude way, figure 3 serves to show how adatoms may interact at higher coverage levels. The adatoms have no
976:
have been shown by both experiment and simulations to take place in certain systems. Since the motion does not result in a net displacement of the adatom involved, experimental evidence for rebound jumps again comes from statistical interpretation of atomic distributions. A rebound jump is shown in
679:
There are four different general schemes in which diffusion may take place. Tracer diffusion and chemical diffusion differ in the level of adsorbate coverage at the surface, while intrinsic diffusion and mass transfer diffusion differ in the nature of the diffusion environment. Tracer diffusion and
787:
817:
is conceptually the most basic mechanism for diffusion of adatoms. In this model, the adatoms reside on adsorption sites on the surface lattice. Motion occurs through successive jumps to adjacent sites, the number of which depends on the nature of the surface lattice. Figures 1 and 3 both display
805:
Diffusion of adatoms may occur by a variety of mechanisms. The manner in which they diffuse is important as it may dictate the kinetics of movement, temperature dependence, and overall mobility of surface species, among other parameters. The following is a summary of the most important of these
1181:
Surface diffusion is a critically important concept in heterogeneous catalysis, as reaction rates are often dictated by the ability of reactants to "find" each other at a catalyst surface. With increased temperature adsorbed molecules, molecular fragments, atoms, and clusters tend to have much
994:
is a process involving an adatom inserting into the surface as in the normal atomic exchange mechanism, but instead of a nearest-neighbor atom it is an atom some distance further from the initial adatom that emerges. Shown in figure 9, this process has only been observed in molecular dynamics
828:
involves exchange between an adatom and an adjacent atom within the surface lattice. As shown in figure 4, after an atomic exchange event the adatom has taken the place of a surface atom and the surface atom has been displaced and has now become an adatom. This process may take place in both
958:
consist of adatom displacement to a non-nearest-neighbor adsorption site. They may include double, triple, and longer jumps in the same direction as a nearest-neighbor jump would travel, or they may be in entirely different directions as shown in figure 6. They have been predicted by
915:
Figure 7. Graph showing relative probability distribution for adatom displacement,Δx, upon diffusion in one dimension. Blue: single jumps only; Pink: double jumps occur, with ratio of single:double jumps = 1. Statistical analysis of data may yield information regarding diffusion
333:
must be smaller than the energy of desorption for diffusion to occur, otherwise desorption processes would dominate. Importantly, equation 1 tells us how strongly the jump rate varies with temperature. The manner in which diffusion takes place is dependent on the relationship between
970:(melting temperature). In some cases data indicate long jumps dominating the diffusion process over single jumps at elevated temperatures; the phenomena of variable jump lengths is expressed in different characteristic distributions of atomic displacement over time (see figure 7).
939:
Figure 9. Long range atomic exchange mechanism for surface diffusion at a square lattice. Adatom (pink), resting at surface (1), inserts into lattice disturbing neighboring atoms (2), ultimately causing one of the original substrate atoms emerge as an adatom (green) (3). Not to
950:
Recent theoretical work as well as experimental work performed since the late 1970s has brought to light a remarkable variety of surface diffusion phenomena both with regard to kinetics as well as to mechanisms. Following is a summary of some of the more notable phenomena:
27:
diffusing across a square surface lattice. Note the frequency of vibration of the adatom is greater than the jump rate to nearby sites. Also, the model displays examples of both nearest-neighbor jumps (straight) and next-nearest-neighbor jumps (diagonal). Not to
995:
simulations and has yet to be confirmed experimentally. In spite of this long range atomic exchange, as well as a variety of other exotic diffusion mechanisms, are anticipated to contribute substantially at temperatures currently too high for direct observation.
763:) or the presence of steps on a surface. One of the more dramatic examples of directional anisotropy is the diffusion of adatoms on channeled surfaces such as fcc (110), where diffusion along the channel is much faster than diffusion across the channel.
1049:” of terrace adatoms onto the cluster leading to a change in the cluster’s center of mass. While figure 10 appears to indicate the same atom evaporating from and condensing on the cluster, it may in fact be a different atom condensing from the 2D gas.
519:
1017:
to islands containing hundreds of atoms. Motion of the cluster may occur via the displacement of individual atoms, sections of the cluster, or the entire cluster moving at once. All of these processes involve a change in the cluster’s
1003:
Figure 10. Individual mechanisms for surface diffusion of clusters. (1) Sequential displacement; (2) Edge diffusion; (3) Evaporation-condensation. In this model all three mechanisms lead to the same final cluster displacement. Not to
86:. While in principle the process can occur on a variety of materials, most experiments are performed on crystalline metal surfaces. Due to experimental constraints most studies of surface diffusion are limited to well below the
298:
758:
Directional anisotropy refers to a difference in diffusion mechanism or rate in a particular direction on a given crystallographic plane. These differences may be a result of either anisotropy in the surface lattice (e.g. a
779:
Figure 4. Model of an atomic exchange mechanism occurring between an adatom (pink) and surface atom (silver) at a square surface lattice (blue). The surface atom becomes an adatom. Not to scale on a spatial or temporal
680:
intrinsic diffusion both refer to systems where adparticles experience a relatively homogeneous environment, whereas in chemical and mass transfer diffusion adparticles are more strongly affected by their surroundings.
1201:
Surface diffusion may be studied by a variety of techniques, including both direct and indirect observations. Two experimental techniques that have proved very useful in this area of study are field ion microscopy and
576:
675:, flux is in the opposite direction of the concentration gradient, a purely statistical effect. The model is not intended to show repulsion or attraction, and is not to scale on a spatial or temporal basis.
668:
977:
figure 6. The figure is slightly misleading, however, as rebound jumps have only been shown experimentally to take place in the case of 1D diffusion on a channeled surface (in particular, the
790:
Figure 5. Model of surface diffusion occurring via the vacancy mechanism. When surface coverage is nearly complete the vacancy mechanism dominates. Not to scale on a spatial or temporal basis.
716:, wherein the terrace is a sharp sample tip on which an adparticle diffuses. Even in the case of a clean terrace the process may be influenced by non-uniformity near the edges of the terrace.
20:
1193:
aside because of the interplay between increased rates of diffusion and decreased lifetime of adsorption, increased temperature may in some cases decrease the overall rate of the reaction.
873:
can occur as the predominant method of surface diffusion at high coverage levels approaching complete coverage. This process is akin to the manner in which pieces slide around in a "
691:), particle interaction is low and each particle can be considered to move independently of the others. The single atom diffusing in figure 1 is a nice example of tracer diffusion.
901:
899:
722:
takes place in the case where adparticle sources and traps such as kinks, steps, and vacancies are present. Instead of being dependent only on the jump potential barrier E
847:
is a physical manifestation of the quantum tunneling effect involving particles tunneling across diffusion barriers. It can occur in the case of low diffusing particle
462:
1035:
involves movement of adatoms or vacancies at edge or kink sites. As shown in figure 10, the mobile atom maintains its proximity to the cluster throughout the process.
657:
631:
605:
1055:
is similar to edge diffusion, but where the diffusing atom actually moves atop the cluster before settling in a different location from its starting position.
222:
867:
surfaces. The phenomenon is unique in that in the regime where the tunneling mechanism dominates, the diffusion rate is nearly temperature-independent.
893:
200:
factor that dictates the probability of an attempt resulting in a successful jump. The attempt frequency ν is typically taken to be simply the
372:, is relatively uncommon and has only been observed in a few systems. For the phenomena described throughout this article, it is assumed that
671:
Figure 3. Model of six adatoms diffusing across a square surface lattice. The adatoms block each other from moving to adjacent sites. As per
898:
530:
905:
in same place it started). Non-nearest-neighbor jumps typically take place with greater frequency at higher temperatures. Not to scale.
192:, moving between adjacent (nearest-neighbor) adsorption sites by a jumping process. The jump rate is characterized by an attempt
687:
describes the motion of individual adparticles on a surface at relatively low coverage levels. At these low levels (< 0.01
897:
755:(111) tend to have higher diffusion rates than the correspondingly more "open" faces of the same material such as fcc (100).
97:
Surface diffusion rates and mechanisms are affected by a variety of factors including the strength of the surface-adparticle
1541:
Metal-Surface
Reaction Energetics: Theory and Applications to Heterogeneous Catalysis, Chemisorption, and Surface Diffusion
1182:
greater mobility (see equation 1). However, with increased temperature the lifetime of adsorption decreases as the factor k
900:
877:". It is very difficult to directly observe vacancy diffusion due to the typically high diffusion rates and low vacancy
896:
895:
894:
1548:
1529:
963:
to exist in many different systems, and have been shown by experiment to take place at temperatures as low as 0.1
456:
corresponds to the spacing between nearest-neighbor adsorption sites. The root mean squared displacement goes as:
1332:
1158:
is a snake-like movement (hence the name) involving sequential motion of cluster sub-units (see figure 11(c)).
739:
Orientational anisotropy takes the form of a difference in both diffusion rates and mechanisms at the various
1203:
712:) such as a single terrace, where no adatom traps or sources are present. This regime is often studied using
698:"choice" but to move to the right at first, and adjacent adatoms may block adsorption sites from one another.
91:
83:
1186:
T becomes large enough for the adsorbed species to overcome the barrier to desorption, Q (see figure 2).
822:
transition states between adsorption sites wherein it may be possible for adatoms to temporarily reside.
1569:
1327:
Structure and
Dynamics of Surfaces II (Topics in Current Physics), W. Schommers, P. Von Blanckenhagen,
819:
672:
419:. For cases where more than one diffusion mechanism is present (see below), there may be more than one
393:
1128:
are those that involve movement of either sections of the cluster or the entire cluster all at once.
433:
94:, and much has yet to be discovered regarding how these processes take place at higher temperatures.
1574:
1134:
occurs when adjacent sub-units of a cluster move in a row-by-row fashion through displacement of a
426:
such that the relative distribution between the different processes would change with temperature.
1061:
refers to the process involving motion one atom at a time, moving to free nearest-neighbor sites.
514:{\displaystyle {\sqrt {\langle \Delta r^{2}\rangle }}=a{\sqrt {\Gamma t}}\qquad {\text{(eq. 2)}}}
16:
Process involving the motion of atoms and molecules adsorbed at the surface of solid materials
1215:
713:
101:, orientation of the surface lattice, attraction and repulsion between surface species and
79:
999:
8:
1227:
978:
838:
818:
adatoms undergoing diffusion via the hopping process. Studies have shown the presence of
752:
636:
610:
584:
201:
134:
130:
55:. The process can generally be thought of in terms of particles jumping between adjacent
1164:
is a concerted displacement of a sub-unit of atoms within a cluster (see figure 11(d)).
727:
709:
705:
326:
102:
881:. Figure 5 shows the basic theme of this mechanism in an albeit oversimplified manner.
1544:
1525:
1328:
1207:
1187:
1014:
142:
122:
68:
1152:
refers to the concerted motion of an entire cluster all at once (see figure 11(b)).
213:
205:
110:
106:
667:
293:{\displaystyle \Gamma =\nu e^{-E_{\mathrm {diff} }/k_{B}T}\qquad {\text{(eq. 1)}}}
1237:
1232:
760:
189:
118:
52:
1142:
of the dislocation followed by what is essentially sequential displacement on a
1071:
1190:
1019:
911:
874:
408:
129:
industries. Real-world applications relying heavily on these phenomena include
1095:
775:
75:
1563:
964:
935:
878:
748:
197:
184:
Surface diffusion kinetics can be thought of in terms of adatoms residing at
153:
138:
126:
98:
87:
29:
19:
1046:
744:
740:
1102:
1078:
1013:
Cluster diffusion involves motion of atomic clusters ranging in size from
924:
157:
Figure 2. Diagram of the energy landscape for diffusion in one dimension.
1135:
1042:
452:. In the most basic model only nearest-neighbor jumps are considered and
429:
48:
78:
surface diffusion mechanisms and rates, the most important of which are
1520:
Oura, K.; V.G. Lifshits; A.A. Saranin; A.V. Zotov; M. Katayama (2003).
1519:
1139:
185:
56:
1211:
1143:
988:
may lead to a variety of final adatom and surface atom displacements.
688:
193:
114:
60:
786:
368:
ceases to be a meaningful barrier to diffusion. This case, known as
121:. As such, the principles of surface diffusion are critical for the
860:
856:
830:
44:
571:{\displaystyle D={\frac {\Gamma a^{2}}{z}}\qquad {\text{(eq. 3)}}}
1176:
607:
for 1D diffusion as would be the case for in-channel diffusion,
960:
864:
834:
40:
24:
1497:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 341-343
1488:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 343-345
1479:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 343-344
1425:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 340-341
1407:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 338-340
1371:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 336-340
1344:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 330-333
1275:
1273:
216:
barrier to diffusion. Equation 1 describes the relationship:
64:
848:
743:
of a given material. For a given crystalline material each
1270:
1214:
gas, as is the case when using He or Ne as imaging gas in
1470:
Oura, Lifshits, Saranin, Zotov, and
Katayama 2003, p. 341
1389:
Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 338
1353:
Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 333
1318:
Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 327
1279:
Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 349
1258:
Oura, Lifshits, Saranin, Zotov, and Katayama 2003, p. 325
1210:(UHV) conditions or in the presence of small amounts of
1118:
Figure 11. Concerted mechanisms for cluster diffusion.
1029:
are those that involve movement of one atom at a time.
639:
613:
587:
533:
465:
436:
of diffusing species in terms of the number of jumps
225:
59:
sites on a surface, as in figure 1. Just as in bulk
1138:. As shown in figure 11(a) the process begins with
318:is the jump or hopping rate, T is temperature, and
204:of the adatom, while the thermodynamic factor is a
1045:” from the cluster onto a terrace accompanied by “
651:
625:
599:
570:
513:
292:
173:is the spacing between adjacent adsorption sites;
105:gradients. It is an important concept in surface
1561:
892:
411:of the logarithm of the diffusion coefficient,
747:plane may display unique diffusion phenomena.
448:multiplied by the time allowed for diffusion,
361:the thermodynamic factor approaches unity and
349:as is given in the thermodynamic factor: when
1177:Surface diffusion and heterogeneous catalysis
169:is the heat of adsorption or binding energy;
39:is a general process involving the motion of
1538:
484:
468:
444:. The number of successful jumps is simply
1296:
1294:
704:occurs on a uniform surface (e.g. lacking
1254:
1252:
884:
770:
1117:
998:
934:
923:
910:
888:
785:
774:
666:
152:
74:Various analytical tools may be used to
18:
1291:
855:, and has been observed in the case of
524:The diffusion coefficient is given as:
1562:
1249:
1524:. Springer-Verlag Berlin Heidelberg.
1008:
800:
396:it is possible to extract both the
13:
543:
497:
471:
258:
255:
252:
249:
226:
14:
1586:
1522:Surface Science: An Introduction
1288:Antczak, Ehrlich 2007, p. 50, 59
1101:
1094:
1077:
1070:
137:used in electronic devices, and
1491:
1482:
1473:
1464:
1461:Antczak, Ehrlich 2007, p. 48-50
1455:
1452:Antczak, Ehrlich 2007, p. 40-45
1446:
1437:
1428:
1419:
1410:
1401:
1392:
1383:
1374:
1365:
1356:
1347:
1196:
562:
505:
284:
32:on a spatial or temporal basis.
1503:
1338:
1321:
1312:
1303:
1282:
1261:
208:dependent on temperature and E
1:
1362:Shustorovich 1991, p. 114-115
1309:Shustorovich 1991, p. 109-111
1243:
1204:scanning tunneling microscopy
766:
734:
84:scanning tunneling microscopy
1443:Antczak, Ehrlich 2007, p. 58
1434:Antczak, Ehrlich 2007, p. 51
1398:Antczak, Ehrlich 2007, p. 48
1112:
1109:
1088:
1085:
180:is the barrier to diffusion.
23:Figure 1. Model of a single
7:
1267:Antczak, Ehrlich 2007, p.39
1221:
148:
10:
1591:
992:Long-range atomic exchange
662:
440:and the distance per jump
1539:Shustorovich, E. (1991).
1416:Shustorovich 1991, p. 115
1380:Shustorovich 1991, p. 111
1300:Shustorovich 1991, p. 109
434:mean squared displacement
1509:G. Antczak, G. Ehrlich.
1039:Evaporation-condensation
981:(211) face of tungsten).
728:Terrace Ledge Kink model
432:statistics describe the
314:are as described above,
1543:. VCH Publishers, Inc.
1516:(2007), 39-61. (Review)
1511:Surface Science Reports
1059:Sequential displacement
985:Cross-channel diffusion
841:throughout the process.
720:Mass transfer diffusion
47:, and atomic clusters (
1005:
941:
930:
917:
906:
829:heterodiffusion (e.g.
791:
781:
676:
653:
633:for 2D diffusion, and
627:
601:
572:
515:
294:
181:
117:, and other topics in
33:
1132:Dislocation diffusion
1027:Individual mechanisms
1002:
938:
927:
914:
903:
789:
778:
751:surfaces such as the
670:
654:
628:
602:
573:
516:
295:
202:vibrational frequency
156:
22:
1335:. Chapter 3.2, p. 75
1216:field-ion microscopy
1126:Concerted mechanisms
741:surface orientations
714:field ion microscopy
637:
611:
585:
531:
463:
223:
131:catalytic converters
80:field ion microscopy
51:) at solid material
1228:Surface engineering
845:Tunneling diffusion
761:rectangular lattice
702:Intrinsic diffusion
652:{\displaystyle z=6}
626:{\displaystyle z=4}
600:{\displaystyle z=2}
135:integrated circuits
123:chemical production
1053:Leapfrog diffusion
1006:
942:
931:
918:
907:
792:
782:
695:Chemical diffusion
677:
659:for 3D diffusion.
649:
623:
597:
568:
511:
327:Boltzmann constant
290:
182:
103:chemical potential
34:
1570:Materials science
1208:ultra high vacuum
1122:
1121:
1009:Cluster diffusion
948:
947:
871:Vacancy diffusion
798:
797:
566:
560:
509:
503:
487:
394:Fickian diffusion
392:. In the case of
288:
161:is displacement;
143:photographic film
69:quantum tunneling
67:surfaces via the
37:Surface diffusion
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1086:(a) Dislocation
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1041:involves atoms “
891:
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801:Adatom diffusion
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685:Tracer diffusion
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214:potential energy
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111:epitaxial growth
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1575:Surface science
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1170:Size-dependence
1150:Glide diffusion
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826:Atomic exchange
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1191:thermodynamics
1183:
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1173:
1167:
1166:
1165:
1159:
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1147:
1120:
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1110:(c) Reptation
1107:
1106:
1099:
1091:
1090:
1087:
1083:
1082:
1075:
1065:
1064:
1063:
1062:
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1033:Edge diffusion
1020:center of mass
1010:
1007:
997:
996:
989:
982:
971:
966:
946:
945:
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932:
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875:sliding puzzle
868:
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823:
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403:
384:and therefore
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188:sites on a 2D
176:
150:
147:
141:salts used in
15:
9:
6:
4:
3:
2:
1587:
1576:
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1571:
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1546:
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1531:3-540-00545-5
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1234:
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1218:experiments.
1217:
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1024:
1023:
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1016:
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990:
986:
983:
980:
975:
974:Rebound jumps
972:
969:
962:
957:
954:
953:
952:
944:
937:
933:
926:
922:
921:
913:
909:
887:
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879:concentration
876:
872:
869:
866:
862:
859:diffusion on
858:
850:
846:
843:
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836:
832:
827:
824:
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557:
551:
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479:
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459:
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406:
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387:
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371:
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348:
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328:
324:
317:
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306:
279:
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270:
265:
244:
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198:thermodynamic
195:
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187:
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172:
168:
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160:
155:
146:
144:
140:
139:silver halide
136:
132:
128:
127:semiconductor
124:
120:
116:
112:
108:
104:
100:
95:
93:
89:
88:melting point
85:
81:
77:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
31:
26:
21:
1557:
1540:
1521:
1513:
1510:
1493:
1484:
1475:
1466:
1457:
1448:
1439:
1430:
1421:
1412:
1403:
1394:
1385:
1376:
1367:
1358:
1349:
1340:
1323:
1314:
1305:
1284:
1263:
1200:
1197:Experimental
1180:
1169:
1161:
1155:
1149:
1131:
1125:
1058:
1052:
1047:condensation
1038:
1032:
1026:
1012:
991:
984:
973:
955:
949:
870:
844:
839:coordination
825:
814:
810:
804:
757:
749:Close packed
745:Miller Index
738:
719:
701:
694:
684:
678:
580:
523:
453:
449:
445:
441:
437:
428:
420:
416:
412:
401:
397:
389:
385:
373:
369:
362:
350:
342:
335:
319:
315:
308:
304:
302:
183:
174:
170:
166:
162:
158:
96:
73:
36:
35:
1504:Cited works
1136:dislocation
1043:evaporating
833:adatoms on
806:processes:
430:Random walk
415:, versus 1/
165:is energy;
49:adparticles
1564:Categories
1333:0387173382
1244:References
1140:nucleation
1113:(d) Shear
1089:(b) Glide
956:Long jumps
916:mechanism.
820:metastable
767:Mechanisms
735:Anisotropy
673:Fick’s law
378:>> k
186:adsorption
57:adsorption
1156:Reptation
1144:concerted
851:and low E
710:vacancies
689:monolayer
544:Γ
498:Γ
485:⟩
472:Δ
469:⟨
388:<<
241:−
233:ν
227:Γ
194:frequency
115:catalysis
92:substrate
76:elucidate
61:diffusion
45:molecules
1222:See also
1188:Reaction
1162:Shearing
861:tungsten
857:hydrogen
407:from an
149:Kinetics
71:effect.
53:surfaces
815:jumping
811:Hopping
663:Regimes
565:(eq. 3)
508:(eq. 2)
325:is the
287:(eq. 1)
190:lattice
90:of the
41:adatoms
1547:
1528:
1331:
1146:basis.
1015:dimers
1004:scale.
961:theory
940:scale.
929:scale.
865:copper
780:basis.
581:where
355:< k
303:Where
212:, the
196:and a
25:adatom
1212:inert
706:steps
65:metal
30:scale
1545:ISBN
1526:ISBN
1329:ISBN
863:and
853:diff
849:mass
724:diff
423:diff
404:diff
400:and
376:diff
365:diff
353:diff
341:and
338:diff
331:diff
311:diff
307:and
210:diff
177:diff
163:E(x)
125:and
99:bond
82:and
979:bcc
813:or
753:fcc
708:or
329:. E
1566::
1514:62
1293:^
1272:^
1251:^
1022:.
835:Ni
831:Pt
145:.
133:,
109:,
43:,
1553:.
1534:.
1184:B
967:m
965:T
730:.
647:6
644:=
641:z
621:4
618:=
615:z
595:2
592:=
589:z
558:z
552:2
548:a
538:=
535:D
501:t
493:a
490:=
480:2
476:r
454:a
450:t
446:Γ
442:a
438:N
421:E
417:T
413:D
402:E
398:ν
390:ν
386:Γ
382:T
380:B
374:E
363:E
359:T
357:B
351:E
347:T
345:B
343:k
336:E
322:B
320:k
316:Γ
309:E
305:ν
280:T
275:B
271:k
266:/
259:f
256:f
253:i
250:d
245:E
237:e
230:=
175:E
171:a
167:Q
159:x
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