a Xi :0907.4499 1 [nlin.PS] 26 Jul 2009
Manusc ip submi ed o Websi e: h p://AIMsciences.o g
AIMS’ Jou nals
Volume X, Numbe 0X, XX 200X pp. X–XX
INTERACTION OF MOVING DISCRETE BREATHERS WITH
INTERSTITIAL DEFECTS
J. Cue as, B. S´
anchez–Rey
G upo de F´ısica No Lineal. Depa amen o de F´ısica Aplicada I.
Escuela Uni e si a ia Poli ´ecnica. Uni e sidad de Se illa.
C/ Vi gen de ´
A ica, 7. 41011 Se illa, Spain
J.C. Eilbeck and F.M. Russell
Depa men o Ma hema ics and he Maxwell Ins i u e o Ma hema ical Sciences
He io -Wa Uni e si y - Ricca on, Edinbu gh, EH14 4AS, UK
Abs ac . In his pape , in e s i ial mig a ion gene a ed by sca e ing wi h
a mobile b ea he is in es iga ed nume ically in a F enkel-Kon o o a one-
dimensional la ice. Consis en wi h expe imen al esul s i is shown ha
in e s i ial diffusion is mo e likely and as e han acancy diffusion. Ou sim-
ula ions suppo he hypo hesis ha a long- ange ene gy anspo mechanism
in ol ing mo ing nonlinea ib a ional exci a ions may significan ly enhance
he mobili y o poin de ec s in a c ys al la ice.
1. In oduc ion. The F enkel–Kon o o a (FK) model, in oduced almos 70 yea s
ago [1], is one o he mos pa adigma ic nonlinea sys ems, whose dynamics has been
widely s udied du ing he las decades (see [2,3,4,5] and e e ences he ein). F om
he poin o iew o condensed ma e physics, i s pa amoun impo ance elies
on he abili y o desc ibe a as numbe o phenomena, including di e en kinds
o de ec s such as acancies (Scho ky de ec s) and, o some ex en , in e s i ials
(F enkel de ec s), which can play an impo an ole in he design o new ma e ials
[6].
As he FK model is basically a one-dimensional la ice o pa icles subjec ed
o a nonlinea pe iodic subs a e po en ial and a nea es -neighbou in e ac ion, i
con ains he basic ing edien s o sus ain localized exci a ions such as opological
soli ons (kinks o an ikinks) o b ea he s. Disc e e b ea he s (DBs), also called
in insic localized modes ( o a e y ecen e iew abou hei p ope ies, exis ence
p oo s, compu a ional me hods and applica ions see [7]), a e exac solu ions o he
dynamical equa ions whose ene gy, in con as wi h no mal ex ended wa e exci a-
ions, is no sha ed among la ice componen s bu ex ends only o e a ew la ice
si es. In his sense, hei spa ial p o iles esembles localized ib a ional modes in-
duced by a de ec si e in a ha monic la ice [8]. Howe e DBs a ise only hanks o
he in e play be ween nonlinea i y and disc e eness and, o ha eason, hey may
occu anywhe e in he la ice gi en su icien ib a ional ampli ude. They a e also
a he uni e sal since hey a e no speci ic o Hamil onians wi h a pa icula o m
and can be ound in la ices o a bi a y dimensions. Mo eo e , heo e ical s udies
2000 Ma hema ics Subjec Classi ica ion. P ima y: 70K75, 74J30; Seconda y:
Key wo ds and ph ases. Mo ing b ea he s, kinks, de ec s, F enkel–Kon o o a model.
1
2 CUEVAS, S ´
ANCHEZ-REY, EILBECK AND RUSSELL
ha e shown DBs a e linea ly s able [9], which implies hey can pe sis o e e y long
imes on op o a he malized backg ound [10]. Thei in es iga ion is no es ic ed
o simple oy models. Apa om indi ec spec oscopic obse a ions [11], DBs
ha e been de ec ed and s udied expe imen ally in such di e en mac oscopic sys-
ems as wa eguide a ays [12], mic omechanical can ile e s [13], an i e omagne ic
s uc u es [14] and Josephson-junc ions [15].
In his con ex , an in e es ing p oblem ha has a ac ed much a en ion in
ecen yea s is he in e ac ion be ween a mo ing localized exci a ion and a la ice
de ec . The p oblem has been add essed wi hin di e en amewo ks: impu i ies
[16,17], la ice junc ions [18,19], bending poin s o a polyme chain [20,21,22],
bu mos s udies assume ha he posi ion o he de ec is ixed and is no able o
mo e along he la ice. O cu en in e es is he in e ac ion be ween la ice de ec s
and mo ing localized exci a ions, which migh esul in mo emen o he de ec .
This is especially ue in he case o in e ac ions a ising du ing i adia ion o solids
by swi pa icles, which usually in ol e he c ea ion o DBs o ei he longi udinal
o ans e se op ical mode ype.
The possibili y o such in e ac ions a ose in he s udy o high ene gy cha ged
pa icles passing h ough c ys als o musco i e, when sca e ing e en s we e pos u-
la ed o c ea e many mo ing highly ene gy DBs. I was sugges ed ha when such
DBs ( he e called quodons) eached he end o a chain, which ep esen s a de ec in
a chain, i migh cause he las a om o be ejec ed om he su ace [23]. This p e-
dic ion was suppo ed by s udies using bo h mechanical and nume ical models [24].
Subsequen ly, i was e i ied by expe imen using a na u al c ys al o musco i e [25].
In he expe imen one edge o a c ys al was bomba ded wi h alpha pa icles a nea
g azing incidence o c ea e mo ing DBs. These p opaga ed in chain di ec ions in
he laye ed c ys al and caused a p opo iona e ejec ion o a oms om a emo e edge
o he c ys al ha was >107uni cells dis ance in a chain di ec ion om he si e o
bomba dmen . As his expe imen was pe o med a 300K i no only e i ied he
p edic ion bu also demons a ed he s abili y o hese mobile DBs agains he mal
mo ion.
O he i adia ion s udies ha e p o ided mo e empi ical signs o he in e ac ion o
DBs wi h de ec s. Fo ins ance, in e . [26] he au ho s p o ide e idence ha , a e
i adia ing a silicon c ys al wi h sil e ions, a pileup o la ice de ec s is accomplished
a loca ions spa ially sepa a ed om he i adia ion si e. The e idence indica ed
ha de ec s could be swep by up o abou 1 mic on om he i adia ed egion.
This e ec was asc ibed o he p opaga ion o highly localized packe s o ib a ional
ene gy, o DBs, c ea ed by he bomba dmen o hea y ions.
Ano he ion-induced, a he mal anspo p ocess was epo ed in e . [27]. In his
case in e s i ial N di usion in aus eni ic s ainless s eel unde A ion bomba dmen
was in es iga ed. I was ound ha N mobili y inc eases in dep hs se e al o de s
o magni ude la ge han he ion pene a ion dep h. This i adia ion-induced en-
hancemen o N di usion is consis en wi h p e ious obse a ions which show a
dependence o he ni iding dep h on ion ene gy [28] and also on he c ys alline
o ien a ion [29], bu no con en ional mechanism o di usion can explain hem. Fo
his eason i was sugges ed ha di usion o in e s i ial a oms migh be assis ed by
highly anha monic localized exci a ions which p opaga e dis ances well beyond he
ion pene a ion dep h.
In e s i ial a oms eside in po en ial wells be ween he la ice a oms. When a
b ea he p opaga es i s ongly dis u bs he la ice locally. I i passes nea an
BREATHER–INTERSTITIAL INTERACTION 3
in e s i ial hese oscilla o y mo ions will dis o he po en ial well con ining he
in e s i ial and will a ec signi ican ly i s mobili y. In e s i ial mo ion consis s o
jumps om one po en ial well o he nex . Since expe imen al measu es deal wi h
concen a ion dep h p o iles, in e s i ial di usion p ocess can be analyzed in e ms
o an e ec i e mo emen along a one-dimensional chain o po en ial wells. Mo eo e
he p esence o an in e s i ial modi ies po en ials in adjacen a omic chains, causing
he spacing be ween he wo nea es a oms in a chain o he in e s i ial o inc ease.
The e o e, in a i s app oxima ion, an in e s i ial can be modelled in oducing and
addi ional pa icle in a one-dimensional sys em and his p o ides he link o he
FK model.
In his pape , using a FK model wi h nonlinea nea es -neighbou in e ac ion, i
is shown ha mig a ion o he dis u bance in a chain caused by an in e s i ial can
be induced by sca e ing wi h a mobile longi udinal mode b ea he . Compa ison
wi h p e ious wo k on acancies mig a ion [30,31] also sugges s ha , acco ding
o expe imen al esul s, in e s i ial mobili y is mo e likely and as e han ha o
acancy de ec s. O cou se, he speci ic cons ain s o a one-dimensional sys em im-
plies ha ca e is needed when a emp ing o ca y o e esul s o highe dimensional
la ices. Ne e heless we hink ha a one-dimensional s udy is a necessa y and use-
ul i s s ep be o e app oaching he p oblem wi h a mo e ealis ic and complex wo
o h ee-dimensional model.
2. The model. As desc ibed in he in oduc ion, he F-K model consis s o a
chain o in e ac ing pa icles subjec o a pe iodic subs a e po en ial. This sys em
is desc ibed by he ollowing Hamil onian:
H=
N
X
n=1
1
2m˙x2
n+V(xn) + W(xn−xn−1),(1)
whe e xnis he absolu e coo dina e o he n- h pa icle. The co esponding dy-
namical equa ions a e
m¨xn+V′(xn) + [W′(xn−xn−1)−W′(xn+1 −xn)] = 0, n ∈Z.(2)
In o de o in es iga e in e s i ial mobili y we ha e chosen a cosine po en ial wi h
he la ice pe iod a
V(x) = a2
4π2[1 −cos(2πx/a)] ,(3)
as he simples , pe iodic subs a e po en ial, wi h he linea equency no malized
o uni y ω0=pV′′ (0) = 1.
Fo he in e ac ion be ween pa icles, we ha e selec ed he Mo se po en ial
W(x) = C
2b2[e−b(x−a)−1]2, x > 0.(4)
which has a minimum a he la ice pe iod aand a ha d pa ha p e en s pa icles
om c ossing. The well dep h o his po en ial is C/2b2while b−1is a measu e o
he well wid h. I s cu a u e a he bo om is gi en by C=W′′ (a), so ha we can
modula e he s eng h o he in e ac ion wi hou changing i s cu a u e by a ying
pa ame e b.
In his sys em, an in e s i ial a om is ep esen ed by a doubly occupied well o he
pe iodic po en ial (see he s able equilib ium con igu a ion in panel (a) o Fig.1).
The ela i e coo dina e o each pa icle wi h espec o i s equilib ium posi ion
can be w i en as un=xn−na. Using hese ela i e coo dina es, he in e s i ial
4 CUEVAS, S ´
ANCHEZ-REY, EILBECK AND RUSSELL
−10 −5 0 5 10
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
un
n
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10−3
EPN
b
Figu e 1. (a) Scheme o he s able equilib ium s a e o he
F enkel–Kon o o a model wi h cosine subs a e po en ial and
Mo se nea es neighbo in e ac ion. The doubly-occupied well ep-
esen s an in e s i ial. (b) An ikink co esponding o he s able
equilib ium con igu a ion in ela i e coo dina es o b= 1 and
C= 0.5. (c) Uns able equilib ium con igu a ion. (d) Peie ls-
Naba o ba ie o he an ikink.
can be isualized as an an ikink [2,3]1as i is shown in panel (b) o Fig. 1. I
is well-known ha an an ikink can be pu in o mo emen as soon as an ene gy
ba ie , he so-called Peie ls-Naba o ba ie (PNB), is o e come. The PNB can
be calcula ed as he ene gy di e ence be ween he uns able and s able an ikink
equilib ium con igu a ions (panels (c) and (a) o Fig. 1 espec i ely) and dec eases
mono onically wi h b(see panel (d)).
I is wo h no ing ha a acancy can be isualized as a kink in ela i e coo di-
na es. I s PNB inc eases wi h he pa ame e band is always highe han he PNB
o an in e s i ial, excep o b= 0 whe e bo h ac i a ion ene gies coincide. This is
in acco dance wi h he expe imen al ac ha di usion o in e s i ials is as e han
ha o acancies, and suppo he idea ha i is necessa y o conside a nonlinea
in e ac ion po en ial in o de o s udy di usion o de ec s, since b= 0 ep esen s
he linea limi o he Mo se po en ial.
1No ice ha in Re . [3] he e ms kink and an ikink a e in e changed.
BREATHER–INTERSTITIAL INTERACTION 5
In ou F-K chain, s a iona y disc e e b ea he s can be nume ically ob ained
using he s anda d me hod o con inua ion om he an icon inuous limi [32,33].
T ansla ional mo ion o disc e e b ea he s can be induced [34,35] by adding a
pe u ba ion ~ =λ(..., 0,−1/√2,0,1/√2,0, ...) o he eloci ies o he s a iona y
b ea he , wi h he nonze o alues a he neighbo ing si es o he ini ial b ea he
cen e . The esul ing DB kine ics is e y smoo h and esembles ha o a classical
ee pa icle. The e o e, he o al ene gy o a mo ing disc e e b ea he can be
es ima ed as he sum o i s ib a ional in e nal ene gy, equal o ha o he s a iona y
b ea he , plus i s ansla ional ene gy, which is equal o he ene gy o he added
pe u ba ion K=λ2/2.
3. Nume ical s udy. In o de o in es iga e in e s i ial mobili y, we ha e gene -
a ed a b ea he cen e ed a si e n=−25, ela i ely a om an in e s i ial whose
le mos pa icle is loca ed a n= 0, and hen launched ha b ea he owa ds i
ollowing he depinning me hod men ioned abo e. Th oughou he pape , we ha e
no malized he la ice pe iod aand masses o uni y and ha e aken C= 0.5 so ha
mo ing b ea he s (MBs) exis in he sys em o a b ea he equency ωb= 0.9.
As a esul o he sca e ing he de ec can be pu in o mo emen leading o
long– ange anspo . We ha e ound h ee well-di e en ia ed egimes depending
on he s eng h o he in e ac ion po en ial. Below a c i ical alue b≈0.83 he
esul o he sca e ing is unp edic able. The dynamics is ex emely sensi i e o
ini ial condi ions ( alue o he pe u ba ion λand ini ial posi ion o he b ea he )
and he in e s i ial can a el o make andom jumps (backwa d o o wa d) o e en
emain a es . Howe e , a ne backwa d mo emen o he de ec is only possible
i he in e ac ion po en ial is s ong enough. In ac we ha e obse ed i only o
alues o b.0.69. An example o a backwa ds a elling in e s i ial is shown in
Fig. 2, whe eas Figs. 3and 4show a backwa ds and o wa ds, espec i ely, hopping
in e s i ial. In his case, he in e s i ial, a e se e al andom jumps, emains pinned
on he la ice. These h ee igu es display h ee panels. Le panel co esponds o
an ene gy densi y plo whe e lines join poin s wi h he same ene gy in ime while
da ke colo indica es la ge ene gy. Cen al panel displays he ime e olu ion o he
an ikink (in e s i ial) cen e o mass. This g aph helps o isualize mo e clea ly he
jumps o he in e s i ial pa icle and he inal oscilla o y s a e a ound an equilib ium
con igu a ion. Finally, igh panel shows a s eak plo wi h he ime e olu ion o he
b ea he and he in e s i ial. I is no ewo hy ha in ou nume ical expe imen s
smalle alues o benhance backwa d mo emen and hopping beha iou o he
in e s i ial pa icle. This la es beha iou is he only obse ed in he ha monic
limi o he in e ac ion po en ial (b= 0).
No ice ha he complexi y o he dynamics is linked o he disc e eness o he
F-K model conside ed [36,37]. In he con inuous limi wi h b= 0 he b ea he -
an ikink in e ac ion is an in eg able and well-known case, and he esul ing scena io
is qui e simple: he b ea he always c osses he an ikink which mo es backwa ds
du ing a b ie lapse o ime [38].
Due o he exis ence o an ac i a ion ene gy o mo e an an ikink in he disc e e
case, in e s i ial mo ion is only ound abo e a h eshold alue, Kc, o he kine ic
ene gy o he inciden b ea he . In he chao ic egime, b.0.83, his h eshold
alue, plo ed in igu e 5, inc eases mono onically in con as wi h he PNB beha io
ound in he p e ious sec ion. On he con a y, o b&0.87 we ind he opposi e
endency: Kcdec eases wi h bindica ing a deep change in he dynamics. Indeed in

6 CUEVAS, S ´
ANCHEZ-REY, EILBECK AND RUSSELL
B ea he pe iods
n
−50 0 50
0
10
20
30
40
50
60
0 10 20 30 40 50 60
−35
−30
−25
−20
−15
−10
−5
0
Xins
B ea he pe iods
10 20 30 40 50 60
−25
−20
−15
−10
−5
0
5
10
xn
B ea he pe iods
Figu e 2. (Le panel) Ene gy densi y plo , showing a backwa d
mo emen o he in e s i ial de ec a e b ea he sca e ing. The
lines join poin s wi h he same ene gy in ime. The da ke colou
he la ge ene gy. (Cen al panel) Time e olu ion o he an ikink
ene gy cen e . (Righ panel) S eak plo . Pa ame e s: K= 0.0220
and b= 0.5.
B ea he pe iods
n
−50 0 50
0
20
40
60
80
100
120
140
160
180
200
0 50 100 150 200
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
Xins
B ea he pe iods
20 40 60 80 100 120 140
−25
−20
−15
−10
−5
0
5
10
xn
B ea he pe iods
Figu e 3. Same as Fig. 2bu o a hopping in e s i ial wi h ne
backwa ds displacemen . Pa ame e s: K= 0.0050 and b= 0.2.
B ea he pe iods
n
−50 0 50
0
20
40
60
80
100
120
140
160
180
200
0 50 100 150 200
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Xins
B ea he pe iods
20 40 60 80 100 120 140
−25
−20
−15
−10
−5
0
5
10
xn
B ea he pe iods
Figu e 4. Same as Fig. 2bu o a hopping in e s i ial wi h ne
o wa d displacemen . Pa ame e s: K= 0.00605 and b= 0.1.
his pa ame e egime, o K > Kc, he in e s i ial always mo es o wa d a e he
sca e ing and, ema kably, i always mo es wi h app oxima ely cons an eloci y.
In his egime, he Mo se po en ial becomes essen ially “ la ” wi h a ha d co e and
he dynamics is domina ed by he epulsi e pa o he in e ac ion po en ial. An
example can be obse ed in Fig. 6. A e he collision wi h he b ea he , in e s i ial
mo ion is clea ly linea in ime. I s eloci y has been compu ed i ing he poin s o
he cen al panel wi h linea eg ession. In he ansi ion be ween bo h egimes, i.e.
o 0.83 .b.0.87, he in e s i ial always emains pinned on he la ice, a leas o
hose alues o λ o which he b ea he p opaga es wi hou signi ican dis o ion.
BREATHER–INTERSTITIAL INTERACTION 7
0 0.5 1 1.5 2
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Kc
b
Figu e 5. Minimum ansla ional ene gy (Kc) o he incoming
b ea he needed o mo e an in e s i ial. In he band 0.83 &b.
0.87 he in e s i ial always emains pinned on he la ice.
B ea he pe iods
n
−50 0 50
0
10
20
30
40
50
60
0 10 20 30 40 50 60
5
10
15
20
25
30
Xins
B ea he pe iods
10 20 30 40 50 60
−15
−10
−5
0
5
10
15
20
xn
B ea he pe iods
Figu e 6. Same as Fig. 2bu o a b ea he wi h K= 0.020
and b= 1.5. The in e s i ial always mo es o wa d wi h cons an
eloci y in he pa ame e egion b&0.87, K > Kc
Fig. 7shows he e olu ion o a pinned in e s i ial o b= 1. In his case he
inciden b ea he possesses a ansla ional ene gy smalle han he c i ical alue Kc
and, consequen ly, he in e s i ial ac s as a wall which o ally e lec s he b ea he .
I is obse ed ha , a e he collision, pa o he b ea he ene gy is employed in
exci ing an in e nal mode o he in e s i ial wi h a equency smalle han ha o
he inciden b ea he . This linea localized mode co esponds o he line below he
phonon spec um shown in Fig. 8 o he in e s i ial s able equilib ium con igu a ion.
No e ha nonlinea localized modes do no exis close o he in e s i ial since he
in e ac ion po en ial is so , and he equency o he linea localized mode is always
below ωb= 0.9.
As men ioned abo e, o b&0.87 a s able in e s i ial p opaga ing mode appea s
i he kine ic ene gy o he inciden b ea he is highe han he h eshold alue
Kc. In his pa ame e egion he Mo se po en ial becomes essen ially a epulsi e
po en ial. In ac , in he limi b→ ∞ i becomes a ha d-sphe e po en ial. Fo
his eason, in his dynamical egime in e s i ial pa icles mo e oughly like ha d
sphe es on a wa y ene gy landscape. A e he b ea he sca e ing, he in e s i ial
pa icle su moun s he ene gy ba ie o he on-si e po en ial well and collides wi h
he pa icle ha occupies he ollowing well ans e ing i s ene gy and momen um
o i . In his way he de ec p opaga es a cons an eloci y o e e . In igu e 9we
8 CUEVAS, S ´
ANCHEZ-REY, EILBECK AND RUSSELL
B ea he pe iods
n
−50 0 50
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40 50
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Xins
B ea he pe iods
10 15 20 25 30 35 40 45 50
−25
−20
−15
−10
−5
0
5
xn
B ea he pe iods
Figu e 7. Same as Fig. 2bu o a b ea he wi h K= 0.0162 <
Kcand b= 1. As he kine ic ene gy o he inciden b ea he is
below he h eshold alue Kc, he in e s i ial emains pinned on
he la ice.
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
b
Linea modes equencies
Figu e 8. Linea modes spec um o he s able equilib ium con-
igu a ion o C= 0.5. ωband 2ωba e depic ed h ough dashed
lines.
ha e plo ed he dependence o he eloci y o his p opaga ing mode on he kine ic
ene gy o he inciden b ea he . One can obse e ha jus abo e he h eshold
ene gy, in e s i ial eloci y inc eases wi h he kine ic ene gy, K, o he incoming
b ea he . Howe e o highe alues o K he in e s i ial eloci y ends o sa u a e
a ound a alue 0.14, wha means ha he in e s i ial pa icle mo es app oxima ely
0.14 2π
ωb≈1 si e on he chain pe b ea he pe iod, independen ly o he coupling
s eng h.
This phenomenon is con i med in igu e 10 whe e we ha e plo ed he in e s i-
ial eloci y e sus pa ame e b o a ixed alue o K. Indeed, o K= 0.045
(dashed line) well abo e he ene gy h eshold, in e s i ial eloci y akes oughly he
sa u a ion alue 0.14 independen ly o he coupling s eng h. In e media e alues
o kine ic ene gy as K= 0.02 (con inuous line) also leads o sa u a ion eloci ies
independen ly o bbu wi h alues lowe han 0.14 and less luc ua ions.
4. Conclusions. We ha e p esen ed nume ical esul s a ising om he in e ac ion
be ween a mo ing disc e e b ea he and an in e s i ial de ec in a FK chain. The
main esul is he exis ence o h ee di e en ia ed egimes depending on he s eng h
o he in e ac ion po en ial. When he in e ac ion be ween neighbo s is s ong he
BREATHER–INTERSTITIAL INTERACTION 9
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
In e s i ial eloci y
K
b=1.25
b=1.5
b=2
Figu e 9. In e s i ial eloci y as a unc ion o he ansla ional
ene gy Ko he inciden b ea he o h ee di e en alues o he
coupling s eng h in he egime (b&0.87). In his egime he
in e s i ial always mo es o wa d wi h cons an eloci y because
he in e ac ion po en ial educes essen ially o a epulsi e ha d
co e.
0.8 1 1.2 1.4 1.6 1.8 2
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
In e s i ial eloci y
b
K=0.02
K=0.045
Figu e 10. In e s i ial eloci y e sus coupling s eng h o a ixed
ansla ional ene gy o he inciden b ea he .
dynamics is chao ic and he beha io o he in e s i ial pa icle is unp edic able:
i can jump backwa ds, o wa ds o emains a es . Howe e , i he in e ac ion
po en ial is weak enough, he de ec mo es o wa ds along he la ice wi h cons an
eloci y. This s able p opaga ing mode had no been obse ed o ou knowledge in
p e ious nume ical s udies conce ning he in e ac ion be ween mo ing b ea he s and
poin de ec s. The e ec is asc ibed o he ac ha he in e ac ion po en ial educes
essen ially o a epulsi e ha d co e. Be ween hese wo dynamical egimes he e is an
na ow in e media e ange o he coupling s eng h in which he in e s i ial always
emains pinned.
Ou o ha pinned egime, he kine ic ene gy o he incoming b ea he s mus
su pass a h eshold in o de o mo e he in e s i ial. This ene gy h eshold has
a non-mono onic beha io . I g ows wi h pa ame e bin he chao ic egime, bu
dec eases wi h bwhen he sys em losses sensi i i y o ini ial condi ions and he