Rev. Mex. Fis. S 45(2) - Revista Mexicana de Física

Rev. Mex. Fis. S 45(2) - Revista Mexicana de Física

REVISTA ¡\IEXICANA
Non-linear
phenornena
DE FíSICA -tS SUPLEMENTO
2, XO-X5
in nuclci: tite antisoliton
OCTUBRE 19l)9
rnodel for fission
lP. Draaycr, A.Ludu, ami G. Stoitcheva
D(!¡JllrfmC!II
oI ?hysics
(/f/{I
ASfmIlOIllY,
80(011 ROlIMe,
LOlIÜÚII/(/
LA 70803-400J,
SfOfl'
UllilTrsify.
USA
Recihido el 10 L1efehrero de 1999; m:eplado el 2~ de ¡lhril de 1999
The Iloll-linear solutions of the l\lodified Kortev.'eg-de Vries (i\1KdV) equatiolls trawl on Ihe lluclear surfnce 01' Illcdium-heavy nuclei and
gCtlCTnlehighly Jeformcd shapes. TIle clloidnl and soliton solulions providc Ihe existence 01"rololls ;lS large amplitudc collective oscillations.
Tlle dynamics is hased on lile nonlinear cquations and Hamiltonian of a realistic Iiquid drop model (LDI\1). The antisolilOIl solutions are oh.
laíned through a gener;ll formulation of nonrelalivistic quantum Ille(hanics in lcrms 01'dcnsity amI currcnl operators. The quantum averaging
01'lhe alltisolilOtl pail"rotation describes symmetric or aSYll1mctric fission modes. The nuclear aSYllllllelry near the scission poinl scems to he
in qualitative agrcelllent with lhe general shapes considcred in all otller phenol11el1oJogical fission Illodels.
Key\\"(//"{I.\':
Fission; SOJitOll;llotllincar; collectivc cxcitatioJls
Las soluciones [lO lin~ales de las ecunciones modificadas KOrleweg-de Vrics (r-.1KdV) viajan en la superficie lluclear de núcleos medianos (1 pesados generando formas altamente deformadas. LIS soluciones cnoidaks y soJit6nicas dan lugar a la existencia de rotones como
oscilaciones colectivas de gran amplitud. La dinámica esl~í has ada en las ecuaciones no lineales y el Hamiltoniano del modelo de gota
líquida realista (LDM). Las soluciones ;llltisolitélnicas son ohtenidas usando una formulación general de la rnednica cuántica no relativista
en tl5rmillos de operadores de densidaú y de corriente. Los promedios (u,ínlicos de 1,1rol,H,.'i<Índe pares anti.solitónicos d~serihen modos
simétricos o asimétricos de fisi6n. La nsimetría nuclear cerca del punto de ruptura parece estar cualitativamente de acuerdo con la formas
generales consideradas en todos los modelos fenomenológicos de fisión.
1k.\("J"il'for('s: Fisión; solitón; no lineal; excitaciones colectivas
PACS: 2 J.60.Ev; 24. I O.Nz; 25.85.Ca; 83. IO.Ji
1. Introdnction
The thcoretical sludy 01" atomic nuclei under extreme conditiollS indultes largc nuclear dcfonnations,
Interest in Ihese
fcatures is completllCnlary
10 cxperimenwl
work al ncw experimental facilities. like radioactive heams. In such nuclci.
lhe oulennost
neulrons amI protons 1110ve with respect to
lhe inne!" shells and the rcsult is a highly deformed shape,
or even super-defunncd
or hypcr-defonned
nuclci can result. l\1orC'over, such deformations
Ilecomc aSYllllllc!ric or
call lead to a "Jleck". This shape is no! COllvex, amI tl](' tradilitmal
paramctrizations
more valid.
In lhe lheory
(sphcrical
harmonics)
are not any-
kal) bUl, fundalllentally
there is the prohlcrn that the radius
hccol1lcs a lllultivalued
fUI1Cliol1 of lhe angle if the nuclear
necK is suniciently cOllslricted,
Tl1e tradilion:ll
tl1eorelical
approaches
use dilTcrcl1t
p;:~ri.lmetril.alions lo descrihe highly dcfonned
shapes, like
l<lngent sph~rcs. llcck coon.iinales, Casinian oval s, derorllled
ellipsoids 111. A convenienl
parallletriz~tion
01" lhe nuclear
shape is a decisive factor ror the succes orthe c~lculatiolls I::n
!-Iowcvcr, wi¡hout exception. lhe known paramctrizalions
are
arlificialy intr(lduced hy pure gcol11elric~1 tllcthods, wilhoul
physie:lI supporL
In Ihe preselll work we int¡'()ducc a moJel
01" fission,
the nuclear
potellti~.¡J energy
is
01"high interesl. The dependen ce of the potclltial ellergy 011
lhe ~hape 01' Ihe nuclear surface allows conclusiol1s ahoullhc
dynalllics 01"Ihe Iission process and informalions
concerning
Ihe harricr. These calculations
can furthcr provide cs!iml.ltes
(JI slahi lil)' against spontaneous
nssion and prediclions 01'oplimal lüsioll pallls rOl' the synlhesis of Ihe superhcavy nudci.
The simplest way lo 1I1ldersland nuclear fissiO:l is P1"O\'ided by Ihe LD!vl togclhcr with shell C('ITeClions. Ilowc\'cr.
l"or such a proccss illvolving Iarge dcrorlllations
olle has lo go
bcyolld tlle harlllonic approxilllations,
Thc expansioll in Illultipoks C;Jllllo1 descrihe nSSiOll throught the sepa ratio n inlO
dauglller Iluclci, IlO maller lhe llulllherofhanllollics
involvcd.
]¡.s not onl)" lhe convcrgcllce
problclll (lhcorclical or 1l11!llC:r-
which can de-
scribe higlJly dcforllled nuclear shapes 011 lhe way to llssion in a dynamic appro~ch. Starting from a Ilonlinear IlUcle;lr hydrodynalllic
Halllillonian
wilh clfcctivc
Skyrme
con-
l:Jel á-inleractions,
plus shcll corrections,
ami hy including
nOllline:lr terms, \Ve obtain a Ilonlincar Schrüdingcr
cquatioll
(NLS) 01"ordcr ,hree Ihal descrihes the lluclear density and
surrace. This equ<llion reduces to a lllodifled KdV (MKtlV)
equalioll having solil011 and antisoli(OIl solulions. The solilon
solUliolls are jusI lhc ro(ons, inlroliuced in ReL 3. The anlisolilon soluliollS, coup!ed in sYI1l111elric pairs. produce all
lhe lltlC!c¡!I' shapcs oblained in olhcr lissil>n IlHH.lcls. Loosely
speaking, such a deforllwlioll is a pair or s)'lllllletric holes in
(he Iluclear surbce wliieh lravel wilh con>;\ant velocily. From
{he (jua:lllllll Illcc!wnic¡¡1 point 01' vic\\', thesc rotating holes
NON-LINEAR
PIIENOr-.1ENA IN NUCLEI: TIIE ANTISOLlTON
'.cut" a chane I of lowel" prohability 01' localizations for nllC\cOIlS. Thc corrcsponLling shape is equivalent to a scparalion
into Iv,'O fragmcnts. ~loreover, hy using quanlUlll mecahnical
I"raglllentalion thcory or a \\lKB approximation for the corre~p()lldillg [1otential vallcys, olle can relale the paralllctcrs of
Ihl:se anlisolitOlls (height, widlh and velaeity) wilb the tlUckar pOIl:ntial. This anlisolilOIl syslelll reprcscnts a nc\\' colkcti\'e cxcitalioll 01"the nucleus towards fission, with a dccp
(JHlllliIH.'ar) physical background, which answers, nol only Ihe
qucslioll "how'!"', huI also the qllcstion "wby'!".
where /' = A,A,j( ..\¡ + A,) is Ihe reduced mass. The resl
01' (he cnefficients in the above cquations are fixed nutlleri<.:al
values, providcd for cxalllplc in Rcf. 4.
The Ilcxt slcp is lhc quanlization ol" lhe c1assical kinetic
cnergy dcscribed by (hc (chargc ami l1Iass) asymlllctry coordinales, '1" = (Z, - Z,)/Z and '/,1 = (A¡ - A,)/A, respectively. By adopting Ihe Pauli prescription [11. a Laplacc
operalor in lhe curvilinear coordinalcsl/z.
'/.-1 can be writlen.
similar lo the Bohr Hamillonian for the {-J - ,. vihralions
/I:!
T=2I31/;,:,/;,:
2. l\\mlels till" nuclear Iission
One 01' lhe l1lost lIscJ lllethoJs in Illodeling lluclear fission
is qU<lnllllll Illechanical I"ragmcnlalion theory where eollec1ive coordinates 01" charge ami mass aSYllllllclry are intro~
duced 11J. In Ihis mode! Ihe polenliai cnergy 01"Ihe syslelll is
delincd as the sllm 01"Ihe liquid drop energies. shell elTeCls,
lile proximily nuclear polential, lhe Coulomh interaclion and
lhe rolalional ellergy due lo the angular 1ll0menl1l1ll
\' =
(,1,. Z,j
¿[I'j.",¡
+ 6U;) + VI' + 1,: + \',
(1)
i=J
I"or tlle LDM descrihes
ulle has
in [;j], wilh empirical
.,
[
\ . = - L0.
Ol,.,ti -
'/3
(/.~Ai
1=1
\l,.'ilh cmpírical
l' (.\'-
'1>(<)=
zf
- (/c--:;-/
4.•
values nl
¡
=
i
-
shell elTects,
(A, - 2Z;)']
(/11-----
•.li
3
al'!'; - (/sl\'A-J/:l,~,.
=
"Yo[I -
z)' (1'1 and
- 4 (( -
2.::;1)' - O.OS52(( - 2::;4)'
{ -3A37cx1'U
Thc shell correclion
.2:) 11
(2:1.2::;11.
(/0 (5)
is describcd
E.:S;
hy [5)
- ~(Y"j:l
5 "
- U"j:l)
. ;-"
(1)
-
whcre .-\1;_1 is lhe lowesl magil: Illlmher closesl lo X (=2¡ or
Ni l. Since we are illterestcd in spolltaneous f1ssion alld !lence
in low spin slatcs, \ve call approximate the 1ll01llenl of illenia
wilh
J(lI,. I!,)
=
~(¡¡,
(4 )
81
r-.lOl)EL FOR FISSI()N
¿):!
t,'2
----¿J'I; 2D11.\lI.\
D"2
DI/~
tl.'2
-----'2.[JIIZIJ..j
¡y
(5)
D'JzD,/z
'
\\'here Ihc lll<lSSpar:Il11CICrS[J can he cakulatcd in the cranking model or in the hydrodynamic
moJc!. From Eq. (2) une
can calclllale Ihe pOlcntial encrgy surface againsl Ihe parallletcrs 01' lile mode!. As lhe nuclcus proceeJs lowards f1SSiOIl,
there are a numher nI' valleys, indicating lhe prcferreJ hinary
Illass spilt-ups: Ihese are usually characterized by lhe forl11a(ion 01"one di1ughter IlUClcllS close to magic nUl11bers. The
main characlerislic in all Ihese represcntatiolls is that Ihe valleys llave an allllost constant prolllc as Ihe separation coordinale incre;¡ses closel' lo scission. This indicales Ihal the lllass
dislrihlllion 01"tlle fragmcllts has hecll dctermined early in the
lission process.
The mass di~trihLlliol1s expectcd are calculared hy solving
the colleclive Schri)cdinger cquation in Ihe aS)'llllllelry dcgrces of frceLlolll, for lile Hamiltonian givcn in Eq. (5). This
model can reproducc lhe experimenlal data resonably wcll, ir
lile aSYllllTlclry is ]Jol cOllpkd wilh the rclativc Illotion.
A similar approach is givcll in lhe Two-Cenlcr Sllell
!\10del lil. This model dilTers from Ihe previolls hy Ihe facl
that lile gcomctric scparatioll is inlrodllced by an anharmonic
(quadralic) ()seillalor potential. Also. the neck paralllclcr is
detcrmined unly indircctly hy inlcrpolatiol1 01' lile cquipo1ential surl"aces will1 a l1armonic oscillator.
3. The antisoliton model
Thc lIlicroseopic Iheories have shoWIl lllat the harmonic approximatiolls llave only a very rollgh valiJily in Ihe limil 01'
very small vibralions. For collcetivc motions wilh larger amplitudes olle has lo lake ¡nlo account cilher dilTerenl anharmonie lerms or a cohercnt comhinalion 01' Ihese, like in nOI1linear syslems.
\Ve llave recently inlrodllced a llonlincar mollel [3. G}, that
is a 1H:\Vtype ofsmfacc nuc!carexcilalion,
calleu a "roten". in
urder to cxplain higher deformaliolls 01' nuclci lhrough emission 01' cluslers. ){OtOIlS are localizcJ waves Ihal propaga le
",ith litlle change in fmm on the surface 01' droplcts, shells.
or huhhks. They hchavc likc solilons, hUI arise from normal
Illodes 01"spllcroids tllat ohey llonlinear dynJl11ics. Our model
descrihes a I1C\Vlarge amplitude coltectivc motion in Iluclei
descrihcd by cqllalions 01' lile KdV type and their Iraveling
\Vave solll(ions. This mollel yields a unirying dynamical pic~
tme 01' these Illodes: solut iOllS si llIulate harlllonie oscilbtions
{hat are dri\'ell ill10 <lnharlllOllic ones by nonlincar Icrllls in
/<('1'. ¡\f('x. P,\-. 45 S2 (19QQ) SO-X)
H2
J.P. IJRAAYER,
lhe Hamiltonian.
and ultimately
cnoidal
wave fonns
ALUIJU.
al' lhe
latlcr dcvclop inlO rolaling solitary waves.
Thc rotan model hrings ane Bew thing inlO lhe physics
01' Ihe liquid drop (hesides
the nove1ty of ohtaining
the KdV
cqualioll 011 <1sphcrc, without gravity). 1I generales a largc
se! 01"highly dcforrncd shapes in a dynamical
way. Tllal is
il explains
lhe Il1cchanislll
for lhe formation
providcs analylical solutions.
Illcntally when lhe amplitudc
01' palterns
and
Rotoos werc ohscrvcd cxpcri01' lhe shapc oscillations
al' a
lImplel bccD.mc suostantial.
Howcvcr, lile rolan model cannot descrihe or cxplain lhe
forrnation
01'
"l1ccks"
Of
con cave sh<lpcS 011 Ihe surfacc.
Thc
solitons are always convexo Howcvcr, nonlinear cquations
also llave i1nlisoliton solulions, lhat is solitons with negative
Hmplilllde.
There
are many
oppourtunities
!luid dynamics COllccpts
car LDM approach was
introduce in lhis paper
with ellcctive an Skynnc
l]lIantization rOfmalism,
rclalivistic Hamiltonian
lo introduce
AND U. STOITCIIEVA
3
_
111 '"
l!--:-L..
2 1.:=1
.
~
.
l."
h.¡¡-.l,d,,,+-
/
.
/1
811I
\' {J l"+ '1""'-)-d,1'
/
.
J' (I(.¡;)U(.I'-y)(I(y)(I"xd. y,
'l
l
¡
=
j¡ (,1')
=
u
-21 [ (1(,1'), -O, '¡'(.>:)
.r1.:
which reduces lhe cOllllllulalion
lhe Heisenherg rcpresenlalion
oc
=
[¡!(I'),j,(.,,)]
,11' /' vW+(.1')vw(x) dI,,,
211l
[¡!{.I') , 1'(.'1)]
to Ihe canonical
'¡I+(.I')W(,I')U(x
- y)w+(y)w(y)d".1;d"y,
" u
llelds
W+(:r), w(x)
are canonically
senting Ihe dCllsity and nllcleon
=
(1(.1')
currenl of the systern
,,11 . [W+(.I')~W(x)
~IJII
D.fA-
- ~w+(:r)w(:r)],
D.rl.:
(7)
are
1
~
= 1[(1(.1'), JI] = - L..
I /
O,Ú(.")
u¡
(1 1)
[J(:I' - y)p(.>:)j,
IJ,
ones
A-=l
= ~['
.(,1')
ir,.I',
= ---
O
--
L D.r
,,=1 n
2
- -(1(,1')-0
1//
where the kinetic encrgy
[
-¡¡-:-Jd1'),
[1'(,1'),(1(.'1)]
= O,
[<¡>(r), <1>(.'1)) = IJ,
hencc providing
a complele
colleclivc
hydrodynarnical
de-
scriplion of this nuclear systclll.
[n lhe semiclasskal
limit [ID\. lhe operutors describing
lhc qllantum lIelds orthe lhcory (in our case lhe operalors p.
ami (I'), and obcying Heisenherg tield cquations plus commut<.llion rclations. are transformed
¡nto c1assical ficlds dcscribing the states 01' Ihe syslelll. This is jusi Ihe application of lhe
COlTcspondcnce
Principie in the qllanllllll f¡eld theory. The
hasic idea is ro tirsl idenlil"y an isolatcd set 01"sta les 01" lhe
system. These slates are usually Ilon-perturhative
solulions
Illonopoles.
instantons. The nexl step is to rcplace the operalms in thc fie]ll operatorial equations with Iheir average
value 0/1 lhcsc spccial slalcs. This procedurc replaces opcr-
.¡;I.:
alors wilh fUllclions in lhe dynamical eqllalions. Finaly, one
uses the sCllliclassical expansion \vhich relates the quantulll
h::vcls with the classical orhits.
Tul.:
O!
iI, J(;I' - y),
01"the associaled c1assical Ilonlincnr fleld cquations.
ExampIes are vaClllll slales, huhhle states or solilons, antisolilons,
,
JI]
r,' '""
21/1.'2
O
=
J
w+(:I')w(x).
The operators /1, j fulflll commutation
relations in lhe Heisenbcrg reprcsenlation.
The corresponding
Hamilton eqllalions
----¡jI
,
III
conju-
gated ami flllfil equa] timc anticommutation
felations. Hefe;l'
is lhe spacc-limc 4-vector. Wc will describe lhe dynalllics of
lhe nuclells in a restricted space 01' collective variables repre-
U¡!{.I')
=
O.r 1.:
+
of lhe fields p, JI.: in
relalions
-i--
]
(6)
[1'('''), '1>(;1')]
where the nuclcon
=
(10)
.
+./
.1,.(.1')
.
As in the roton molle!. we restricl ourselves to irrotational
motion and introduce a \'elocily pOlential operalor III(:r) 01"
the form [!} J,
'/11
JI
.
.
which is similar lo lhe hydrodynamic
Hamiltonian
in Rcf. 3.
\Vilhin lhis equivalence,
(hc equation 01"mol ion Eqs. (8) are
fonnally jusI Ihe conlinuily ami ElIlcr opcrator equalions ror
a quanlulIl Illlid.
nonlinear
in nuclear physics. Sincc the nonlinby and Iarge presentcd in [3, GJ, we
a quantum hydrodynamic
approach
intcfaction 18J. In the usual seconda nucleus can
describcd by a nonwith a local lwo-body potenlial
in Eq. (6) becomcs
Finally, lhe Hamiltonian
U(.r - y)(I(y) dly,
XI.: .
tensor operalor
is
(8)
Consequcnlly,
Eqs. (8) can he reduccd, in the case of irrotalionalllO\\'
Eq. (11), lo a nonlincar f1cld equalion for the
corresponding
classical lields. 1'(.1:), (fl(x). In this serniclassical Iimil. hy using Ihe following sllhstilulion
for lhe local
densily and the vc]ocity potential
1'(.1', t)
=
1,,(.1',1)12,
"
<1>(,1',1) = -argH,r,I)],
/11
Rc!'. Mex. PÚ. "¡S S2 (1 t)t)t» ~0--X5
( 12)
I':ON-LlNEAR I'IIENOMENA
\Ve can reduce Eqs. noS) (Ihrough
lo i.INonlinear
Schrüedinger
VII
=
il,DI
Ihe irrolaliollalily
equation
,,'2
--o -6.11
2m
83
IN NUCLEI: TitE ANTlSOl.lTON MOIJEL 1"01{rlSSION
i
condition)
(NLS)
_"
(lO )
+ UlI"l-jll.
D [/}]
Here \ve have introLluced
as a non-linear clTcctivc pOlen(ial Iike in (he Hartrec-Pock
approach [1). \Vilh Ihis choice
D[p] is Ihe uniquc tCfln rcsponsible for Ihe Ilonlinearilies
in lhe dYllamical cquation. The cOl1ll1lutalioll rela(ions he.
l\Veen the quantulll
licIos "ecome a linear rcprcsenwlion
of the Heisenherg
H( 1) Lie algehra gcneraled hy Ihe 11elds
1'(.,.). '1'(.,.).
In urde!" lo reduce Eqs. ( 12) and ( 13) lo a sol \'ahle nuc kar
Illodel
\Ve
choose
a J-fullclion
f-[(jURE 1, Two antisolilon solUlion on a sphcrical surface.
intcraction,
U (.,. - y) = -k / J(.,. - .'1),¡3.,. ,¡'y -; (¡.
( I~)
whcre fmm no\\.' on \Ve denole by _f o11ly Ihe space component
of (he -1--\'CCIOI".
Consequently.
\••...
e oblain the NLS cquatioll
DIJ'
¡/'-.-
DI
101'the d:'namics
=
r/"l
( 15)
211I
Herc \Ve dclined
\}"J(,r.
t)
=
1/'2(.1". I)/p
where P.v is the nuclear matter density. If \Ve
add l1igher ordcr lerms in dcnsily in Ihe á-inleractioll
wc
will ha\'(' higher onler Ilonlinearities
in the NLS, respectivcly.
Similar cqualions
have bcen succcssflll1y used lo describe
olher nonlincar systcllls (plasma, solids. cte.).
The mos! generallocalizcd
stahle solutioll nf Eq. (15)
u(.r. f)
=
('osh
ole
Po ,('eh (21'0.1' - SPJk"t
Tile antisolitons
iS;1Il ;lI1lisolitol1 wilh negativc amplitudc -(/ :S (J ;IIlJ vclocily
11. Also. lJ. \ are free paramc!ers
in the sollltion. Thc '//(:/:.1)
sollltion is relatcd \vilh lhe Iluclear dcnsily in ¡his model, by
Eqs. (X). (10), ami (12). The support (lf lhe dCllsily 1) (01"
(lf the funclion
/1) along a givell direCliolJ gives lhe size 01"
Ihe nuclclls \'(:r,\'II.'> thal coordinale.
\Ve call plot this density
agains{ olle space coordinatc
amI find out the prolilc 01' Ihe
nuclear malle!" \'cr,'Ill,\' that coordinale.
By choosing difcrcll(
plOlS, along úiffcrcnl directions. \Ve havc ti picture 01' the nU
have spccial
( 19)
lhe amplitude
=
shape-motion
of Ihe
Jo
is a free
1/2Po and
dependenee.
The larger lhe amplilude the narrower the width and lhe largc
Ihe veloeily. This rclalion can he uscd to cxperimenlally
distinguish them from other linear modes. Sincc this solution
is \'ery wcll localized, any linear combination
01' soliton or
antisoliton, shifted wilh a distance larger than 2£, is still an
nate l." -;
(16)
describing
+ Jo).
=
solulion.
Equalioll
il",'f /S'" + ir",'f /2", + i\)
(1(,1' lJ + f¡¡tl /2/1/)
solution
soliloll/anlisolilon
(l~J > <JI"< O, rcspcclivcly)
and
phase faclor, The solulion has a half-width
£
travels wilh Ihe velocity 11
elPo.
approxilllatc
flfl
-1''''-,¡¡:;;;
<"I'[il,.,./2 x ---------------.
=
1'( .•.. f)
soliton/anlisolilOn
\l,'herc J~) is :1 free paramelcr
"
--6.~'- ki~'I-'~.
nf the systcm.
with a traveling
idenlieal
(15) can he projectcd
on olle sphcrical
coordi-
1')
¡:l]. !I solution 01' Eq. (15) descrihcd hy lwo
;IIHisolilollS shifted with 1r is prescnted in Fig. l.
Frolll the c1assical point of view, this solution represents
lwo idenlical holes in (he surfacc (aclllally two gaps in densily) placed in oppnsilioll and moving wilh lhc SUIne angular \'elocity. FWIlI the quanllllll mechanics point of view, Ihe
prohahi Iity nI' localizalion of thcse antisolitons
is equally dis.
lributed on their whole path, which is a circle. Hence. lhe
average quantulll ellecl of this solulion is a narrowing of Ihe
surbce precisely lInder lheir path. This is equivalenl with the
occurencc
of a lIeek. Slatus:
RO
4
clear surface.
\Vc can oblain
lile sallle soliloll/a!llis(llilcll
Eq. {15) hy inlrodueing
'!'( .•.. f)
solution
frolll
a functionallransform
=
I'(,".I),.,S("'I.
( 17)
where P ami S are real functions, In addition, sincc lhe pa!'lieles should he free al illfinily (we take into ¡Kcou¡ll only
loca1i¡;ed solUliolls) \Ve can considcr S lo be a phase fae.
tor, S(.r. f)
in Eq. (15)
=
\Ve
1...1: - wl + So. Introducing this suhSlitUllOll
obl<lin ti l\1KdV equalioll for
r
01'
.,0[,
0"1'
-O - GP-- - "-O .¡
i
.1'
.Z"
O
=
(J.
(1 X)
DL'pcnding Oll Ihe tlntisoliLon parameter
Po and on lhe
shift 6:.<!J hct\'.'eell the Iw() anlisolitons (ir it is not quite tr) we
can describe dilTerenlliuclcar
shapcs. \Ve can relate lhe nonlincar paramcters 01' Ihe moJel \Vi:ll the typicnl pararnclcrs
requirctl U describe lhe lission process in Iraditional models.
The elongai!on coordinate, which describes the length of
Ihe l1lajor semi-axis at Ihe hegining 01' the tission, and approaches lhe dislance be(\Veell lhe separated fragments
h~ rclaled lo the half.width 01' the nntisolitoll pair, L.
can
The n~ck coordinale. \Vhich describes the thickness of Ihe
neck bct\Veen lhe fragmcnlscJn
be rciared with the amplilude
[lu 01"r!le solitoll or <Inlisoliton pairs and al so wilh the \'clocity
l.' which controls
lhe probahility oí' t1elocalization.
/«('1'. Ate.\'. "'1.\. 45 S2 (19')9) XO-H5
J.P, DRAA"ER.
A.LUDU. AND G. STOITCflEVA
o
'[
8
0.6
<
0.4
0.2
0.1
0,2
O,)
0,4
0.5
0.6
0,7
d
J. SChCm:llic
FIGURE
harrier
d :::: (l/no
lúrlll(ltion
FUjURI'.
l.
Cross-scctions
of Ihe !wo-anlisoliloll
solulioll
(lll a
spherc. Wilh conlinuolls
line are Jrawn Ihe twn-anlisotitolls
shifled
Wilh ¡r. \Vilh tbshed tille thcrc are (wn anlisolil()[]S \••.
'ith ti ditlerent
\hifl. in (lrdcr to simulate
Lhemy I:tnguage,
Ihe rnass assymetry.
the h :lxís represenb
cnordinalc
and
Last row shows diffcrellt
snliloll s(llulions.
Thc I"ragmcntation coordina!e. which Illeasures Ihe dcvi:Ilion rrom sYlllllletry in Ihe mas." distribution is also rclalcd
to ti cOlllhination 01' the all1plitutlc and half-width.
In Fig. :2 we present so me shapes in a suitablc
paramctril.alion.
AlI llgures in the frame are cross-seclions
01";¡ Iwo antisolilon solution 011a sphere. Iike in [oigo l. The
cOlltillllOUS line represcnts t\VO:lnlisolitons that are shifted by
ir amI Ihe L1ashed line t\VO antisolitons
\vilh a dillerent shin
(hut same amplilude amI half-\',.:idth) ro simulale Ihe mass assVlllctrv. Inlhe fra~Il1entation Iheory language, the b axis rcpr~scllts.lhe elongalion coordinalc am!lhe (l axis rcrresents lhe
net:k. coordinate, \Ve llave round a pretty good til hetwccn all
IhL' tradilionalllllclear
shapes anl! the two anlisolilons 11 J. In
IhL' lower row we present, rO!" comparison, solitoll solutions
lrolOlls) \vith positive arnplitude.
A proper <]llallli/.alion 01' lhe anlisolilOns amI ohtaining 01"
lhc cxac( \Vavcfllllctions 01"Ihe rcslll(ing Schrikdinger e<]lIalion are the ncxt slep lowards experimental comparisoll. Once
IIlL' ~urrace is givcll by Ihe anlisoliton solulion (in olle O!"
lll;!ny pairsl. \VCC.Hlt:alclllate the total nuclear cllergy by llSing El]. (2) I"or aH Ihe shell elTect corrcctions. \Vith pinned
illilial ami Hnal sta(es (the lission t:hallllel 01":1 givcn parent
Illlckus) in lhe -"pace of lhe parame(ers (/)0. L, F) \Ve can
calt:lIlatc lhe <]u:lntum penetrahility by using lhe Gamov formula
s = "X!,
[-
f,(~',::,',"'' 'V¡N[
drop
fission
(lhe nccl\ in Ihe traditional
undcforllled
riel" inlhe
ll11clells. Thc conlinuous
tr;HJitinnallllcorY.
case)
amI Ihe radius
linc descrihes
The dashcd
nI" rhe
Ihe lission
line descrihcs
har-
Ihe haricr in
lhe alllisolitolllllOlkl.
In Ihe fraglllenLalioll
lhe elongalion
Ihe (/ axis rcprCSCnlS Ihe !lec!,:. coonJinalC.
01' Ihe liquid
rerrcsclltatinJl
the defonnation
in arhilrary llnits. The parameter
rcpresClllS the fi]tio hclwccn Ihe arnplitude 01' thc de-
\'CI"SIIS
Po( T), L( T). I( T)I liT].
(20)
which \vill providc lhe Iil"etime rOl' spontancous fission. The
pOlential valky along lhe palh, in lile two antisolit{lll molle!.
.'\hould llave an additiol1alminulllim
",ilb respect lo the same
path in lhe LDP. Fig. 3. The t¡ssion probability is given by the
ralio 01"Ihe 1\\'0 \vave ampliludes in the corresponding \vells
cvaluated \l,'ith the help of Ihe harricr penctrability hctween
these two minima.
4, Conc!nsions
The antisoliton llloLlel introdllced in this papel'. brings correclion to lhe fission t:alculatioll wilhin the LDM. In general,
by lhe inlroc!lIclion 01' arbitrary paralllelers to describe Ihe
gcomelry of lhe fissioll, one can obtain a good lil \Vith the
experimcllL HowcveL thcse parameters are on1y abslract geolllelrical cntilies artilicially introduccd ami wilhout a physical background. Thc roton ami antisoliton moJel rreJicl the
same shapes. slarting I"rolll a physical Hamillonian 01' the
LD.i\-l or nuclear hydrodynamics \vith cllective Skynne contact ¡)-inleractions, It has beell shown that, in a certain 01'del' 01"approximalion, lhese Hallliltonian reduces lo Ihe NLS
and MKdV (01' KLlV) equations respeclively, allowing the existence of rolon sollltions (solitons 011 the nuclear surrat:c)
and anlisolitons. Ncxl. by quantil.ing the stalionary motion
01' an anlisolilon pair. olle can "cut" a <]uantum channel in
Ihe initial Illlcleus. hencc produt:ing a suplimentary valley in
the potential energy protilc. experiemelllal resulls. The anlisoliton modeL which is complementary
lo the roto n moJel.
can be also used in various nonlincar dynamical rhenomena
in hC,lvy-ioll collisions besides othcr semiclassical methods
(TDI-fF. Ihe dynalllical TIHllllas-Fermi lheory, ctc.)
Acknowledgmcnts
Supporlcd by Ihe U,S, National St:iellcc Foundation through
a regular grant. No, tJ603006. and a Coorerative Agrcclllcnt,
No. EPS-95501R l. lll<ltincludes malching fmm the Louisiana
Board 01"Regcnls Suppon Fund .
R('I'. MeT. Fís. -15S2 (199l)
~O-X.5
N()N~L1NEAR PIIENOMENA
l. P. Rin~ ami P. SdlU(k. "'h(' Nuclmr
(Srrinf~w-Vt'r1;¡g. Ncw York 19RO).
./ :\. Bohr ami lA. \\'hedcr./JhYJ.
X5
IN NUCLEI: TtIE ANTIS<H.IT<)i\J M<)1)EL F<lR FISS10N
Afafly-lJody
MrlllOtI.\ i/ll'hy.\Ín',
G22. Olohnrt. lO-IR July. 19(8).
in print; A. Llll!u ami lP. [)raaycr.l'hy.~ictl [) 123 (1998) 82.
(lrl'liwl
Prohlem.
i. l'••.1.G. ~l11stara. U. \loscl.
Rel'. 56 (1939) 426.
~lI1tlH.\V. Schmidt. PhYJ. Rel'.
e7
(1 tJ7J) 42<).
:1 A Llldu :Ind lP. Draaycr. I'hn.
R('l~LA.'ll.
10 (199R) 2125.
8. T.II. K. Skyrmc. NI/el.
R. K. GlIpIa. S. Kumar. W. Schcid. and W. Grcincr. 1. I'hys. G:
NI/el. Hm, j'hn. 2-t (1998) 2119.
.J
\V.D. l\lYl'rs and W.J. Swiatccki. Nucl. Phys Xl (1966) 1.
(j
¡\, Ludll and lP. Draaycr. ProcceJings
!tlt. COf!f 011 Gral/JI The-
HCI'.
IJhy.~. t) ( 1959) 615.
9. VG, Kartavcnko. A. LlIdu. A. Sandulcscu. nnd W. Grciner,
./. Moti. I'hl".\. f 5 (1t)t)6) ~29.
lO. R. Rajaraman. So/lto"s (/Iuf
terdam. 1t)X7) chartcr V.
l'vfe.r. Fi.\'. 45 S2
(1l)t)(»
XO---XS
ImttUltOfls.
(North-Holland.
1111.
Ams-