Search for hyperheavy toroidal nuclear structures formed in heavy ion collisions
Anna Sochocka and Roman Płaneta,
M. Smoluchowski Institute of Physics, Jagellonian University, Cracow, Poland
1
This idea was introduced by Wilson (1946) and Wheeler ( 1950 ). Theyproposed nuclei with new exotic types of topology and investigated the stability of toroidal nuclei
Motivations:Motivations:
Stability of systems with exotic shapesStability of systems with exotic shapes::
Siemens and Bethe showed that some bubble nuclei with sufficiently large charge can be stable against breathing deformation ( monopole oscilations) P.J. P.J. Siemens andSiemens and H. H. BetheBethe, , PhysPhys. . RevRev. . LettLett 18 18 (1967(1967) ) 704704--706 706
Wong showed that as nuclear temperature increases, the surface tension coefficient decreases and the Coulomb repulsion is pushing nuclear matter outward leading to the formation of exotic nuclei
C.Y. C.Y. WongWong, , PhysPhys. . RevRev. . LettLett 55 ( 1985 ) 55 ( 1985 ) 19731973--1975 1975
For heaviest nuclear systems (A > 300) a pocket of the potential energy even at zero angular momentum appears for toroidal shape
C. Fauchard, G. Royer, Nucl. Phys. A 598 ( 1996 ) 125 - 138
The theoretical analysis of properties of super-heavy nuclei do not predict any long living nuclei with compact shapes beyond the island of stability (N ~ 184, Z ~ 114).
Liquid drop model with shell corrections and Hartree – Fock – Bogoliubov theory with the Gogny D1S force calculations have shown that metastable islands of nuclear bubbles can exist for nucleon numbers in the range A=450-3000K. Dietrich, K.Pomorski Phys. Rev. Lett. 80, 37 (1998)
J. Decharge et al. Nucl. Phys.A 716, 55 (2003 )
Predictions of the HFB model with the Gogny D1S force
bubbles
semi - bubbels
ordinary nuclei
typical density profiles corresponding to the aboveconfigurations
The lightest semi bubbels are foreseen around mass A300, while the true bubble appear at A400, the lighter nuclei prefer ordinary solution
J. Decharge et all. Phys. Lett. B (1999) 275 - 282
Q 2 - quadrupole moment
RMSR – root mean square radius
d – tube radius
Torus is another topology which is investigated
M.Warda, Int. J. Mod. Phys. E 16, 2 (452-458), 2006
Minimum potential energy for the toroidal shape
RMSR
d
Prediction for the toroidal shapes
The energy of the toroidal minimum decrease relatively to the potential energy of the spherical configuration with increase of the mass of the system For Z>140 , the global minimum of potential energy corresponds to the toroidal shape
M. Warda, poster on XIII Nuclear Physics Workshop in Kazimierz 2006
Dynamical model predictionsDynamical model predictions::
BUU transport calculations showed that exotic nuclear shapes
may be created in central heavy ion collisions at intermediate energies L. G. Moretto et al.L. G. Moretto et al.,, Phys. Rev. Lett. 78 ( 1997 Phys. Rev. Lett. 78 ( 1997 824824 - -827827))
Lien-Ven Chen et all. Phys. Rev. C 68 (2003 ) 014605
BUU calculations
zbeam
direction
x
y
x
Boltzmann – Uehling – Uhlenbeck model
The BUU transport equation for the nucleonic one-body density distribution function f = is given by: tprf ,,
212133
221212
1223
23
3
21111
2
1,,
ppppffffffff
d
ddpdpdtprfU
t prr
d /d - nucleon-nucleon cross section
v12 - relative velocity for the colliding nucleons,
U - mean-field potential consisting of the Coulomb potential and a nuclear potential with isoscalar and symmetry terms.
The potential field is approximated by
,1, ,00
asypnczz VVBAU
0 - normal nuclear matter density,
, n , p - nucleon, neutron, and proton densities,
z - equals 1 or -1 for neutrons or protons, respectively.
= (n - p) /( n+ p) – asymmetry parameter
EOS A[MeV] B[MeV] K[MeV]
STIFF -124.69 74.24 2 380
SOFT -356 306.1 7/6 200
Simulation results for central collisions of Au+Au
BUU calculations
y
y
xx
zbeamdirection
E=15MeV/nucleon
E=23MeV/nucleon
E=40MeV/nucleon
K=200MeV
y
z
x
Simulation results for central collisions of Au+Au
Ksym =-69MeV – blue line
Ksym =61 MeV – red line
K=200 MeV K=380 MeV
flat bubble
toroid
toroid
flat sphere
disc
toroid
Central density ( x=0, y=0, z=0 )
E=15 MeV/A
E=23MeV/A
E=40MeV/A
BUU calculations
y
x’
Simulation results for non-central Au+Au at 23 MeV/A
z
xy
x
y
zx’
z’
z’
beamdirection
K=200 MeV
x’
b=1.25 fm
b=3 fm
b=8 fm
Time = 200 fm/c
Results for central collisions of 124Sn+124Sn
y
x
z
x
E=35MeV/nucleon
E=50MeV/nucleon
E=25MeV/nucleon
K=200MeV
BUU calculations
Decay characteristics for non compact nuclear objects (dynamical model predictions)
more of intermediate mass fragments ( Z > 3 ) should be generated than would be expected for the decay of a compact object at the same temperature
enhanced similarity in the charges of fragments
ETNA – Expecting Toroidal Nuclear Agglomeration
Flow diagram
Drawing of fragments:
•Gaussian distribution
Established : Zi , Ai ; i = 1,N ( N=5 )
All the fragments are placed in ball, bubble and toroidal configuration with additional condition: Rij > Ri + Rj + 2fm
ACN = AT + AP
ZCN = ZT + ZP
-minus preequilibrium nucleons Partition of the available energy:
Eava = ECM + Q –ECOULOMB
Acceleration in mutual Coulomb field
Detection of particles in the CHIMERA
detector
, detector number rand ,rand
Non - central collisions are taken into acount up to give impact parameter b
Ethr=1 MeV/A
Definition of sphericity and coplanarity
From the Cartesian components of fragment (Z 5) momenta in the centre of mass one may construct the tensor
n
n
n n
nj
ni
jip
p
pp
F
,
where p(n) i is the i-th Cartesian momentum component of the n-th
particle, and is the n-th fragment momentum vector. For eigenvalues t1 < t2 < t3 of the tensor F one dehines the reduced quantities:
j j
ii t
tq
2
2
Then sphericity and coplanarity parameters are defined as:
312
3qS 122
3qqC
Conclusions
Microscopic models of the nuclear system predict that for Z>130 the exotic shapes ( bubbles, toroids ) corresponds to the stable configuration of very heavy nuclear matter
The threshold energy for toroidal shapes formation decrease with increasing mass of the system ( BUU predictions )
This threshold energy depends on the stiffness of the nuclearequations of the state ( BUU predictions )
Preliminary predictions of ETNA code indicate that at 23 MeV/Athe proposed signitures able to distinguish between different freeze-out configurations
Comparison with other dynamical models in progress
Conclusions
Przewidywania modeli mikroskopowych wskazuja na egzotyczne ksztalty dla systemow o duzych masach bedacych w rownowadze
Energia progowa na formowanie sie toroidalnych ksztaltow maleje wraz z rosnaca masa zderzajacych sie jader
Dla rownania stanu ksztalty toroidalne tworza sie przy wyzszych energiach w porownaniu dla przewidywan dlamiekkiego rownania stanu
Vlasov Boltzman Langevin
Characterizaton of the dynamical models
Vlasov model – paricles experience only the self – consistent effective field, leading to a singledynamical trajectory
Boltzman model – various possible outcomes of the residual collisions are being averaged at each step, leading to a different but still single dynamical trajectory
Langevin model – various stochastic collisions outcomes to develop independently, leading to a continualtrajectory branching, corresponding ensemble of histories
A.Sochockag*, C.Agodia, R.Albaa, F.Amorinia, A.Anzalonea, L.Auditored, V.Barane, I.Berceanue, J.Blicharskaf, J.Brzychczykg, B.Borderieh, R.Bougaulti, M.Brunoj, G.Cardellab, S.Cavallaroa, R.Coniglionea, M.B.Chatterjeek, A.Chbihil, J.Ciborm, M.Colonnaa, M.D’Agostinoj, E.DeFilippob, R. Dayraso, A.DelZoppoa, M.DiToroa, J.Franklandl, E.Galicheth, W. Gawlikowiczg, E.Geracij, F.Giustolisia, A.Grzeszczukf, P.Guazzonip, D.Guinetq, P.Hachaju, M.Iacono-Mannoa, S.Kowalskif, E. La Guidaraa, G.Lanzanòb, G.Lanzalonea, C.Maiolinoa, N.LeNeindreh, N.G.Nicolist , Z.Majkag, A.Paganob, M.Papab, M.Petrovicie, E.Piaseckir, S.Pirroneb, R.Płanetag, G.Politib, A.Pope, F.Portoa, M.F.Riveth, E.Rosatos, F.Rizzoa, S.Russop, P.Russottol, D.Santonocitoa, M.Sassip, K.Schmidtf, K.Siwek-Wilczyńskar, I.Skwirar, M.L.Sperdutob, L.Świderskir, A.Trifiròd, M.Trimarchid, G.Vanninij, G.Verdeb, M.Vigilantes, J.P.Wieleczkol, J.Wilczyńskic, L.Zettap, and W.Zipperf
a) INFN, Laboratori Nazionali del Sud and Dipartimento di Fisica e Astronomia, Università di Catania, Italy
b) INFN, Sezione di Catania and Dipartamento di Fisica e Astronomia, Università di Catania, Italy
c) A. Sołtan Institute for Nuclear Studies, Swierk/Warsaw, Poland
d) INFN, Gruppo Collegato di Messina and Dipartamento di Fisica, Università di Messina, Italy
e) Institute for Physics and Nuclear Engineering, Bucharest, Romania
f) Institute of Physics, University of Silesia, Katowice, Poland
g) M. Smoluchowski Institute of Physics, Jagellonian University, Cracow, Poland
h) Institute de Physique Nuclèaire, IN2P3-CNRS, Orsay, France
i) LPC, ENSI Caen and Universitè de Caen, France
j) INFN, Sezione di Bologna and Dipartimento di Fisica, Università di Bologna, Italy
k) Saha Institute of Nuclear Physics, Kolkata, India
l) GANIL, CEA, IN2P3 – CNRS, Caen, France
m) H. Niewodniczanski Institute of Nuclear Physics, Cracow, Poland
o) DAPNIA / SPhN, CEA – Saclay, France
p) INFN, Sezione di Milano and Dipartimento di Fisica, Università di Milano, Italy
q) IPN, IN2P3 – CNRS and Universitè Claude Bernard, Lyon, France
r) Institute for Experimental Physics, Warsaw University, Warsaw, Poland
s) INFN, Sezione Napoli and Dipartamento di Fisica, Università di Napoli, Italy
t) Department of Physics, University of Ioannina, Ioannina, Greece
u) Cracow University of Technology, Cracow, Poland
* Corresponding author, e-mail: [email protected]
CHIMERA - ISOSPIN Collaboration
Outlook
Incorporation of angular momentum into the ETNA code
Additional calculation with BUU code
Introduction of novel signatures of exotic shapes
Test of signatures for systems with different masses:
Au+Au @ 40 MeV/nucleons; INDRA, GSI
U+U @ 24 MeV/nucleons; INDRA, GANIL
Sn + Sn @ 35 MeV/nucleon, CHIMERA, INFN-LNS
Beam direction
Main axis of events
Definition of sphericity and coplanarity
flow
n
n
n n
nj
ni
jip
p
pp
F
,
where p(n) i is the i-th Cartesian momentum component of
the n-th particle, and is the n-th fragment momentum vector.
From the cartesian components of fragment Z 5 momenta in the centre of mass may construct the tensor
Events selection for central collisions
events located in „3” are well measured events :
120 Ztot ( ZP+ZT =156)
0.8 Ptot II /Pproj 1.1
= 93mb
J.D Frankland et al., Nucl. Phys. A 689 (2001),905-939
II
Total reaction cross sectionR = 6500 mb
Definition of TKE
TKE – total mesured c.m kinetic energy of detected charged products
TKE = EC.M + Q - Eneutron - E
Where EC.M , Q, Eneutron, E are the available centre of massenergy, the mass balance of the reaction and total neutron and gamma ray kinetic energies, respectively
The average kinetic energyof the largest fragment is smaller than energy of theother fragments and show maximum for Z30-35
G.Tabacaru Nucl. Phys A 764 ( 2006 ) 371-386
simulationdate
Ftotal
Results
simulationdate
G.Tabacaru Nucl.Phys. A 764 ( 2006 ) 371-386
In the region Z=15-25 the heaviest fragment, Zmax, has always the lowest average kinetic energy
G.Tabacaru Nucl.Phys. A 764 ( 2006 ) 371-386
i ) Zi,j5
ii ) 5 Zi,j 20
iii ) Zi Zmax
Black line – experimental dataRed symbols - dynamical simulation
The one body density evolution calculated in a Boltzmann-Nordheim-Vlasov approach (BNV) up to 40 fm/c (the instant of maximum compression) after Brownian One Body (BOB) dynamics
BOB simulation
35L [hbar]1500
L
2,6 mb
Sharp cut off approximation
Experimatal event selection
Here is the place for other event geometries
02
2209
symsym
eK
2
0
0
0
00 183
symsymsym
KLee
symeee 20,,
pn
Binding energy per nucleon e( , ) as a function of density and isospin asymmetry parameter :
Where:
N - density of neutron
P - density of proton
Experimental observables
)(1 redvR
),(
),(
21
212
ppY
ppYC
b
where:where:
),( 212 ppY
- two particle coincidence yieldtwo particle coincidence yield
),( 21 ppYb
- background yield obtained by event mixingbackground yield obtained by event mixing
jiij vvv
- relative velocityrelative velocity
v red =ji
ij
ZZ
v
- reducereducedd velocity velocity
--
Space distribution of fragments fordisc and torus configurations; ( = 0/3 )
x
y
beam direction
z
Au+Au at 15 MeV/nucleons
General decay characteristics for Au + Au reaction at 15 MeV/nucleons
Granulation of the CHIMERA detector taken into account
Common temperature
Planarity is able to disantangle between ball, disc and toroidal shapes for the heavy Au + Au system and unable for the lighter system
Simulation predictions
Noticeable differences in 1+R function are observed for the heavier system, for the lighter system are less visible
Simulation predictions
)(1 redvR
),(
),(
21
212
ppY
ppYC
b
),( 212 ppY
),( 21 ppYb
jiij vvv
v red = ji
ij
ZZ
v
where:where:
- two particle coincidence yieldtwo particle coincidence yield
-background yield obtained by event mixingbackground yield obtained by event mixing
- - relative velocityrelative velocity
- reducereducedd velocity velocity
Definition of 1+R correlation function
Summary and conclusions:
preliminary simulations with ETNA code were performed
observables discriminating different exotic shapes were found ( 1+R, planarity) for heavy Au + Au system, for lighter Sn + Sn system discrimation is less obvious
it is necessery to performed additional simulations for more realistic mass distribution ( experimental data )
simulations with dynamical models are necessery in order to rushed more light at the dynamics of exotic systems formation
Invariant velocity plots
E = E*
Energia wzbudzenia
Energia ruchu termicznegoE = ETh
Eavailable = E*Eth=0
Eavailable =Eth
E*= 0
Hachaj prescription
BUU predictions for central collisions of Mo + Mo at 75 MeV/nucleon
K = 200 MeV K = 540 MeV
20fm/c
60fm/c
120fm120fm/c/c
180fm/c
Common temperature of thermal motion and fragments excitation
Temperature T
kTEaTETETEE ThN
i
Thiiava 2
3,*;)()(* 2
1
Common temperature of thermal motion and fragments excitation
Hachaj prescription Gaussian distribution
16O + Au _60MeV
We observed energy spectrum for oxygen in reaction 16O + Au at 60MeV
Elastic scattering
Elab_p vs yield abs(theta_p-74o ).lt. 4o
Detector 850; 305m Si; =74o
58Ni + Au _100MeV
We observed energy spectrum for nickel in reaction 58Ni + Au at 100MeV
Detector 850; 305m Si; =74o
Elab_p vs yield abs(theta_p-74o ).lt. 4o
Elastic scattering
16O + Au _100MeV
We observed energy spectrum for oxygen in reaction 16O + Au at 100MeV
Elastic scattering
Detector 850; 305m Si; =74o
Elab_p vs yield abs(theta_p-74o ).lt. 4o
Detector 850; 305m Si; =74o
16O + Au at 60MeV
E160 el. scater. = 52.25MeV
E 160 el. scater. = 88.02MeV
16O + Au at 100MeV
58Ni + Au at 100MeV
E 58Ni el. scater. = 88.02MeV
Calculations
Alpha line
Desilpg 218.31 272.3 Time 2565.1 2599.5
Carbon line
Desilpg 218.31 272.3 Time 2565.1 2599.5
3H punch through
Desilpg 205.62 208.13Time 2605 2612.5
Detector 850; 305m Si; =74o
Au + C at 15MeV/A
Experimentaldata
Simulation results for central collisions Ar + Sc at 80 MeV/nucleon
10fm/c 50fm/c 100fm/c 150fm/c
x
y
zbeam direction
BUU calculations
D.O. Handzy et al. Phys. Rev. C 51, 2237 (1995)
Decay characteristics for non compact nuclear objects ( model predictions)
more of intermediate mass fragments ( Z > 3 ) should be generated than would be expected for the decay of a compact object at the same temperature
enhanced similarity in the charges of fragments
suppressed sphericity in the emission of fragments
Detector 850; 305m Si; =74o
Au + Au at 15MeV/A
Au fission
Desilpg 262.77 637.47Time 2339.3 2479
Alpha punch through
Ealpha = 24.7 Mev
Experimentaldata
BUU equation is solved by test – particle method
Each nucleon is replaced by N test- particles
NA - number of nucleons A nucleusNB - number of nucleons B nucleus
NB * NNA * N
A B=+ ( NA +NB )*N
( r ) = N’/[N(NA + NB )]( r )3
N’ - number of test particles in small volume ( r )3 around the point
Test particles collide with a cross section nn/N
r