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ENERGY STORING IN COMPRESSED NUCLEAR MATTER
E. Hefter, K. Gridnev, S. Saad, V. Semjonov, V. Subbotin
To cite this version:
E. Hefter, K. Gridnev, S. Saad, V. Semjonov, V. Subbotin. ENERGY STORING IN COM- PRESSED NUCLEAR MATTER. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-241-C6-243.
�10.1051/jphyscol:1984628�. �jpa-00224230�
JOURNAL DE PHYSIQUE
Colloque C6, supplément au n06, Tome 45, juin 1984 page C6-241
ENERGY S T O R I N G I N COMPRESSED NUCLEAR MATTER
E.F. ~ e f t e r * , K . A . Gridnev, S. Saad, V.M. Semjonov and V.B. S u b b o t i n
NIIF, Leningrad S t a t e University, Leningrad, U . S . S . R.
and I n s t i t u t für Theoretische Physik, Universitut Hannover, AppeZstrasse 2, 0-3000 Hannover 1, F.R.G.
Résumé
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On r a p e l l e que l e s a n a l o g i e s a v e c l a dynamique des f l u i d e s c o n d u i s e n t à une é q u a t i o n d e S c h r o d i n g e r n o n - l i n é a i r e (NOSE), q u i modèle d e s c o l l i s i o n s é l a s t i q u e s d ' i o n s l o u r d s . On montre que l ' é n e r g i e é l a s t i q u e d e compression s t o c k é e dans l e s r é g i o n s s u p e r f i c i e l l e s d e s i o n s e n c o n t a c t p r o d u i t une f o r c e de r é p u l s i o n . Des r é s u l t a t s numériques o b t e n u s e n r é s o l v a n t l a "NOSE" c o n f o r t e n t c e t t e méthode. On a t t i r e l ' a t t e n t i o n s u r l e f a i t que l a même p h y s i q u e e s t apparemment e n c o r e v a l a b l e à d e s é n e r g i e s r e l a t i v i s t e s .Abstract
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It is recalled that fluid dynamical analogies lead to a nonlinear Schrodinger equation (NOSE) modelling elas- tic collisions of heavy ions. The elastic compressional ener- gy stored in the surface regions of the touching ions is shown to give rise to a repulsive force. Numerical results obtained upon solving the NOSE support the approach. Attention is drawn to the point that the same physics do apparently also work at relativistic energies.The angular distributions of the differential cross-sections of elas- tically scatter@d heavy ions, da/dQ,are "anomalous" in so far as one observes a distinct rise of da/dQfor backward angles (in contrast to the "normal" almost exponential fa11 off). It has been noted that this anomalous large angle scattering (ALAS) may be accounted for by sup- plementing the usual optical model potential by a repulsive term, Vr:
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V~~~ - vom + Vz (ESP-effective surface potential). ( 1 ) The effect of V in liaison with V is to produce a potential pocket in the surface rregion which facil??ates the description of dU/dQ for backwarü angles (see e.g. /1/ and references).
The of the simple potential model /1/ lead on one side to calculations within the microscopic resonating group method (see /2/ and references) and on the other side to an attempt to model such collisions in terms of liquid drops with diffuse surfaces /3,4/. The physical picture in- voked in this semi-classical model is that of two drops compressing each other elastically in their respective surface regions as they ap- proach each other. The elastic energy stored in the compressed nuclear matter gives then rise to a repulsive .spring-type force providing the wanted repulsive potential Vr. Putting this picture into practice and using the fluid dynamical interpretation of the wavefunction / 5 / we ended up with a nonlinear Schrodinger equation (NOSE) / 3 / . Approxima- ting the nonlinearity in the spirit of (1) by phenomenological expres- sions iead to very nice results / 6 / . Kowever, in the meantime we re- duced the NOSE to its counterpart for the wavefunction of relative motion, X,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984628
C6-242 JOURNAL DE PHYSIQUE
Fig.1: Experimental and theoretical differential cross-sections for elastically scattered 9 ~ e ions (E=27 MeV) incident on 160 are displayed. Top: K is varied from O MeV to 350 MeV. Bottom:
Experimental data (points) are compared with computations based on K=O MeV (broken curve; i.e. optical model) and on K=270 MeV
(full curve)
.
Fig.2: As bottom of Fig.1 but E=20 MeV and K=270 MeV.
1 2 ~ ( 9 ~ e . 9 ~ e ) 1 2 ~
I I I < 1 I l
O" 60' 120D lEOD
SCATTERING ANGLE (c.rn.1
Fig.3: As bottom of Fig.1 but E= 30.6 MeV and K=270 MeV.
SCATTERING ANGLE (c.rn.1
50' 100' 150'
SCATTERING ANGLE (c.rn.1
fi2 2 2
-
2n; Vx +
Vomx+
C-1x1x
= E X with C=K/9>0and solved it numerically. The notation is standard with K=9C denoting the compressibility of nuclear matter. For C=O (2) is just the usual optical model equation with Vr=O. C#O implies that Vr will also be dif- ferent from zero. In the case of nucleon-nucleus scattering, the com- pression of nuclear matter will be negligible, a point correctly re- flected by (2): We apply (2) in such a way that we take the parameters of the optical model from the literature, hence, C is the only adjusta- ble parameter available to improve the correspondende between theory and experiment. In the "normal" case, Say, of nucleon-nucleus scat- tering the V yields adequate da/dQ implying C=O. However, in the case of collidingomheavy ions Vom leads only for forward angles to an ade- quate description; the anomalous rise for backward angles requires CZO.
It turned out that an increase in C leads also to an increase in da/dR for backward angles giving also rise to some changes in its structure.
An unexpected feature is that £rom a critical value of about K=9C=
260 MeV onwards du/dR does no longer rise but it just fluctuates around this critical value. We did not yet find a satisfactory interpretation for this behaviour. But it is interesting that this critical value is very close the one required for the description of the experimental do/dR. From Our analysis of more than ten cases we obtain the (average) value K=250 MeV (+IO%, -15%) for the compressibility modulus of
(finite) nuclear matter
--
a result which is not just in line with literature results but also with the numbers due to older calculations of ours with phenomenological substitutes for the nonlinearity of (2), see e.g. / 6 / .Turning towards relativistic heavy ions, let us recall that it has re- cently been shown /7/ that systematic discrepancies between experimen- ta1 data and cascade model calculations may be explained invoking the picture that part of the available kinetic energy of the projectile goes into the elastic compression of nuclear matter. At a later stage of the reaction process this energy is to be released.
--
The amount of compressional energy required is compatibelwith
compressions in- volving only the surface regions of the participating ions. This is readily verified by taking Eq.(6) of /4/ and by inserting into it reasonable numbers for mass numbers,radii and separations between the two ions; the resulting numbers compare nicely with the equation of state given in Fig.2b of /7/. A more detailed discussion is to be given elsewhere.REFERENCES
/1/ BAZ A.I., GOLDBERG V.Z., GRIDNEV K.A., SEMJONOV V.M. and HEFTER E.
F., Z. Physik
A280
(1977) 171; DARWISH N.Z., GRIDNEV K.A., HEFTER E.F. and SEMJONOV V.M., Nuovo Cim. A42 (1977) 303./2/ SUBBOTIN V.B., SEMJONOV V.M., GRIDN-K.A. and HEFTER E.F., Phys. Rev. C28 (1983) 1618.
/3/ DELION D.S.,R~DNEV K.A., HEFTER E.F. and SEMJONOV V.M., J. Phys. G4 (1978) 125.
/ 4 / DELION D . Z , GRIDNEV K.A., HEFTER E.F. and SEMJONOV V.M.,
Z. Physik A297 (1 980) 1 1 5.
/5/ SPIEGEL ~ . K ~ h y s i k a
p J
(1980) 236;MADELUNG E., Z. Physik
40
(1926) 322./6/ GRIDNEV K.A., DARWISH N.Z., DEMYANOVA A.S., SEMJONOV V.M., SUB- BOTIN V.B. and HEFTER E.F., Izv. Ak. Nauk SSSR, ser. fiz.
42 (1978) 2361.
/7/
