Institute of Physics, Technical University of Wroclaw, Wyb. Wyspianskiego 27, Wroclaw, Pol and

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Everett Taylor
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1 06beAiHeH~blR ih CTiTY T laephblx nccne ~ o s a ~ n f i AY6Hii L. Jacak l, W. NawrockaZ, R. C. Nazmitdinov SIMPLE MODEL OF A RAPIDLY ROTATING HOT NUCLEUS Submitted to "Physics Letters B' Institute of Physics, Technical University of Wroclaw, Wyb. Wyspianskiego 27, Wroclaw, Pol and * Institute of Theoretical Physics, University of Wroclaw, Cybulskiego 36, Wroclaw, Pol and

2 The study of thermal effects in finite nuclear systems has been thz subject of many publications in the last 5 years. Among them the various thermal mean field approximations have been done [I], temperature-dependent collapse of the pairing correlations has been Investigated 2, the RPA solutions at TfO have been obtained [3,[2bl and finite-temperature HFB cranking equations have been derived and applied [4. The model of the cranked harmonic oscillator has been solved (5 In the sem~classical limit, which imposed,however, some rzstrtiction on the value of angular velocity. Further the same problem has been analyzed in [6 at the arbitrary value of the cranking angular velocity w. Temperature dependent. many fermions problem in the spherically symmetric harmonic oscillator well has been studled in [7. In the present work we examine the system of A nucleons in the cranklng deformed harmonic oscillator potential as the grand canonical ensemble which is described by the temperature T and the chemical potential p.we treat the nucleus as a system with the heat source which one can identify' with the experimental instrument, say a particle beam used to excite the targed plus other particles - - products of the reaction ( photons, nucleons. etc.). Our task is to investigate the properties of nuclei such as deforlnation parameters. angular momentum, moment of inertia, level-densities, etc. as the functions of temperature and cranking velocity O. In this approach we have to remember that for the finite nuclei the statistical fluctuations of the number of particles (and other quantities also) are important even when the system is not near to the ett~77.,. < ;,<.. q....,, -- ''.,.. ', 26..:,,.:. ;,,.;;..,

3 critical point which is the consequence of application of grand canonical ensemble without the thermodynamic limit. This fluctuations have been investigated in [I], it has been shown that In the mass region A>00 the relative particle number fluctuation ba/a is of order of 5% ( the energy fuctuation is of order of 3%. This problem has been discussed also more recently with the similar results. 2.Formulation of the problem Let A independent nucleons move in an assymetric harmonic oscillator potential well which rotate with frequency LJ around the x-axis in the body - system. The hamiltonian has the form: A H; - E (h;);, () I - h$ is the single particle hamiltonian hgw = ho - nutx ~ m + ~ ( ~ x 2 + W ~ y 2 t w : z 2 ) - n, u ~, (2) is the angular momentum along the axis of rotation. Diagonalization (cf. 8 ) of hamiltonian (2) gives the one - body wave.functions In,n+ n-> n,, n,,n- are the quantum numbers of the new normal modes. and the one - pariicle energies of htare: and p- 'Pa represents the occupation numbers of the one - particle states = 0 or ). In the framework of the thermal approach the statistical averaging over the grand canonical ensemble has to be performed. In this way the dependence on the fixed quantum conflguratlon is removed. Instead of that the average value of particle number agrees with the number of nucleons A. The grand canonical potential fi s expressed by the partiticn function 2: p = T Q-- y I InZ. (7) and Z - Tr (exp(-/3(:; -)J;)). (8) The occupation number of the individual fermlon level Parameterp is determined by the condition A = a ebka-p+l ' (0) Assumption that the system is in the thermal equilibrium leads to the following system of equations which are equivalent wlth the condition ror the extremum of potential Sl : These three conditions are not independent. They are connected via the w,,? + u,' /2 u: = 2 + O ~ * ~ ( ( ~ ~ - L J, ) +8w (wy+w,)). (4) The total energy of the system depends on the given configuration : E*- C 8:: pu - fi wxrx + n~+r+ + nu-r-. (5) 0.z*- z(n(+~)pu. Y-+,-vX* L 6 u potential volume conservation condition: 3 W,' WY'WZ = Wo. (2) This last condition displays, in the slmplest manner. the consequences of nuclear forces. As ~t has been shown in [ 7 ( ~ ~ could also depend on temperature. For the sake of simplicity we assume uo to be constant. After elementary calculations we find that equations () together with the condition (2) are equivalent to the following equations: <...> means the statistical averaging.

4 critical point which is the consequence of application of grand canonical ensemble without the thermodynamic limit. This fluctuations have been investigated in, it has been shown that In the mass region A>00 the relative particle number fluctuation AA/A is of order of 5% ( the energy fuctuation is of order of 3% ). This problem has been discussed also more recently with the similar results. 2.Formulation of the problem Let A independent nucleons move in an assymetric harmonic oscillator potential well which rotate with frequency w around the x-axis in the body - system. The hamiltonian has the form: A H; - 5 (h2li. () h$ is the single particle hamiltonian and pp represents the occupation numbers of the one - particle states 'Pa = 0 or ). In the framework of the thermal approach the statistical averaglng over the grand canonical ensemble has to be performed. In this way the dependence on the fixed quantum configuration is removed. Instead of that the average value of particle number agrees with the number of nucleons A. The grand canonical potential s expressed by the partitlcn function 2 : R -- g ~ n z, (7) p= and Z - T r (exp(-p(i; (8) The occupation number of the individual fermlon level Parameterp is determined by the condition is the angular momentum along the axis of rotation Diagonal ization (cf.[8) of hami ltonian (2) gives the one - body wave.functions In,n+n-> n,, n,,n- are the quantum numbers of the new normal modes. and the one - particle energies of h;are: Assumption that the system is in the thermal equilibrium leads to the following system of equations which are equivalent wlth the condition for the extremum of potential R : These three conditions are not independent. They are connected vla the a,' + a,' /2 O: = 2 +~~*~((cd,?-u,) +80 (wy +w,)). The total energy of the system depends on the given configuration : EWP c a C ( n l ( + ~ ) ~ r r. P t,ayrx +nw+r. +no-r-, r'+,-lx* (4) potentla volume conservation condition: 3 w;wy'w, = Wo. (2) Thls last condition displays, in the slmplest manner. the consequences of nuclear forces. As lt has been shown in [7Io0 could also depend on temperature. For the sake of simplicity we assunre uo to be constant. After elementary calculations we find that equations () togethzr with the condition (2) are equivalent to the following equations: w,z<x2> = 'd; < Y2> - 0: < z2>. <...> mzans the statistical averaglng.

6 Flg.. The axial solutions for frequencies w y-*z as functions of W at several temperatures. 3.Numerical results The model used for the computer calculations conslsts from A=00 and LJ~=~~A-~/~M~V. We put 5-k-, thus all variables are taken in the energetical units Oo. NOW we present the preliminary numerical results. More detailed and comprehensive analysis will be published else in near future. In Fig. the solutions wy-w, of the system (8) are presented as the functions of w. One can observe the sharp temperature dependence ( esspecially, in the small temperature regjon ) - which displays the fact that near T-0 the nucleus conserves almost spherical shape even at nonzero angular velocities W.This is no case in higher temperature. This behaviour is connected with the sharp Pauli principle in the low temperature limit. In Fig. 2 the corresponding solutions for p as the functions of w are depicted. It is characteristic that the chemical potential weakly depends on temperature. In both Figs and 2 the ' line wy = w is also plotted Along this curve the frequencyw-in the axial case: w-=iwy-wj changes its character. which imposes constraints onto solutions.. In order to find the nonaxial solutions the more thorought analysis is necessary. We confine here to employing of the condjtion (9) only.the bifurcation points on the curves wy=uz can be found as the solution of Eq.(9) together with Eqs(l8a.b). In Fig.3 the interpolated curve of bifurcation points versus temperature is plotted. I 0 I I 02 I I 0.4 I I 06 I w lwo Fig.2. The chemical potential p at varjous temperatures. versus u at which the axial solution 008 splits into axial and nonaxial solutions. 0 T/ w, 3 05

7 U.blose, P.G. Zint, K.Passler, ~iucl.ph:fs.,a232 (974) 252 G.Sauer, H. Cher~dra arid U.L.ose,.iucl.Pn;.s., ~264 (76) 22 2 a) A.L.Goodrnan, Nucl.Phys.,A352 (98) 30, 45 j) A. v.~~,natyuk, Statistical Properties of Excited Nuclei. (Energoatornizdat,?:Aoscow, 983) c) A.L. Goodman, Phys. Re-f., C33 (986) 222 P~CCMOT~~HH HYKnOHLd, ABkiX~HeCR B IIOTeHqHane ahh30- TponHoro rapmohuseckoro ocrqmnrtopa, BpaqawerocR c 60nbqo# yrno~o# CKO~OCT~H) 0. Ca~ocornaco~aa~oe pemeaae 3a- AarIH II03BOnHeT HCCnepOBaTb II3MeHeHHe II ApyrkiX XapaKTepkiCTHK TeMnepaTypbl T H 0. 3 O.Civitarese, ii.a.broglia., C.H.Dasse, Ann. of Phys., 56 (984) 42 S.M.Fedotkin, I.iJ.Idilcneilov, H.G.Nazmitdinov, Phys.Lett., B2 (983) 5 P.Zing et al, Nucl. Phys., A49 (984) 26 4 K.Sugawara-Tanabe and K.Tanaoe, Progr. Theor.Ph;.s,,76 (986) 356, 272 K.Tanabe, Phys. He7r., C37 ( 908) T.Troudet and H.Arviev, Z.Phys.,A2Y (9.75) 83 Ann, of Phys.,34 (98) i( G.dipko, J. P. 3lniz:it and II. Kessis,ilea7/:j Ions Hich Spin States at:d ;ucleer Structure (IAEA, Vienna (975, v., p.445).: 0 L.I.iirenberg, Topics in Nonlinear Functional Anelysis (New York, 9'74) Jacak L., Nawrocka W., Nazmitdinov R.G. E Simple Model of a Rapidly Rotating Hot Nucleus Nucleons moving in an anisotropic harmonic oscillator potential are considered. The whole system rotates with a large angular velocity 0. A self-consistent solution of the problem makes it possible to study the variation of nuclear shape and other nuclear properties as functions of o and T. The investigation has been performed at the Laboratory of Theoretical Physics, JINR. Received by Publishing Department on August 3, 989. Preprint of the Joint Institute for Nuclear Research. Dubna 989

Observables predicted by HF theory

Observables predicted by HF theory Total binding energy of the nucleus in its ground state separation energies for p / n (= BE differences) Ground state density distribution of protons and neutrons mean