A simple model of a rapidly rotating hot nucleus

Volume 238, number 2,3,4

PHYS1CS LETTERS B

5 April 1990

A SIMPLE MODEL OF A RAPIDLY ROTATING HOT NUCLEUS W. N A W R O C K A Institute of Theoretical Physics, University of Wroclaw, Cybulskiego 36, PL-50-205 Wroclaw, Poland

R.G. N A Z M I T D I N O V Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Head Post Office, PO Box 79, SU-IOI O00 Moscow, USSR

and L. J A C A K Institute of Physics, Technical University Qf Wroclaw, Wyb. Wyspianskiego 27, PL50-370 Wroclaw, Poland

Received 24 July 1989; revised manuscript received 22 january 1990 Nucleons moving in an anisotropic harmonic oscillator potential are considered. The whole system rotates with a large angular velocity ~). 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.

1. Introduction The study o f thermal effects in finite nuclear systems has been the subject o f m a n y publications in the last 15 years. A m o n g them various thermal mean field a p p r o x i m a t i o n s were done [ 1 ], the t e m p e r a t u r e - d e p e n d e n t collapse o f the pairing correlations was investigated [2,3] the RPA solutions at T = 0 were o b t a i n e d [4,3] and finite-temperature H F B cranking equations were derived and applied [ 5 ]. The model o f the cranked h a r m o n i c oscillator was solved [ 6 ] in the semiclassical limit, which imposed, however, some restriction on the value o f the angular velocity. Further, the same p r o b l e m was analyzed in ref. [7] at an arbitrary value o f the cranking angular velocity. The t e m p e r a t u r e - d e p e n d e n t m a n y - f e r m i o n p r o b l e m in the spherically-symmetric harmonicoscillator well was studied in ref. [ 8 ]. In the present work we examine the system o f A nucleons in the cranking d e f o r m e d harmonic-oscillator potential as the grand canonical ensemble which is described by t e m p e r a t u r e T and chemical potential ~. We treat the nucleus as a system with a heat source which one can identify with the experimental instrument, say a particle b e a m used to excite the target plus other particles - products o f the reaction (photons, nucleons, etc. ). O u r task is to investigate the properties o f nuclei such as d e f o r m a t i o n parameters, angular m o m e n t u m , m o m e n t o f inertia, level densities, etc., as functions o f t e m p e r a t u r e and cranking velocity ~. Note also that in order to prevent the stability o f the shell structure, the investigated region o f t e m p e r a t u r e should be limited. As it was shown, in ref. [ 1 ] the t e m p e r a t u r e weakly influences the shell model up to temperatures T o f the order o f 3 - 4 MeV (for heavier nuclei). In this a p p r o a c h we have to r e m e m b e r that for finite nuclei the statistical fluctuations o f the n u m b e r o f particles ( a n d other quantities also) are i m p o r t a n t , even the system is not near the critical point, which is the consequence o f the a p p l i c a t i o n o f the grand canonical ensemble without the t h e r m o d y n a m i c limit. These fluctuations have been investigated in ref. [ 1 ] where it has been shown that in the mass region A > 100 the relative 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )

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particle number fluctuations AA/A are of the order of 5% (the energy fluctuations are of the order of 3%). This problem was discussed also more recently with similar results.

2. Formulation of the problem

Let A independent nucleons move in an asymmetric harmonic-oscillator potential well which rotates with frequency m around the x-axis in the body system. The hamiltonian has the form A

H~=

~ (h~)i,

(1)

i:!

where h~ is the single-particle hamiltonian h~J=ho_hOglx= ~m l 2 2+~oyy 2 2+cozz 2 2)-hoglx . p2 + ~rn(og~,x

(2)

lx is the angular momentum along the axis of rotation. Diagonalization (cf. ref. [9 ] ) of hamiltonian (2) gives the one-body wave functions [nxn+n_ ) where nx, n+, n are the quantum numbers of the new normal modes, and the one-particle energies of h~' are e~=e(nx, n+,n

)=hmx(nx+½)+htn+(n+ + ½ ) + h m _ ( n _ + ½ ) ,

(3)

where o~z+ = ½(a)~+0~2)+0~2+½ [(a)y-2 m2)2+8m2(oj~+o~2)],/2

(4)

The total energy of the system depends on the given configuration: E°~= ~. e~p,~=htoxZx +hm+Z+ +ho)_ Z_ ,

(5)

ot

where Z,,= ~ (n,~+½)&,,

tc=+,-,x,

(6)

and p~ represents the occupation number of the one-particle states (p~ = 0 or 1 ). In the framework of the thermal approach statistical averaging over the grand canonical ensemble has to be performed. In this way we remove the dependence on the fixed quantum configuration. Instead, the average value of particle number agrees with the number of nucleons A. The grand canonical potential (2 is expressed by the partition function Z: (7)

(2= f l l n Z , where fl= 1 / k T a n d Z = Tr{exp [ - fl(/4~' -/2.N) ]}.

(8)

The occupation number of the individual fermion level is 1 01nZ

1

fl 0e~

exp[fl(t~-/2) ] + 1 '

(9)

The parameter/2 is determined by the condition A=

132

Y

1

exp[fl(ea--/2) ] + 1

.

(10)

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PHYSICS LETTERSB

5 April 1990

The assumption that the system is in thermal equilibrium leads to the following system of equations which are equivalent with the condition for the extremum of the potential £2: 0t2 =0,

OoJ--7

0£2 _0,

&Q=0

0%

O~o_.

"

( 11 )

These three conditions are not independent. They are connected via the potential volume conservation condition

oo,.corO)z=O.) 3 .

(12)

This last condition displays, in the simplest manner, the consequences of nuclear forces. As it was shown in ref. [8 ], O9ocould also depend on the temperature. For the sake of simplicity we assume % to be constant. After elementary calculations we find that eqs. ( I 1 ) together with the condition (12) are equivalent to the following equations: 2 ~o~ =o~ =o~ ,

(13)

where <... ) means statistical averaging. We have

_ ,

(14a)

mfQv

1 [ + = ~mm

oJ+

1 [

=~L

22
To calculate ( x 2 ) ,

2 +4092 ( ( n + +½) + 09~ - -~Y

+

~+

~-

( n=--l- ½) ' I f , o)_ /j
.

(14b) (14c)


( n x + ½ ) = .x..+~",.- (n:,+½)exp[fl(e

=

E

........

(n++½)

_lt)]+l

,

1

exp[fl(e.-/z) ] + 1 '

(15a)

(15b)

1

(n_+½)=,,x,,+Z,,_ ( n - + ½ ) e x p [ f l ( ~ _ # ) ] + 1 "

(15c)

Unfortunately, these sums cannot be calculated analytically. Note, however, that is a similar summation problem as in the calculations of the quantum Hall effect or in the theory of the de Haas-van Alphen effect [ 10]. In spite of these two cases, in our present problem we have no small parameter (since in our case T/lt~ 1 ). The explicit summation in eqs. (15) is possible in the high temperature limit (when we deal with the Boltzmann distribution). This limit is not interesting, however, since with growing temperature the nuclear shells and other quantum effects disappear. The selfconsistent system of eqs. (13) together with the condition (12) and (10) resolve themself into the following system of equations: f ( ~ , , ~o., 2 - o J .v

=0,

(16a)

A ( % , % , / ~ ) =Or -~o~ = 0 ,

(16b)

f3(%, , O z , ~ ) = A - F . f , = 0 .

(16c)

o!

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As independent variables we choose my, 09z and/t(09x = 093/09y09~ owing to condition ( 12) ). The explicit form of the system (16) is as follows: 09x_0921 I < n + + ~ ) -

(o)2-o)2)

~

I+

a

+09y2 +09~ +4092 ( < n + + ½)

09_

4092+09y2+09~'~ .-TT---T~.2 "J 0 9 + + 09+ --09_

09~+ _ , o 2

I--

09+


409~+m~+09='~
(17a) ,

(17b) '

09o~

09~-

A= Zf-,

+
09+

(17c)

093, 0 9 z

From the above equations it follows that there exist two types of solutions: ( 1 ) % = 09z: the axial rotational case; (2) 09y~ 09z: the nonaxial case. For the first case the system ( 17 ) attains the form of only two equations: 09:~
(18a)

A= ~f,.

(lSb)

One can consider also the limit %-'09z (09y¢ 09z) for eq. (17b), which leads to the following condition:
--

09y2 _ 209y09+ 2092 = 0 . Imv-091 2%09

(19)

This condition, together with eqs. (1 8a), (1 8b), determines the position of the bifurcation point on the curve % = 09~ versus 09 at each value of T, i.e., the location of the point where the axial solution splits into axial and nonaxial solutions. From the other side, these bifurcation points can be found as the solution of the sufficiency condition for bifurcation [ 1 1 ], applied to the system ( 1 6 ), i.e., det(l--{fk}) =0,

(20)

where Ofl

af2

OY3

009y 009y 009~ {fk}= of, af= oA

(21)

009~ 009z 009~

of,

of=

oA

O#

8#

8#

Finally, note that the averaged angular momentum of the nucleus is

(

____ 09+

< L > = + \

(n +½)) 09_

09+

409 3 tu+ -- _

=--~o~

409 + 092+ _092_ (09_--
( 09+

(n__+½)'] 09_

/ (22)

The two first terms in the above equation give the "rigid body" dependence: rn09( ), while the third is the deviation from the "rigid body" formula. Note that in principle the angular momentum is an observable characterizing the state of the system and therefore 09 is rather determined from the constraint (lx> = L In this paper all calculations are made, however, for a given 09 and versus 09 is determined by eq. (22).

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3. Numericalresults The model used for the computer calculations has A = 100 and coo= 4 I A -1/3 MeV. We put h = k = 1, thus all variables are taken in energy units coo. Now we present preliminary numerical results. A more detailed and comprehensive analysis will be published elsewhere in the near future. In fig. 1 the solutions co~= coz of the system (18) are presented as functions of co. One can observe a sharp temperature dependence (especially, in the small temperature region) - which displays the fact that near T = 0 the nucleus conserves an almost spherical shape even at nonzero angular velocities co. This is not the case at higher temperatures. This behaviour is connected with the sharp Pauli principle in the low temperature limit. In fig. 2 the corresponding solutions for # as function of 09 are depicted. It is characteristic that the chemical potential depends weakly on the temperature. In fig. 3 the angular m o m e n t u m (/x) is plotted as function of co for several temperatures. The "rigid b o d y " dependence (cf. eq. (22) ) is also indicated. One can notice that the q u a n t u m originated deviations from the "rigid b o d y " dependence manifest themselfat small angular velocities and low temperatures. This once more displays the quantum effect due to the blocking of states by the Pauli principle at low temperatures. In both fig. 1 and fig. 2 the line coy=co is plotted also. Along this curve the frequency co_ in the axial case ( c o = Icoy-o)l ) changes its character, which imposes constraints onto solutions. In order to find nonaxial solutions a more thorough analysis is necessary. We confine ourselves here to era-

T=0.5 Wo

j - - - - - -

f T = 0.3 coo /

/

1.0~

T " 0.1 COo T ~ 0.05 w0

i

"2

ii

_

_

l

j/

T =

03 ~o

/

/

X

/

:a.

0.5

/

O~

/ /

5t

/ / A

i

=

100

/ / !

0

A = 100

0.2

0.4

0.6

1-

~/~o

Fig. 1. The axial solutions for frequencies oJy=o9z as functions of ~oat several temperatures.

0.2

0~4

06

/co o Fig. 2. The chemical potential/z versus oJ at various temperatures.

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Volume 238, number 2,3,4

200 t .

.

.

PHYSICS LETTERS B

iI

.

5 April 1990

-

A = 100 1601

r

T = 0.5 o~o

[ .... %q /

~_

,v/ fl

/

// T= O.lw ° //-7--

--

,,/; [

// / /

80)

,/

//ST= .

.

3

/ /

.

4/

.

0.05 i..a0 .

/

0.12

//J7 ~

0

o.1I

body .-"

0.1

<

L

0.2

[×>

03

0,4

0.08[----0

T/¢.,o °

0.5

q..) / 60 0

Fig. 3. The averaged angular momentum (l~) versus rotational frequency ~oat several temperatures.

Fig. 4. The line of bifurcation points at which the axial solution splits into axial and nonaxial solutions.

p l o y i n g c o n d i t i o n ( 1 9 ) only. T h e b i f u r c a t i o n p o i n t s on the c u r v e s coy= oJ~ can be f o u n d as the s o l u t i o n o f eq. ( 1 9 ) t o g e t h e r w i t h eqs. ( 1 8 a ) , ( 1 8 b ) . In fig. 4 the i n t e r p o l a t e d c u r v e o f the b i f u r c a t i o n p o i n t s versus t e m p e r a ture is plotted.

References [ 1] M. Brack and Ph. Quentin, Phys. Scr. 10A (1974) 163; U. Mosel, P.G. Zint and K. Passler, Nucl. Phys. A 232 (1974) 252; G. Sauer, H. Chandra and U. Mosel, Nucl. Phys. A 264 (1976) 221. [2] A.L. Goodman, Nucl. Phys. A 352 ( 1981 ) 30, 45; Phys. Rev. C 33 (1986) 2212. [ 3 ] A.V. Ignatyuk, Statistical properties of excited nuclei (Energoatomizdat, Moscow, 1983 ). [4 ] O. Civitarese, R.A. Broglia and C.H. Dasso, Ann. Phys. (NY) 156 (1984) 142; A.N. Ignatyuk, I.N. Mikhailov, L. Molina, R.G. Nazmitdinov and K. Pomorsky, Nucl. Phys. A 346 (1980) 191. [ 5 ] K. Sugawara-Tanabe and K. Tanabe, Progr. Theor. Phys. 76 ( 1986 ) 356, 1272; K. Tanabe, Phys. Rev. C 37 ( 1988 ) 2802. [6] V.G. Zelevinsky, Sov. J. Phys. 22 (1975) 565. [7 ] T. Troudet and R. Arvieu, Z. Phys. A 291 (1979) 183; Ann. Phys. (NY) 134 ( 1981 ) 1. [8] H. Sato, Phys. Rev. C 36 (1987) 785,794. [9] G. Ripka, J.P. Blaizat and N. Kassis, Heavy ions high spin states and nuclear structure, Vol. 1 (IAEA, Vienna, 1975) p. 445. [ 10] A.A. Abrikosov, Theory of metals (Nauka, Moscow, 1987). [ 11 ] L. Nirenberg, Topics in nonlinear functional analysis (New York, 1974).

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