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Statistical theory of hot nuclei and high spin states

M RAJASEKARAN, N A R U N A C H A L A M , T R RAJASEKARAN and V DEVANATHAN

Department of Nuclear Physics, University of Madras, Madras 600025, India

Abstract. The hot rotating compound systems formed in heavy ion collisions are studied using the statistical theory with a view to determine the spin and temperature dependence of nuclear shapes. Shape transitions are observed for these systems at particular spin values.

The neutron and proton separation energies for heavier high spin systems have been evaluated. Results are presented for tvVo°Yb and 19~pt.

Keywords. Hot nuclei; high spin states; heavy ion collision; statistical theory.

PACS Nos 21.10; 24.60; 27.70

1. Introduction

In heavy ion collisions, the compound nuclear systems of high spin are formed with high excitation and temperature and as remarked by K h o o (1981) it will be exciting to explore the spin and temperature dependence of nuclear shapes. A statistical theory (Moretto 1973 alid Ramamurthy et al 1970) has been developed to study such highly excited hot rotating compound nuclear systems formed in fusion reactions.

Calculations have been performed (Rajasekaran et al 1987) for the hot compound nuclear systems of ~2C + 12C, t 6 0 + t60, 180 + 180 and 4°Ca + 4°Ca. The very high spins populated in these reactions are found to exhibit such phenomena like back- bending and Yrast traps. Shape transitions are observed for these systems at particular spin values (Rajasekaran et al 1987). The constant entropy lines are drawn for these systems and they are found to be almost equally spaced. The experimental data on the angular momentum limitation on fusion probability are reproduced at higher entropy values. The single particle level density parameter which is an important input in the nuclear level density calculations is found to depend strongly on the spin, temperature and deformation (Rajasekaran et ai 1988a).

Introducing the isospin degree of freedom, a new method of extracting the neutron- proton asymmetry parameter is proposed (Rajasekaran et al 1988b). The neutron- proton asymmetry is found to depend very strongly on deformation and isospin at low temperatures. The single particle level density parameter a is extracted as a function of temperature for various isospins of the system. At large temperatures, the empirical value of a " A / 8 is reproduced. The excitation energy versus isospin plot for constant entropy of the system exhibits pockets similar to Yrast traps in high spin states of highly excited nuclei.

Here, we present some results on heavier high spin systems like

t70VhTo__,,

and t79~Pt.

We have analysed the behaviour of these systems at very high spins I _~ 70 h. Shape transitions as in the case of N = 88 isotones, are observed. The neutron and proton 515

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separation energy has been calculated as a function of spin and excitation energy of the system, since it is an important input to obtain the neutron emission spectrum (Rajasekaran et al 1988c).

2. The statistical theory

The compound nucleus that is formed in heavy ion collisions will have no memory of the original reaction that is responsible for its formation. With the increase in the excitation energy of the compound nucleus, the lifetime becomes shorter and shorter as more and more channels open for decay. The spacing between the levels is progressively reduced and the nature of the excitation becomes very much complicated. In such a case, the statistical approach will allow a comprehensive description of the average behaviour of the compound nucleus.

The statistical properties of the system are contained in the grand canonical partition function

Q(az, aN, fl, ~,) = Y'. exp (azZi + aNN~ -- flEi + YMI).

ZI,N~,E~,M~

(1)

The partition function Q has a simple physical interpretation in terms of grand canonical ensemble. In such an ensemble, the probability of finding the nucleus with Zi protons and Ni neutrons in the angular momentum state M~ and the energy state E~

is proportional to e x p ( a z Z ~ + a N N ~ - f l E ~ + 3 , M ~ ) . If we treat the variables as continuous, we can express the partition function as the Laplace transform of the nuclear level density p(Z, N, E, M)

¢

Q (az, aN, ,8, ~,) = | p (Z, N, E, M) exp (az Z + aN N --

a /

+ yM) dZ dN dE dM.

(2)

Consequently, the nuclear level density is the inverse Laplace transform of the partition function and can be evaluated using the saddle point approximation.

_ 1 ~J®

p(Z, N, E, M) - ~ j _ ,~ Q(a z, a N, fl, y)exp ( - a z Z - a s N + fiE

- ~M) daz das dfl dy, (3)

=

eS/(2n)z D I/2, (4)

where S is the entropy of the system,

S = In Q - a z Z - aNN + [3E - y M ,

(5)

and D is a 4 x 4 determinant involving the differentials of In Q with respect to the Lagrangian multipliers.

Rajasekaran and Devanathan (1982) have given an alternative way of obtaining the nuclear level density in a finite dimensional phase space. Starting from Boltzmann's definition of entropy

W(E) = e s, (6)

(3)

where W is the total number of configurations available for a given energy E of the system, we obtain the nuclear level density as

p(E) = OW/OE = eS(OS/OE) = tie s. (7)

The last step in (7) follows from (5)

os/oE

= t (8)

since

(O In Q/Oaz) = ( Z ), O In Q/daN = < N ),

O In Q/Ot = - < E >, 0 In Q/07 = < M >. (9) The nuclear level density given by (7) is not normalized and the definition of partition function can be used to normalize it.

f p(E)exp(azZ + aNN -- t E + ?M)dE

Q

_IeS(OS/OE) exp(az Z + auN - fiE + 7M)dE

C

re'* a(OS/OE) dE. (1 O)

C

The normalization factor is given by

/f:os

C = l - ~ d = l/Sm,,. (II)

Thus we obtain a simple formula for the nuclear level density which is valid for a finite dimensional phase space

p(E) = eS/Sm,,. (12)

If No is the number of states and N is the number of particles, then

SIn,, = In No!/[N!(No - N)!]. (13)

3. Hot rotating nuclei

The Lagrangian multipliers az, ~'s and ? in the partition function (I) conserve the proton number, neutron number and total angular momentum along the Z-direction for a given temperature T = l / t as shown in (9).

In terms of single particle occupation probabilities, energies and spin projections,

< Z ) = ~nff = ~ 1/rl + e x p ( - a z + t e z+Tmz)],

< N ) = ~ n i = E 1/[1 + e x p ( - a s + t e ~ + T m r ) ] ,

N N

nN. m .N

< M ) = Z n Z m Z + z., , ,,

(14) (15)

(16)

(17)

(4)

and with

S = Sz + SN,

(18)

s z = - Y~ n, ~ i n n,~ + (1 - ~,~)In (1 - . f ) ,

S N = - -

~n~lnn~ +

(1 -- n~)ln (1 -- n~). (19)

Excitation energy E* of the system for an angular momentum state M is given by

E*(M, T) = ( ~ nZeZ - ~l ~z) + ( ~ n~e~- ~ e~),

(20) Rajasekaran

et al

(1987) used the single particle levels e z with spin projection m z and e~ with spin projection mr obtained by diagonalizing the triaxially deformed Nilsson hamiltonian.

4 . H o t i s o b a r i c n u c l e i

The statistical theory has been extended to the study of hot isobaric nuclei by Rajasekaran

et al

(1988a). The partition function and the corresponding expectation values of nucleon number, energy and isospin are given below

Q(~, fi, y) = ~ exp

(~N~ - fiE, + yT~),

(21)

( N ) = A = ~ l n Q =~,n~ + + ~ n i , (22)

E ) = - ~-~ In Q = ~ (n[ + n f )~, (23)

(

( ~ ) = ~ In Q = ~ n~ + z~+ + ~ n 7 ¢~-. d (24)

The single nucleon occupation probabilities n~ + and n F with isospin projections Zz + = + 1/2 and ~ z = - 1/2 are given by

ni + = 1/[1 + e x p ( - ~z + fiei + ~'~ )], (25)

nf

= 1/[1 + e x p ( - ~z + fiei + VZz)]. (26) The excitation energy and entropy are obtained from the single-nucleon occupation probabilities and energies.

A

E*(z, T) = ~i (n + n; )e, - ,= lZ e,,

(27)

S(~,E*) = S + + S-,

(28)

with

s + = - Z [,,, + I n . , + + ( 1 - . , + ) I n ( 1 - n,+)].

S- = - ~ [hi- In

nF +

(1 - nF ) In (1 -

nf

)]. (29)

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The neutron-proton asymmetry parameter has been studied as a function of temperature, deformation and isospin. Also the single-particle level density parameter has been studied as a function of temperature and it is found to reach a constant value of A/8 at higher temperatures for all isospins. For further details on this subject, the reader is referred to Arunachalam (1988) and Rajasekaran et al (1987,

1988a, b, c).

5. P r o t o n a n d n e u t r o n s e p a r a t i o n e n e r g y

The proton separation energy Sp and neutron separation energy S, can be obtained from the free energy fl of the system. The neutron separation energy can be obtained using the equation

S,, = ~D/~N, (30)

where ~ = - T In Q(a,//, 7). Since fl is the function of T and angular momentum M, S.

is also a function of T and angular momentum. S. can be written explicitly in terms of the single particle occupation number ni as follows.

S, = TN/[r,(I - ni)nt]. (31)

A similar equation for proton separation energy can be written.

The neutron-proton separation energies have been calculated as a function of T and spin of the system. In figure l(a), the proton separation energy Sp for tTOVkTo__~, is plotted as a function of angular momentum M for T ranging from 0"2 to 1.0 MeV.

At low temperatures for angular momentum M " 35 h to 40 h, the value of Sp is approximately 6 MeV. It is nearly 4 MeV less than the value at zero angular momentum. It is therefore anticipated that the proton emission should be relatively higher at these angular momentum.

From figure lb, using the same argument as above, it is obvious that the neutron emission should be relatively higher for M > 45 h.

12

~q Q.

1 ~ b

T (MeV) I:'~L-w~'.~-~.. I ~ ... I..-..I ... 1 . 0 I - , . . ~ " = ~ ... "/"'~_ _/_ ..L ~ --- ~ 0 . 7

I ~ " - , , ~ . . . . - - / - ~ t ...__.~o.5 ] ~ " . , " ~ . . . I .~"r-"-T °

8

I--

~ /

-/=-

' ~__

. . .

0 2 0.3

4 | i I [ I

0 2 0 4 0 6 0 8 0

M ( ' f i )

(a)

(6)

b

17o

t5 - A '°Yb

:E ~ . ~ - ~ ' ~ - ~ . ~ - T(MeV) (b)

c 1 . 0

t/) 10 - . . . . 0 " 7

- ~ O . 5 0.3 0 . 2

5 I I I I

0 2 0 4 0 6 0 8 0

M ('hi

Figure 1. (a) Proton separation energy Sp as a function of spin M and T for =~°Yb.

(k) Neutron separation energy S, as a function of spin M and T for l~o°Yb.

12

<0

:E (x 8 tn

A

/ / ~ T (MeV}

-,... " ' . . . . . . , . o

=~--J"="k--...==.-....=....y" 8'.~

- / ~M ~ ' ~ 0.3

/ ~ "~.---.---.~"

- ' I L ~ . ~ . . . I ~ ° ' 2

I I I !

0 2 0 4 0 6 0 8 0

M (t5)

15

t- 03

10

Figure 2.

T (MeVl

~ ' ~ 194

_ \

0.7 ~c_"-- ~ ' ~ ~ . • - - , ~ . ~ , - ~ . - - ..-w.-.~.-'=.-- I . . .

. . . . ~.";"""0.5

I I I I I I I I

0 2 0 4 0 6 0

M (1~)

(a) As in figure l(a) for =~Pt, (b) As in figure l(b) for '9~pt.

(b)

Figures 2(a) and 2(b) depicts the results for 19~pt. The neutron separation energy shows a slight dip at M - 30 h at all temperatures. However, S, ~- 11 MeV for all angular momentum states. Hence, the neutron emission in 19~pt is possible only at very high excitations.

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Acknowledgements

This article is dedicated to Prof S P Pandya with whom the Madras Nuclear Theory group had close association. One of us (VD) is grateful to Prof J C Parikh for his kind invitation to participate in the symposium organized in honour of Professor S P Pandya.

References

Arunachalam N 1988 Statistical theory of hot nuclei and high spin states, Ph.D., University of Madras Khoo T F 1981 Phys. Scr. 24 283

Moretto L G 1973 Nucl. Phys. A216 1

Rajasekaran M, Arunachalam N and Devanathan V 1987 Phys. Rev. C36 1060

Rajasekaran M, Arunachalam N, Rajasekaran T R and Devanathan V 1988a Phys. Rev. 38 1926 Rajasekaran M, Rajasekaran T R and Arunachalam N 1988b Phys. Rev. C37 307

Rajasekaran M and Devanathan V 1982 Phys. Lett. 11113 433

Rajasekaran M, Rajasekaran T R, Arunachalam N and Devanathan V 1988c Phys. Rev. Lett.

Ramamurthy V S, Kapoor S S and Kataria S K 1970 Phys. Rev. Lett. 25 386

References

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