Prediction of a new phase in deformed nuclei

Volume

97B, number

PHYSICS

3,4

15 December

LETTERS

1980

PREDICTION OF A NEW PHASE IN DEFORMED NUCLEI * K. TANABE ’ and K. SUGAWARA-TANABE Physik-Department, Received

Technische

16 September

A possible fist-order > 0.5 MeV) is indicated

2

Universitiit Miinchen, D-8046 Garching, West Germany

1980

phase transition across the “zero-rotation” line in the low-spin and high-temperature by the solution of the cranked temperaturedependent HFB equation for a deformed

In this letter we study in some detail the average properties of deformed nuclei in the not too highly excited region, where the effects of the pairing interaction and the deformation are still important. The Hartree-Fock-Bogoliubov (HFB) approximation with the cranking model has been very useful for a realistic understanding of the yrast and low-lying levels [l-3]. A possible method appropriate for our purpose will be to extend the framework of the HFB approximation by introducing a temperature T,as the parameter to describe the effects of excitations on the mean fields in a statistical manner. In developing the formalism with the cranking model we have to notice some inadequacies of the theory based on the simple Fermi distribution for quasiparticles. First, it results in too large a state density due to the unlimited mixture of the space parity and the number parity. Secondly, to conserve the correct number parity the solution of negative energies for a few quasiparticle states must be chosen at an angular frequency w greater than a certain value [4]. Therefore the temperature-dependent (or thermal) HFB (THFB) equation with Fermi distributions fails to reduce to the HFB equation in the limit of T = 0 because not all the distribution functions vanish in this limit. To solve these problems, we have introduced in ref. * Work supported

by the Bundesministerium fur Forschung und Technologie (BMFT). ’ Present address: Department of Physics, Saitama UniversitY, Urawa, Saitama, Japan 338. 2 Present address: Department of Physics, University of Tokyo, Tokyo, Japan 113.

region (kT nucleus.

[5] the operators projecting space parity and number parity of the quasiparticle. The set of the cranked THFB equations with constraints is derived from a generalized formalism which assumes neither any special functional form for the quasiparticle distribution functionfP, nor for the entropy S. The THFB equation which we solve,

(1) is similar to the ordinary HFB equation [2,3] except that the definitions of the density and the pairing matrices involve the temperature: Pkl = ]B*(l -f)BT

+-@t+]k[ ,

@a)

Kkl = [B*(l

tAfe+],

(2b)

-f)AT

.

It is shown that eq. (1) not only turns into an HFB equation at T = 0,but also includes self-consistently the blocking effect due to our projections. A numerical analysis is carried out for the typical deformed nucleus 164Er. The pairing-plus-quadrupole force of the form

H i,t=~xQ'Q+tGP'P

(3)

is taken to be the same as in ref. [6], and the exchange terms are neglected. The following force constants [7] are chosen as the parameter set which reproduces well the yrast level scheme: x,, = xnn = -0.0418, x,, = -0.1100, G,, = -0.220, G,,,, = -0.198, G,_,_ = -0.173 and G,+,_ = -0.185 in MeV. The spherical 337

Volume 97B, number 3,4

PHYSICS LETTERS

15 December 1980 q

Nilsson bases of two major shells up to the li 13/2 level are employed for the single-particle states. The solution to eq. (1) is obtained by minimizing the following thermodynamical potential (or simply free energy) F with the use of the steepest descent method: F

(H) - Xp(Z)

- Xn(N)

- co())()-

ST,

5O

fit: 0÷~ z,0

(4) 2"

with the following constraints for the proton number, neutron number and the angular m o m e n t u m , respectively: (N>= A - Z,


()X)= I ,

(5)

where )tp, Xn and w are introduced as Lagrange multipliers. The result shows the existence of a solution with vanishing angular frequency, i.e. w = 0, for finite values o f / i n the region of temperatures greater than a critical value T c. The anomalies are observed in the vicinity of the "zero-rotation" line in all three diagrams'(figs. 1 - 3 ) . In fig. 1 the m i n i m u m of the isothermal curve for the excitation energy is for low temperatures located at I = 0, but moves away from I = 0

E

i

t /

30

/

20

UA~--

I

i

~---"'~i

~'~

i

i

2

5

4 kT (MeV)

Fig. 2. Dimensionless specific heat for positive parity state of given angular momentum. The dashed line is the same as in fig. 1.

t '

(MeV)

/,,"

51

-

F0 (MeV) -1

-2 X\

"'

575

-3

2

~

I 6

I

-4.

2

4

6

I

Fig. l . I s o t h e r m a l curve o f the e x c i t a t i o n energy f o r positive

parity state versus angular momentum. The dotted line connects the minima of the curves. The dashed line stands for the zero-rotation solution. 338

Fig. 3. Isothermal curve of free energy for positive parity state versus angular momentum. The dotted line connects the maxima of the curves. The dashed line is the same as in fig. 1.

Volume 97B, number 3,4

PHYSICS LETTERS

at the critical temperature to larger values o f / w i t h increasing temperature. The minimum is located near, but not on the zero-rotation line. This discrepancy must be related to the gradual changes of the structures of both proton and neutron shells, whose changes do not necessarily occur simultaneously. Correspondingly a slight shoulder is seen in addition to the anomalously large peak of the specific heat in fig. 2. The specific entropy (dS/dT)/k 2 also shows a quite similar enhancement at the same position, its peak value reaching 85 MeV -1 in the 0 + state. In fig. 3 each isothermal curve for the free energy has its maximum close to the intersection with the zero-rotation line. This fact together with the drastic enhancements of physical quantities like specific heat and specific entropy indicates the existence of a first-order phase transition associated with the solution of vanishing angular frequency. A similar result is also obtained from the solution for the case of mixed statistics, which allows mixtures of space parities and number parities. Since the nucleus is a finite system, and moreover only a finite number of single-particle levels is taken into account in our calculation, the peaks o f the specific heat and the specific entropy are broad and of finite height. This is in contrast to a phase transition in infinite systems. Therefore, the phase transition takes place in a region where the coexistence of the two phases is expected, and consequently a sharp line which separates the two phases cannot be defined. The critical temperature is predicted to be kT c = 0.5 MeV for the positive parity state (0.45 MeVfor the mixed statistics), and the corresponding excitation energy for the 0 + state is E c = 3 MeV (roughly the same energy also for the mixed statistics). Since co becomes negative in the region o f the new phase, the direction of the angular momentum is opposite to that for rotation. Therefore, it may be called the anti-aligned phase (or antiferromagnetic phase) in contrast to the aligned phase (or ferromagnetic phase). The new phase must be caused by the dominance o f the entropy force (i.e. the entropy term - S T ) as the manifestation o f the temperature effect in competition with the Coriolis force (i.e. the cranking term -co(Jx)). The sudden increase of entropy in the phasetransition region is clearly linked with the decrease of the gap values. In our solution at kT = 0.5 MeV, the gap value averaged over the proton shells is about 60% of its ground-state value of 0.80 MeV, and the one

15 December 1980 I

fl-t

I

I

I

I

I

r a p .

p-- -- n* ......... n-

/f

.....

10-1

/ i /

/// 10 -2

1.0

p

z

I ¢/ll

I

I

I

I

t I

I

I

[

I

Ep. (MeV) 05

I

0.2

0.4 0.6 kTlMeV) Fig. 4. The lowest quasiparticle energy and the corresponding distribution function in the 2+ state. The four curves correspond to the lowest quasiparticle energy in each of the four shells, the positive or negative parity proton shell and the positive or negative parity neutron shell. averaged over the neutron shells is about 70% of its ground-state value o f 1.04 MeV. As is seen in fig. 4, the rapid increase of the quasiparticle distribution function associated with the decrease of the quasiparticle energy in this region gives rise to a large negative contribution of the entropy term to the free energy, which is large enough to compensate the positive contribution o f the cranking term in the new phase. It will be seen in the follo~,ing that the Coriolis term even helps to decrease the sum over the quasiparticle energies in the new phase. Since co is small in the phase transition region, we retain only the term linear in co and neglect all terms coming from the nonlinear effect. Then as an indicator we consider the rotational contribution 6Eu(co ) to the summed quasiparticle energies approximated as

6Eu(~ ) ~ -coTr {Jx[1 - 2p + 2 (A fA ? - B *fBT ) ] ) I,t

2wI,

(6)

which is obviously negative in the new phase. To obtain the last line, we have neglected the third term in 339

Volume 97B, number 3,4

PHYSICS LETTERS

the trace because the functions f are small except for those corresponding to the lowest quasiparticle energies, and because o f the random phases of those terms. Our theory is clearly invariant under the simultaneous inversions w -+ - c o and I ~ - I . Therefore, any physical quantity must have its local maximum or minimum at I = 0. At low temperatures our free energy takes its maximum at I = 0, where the invariance of the theory under the full three-dimensional rotation holds. When the free energy is stabilized b y the increase o f the entropy at a temperature greater than the critical value, the unstable maximum point has to move away from this axis and the domain o f the new phase spreads with increasing temperature. Therefore the maximal symmetry holds for the zero-rotation line for nonvanishing values o f / . Along the zero-rotation line the three generators of rotation ix,J._( = J y +-iJz) are conserved. The non-zero value o f / i n this solution implies that the angular m o m e n t u m density operator corresponds to such a local field quantity whose Goldstone commutator with these generators has nonvanishing statistical average, i.e.

[)_+,

= -+2dx(x)

o.

(7)

Thus, the suitable choice o f the order parameter for the phase with maximal symmetry must be the vol-

340

15 December 1980

ume integration of the quantity in (7),

Io(T ) : (JX)to:O , which vanishes at T ~< T c and becomes finite at T > T c. To discriminate the anti-aligned phase from the aligned phase, the convenient choice of the order parameter must be r / = I - I 0 ( T ). Then, r~ < 0 in the anti-aligned phase. Detailed accounts of the theory [5] and. the numerical analysis [7] will be published elsewhere. The authors wish to thank Professor H.J. Mang for his valuable discussions and his warm hospitality extended to them. They are also grateful to Dr. B. SchiJrmann for his careful reading o f the manuscript.

References [1] R.A. Sorensen, Rev. Mod. Phys. 45 (1973) 353. [2] H.J. Mang, Phys. Rep. 18 (1975) 325. [3] A.L. Goodman, Advances in nuclear physics, Vol. 11 (Plenum, New York, 1980). [4] B. Banerjee, HJ. Mang and P. Ring, Nucl. Phys. A215 (1973) 366. [5] K. Tanabe, K. Sugawara-Tanabe and H.J. Mang, to be published. [6] J.L. Egido, H.J. Mang and P. Ring, Phys. Lett. 77B (1978) 123; Nucl. Phys. A334 (1980) 1. [7 ] K. Sugawara-Tanabe, K. Tanabe and H.J. Mang, to be published.