Backbending: Coriolis antipairing or rotational alignment?

I.D.2 [

Nuclear Physics A256 (1976) 106--126; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

BACKBENDING: C O R I O L I S A N T I P A I R I N G O R R O T A T I O N A L ALIGNMENT?. A. FAESSLER t, K. R. SANDHYA DEVI, F. GROMMER, K. W. SCHMID and R. R. HILTON lnstitut fiir Kernphysik, der Kernforschungsanlage Jiilich, D-SI 7 Jiilich, West Germany

Received 25 June 1975 (Revised 2 October 1975) Abstract: A theoretical description of the ground-state bands in even-mass nuclei is given which

includes the possibility of Coriolis antipairing (CAP) and rotational alignment (RAL) in the explanation of backbending. The method is based on a combination of Hartree-FockBogoliubov, cranking model and particle number projection and applied to the high (=< 18/~) and very high ( ~ 80 ~) spin states of 162Er. The results indicate that the nucleus behaves according to a combination of CAP and RAL effects. Pairing for all neutron pairs near the Fermi surface is markedly reduced although the two i~ pairs nearest to the Fermi level are somewhat more affected. However the dominant contribution to the total angular momentum for spins greater than 12 ~ comes from one i~ pair, (with large components ~2 = ½, ], 6). For the very high spin states (30 ~ ~ J --< 80 ~) the nucleus becomes triaxial and well defined shape isomers appear.

1. I n t r o d u c t i o n

The anomalous behaviour of the moment of inertia ("backbending") at high spins 1-3) has been described by two competing microscopic models, the Coriolis antipairing 4, 5) (CAP) and the rotational alignment (RAL) effects 6). In both models backbending is due to the intersection of the ground state and an excited band with larger m o m e n t of inertia. The nature of the excited band for CAP is the unpaired (or at least neutron unpaired) band, whilst R A L assumes a two quasiparticle state of two i~ neutrons aligned along the axis of rotation. Both models are able to describe qualitatively and in several cases even quantitatively backbending in rare earth nuclei 7,s). However more detailed studies on R A L 9) and the CAP Io. 11) models show that the quality of agreement varies f r o m nucleus to nucleus depending strongly on the single-particle level scheme and also in special cases on the force parameters of the two-body interaction 1 o). In attempting to describe backbending one obvious improvement is the combination of the CAP and R A L effects. The data appears to suggest both effects play an important role: the linear increase of the moment of inertia with the square of the rotational frequency for lower ( < 10 h) angular momenta, phenomenologically well described by the V M I (variable moment o f inertia) model 12), can be described microscopically within the framework of the t Also: University of Bonn, D-53 Bonn, West Germany. 106

BACKBENDING

107

CAP effect 9,13,14). The decrease of the moment of inertia near spins 18 + and 20 + (downbending) which is found in several nuclei has been explained 9,14) using RAL. Major differences between CAP and RAL do not exist from the standpoint of the correlation energy even though in one model all and in the other model only one pair is broken since the total pairing correlation energy for the neutrons is about 2.4 MeV, whilst the correlation energy of one pair near the Fermi surface may already be 1.8 MeV. Therefore one does not need much more energy to break all the pairs than that required to break the pair closest to the Fermi surface. The more important difference may be seen in the fact that RAL aligns the decoupled particles along the axis of rotation. A description of this effect is only possible if one does not require time reversal symmetry and allows for the mixing of different angular momentum projections along the intrinsic symmetry axis, either explicitly 15) or implicitly 16), by allowing for an asymmetric deformation 7. With both methods Kumar 1~) and Mang 16,19) are able to describe both CAP and RAL. The work of Mang and collaborators 16,19) is an especially important step in the construction of a unified CAP and RAL model. It combines the self-consistent Hartree-Fock-Bogoliubov approach with tb.e cranking method for determining the average angular momentum of the state. The particle number is conserved only on average in this approach. This is a serious drawback. Recent studies of the Jiilich group 1o) showed that good proton and neutron number is essential for explaining the behaviour of the moment of inertia at high spins. Even projecting out admixtures with two protons ( Z + 2) or two neutrons (N+ 2) more or less than the desired nucleus turns out to be insufficient. Satisfactory results were only obtained if one projects out the admixtures of Z_+2, Z + 4 , ( Z + 6 ) and N+2, N+4, (N+6) protons and neutrons respectively. This sensitivity may be understood in the following way: The admixtures of the different neighbouring nuclei may vary from angular momentum to angular momentum. Suppose the admixture of the (Z+4, N) nucleus increases by 0.5 ~ and that of the ( Z - 4 , N) nucleus decreases by 0.5 ~ between angular momentum J and J + 2 . This implies an error of 4 x 8 MeV x (0.005 +0.005) = 0.32 MeV in the transition energy A E = E j + 2 - E I . Since the difference between backbending and no backbending may often consist of a 50 keV change in transition energy, the above uncertainty appears to represent a severe drawback in any description not possessing good particle number. However this error may be reduced if one adjusts the average particle number for each angular momentum separately. The aim of this publication is twofold: (i) The work of refs. a, lo,11) will be extended to allow for violation of time reversal symmetry and asymmetric deformations y. In order to solve the problem numerically one is forced to use in place of strict angular momentum projection the cranking approach. The advantage over the work of Mang et al. 16,19) resides in the use of particle number projected wave functions which, as discussed previously, are essential for any reliable theoretical description of backbending. [Further advantages are the larger single-particle basis and the method used for determining the cranking frequency o~, which works even in the

108

A. FAESSLERet

aL

backbending region. A drawback with respect to the work of Mang consists in the variation of potential and pairing parameters only. This is the price one has to pay for varying the total energy after particle number projection.] (ii) The second purpose of this work is to study the shape of the nuclear potential surface at very high spins (~> 30 h) for which pairing has totally disappeaied. Interest in such a study stems from the possibility of transferring such large angular momenta to target nuclei using heavy ion accelerators. However one should keep in mind that as soon as one reaches the fission barrier (J ~ 80-100 h) this description of the nuclear potential must fail, since it is restricted to a one center deformed oscillator potential. In sect. 2 we will outline the theoretical model, which is based on the particle number projected Hartree-Fock-Bogoliubov (QHFB) approach. The cranking model is used to conserve the angular momentum on the average. The pairing plus quadrupole force Hamiltonian is employed. In sect. 3 the results for the high-spin states (<- 20 h) are discussed. A study of the very high spin states (30 h < J < 80 h) is presented in sect. 4, The main conclusions of the work are summarized in sect. 5. 2. Theory The aim of this section is to develop a particle number projected Hartree-FockBogoliubov (HFB) theory with the inclusion of the cranking term. Since HFB equations with particle number projection are not available (and would also be numerically difficult to solve) one is forced to minimize the particle number projected expectation value of the many-body Hamiltonian directly. To reduce the number of variational parameters we introduce a model Hamiltonian Ht which produces a trial HFB wave function depending on a few parameters. These are then varied to find the minimum of the particle number projected expectation value of H for a given average angular momentum. 2.1. THE MODEL HAMILTONIAN We construct the Hartree-Fock-Bogoliubov (HFB) trial wave function with the help of the following model Hamiltonian:

Here In>, I~;>. . . . are the single-particle basis states [eq. (12) and (13)] to be specified later. The single-particle Hamiltonian -- 8:az, ~-~,hogo
st. r(r22(s,

Y2_ (s,

(2)

is defined with the help of spherical single-particle energies and the deformation parameters//and 7 [see e.g. Bohr and Mottelson 17)]. The radial coordinate is given

BACKBENDING

109

in units of the oscillatorlength b ---(h/moo)*,which is connected with the oscillator energy he% = 41.2A -÷ MeV. The parameters /(2Z/A)~ ate = [ ( 2 N [ A ) ÷

for protons for neutrons,

(3)

have been introduced by Kumar and Baranger is) to ensure equal radii for protons and neutrons. The second term in eq. (1) is the pairing force with the parameters Gpand Gn. The rotation around the intrinsic x-axis is enforced by the constraint -o~Jx. The Lagrange multiplier co (given in energy units) is determined by the condition t 9) = x/~(J+ 1).

(4)

[For the final determination of coj see eq. (22).] The wave function [HFB) is defined as the quasiparticle vacuum IHFB) = YI a#10).

(5)

all #

The quasiparticles are given by the transformation a + = ~ {Cq+A,,,~, + CaB,,~}.

(6)

gl

The Euler-Lagrange equations for the coefficients A,,~ and B,,~ are determined by minimizing the energy,

E~, = (HFBIHt-2~,IHFB)/(HFBIHFB),

(7)

for a given multipliero~ (cranking frequency). The Lagrange multipliers2~ are determine d by conserving the number of protons Np and neutrons N. on the average = N,.

(8)

Here we use ] ) as an abbreviation for the state [HFB). Minimization of eq. (7) leads to the HFB equations

-eo(aJj~l/;) = -co(alLI/;>-)~,o~,,~,

= - = -G,a~, -r,~>o
00)

The single-particle basis states I~> and I-a> will be defined in the next part of this section. The contribution of the pairing potential to the HF field is neglected as is the usual practice within this model. The solution of eq. (9) yields the trial function (5) depending on the parameters #, y and the pairing gaps Ap and A. of the model Hamiltonian (1).

110

A. FAESSLER et al.

2.2. CHOICE OF THE SINGLE-PARTICLE BASIS Due to the cranking term - c o J x the Hamiltonian (1) and hence also the solution of the corresponding HFB equations (9) are not invariant under time reversal symmetry. However isospin and parity are still good quantum numbers for the single-quasiparticle states. This simplifies the solution of eqs. (9) by reducing the H F B matrix to four square sub-matrices having no common matrix elements. A further symmetry of/art is that of invariance under rotations through an angle rc around the intrinsic x-axis. One verifies easily:

[u,, e'~" ] = 0.

(11)

This symmetry allows a further reduction of the H F B problem, if one introduces eigenfunctions of exp(ircJ=). We denote the single-particle eigenfunetions of this operator by la>, I/;>. . . . I - a > , I - t ; ) . . . . . The spherical shell-model states are characterized by la) - I z n / j t 2 ) o , I~) = (--)l'+J'-a°lznlj--f~)a.

(12)

The state la> and its conjugate are given by *

la>

= x/~la)+Pol~>),

I-a> = x/~Po(la>-Pol~>),

(13)

with P . = ( _ ) z . - a . + , . The phase factor Po has been included in the definition o f I - a > so that the pairing interaction in this basis [see eq. (1)] has a similar form to that in the spherical shell-model basis (12). One may verify as a consequence of the e ~"sx symmetry that one has in this basis the following relations: < a i r - cojxl - ~> = < < - air-

An, s --- 0,

air-

o~jxl [~> = O,

o,j=l -/;> = , As,-5 - ( ~ - ~ ; I A ) = - A , . 6 ~ , ~ .

(14)

Here A, is the pairing potential which is state independent for a constant pairing force. The density matrix p and the pairing tensor x are defined by the expressions

p'.,5 =

=

EBb,

_~B5,* - . ,

,¢~, -5 = = Z A~,,n*5,,.

(lS)

a{

Eqs. (14) immediately give the relations P-',-5 = P-~,5 = 0, tea,5 = x_~,-5 = 0. t The same symmetrization was found independently by Goodman 2~).

(16)

BACKBENDING

111

These equations restrict the quasiparticle transformation (6) so that A~, _~ = A_~,~ = Bs,~ = B_~,_~ = 0,

(17)

and so the summation in eq. (15) is restricted only to "= > 0". In eqs. (12) to (16) the single-particle and quasiparticle symbols [a), [~) and [~) are restricted to the non-time-reversed or non-conjugate states. The time reversed [~) and conjugate states [ - ~ ) , [ - ~ ) are indicated explicitly. Due to the violation of time reversal symmetry for non-zero co the quasiparticle energies for [=) and [ - ~ ) are not degenerate. Using eqs. (14) and (17) one can reduce the HFB equations (9),

Here A and B stand for the non-zero components (A~,~, B_~=) in eq. (9), A, the pairing gap parameters Ap and An and I is the unit matrix. The F_ and F+ are defined with the help of eq. (I0) as F-cojx and F+oJjx, respectively. Eq. (18) has only half the dimension of eq. (6). The remaining quasiparticle states are defined by the transformation coefficients

= (B - *

A,,- ~

(19)

and the energy

+

-(El

< 0).

The HFB equations (18) in general yields the same number of positive (E~+) as negative (E~') quasiparticle energies. Due to the violation of time reversal symmetry by the cranking term the relation E~+ (co = 0) = - ~ (o~ = 0) is not valid for frequencies ~o # 0. The energy E~+ is therefore in general different from the quasiparticle energy E_+~ of the conjugate (but not time reversed) state 1 - ~ ) . [It may happen ~6) that one or more quasiparticle energies E~+, E_+~ become negative for some cranking frequencies to. This indicates that for higher angular momenta one or more quasiparticle states lie lower than the H F B state. In an even-mass nucleus this would imply that the 0 q.p. band is crossed by a 2 q.p., which takes over the role of the yrast band. ] Even though we do not have the usual quasiparticle degeneracy it can be shown that for doubly even nuclei with 0 q.p. states the density matrix p has doubly degenerate eigenvalues, so that the Bloch-Messiah theorem 2o) is still applicable. That is, one can write the H F B wave function in the so-called canonical basis Ji), [k) . . . since

(p+)~,s = E 0

(P-)a,s = E ( - g l - i ) v ~ ( - i [ - ~ ) , i>0

b+ = z,~>oC+(ali)'

b+-" "--"r,>o )-" C+-r'(-a[-i)"

(20)

112

A. FAESSLER e t al.

2.3. N U M B E R PROJECTION

The solution of the HFB equations (18) serves only as a trial function ]HFB, 09, fl, 7, Ap, A.),

(21)

defined by the model Hamiltonian H t of eq. (1). The trial function (21) is characterized by five parameters: cranking frequency to, deformation parameters fl and y and the state independent pairing potentials Ap and An for the protons and neutrons respectively. The cranking frequency is determined by the condition:

('J~'P(~") - ~/J(J + 1).

(22)

Here ~p and ~n are the particle number projection operators onto good proton and neutron numbers respectively, with 1 ~'2, = ~ J o d~0 exp {½i(/9,-N,)q~},

(23)

where N, = (Np, N.) are the number of protons and neutrons and 19, is the corresponding particle number operator. The parameters fl, 7, Ap, A. of the trial function (21) are determined by minimizing the particle number projected energy expectation value for a definite angular momentum J. This is fixed on the average by the Lagrange multiplier to. [For every set of parameters fl, 7, Ap, A. a new toj has to be chosen to ensure that eq. (22) is fulfilled.] Thus Ej(/~, :,, A,, A.)

(HFB, tos, P, 7, A,, A.IH~,~.IHFB, tos, #, 7, Ap, A.) (HFB, tos, fl, r, Ap, A.IQpQ.II-IFB, tos, #, 7, A,, A.)

=

(24)

= minimum. Eq. (24) then represents a minimization after particle number projection and H is now the true two-body Hamiltonian (25) and not the model Hamiltonian Ht ' defined in eq. (1). We choose for the many-body Hamiltonian the pairing plus quadrupole force expression of Kumar and Baranger is).

H

Z

= a

+ G,.6,.,,,C~ + C_~,C_~Cs

ab>O

-½ Z z~..zt,C~C~C~C; • abed#

(25)

The quadrupole force constant X,.X,b ~r ~r is defined as ~/~,.~,b. The quantities ~. are defined in eq. (3). The numerator of the energy expression

BACKBENDING

113

(24) are functions of the variational parameters fl, 7~ zip, zi~: = X

i

~,,,~P,(N,,- 2)

-2 ~ 6.... ~G,. ~ %u,<-al-i><~lk>vku,<-~l-k>P,,(N,,-2) ik>O

ab>O

- ~ ik

f2" d

= Jo ~o H

i>0,zi=t

x~ Z*.~v~,?Pa(N,,-4),

(26)

(u*/+v*/+2u~v2 cos ~o)~r (27)

The quantitiesPi(N.,-2), Pik(N.,--2)~.,,. k and P~k(N.,-4)b...k are defined in the same way as <(~p(~.> in eq. (27) except that in products and sums the terms with i or i,k are dropped and N,, is replaced by N , , - 2 or N,~-4. In the case of the proton-neutron quadrupole force one has V,k(N,--4)6,,,,~ =

P,(N,,-2)P,(N,~-2)/<~,(~.>.

(28)

In expressions (26) to (27) the normalization factors have been dropped since the total energy (24) is explicitly normalized. 2.4. T H E M O D E L

The high and very high spin states are described by minimizing the total energy (24) defined with the many-body Hamiltonian (25). The trial functions are HFB solutions of the model Hamiltonian (1). They depend on the cranking frequency co, the deformation parameters/1 and 7, and the pairing parameters Ap and A.. These wave functions are projected onto good proton and neutron numbers using (23). The expectation value (24) of the total Hamiltonian is then minimized as a function of ~' 7, Ap and A, for a given average angular momentum (22). The average conservation of the angular momentum (22) defines a cranking frequency o j for every J and every set of parameters t, 7, Ap, A,. The numerical procedure adopted is summarized in the following. (i) Choose meshpoiuts for the variational parameters t, 7, Ap,/I,. (For example, five values for every parameter.) (ii) Solve for the meshpoiuts defined in (i) the HFB equations (18) for different Lagrange multipliers (angular velocities) to. Conserve the particle number of the average by choosing 2p and 2~ according to (8). (iii) Determine the angular velocities c0j for all angular momenta (0 ~ o j ~_ 0.3 MeV) with the help of the average conservation of the particle number projected angular momentum expectation value (22) for the meshpoiuts defined in (i).

114

A. F A E S S L E R et aL

(iv) Calculate the particle number projected expectation value (24) of the total Hamiltonian (25) for the meshpoints of the parameters 7, fl, Ap, A, for all angular momenta (or oJs). (iv) Search for the minimum of the energy expectation value (24) for a given average angular momentum defined by (22). In calculating the very high spin states (J > 30 h) we assumed that pairing is reduced to z e r o (Ap = A n = 0 ) .

3. High spins As a single-particle basis we used the oscillator shells N = 4 and 5 for protons and N = 5 and 6 for neutrons. This basis may contain up to 170 nucleons. The inert core is 1~o,-.4oLr7o. This basis is identical with the one used by Baranger and Kumar 18). The pairing and quadrupole force constants are identical to that used in ref. 1o), Gp = 23//1 MeV, G. = 18[AMeV, Z = 73A- 1.4 MeV,

hcoo = 41.2 A -~ MeV.

(29)

The spherical single-particle energies are taken from r~f. 18) p. 535, table 1. The quadrupole matrix elements for the upper proton and neutron shells are reduced by factors of T~~~ and ~3 respectively. This amounts to reducing the mean square radii of the upper shells to those in the lower shells ~s).

TT,

O--EXPERiMENT

1162,_ [ I [__681=rg4 &--THEORY(tB,An,ZX p)

-160

RY (p,Y,Z~n,A7

,~ 120 W

z__ O

-p

W O

~ 40

o0

--

I

a0~

I

(108

I

o.12

(ROTATIONALFREQUENCY x l~)2[MeV=] Fig. 1. Twice the moment of inertia of 2e2Er as a function of the square of rotational frequency. T h e circles indicate t h e experimental values f r o m a n g u l a r m o m e n t u m 2 + to 20 + . T h e triangles a n d squares give t h e theoretical values u p to 18 +. F o r t h e triangles t h e a s y m m e t r i c d e f o r m a t i o n ~, w a s fixed to z e r o .

BACKBENDING 1 20

I

I

I

I

115 I

I

-

1

~

-

162 -=E :D

16

o 12 - -

--

¢Y

z lit

"~ 0.04

I

I

I

I

I

I

I

0.08

012

016

0.20

0.24

0.28

0.32

CRANKING ANGULAR VELOCITY w [MeV]

Fig. 2. Average angular momentum projection along the rotational axis for good particle numbers as a function of the crankitlg angular velocity. For every average angular momentum projection the cranking frequency is iterated for the deformation parameters ~ and 7, and the pairing gaps ,4p and ,4~, which roinin3i~ the particle number projected expectation value (24) of the many-body Hamiltonian (25). This procedure guarantees a unique function of co against the angular momentum even in case of backbending.

t ts

l 162.. 68tr94

:~

I

[ ~

I

I

I

An

_

CL .< 1.o

z

~: 0.5

o. o

I 4 8 12 16 ANGULAR MOMENTUM

I

I

20

24

J[l~]

Fig. 3. Pairing gap parameters 3 r and ,4= of 1621~.r as a function of the total angular momentum. These parameters are determined for every angular momentum by a minimization of the particle numbocr projected expectation value (24) of the many-body Hamiltonian.

Fig. 1 shows the usual backbending plot (moment of inertia versus square of the rotational frequency). The rotational frequency is defined here by the excitation energies using the so called Copenhagen convention 22). The experimental critical angular momentum at which backbending occurs is 14 h. Our calculation yields Jo = 12h. Fig. 2 shows the variation of the total angular momentum with the cranking frequency (22) at the energy minimum of (24). To have a uniquely defined function

116

A. FAESSLER et al. I


I

I

I

I

I

162

I--

--~

LOWER MINIMUM(prolate}

O6 - - - - - Z _. O

,2nd MINIMUM (oblate}

68Er9z,

<~ 0X, O LL 02 ILl a

I

1

I

I

I

I

20

1.0

60

80

100

120

ANOULAR

MOMENTUM Jib]

Fig. 4. Deformation parameter /~ for 162Er as a function o f the total angular momentum. The angular momenta above 20 + are calculated without the inclusion o f pairing. The decrease between J = 8 0 / / a n d 100 ~ may reflect the basis size which is too small for the description at these angular momenta. At the maximum possible total angular momentum ~r _-- 154 ~ o f this basis the deformation is ~ = 0.03. The dashed line indicates a shape isomer with a negative deformation. This result does not include the effect o f Coulomb repulsion between protons. Fig, 16 shows dramatic changes for the very high spin states when Coulomb energy is included.

I,LI UJ

'

$

60

I

162Er --

68

"

i

'

I

'

gz,


o~ 1.0

LOWER MINIMUM {PROLATE) .....

SECONDMINIMUM {OBLATE)

.l-Ltl "S" ~r <

20

30 60 90 ANOULAR MOMENTUM J [1~]

Fig. 5. Asymmetric deformation 7 as a function o f the total angular momentum. The shape isomer with 7 = 60* above J ----80 h may be connected with the too small size o f the basis for these high angular momenta. The above curve does not include the effect o f the Coulomb foree as already mentioned in fig. 4,

we consider in the numerical calculation co as a function of angular m o m e n t u m J. The values of the variational parameters Ap, A,, p and y at the energy minimum are plotted as functions of J i n figs. 3 to 5. The deformation parameters which determine the shape of the potential are given up to angular momentum J -- 100 h (see below). The asymmetric deformation y below angular momentum 20 h is smaller than 10 °. It reaches its maximum value of 14 ° at an angular momentum of 30 h. [This result is calculated without inclusion of the Coulomb energy. For a discussion of the results with inclusion of the Coulomb energy see sect. 4.] The fact that we display

BACKBENDING

I

I

I

NEUTRONS

I /'~Ts(m

117

I

I

//)13/~16o61 .//.~'3/2Is01]

I

PROTONS

,"

/

/

...-~c/

~

9/2-[505]

82

sr.,-tsm

/" "~. {lrz-lSOSl " , ,

" ~ Vz'lr~ol V2-ts3ol

I-_I

/

. llq-~

.

/

/I

0

11/215o51 - ~11,00t

~

D

[50 1

.

.

_

_

5/2.~z.021_

.

...

.

.

.

.

.

-

s,~.{+.,3~-

._

"%,,

/

--

3

~p~"

.

" [ ~ ~ - - - - - = : = =

I

,

~ 1/2-1541]

I 0.1

I q2

I (~3

I 0 OEFORHATION

I 0.1

I 0.2

I

-

o.s

I~

Fig. 6. Nilsson level diagram for rare earth nuclei. These single-particle energies are calculated with the spherical single-particle energies is) used in this work. The single-particle Hamiltonian corresponding to the above energies is given in eq. (2) if one puts the asymmetric deformation 7 equal to zero. t

I

--I

~62Er

1.0

68

~_

r

....

.-

0.4 --

i

{Ti 13/2o~ =1/2|

[Ti 1312. Q =3/2)

~-,~,

0.6 --

o z =_

I

94

,,._

~-.

0,2

I 4 8 12 16 ANGULAR MOMENTUM J [~]

20

Fig. 7. Pairing potential (31) as a function of the total angular momentum for different neutron single-particle levels in 162Er containing, for low angular momenta, mainly an i ~ admixture with projection D on the "symmetry" axis. The D = ~ level (solid line) lies nearest to the Fermi surface. It is followed by the 1'2 = ~ level (line with paired circles). All neutron pairing potentials are reduced quite drastically with increasing angular momentum (CAP).

118

A. FAESSLEK e t z_ I 162 = "T 10 ~- 68 '-r 9/" ~- ' | | 0 . 8 ~

aL

T i 13/2 ~q=512 .....Ai-i (At =parameter) ----

&i-i (Az eq.(30))

A

o.~

0.2

l 8 12 16 ANGULAR MOMENTUM J [h]

20

Fig. 8. Pairing potential, for the i.~, .O = ~ neutron level, calctdated using in cq. (31) the value

ofzl~ obtained (a) after minimising the nurnber projected expectation value (24) and (b) employing eq. (30).

in fig. 3 state independent pairing parameters

As = G~ ~ (cTlk)ukv~(-dl-k),

(30)

kd>O

does not indicate that the pairing potential in the canonical representation (20) A,_, = -

~

<-il-a>

~

<-al-k>ukvk

k>O

ad>O

= G~ ~ <-i[-a> ~ <--al--k>ukvk= As ~ < i l a > < - i l - a > , a>0

dk>O

(31)

a>0

is also state independent. The state independence of As is only due to the state independence of the pairing force. In order to further the discussion it is convenient to introduce a Nilsson level scheme. In fig. 6 the single-particle energies are given for the one-body part (2) of the model Hamiltonian (1) with 09 = 0 and V = 0. Fig. 7 shows the four lowest pairing potentials A i, -i for neutrons with positive parity. They correspond mainly to i~ orbitals. The projection ~ given, is the main admixture (see fig. 15 below) for angular momenta up to 12 h only. The pairing potential A~, -t for the (vi~f2 = ~) level is reduced most strongly with increasing of total angular momentum. The expression for the pairing potential (31) is given without particle number projection. But the parameters A, are determined by minimizing the particle number projected total energy (24). One therefore finds an

BACK.BENDING

119

inconsistency in introducing expression (30) into eq. (31). Fig. 8 shows that for large pairing the difference is largest, while it disappears altogether for small pairing. This is in agreement with the fact that particle number projection is more important for strong pairing. I ~[ "- 1.0

--

<

I

1 - -

162= . . . . . 68 ~ ' 9 4

l I tit.hill2.9=712) (rc hg/2,Q=312 |.

- , - . - ( ~ , h %.~=%1

0.8

0.6

o

-"

0./,

0.2 ~

I 4

I

I

I

I

8 12 16 20 ANGULAR MOMENTUM J {'h ]

Fig. 9. Pairing potential (31) as a function of the total angular momentum for neutron a n d proton levels in l~=Er. The pairing potentials of levels containing, at low angular momenta, mainly h~ = t and ~9 = t neutrons show a decrease which is roughly the same as for the neutron i@ = ½ and D = J levels (see fig. 7). The pairing potential of the h@ K2 = ~ level shows a remarkably weaker decrease with increasing total angular momentum. i

~I

3 2

--

~

I 164 = r

[

~

,

68 L..94

' ~ ~

i

(Yi13#2,~= 512) (Ti1312.£Z=312). {Yi 13/2, £~=712)-(Ti13/2,9= 1/2 )

_~-----.-~-~__~-=_.

w

-1

I

I

I

1

I

4

8

12

16

20

ANGULAR MOMENTUM J I n ]

Fig. I0. Quasiparticle energies in 16~Er for some i ~ neutron levels corresponding to the lowest energies for this parity and charge as a function of the total angular momentum. Due to the lack o f time reversal symmetry quasiparticle ([=> solid line) and conjugate ([--,,> dashed line) states are not degenerate. The negative quasipartiele energy for ~i@, t2 = t means that a hypothetical odd-mass nucleus with the same particle number has a lower total energy than ~¢2Er for these angular momenta.

120

A. F A E S S L E R e t al.

I 1.2 -

I

I

166~Erg4

I

(Yf~2'~=312)

. . . . . . >. z'"e: 0.8 -"N..

.....

-

1Th1112,~=11/21 {Th9/2 ,•=5/2 )

Lu hl

o

o

iiii 4

I

8 12 16 20 ANGULAR MONENTUM J ['h]

Fig. 11. Negative parity neutron quasiparticle energies as a function of the total angular momentum in ~6ZEr. The quantum nmnbers indicate only the main admixture for small angular momenta. Fig. 9 demonstrates that the pairing potential of the h i neutrons is reduced similarly to the vi e fl = ½ or f~ = ½states. The pairing potentials for the protons (see fig. 9 for the ~h÷ f2 = ½ state) are not as strongly reduced as those for neutrons. Figs. 10 and 11 illustrate the variation of the quasiparticle energies with the total angular momentum for the lowest neutron quasiparticle energies with positive and negative parity respectively. One sees that quasiparticle energies E~+ and E_+= for conjugate states are degenerate only for angular momentum zero. The energy E~¢, ~=~ becomes negative in the backbending region. This means that 1 q.p. state built on this level in a nucleus with the same average particle number (but of odd number parity) would have a lower energy (gapless superconductor). The strong reduction of all neutron pairing potentials shown in figs. 7 and 9 could indicate that backbending may be described by the CAP effect. However a totally different picture is found if one checks the contributions of the different q.p. states in the canonical representation to the total angular momentum. At angular momentum 16 h one finds for protons (Tr) and neutrons (v) of different parities the following particle number projected contributions: Jtot.t

=

16h,


(J~,~-Q~)/(Q~) = 1.6h,



/
(32)

The largest contribution comes from the neutrons with positive parity. Figs. 12 and 13 show the various contributions of the states and conjugate states to the total angular momenta. At angular momenta above the backbending region, the particles in the

BACKBENDING

121

vi~ f2 = ~ contribute the largest part and show almost full alignment [(ix, vi,~, ~ =t) 10.5 h as compared to 12 h]. This state and its conjugate state are both almost totally occupied (see fig. 14) for the high spin states. This guarantees the maximum angular momentum and energy gain from such alignment. The nature of the two i~ neutron states with g2 = {([1)) and ~2 = ~([2)) is shown in fig. 15. Up to angular momentum 12 h these quantum numbers determine the nature of the single-particle states in the canonical representation (20). Above angular momentum 14 h the character of these states is totally changed. I A 10

----

~8 x

1

I

I

162 68 1- r94

--

~- 6 _

Ti 13/2.Q=7/2 Ti 1312 ,£t = 5/2

Ti 13/2,~=3/2 non c o n j u g o t e s t a t e s

Z LU

=E C)

Z

n~ <

z

2

/, 8 12 16 20 TOTAL ANGULAR MOMENTUM J[l~]

Fig. 12. Angular momentum component along the rotational axis for different quasiparticle states in the canonical representation for the neutron i 9 levels as a function of the total angular momentum. The s.p. contribution to the total angular momenta is calculated with particle number projection. The level which contains mainly an i~t ~ = t admixture, shows alignment above the backbending region (RAL).

The situation found in this section may be characterized in the following way: For angular momenta up to 12 h we see only the Coriolis antipairing (CAP) effect [refs, 4, s)]. The pairing potentials of a large number of neutron pairs are markedly ieduced and the total anglar momentum is composed of comparable contributions from different pairs. In the backbending region the rate of decrease of the pairing potential for the i¢ neutron pair lying nearest to the Fermi surface is somewhat larger. This pair is aligned along the total angular momentum and contributes more than 50 % of the angular momentum just above the backbending region [rotational alignment 6, 7) RAL]. This means that CAP is responsible for the linear increase of the moment of inertia at low spin states ( < 12 h). CAP still continues for higher angular momenta but the decisive impulse for backbending is due to RAL. The slight decrease of the moment of inertia above the backbending region which has

122

A. FAESSLER I

I

et al.

I

1

I

1~ Er94

lO A

.

~a

.

.

T, i 13r2,f~ = 712

.

- -

Y . i 13r2. f l = 512

e....o.-.o y , i -3r

13/2, f~ = 3/2

conjugate states

n~

~2

4

8

12

TOTAL ANGULAR

16

20

MOMENTUM

Jl'h]

Fig. 13. Angular momentum component along the rotational axis for conjugate quasiparticlestates (see fig. 12).

1.0

% oB .¢ m onn Z

o

06

--

~ •

0.4

__

................. Y. .h9/2,5/2

8 8 0.2

Ii

/ "/ ,

~i/.-/.f"'X..X

--

_

0

5

/\

Y i13/2 ; 7/2

I O

I

I &

I

I 8 ANGULAR

I

I

-

I

_

I

12 MOMENTUM

16 J

--

~,

.. . . . . . . . . . . .

_

.

\yi1~2;7/2-

i~" /i

j

_

V i13/2

;3/2

I

I

--

20

['hl

Fig. 14. Occupation probabilities in the canonical representation o f neutron states lying nearest to the Fermi surface. One notices the drastic changes in the region o f backbending d~r ___ 12 + to 16 +. The quantum numbers indicate only the main admixture for low total angular momenta.

been found in some nuclei is also explained by RAL. After the first pair is aligned the energy may again increase faster with angular momentum. This is reflected by a decrease of the moment of inertia. Although particle number projection seems to be decisive for the quantitative results it does not seem to affect these qualitative features 16).

BACKBENDING

1,0 _ 1

I

I

I

I

t

12 3

I

I

~ 1 2 ) 2

o8

.............

I 162

. . . . ~~

68

Er

I

I_

9t.

~0,6

~

Ti 1312 odmixtures of non ~ ~ ( 7 canonical repres.

0./. 0.2

"~ /

2

tl2}

~

. ~ _ . . - ~ . - ~ - ~ . i _ 1.#~2}1

I 0

I

I

I

]

I

)

2

~"

~ \ . f ' ~ "'~'13/211 // ~ "

-
~C:>'

I

z, 8 12 ANGULAR MOMENTUM J [ ~ ]

1

"- . . . . .

[

I 15

v/2h

I

I 20

Fig. 15. The square of the i ~ mixing amplitude for the two positive parity neutron levels lying ncarost to the Fermi surface. The levels [1> ([2)) are the non-conjugate canonical states which contain mainly an i~/, D = t (1"2= 5) admixture at low angular momenta. This graph displays the change of character of the wave function around the area of backbcnding.

4. V e r y h i g h s p i n s

The above calculations in t62Er are extended here to very high spin states (J < 80 h). For these calculations we assume that all pairing correlations have disappeared. The variational parameters used in minimizing the total energy (24) are now only the deformation parameters fl and 7. Due to the lack of pairing, particle number projection is not needed. The cranking frequency coz is therefore determined by the expectation value (4). The Hartree-Fock Slater determinant is formed from the solution of the model Hamiltonian (1) without pairing. The particles always occupy the lowest single-particle levels. The single-particle basis and the parameters of the pairing plus quadrupole force are unchanged [see eq. (29) and text nearby]. The high spin states (Jr < 18 h) showed only small changes in the deformation parameters fl and ~. This is not to be expected for the very high spin states. One is therefore obliged to include the effects of Coulomb energy. Here we used an expression first given in the paper of Carlson 23). It represents an exact expression for the Coulomb energy appropriate to that for a triaxial ellipsoidal shape with constant charge density. Between angular momentum 80 h and 100 h one expects that fission is induced by the centrifugal force 24). This naturally enough cannot be described within a basis given by a one centre oscillator potential. One should therefore anticipate that before fission starts the limitation of the model will affect the quality of the results. Under the assumption of a rigid-body moment of inertia the liquid drop model 24) yields for 162Er an oblate deformation having rotational symmetry about the rotation axis (x) with y = - 6 0 °. [The problem of having a moment of inertia for an

124

A. FAESSLER et al.

Z=68

A.-282.2 NeV B.-281.9 NeV

Z : 68

A=162 J =30

c.- 2ere Nev

A=162 J =60

?

D.-2769 NeV

%~

A.-2r~.0 NeV B.-262.5 MeV ~(~

F.-2776 HeY G- 279.0NeV

C.-258.0 NeVD..257.9 NeXt ~,

F.-264./, HeY G.-262.0 NeV

N

-t0

V'- o --

'

¥o

Fig. 16. Deformation energy surfaces in a62Er for total angular momenta .pr = 30 + to 60 + as a function of the Bohr-Mottelson deformation parameters/~ and 7'. The equienergy lines (given in MeV) are renormalized relative to the lowest minimum having positive 7'. The total energies of the minima are indicated relative to a a~°ZrTo core. One should notice the appearance of many well defined shape isomers. These results contrast sharply with the results obtained without the inclusion of Coulomb energy for the very high spin states (see figs. 4 and 5). axis of rotational symmetry is not discussed by these authors 24).] Within this model the deformation fl increases from fl = 0, 7 = - 6 0 ° to larger//-values until the instability point is reached at around angular m o m e n t u m dt = 80 h. The nucleus then becomes triaxial [ - 6 0 ° < 7 < 0°] until, at the fission barrier at angular momentum around Jr = 92 h, it again assumes an axially symmetric shape with symmetry axis perpendicular to the rotation axis. The qualitative inclusion of pairing in this picture prefers positive ?-deformation. This may be understood within the framework of a two fluid picture in which only those nucleons outside an axially symmetric core participate in the rotation o f the nucleus. The moment of inertia for positive ?-deformations (x-axis larger than y-axis) is then larger than that for negative ?-deformations. Positive ?-deformations are therefore favoured with pairing present because they lead to a lower total energy. For higher angular m o m e n t u m for which pairing disappears the liquid drop model again predicts 7 = - 6 0 ° until the instability point is reached. These simple considerations are drastically changed if shell correlations are included. Fig. 16 shows the results of our calculation in 162Er for the deformation

BACKBENDING

125

energy surface at J = 30 h and the minima are indicated by capital letters. The deepest energy minimum (A) at angular momentum 30 h is quite flat and zero point oscillations may be quite substantial. However one also finds well defined shape isomers (C, D, F, G) at 1.2, 5.3, 4.6 and 3.2 MeV excitation energy relative to the yrast state. The situation is similar for higher angular momenta. Fig. 16 also shows the deformation energy surface for J = 60 h for which our single-particle basis is still reliable. The ground-state minimum (F) is again very flat and the zero point oscillations may wash out the three minima F, A, B. However one also finds well defined shape isomers (C, D, E, G) with energies 6.4, 6.5, 7.4 and 2.4 MeV above the deepest energy minimum *. One has to be careful in going to higher angular momenta with the single-particle basis used in these calculations as one is unable to describe fission. The maximum possible angular momentum contained in this basis is J = 154 h for 162Er. This configuration corresponds to an intrinsic quadrupole moment of Qo = 77fm2 or fl = 0.03. [These estimates are calculated in the asymptotic representation with ha~ = 41.2 A -+ MeV and R = 1.2 A ÷ fm.] 5. Conclusion

The Hartree-Fock-Bogoliubov (HFB) method has been extended with the help of the cranking model to include simultaneously the Coriolis antipairing (CAP) and the rotational alignment (RAL) effects. The advantage over the work of Mang and coworkers x6) are threefold: (i) The H F B function is particle number projected before the variation. (ii) The cranking frequency toj is considered as a function of the total angular momentum and not vice versa. This guarantees a unique relationship between angular momentum and cranking frequency. (iii) We consider also the very high spin states. [The results seem to be trustworthy up to angular momentum J = 80 h. For higher angular momenta the single-particle basis is too small.] We find in 162Er a strong reduction of the pairing potential of all neutron pairs with increasing angular momentum, (CAP), although the two pairs i~ fl = ~ and especially t2 = ~ show the strongest reduction. On the other hand the i~ t2 = pair contributes 58 ~ of the total angular momentum J = 16 h (RAL). It appears therefore that the linear increase of the moment of inertia at low angular momenta (J =< 12 h) is determined by the CAP effect. In the backbending region both CAP and R A L contribute to the increase of the moment of inertia. However the decisive impulse for backbending is due to R A L in 162Er. The deformation energy surfaces at very high spin states (30 _< J _ 80) show well t Studies by Bengtson et aL 25) and Neergaard and Pashkevich 26) of the deformation energy surface for very high spin states, as calculated within the basis of the Strutinsky method have been completed very recently. The work of Bengtson et al. was however restricted to positive ~-values in their notation and negative ~-values in the notation of this work.

126

A. FAESSLER et aL

defined shape isomers. It should be m o s t interesting to o b t a i n e x p e r i m e n t a l c o n f i rm at i o n o f the existence o f such states by m ean s o f the heavy ion accelerators currently u n d e r construction.

References 1) A. Johnson, H. Ryde and S. A. Hjorth, Nucl. Phys. A179 (1972) 7S3 2) H. Beuschor, W. F. Davidson, R. M. Lieder and C. Mayer-B0rocke, Phys. Lett. 40B (1972) 449 3) P. Thieberger, A. W. Sunyar, P. C. Roger, N. Lark, O. C. Kistner, E. der Mateosian, S. C. Cochavi and E. A. Auerbach, Phys. Roy. Lett. 2,8 (1972) 972 4) B. R. Mottelson and J. G. Valatin, Phys. Lctt. 5 (1960) 511; J. G. Valatin, Lectures in theoretical physics, vol. 4, ed. W. E. Brittin, B. W. Downs and J. Downs (Interscience, New York, 1962) p. 89 5) A. Faessler, W. Gr¢iner and R. K. Sh¢line, Nucl. Phys. 62 (1965) 241 6) F. S. Stephons and R. S. Simon, NucL Phys. A138 (1972) 257 7) F. S. Stephens, P. Kleinheinz, R. K. Sheline and R. S. Simon, Nucl. Phys. A222 (1974) 235 8) A. Faessler, L. Lin and F. Wittmann, Phys. Lett. 44B (1973) 127; A. Faessler, F. Grtimmer, L. Lin and J. Urbane, Phys. Lett. 48B (1974) 87 9) J. Damgaard and A. Faessler, Nucl. Phys. A243 (1975) 492 10) F. Grfimmer, K. W. Schmid and A. Faessler, Nucl. Phys. A239 (1975) 289 11) I. Morrison, A. Faessler, F. Grfimmer, K. W. Schmid and K. Goeke, Phys. Lett., to be published 12) M. A. J. Mariscotti, G. Scharff-Goldhaber and B. Buck, Phys. Rev. 178 (1969) 1864 13) J. Krumlindo and Z. Szymanski, Ann. of Phys. 79 (1973) 201 14) J. Damgaard and A. Faossler, Phys. Lctt. 43B (1973) 157 15) K. Kumar, to be published and private communication 16) B. Banorjee, H. R. Dalafi, P. Ring and H. J. Mang, Prec. Int. Conf. on nuclear physics, Munich 1973, vol. 1 (North-Holland, Amsterdam, 1973) p. 198 17) A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 (1953) no. 16 18) M. Baranger and K. Kumar, Nucl. Phys. A l l 0 (1968) 490, 529 19) B. Banerjee, H. J. Mang and P. Ring, Nucl. Phys. A215 (1973) 366 20) C. Bloch and A. Messiah, Nucl. Phys. 39 (1962) 95 21) A. Faesslor, F. Grfimmer and A. Plastino, Z. Phys. 260 (1973) 305; A. Faessler, F. Griimmer, A. Plastino and F. Krmpoti~, Nucl. Phys. A217 (1973) 420; A. Faessler, F. Griimmer, F. Krmpoti~, F. Osterfeld and A. Plastino, Nucl. Phys. A245 (1975) 466 22) R. K. Sheline, Nucl. Phys. A195 (1972) 321 23) B. C. Carlson, J. Math. Phys. 2 (1961) 441 24) F. Plasil, Prec. Int. Conf. on reactions between nuclei, Nashville 1974 (North-Holland, Amsterdam, 1974) p. 107 25) R. Bongtson, S. E. Larsson, G. Leander, P. MSller, S. G. Nilsson, S. Aberg and Z. Szymanski, preprint 26) K. Necrg~rd and V. V. Paskevich, Proc. Syrup. on highly excited states in nuclei, Jfilich 1975, vol. I, ed. A. Faessler, C. Mayer-BSricke and P. Turek, p. 59 27) A. L. Goodman, Nucl. Phys. A230 (1974) 466