Influence of quadrupole pairing on backbending

Nuclear Physics A29S (1978) 86-110 ; © North-HoUond Publiahlnp Co., Mtrterdarn Not to be reproduced by photoprlnt or mkro6lm without written pamiYion flrom the publlrhar

INFLUENCE OF QUADRUPOLE PAIRING ON BACKBENDING t M . WAKAI't Institut Jtir Kernphysik der Kernforschungsanlage Nilich, D-51701G1rch, West Germany and AMAND FAESSLER ttt Physics Department, State University of Nen~ York, Stony Brook, NY 1179, US.9 Received 1 July 1977 Abstract : The influence of the difFerent quadrupole pairing forces oc Y~ (m = 0, 1, 2) and the spindependent particlo-hok force on backbending (BB) is studied . A cranked Hartree-Fock-Bogoliubov approach with particle number projection before the variation of the important degrees of frcedom is used . To discuss the numerical results qualitatively perturbative formulas for the moment of inertia and the gap parameters are given . The results are the following : (i) The quadrupole pairing Y=, is not affecting the backbending . (ü) The' Y2° pairing is reducing the moment of inertia at low angular moments by about 20 ~ . This just cancels the increase of the moment of inertia by Yz , pairing at low angular moments . (iii) The Y~, and Yz° pairing together shift the backbending point to higher angular moments and better agreement with the experimental data. (iv) A spindependent ph force does not affect the moment of inertia at low angular momentum . But above backbending it reduces the moment of inertia by about 13 ~ to the correct experimental value if a strength parameter adapted in ~°sPb is used .

I. Introducdon The anomaly of the moment of inertia called backbending has been described in the literature as due to the Coriolis antipairing t - s) (CAP) or the rotational alignment effect (RAL). In the meantime it seems that cranked Hartree-Fock-Bogoliubov theories with particle number projection before the minimization of the important degrees e " t ~ of freedom are so reliable that they can decide between the two effects as the real cause for backbending : They seem to indicate that the important push for backbending is due to RAL. But one also has a strong CAP effect . If one freezes the pairing correlations to a constant value, RAL can only produce upbending but no backbending. In spite ofthe large progress in the description of backbending, a truly quantitative method is not yet available. One deficiency which troubles all microscopic theories t Work partially supportoll by USERDA contract E(11-1}3001 . tt Permanent address : Department of Physics, Osaka University, Toyonaka, Japan . m Permanent address : Institut für Kernphysik der Kernforschungsanlage Jûlich, D-5(70 Jülich, West Germany . 86

87

BACKBENDING

is the fact that the backhanding starts at a too small total angular momentum . One possible reason for this may be that practically all theoretical methods use the monopole pairing force only . [Goodman uses a realistic interaction and includes therefore also higher multipole pairing. But since he does not renormalize the force for the small single-particle space used, he finds that nuclei backbend at even smaller rotational frequencies.] This means that a pair feels no further resistance to alignment ifit is bent from the antiparallel situation in monopole pairing to an angular momentum two pair. This changes ifone includes quadrupole pairing. Then one expects that the alignment is retarded and baCkbending sets in a higher total angular momentum . This effect is influenced by all parts of the quadrupole pairing and not only the term proportional to YZt , (defined in the intrinsic frame) which has been mainly considered till now in deformed nuclei "-' 3). An additional motivation for considering quadrupole pairing in the backhanding region lies in the fact that the modification of the pairing correlations is described in lowest order (second order in the cranking term) by quadrupole pairing proportional to Yzti . This has first been pointed out by Migdal ") . Consequently, the correction to the moment of inertia (which increases its value) is called the Migdal term . This term should not be mixed up with the CAP effect ' - s) which represents (in fourth order in the cranking term) an interplay between rotation and primarily the monopole pairing correlations . Moments of inertia have been extensively calculated ' a.'s) using the lowest order expression by Belisev ' They turn out to be too small by about an average of 20 ~ if gap parameters from odd-even mass differences are used. This could be corrected by the Migdal term which increases the moment of inertia on average just by the missing 20 %. But in addition to the Migdal particle-particle term there exists also a ph contribution to the moment of inertia derived also by Migdal "). This term has been calculated by Birbrair and Nikolaev "), and by Kammuri and Kusuno ' Birbrair and Nikolaev ") used the Migdal interaction and found a negative contribution of about 15 ~ canceling by a large part the contribution of the pp Migdal term. Kammuri and Kusuno' ) adjusted the ph force so that the ph Migdal term gives all the corrections needed to bringthe Belisev formula in agreement type opposite with experiment . They need an attractive ph interaction of the to the information about this part of the force from other data Speth and coworkers' ) calculated with the Migdal force the pp and the ph Migdal contribution to the moment of inertia and found that the pp part contributes x 15~ and the ph part -15 %, so that their net effect is negligible . The purpose of this paper is not so much to produce numbers to be compared with experimental data. Its main aim is to study the influence of quadrupole pairing and the spin-dependent ph force (due to the symmetries of the cranking term only the part proportional to v of the ph force can effect the moment of inertia in the lowest order) on the backhanding (BB) behaviour. This will be done in three steps (i) In sect. 2 we extend the cranked Hartree-Fock-Bogoliubov theory with particle 9)

6 ).

8).

8

Q

~

Q

2 °) .

9

Z°)

~

Q

8R

M . WAKAI

AND

A. FAESSLER

number projection e) so that quadrupole pairing can be included . (A ph force proportional to does not modify the formulas . Their matrix elements are exactly ofthe same type as the ones of the quadruple-quadrupole force.) (ü) For the purpose of qualitative discussions we derive perturbative formulas for the influence of the different quadruple pairing forces (proportional to Yz~~~) and the ph force on the moment of inertia and the pairing gaps . The perturbation approach is arranged so that one starts from quasi-particles which diagonalize the self-consistent deformed potential and monopole pairing but not cranking, quadrupgle miring and the ph force. This perturbational approach solves the HFH equations in a similar way as in ref. 's) but gives the formulas also for other than the Yz, quadruple pairing. (iii) In sect . 4 we shall present numerical results for the cranked HFB method with particle number projection given in sect. 2. We shall discuss the influence of the different quadruple pairing forces and the a ph force separately. Finally, we shall summarize the main conclusions in sect . 5. Q

Q

~

~

Q

Q

Q

~

Q

~ Q

2. Theory The detailed theoretical method is described in ref. e). Here we shall only give a short outline. We describe the rotational states with the help of the HFB theory using the following cranking Hamiltonian .;~° : H-

Lr ( E. - ~)Ca a

Ca + 2 ~

vabedCa Cb

CdCn

where C;, Cb are the creation and annihilation operators of spherical single-particle states ~a), fib) . . . characterized by ~a) _ ~tnljS2~~

(3)

We choose for the interaction the monopole-quadrupole pairing-plus-quadruple force Hamiltonian iV = -â ~

G~o~(T)Pôo(T)ßoo(t)- 4 ~ 4nG'z'(r)~Z~(a)~z,~(T)-i ~

The operators boo, ßiß ~z~ are defined as follows : pôo(T) _ ~Ca Ca aT. .o

XfX}¢i~(i~z~(T).

BACKBENDING

89

where the state ~~i denotes the time-reversed state of ~a~(~â) _ ( - ~°+~°-~"~sn1j- fa~a). The constants G~ °~ and G~z~ are the strength constants of the monopole and quadrupole pairing forces. The quadrupole force constant X~X~ is defined as X a~aT, [ref. z')], where

a~

_

(2Z/A)}

for protons

- {(2N/A)}

for neutrons.

It is notable that the contribution from the radial part 1(r) in the matrix elements zz)] .
The creation operators of quasi-particles aâ are given by the transformation

The coefficients A and B are determined by minimizing the energy

The Fermi energy ~ is introduced to conserve the number of nucleons on the average, <10r >

=

(10)

NL

Minimization (9) leads to the following HFB equations (1l) The quantities

r

and d are defined by rah = (Ea - ZNab +(,, ( Varbd -

~, a

VacJ6)~Cc Cd~+

da6 = ~ Vabcd~CJCc~~ ~, r

(12)

If we neglect the contributions of the pairing Hamiltonian to r and of the quadrupole force to d, these quantities are given by the following expressions : raa-(Ea-~Nab-ai~tcuo~a~rz[Yzoß~sY+Ji(Yzz+YZ-z)ßsinY~~bi,

(13a)

M . WAKAI AND A. FAESSLER

90

where dcob.o~ _ ~ d~o .o~(T~a, 6S r, r,, a, r

dcz .~~ _ ~d cz, ~~(T)
(14)

Here the constants ß, y, d~°" °~(s) and d~ z, ~~(r') satisfy the self-consistency conditions r

d~o.o~(T ) d~z .~i( T ) =

42

r _ iGio~( T - 2 )~

Gcz~(T ) ~ ~äLÏ(r)Yz~I6XCbCaibr u. i,

(16a)

~Co C-)J c r. r,' . tar.ry~

(16b)

We can see easily that the ~rameters ß and y are identical to the usual quadrupole deformation parameters z3) and T is the deformed Nilsson single-particle Hamiltonian z`). The parameters d~°" °~ coincide with the energy gap din the theory of BCS zs). The quadrupole pairing leads us to introduce new parameters d~z" ~~, the quadrupole pairing gaps, which tell us how strongly a pair of particles in a nucleus is coupled to J = 2 states . In driving eq. (15), we can assume matrix elements of related quantities are real by choosing appropriate phases . As was discussed formerly in refs. s, ze), the HFB state should be invariant under the rotation Rx(n) = exp (in lx) through an angle n around the intrinsic x-axis . Using this symmetry, one can verify the following relation (17) This symmetry allows us a reduction of the HFB problem . We now introduce new single-particle states laj, Ib> . . . and their conjugate states I-a~, I-b> . . ., which are eigenfunctions of Rx(n)

with e This symmetrization has been used in Jûlich for several years (see ref. )). Indepenze) . . dently it has been found and published by Goodman

91

BACKBENDING

We have in this basis the following relations : (r-wjxh-a = (r-~j~-~ =

0.

d~=0 .

By using these relations, we can reduce the dimension of the HFB eq. (11), so as discussed in detail in ref. s). The solution of eq . (11) is a function of the parameters c~ ß, y, d ~° " °'(z), d "~~(a). Values of the parameters are determined by the self-consistency conditions (15) and (16), or by minimizing the total energy as a function of these parameters . In our treatment, the solution of eq . (11) serves, however, as a trial function . From this we project out a state with good numbers of protons and neutrons, by which we describe the rotational states e). The cranking frequency co is determined by the condition I Js~p~nl I~p4 n I

~ =

(20)

J(J+1),

where ¢p and ~n are the particle number projection operators onto good proton and neutron number, respectively. Using condition (20) to determine the average angular momentum, one has to proceed carefully. If one employs co as the independent variable, < Js) is a multivalued function in the backhanding region . Thus we use
.o>~d(2 .w)) _ ~ IH~ P~ o I ~ = I1:p~nl

minimum.

(21)

We express the HFB state in a BCS type form according to the Bloch-Messia theorem [ref. sa)]

It makes the calculations of eqs. (20) and (21) very easy . In our case, the canonical basis h~, I - i~ is given as follows a)

92

M . WAKAI AND A . FAESSLER

The numerator < IH¢p~I ~ and the denominator < I~p¢o l sion (21) can be written -

~STaTbGioQi

of the energy expres-

~

~ ~ali~uii'i`-âl-~iC~Iki~iki'k<-~I-kiPrk(NT,-2)

i .k>o

aa

ST~Tk4rzG~zi~il YzMI - i)urt'r~kl( - ~Yz -ml -K~ukt'kPik( NT~ - 2)

i.k>o

-~ ~ Xi/.~kC~IQmI~)
(24)

i. k x~

fz o

IYp~nl ~ _ ~~p(Np)~C~n(

(25)

~L

.>o

L~ =T

x cos ~ ~ tan- ` ~_ .__ . k>o Tk = T

The quantities P~(N), Pt~(N) i, j in eq. (26), respectively .

ST~T~

sin

~z zz} - zN~P ~ coscp+uk/c'k J

(26)

can be obtained by dropping the terms with i and

3. Perturbatianal ap~oach In this section, we discuss the effect of the quadrupole pairing force on rotational states, qualitatively. For this purpose, we apply a perturbational approach to solving the HFB equation without the number projection. The effect of the other type of force, the Q ~ Q type interaction, is also discussed at the last part of this section. Only one kind of nucleon is considered . The extension to a system consisting of protons and neutrons is straightforward. One could solve eq. (11) perturbationally along the linés of ref. z9). But this method is complicated in our case due to the presence ofquadrupole pairing and we take another procedure ' z ~ ") in this section. When we neglect the contributions of cross terms (i.e. pairing contributions for the self-consistent potential and contributions from the quadrupole force to the pairing potential), eq. (9) leads to the following relation : ~(Ea -~b
Lr ( `Ca

Ca

~a~CâCa~ + ~ C~o~IS~Ca

Ca ~)

-âGlz> .4n~(~~z~is~~z~i+~~z~iS~ßz~i)-wS
(27)

It is convenient to expand the single-particle states lad into the eigenstetes la> of

BACKBENDING

93

the self-consistently determined deformed Hamiltonian T (eq. (13a)),

By using the conditions (1~ which define the parameters d ~° "°~ and d ~Z " ~~, we obtain the following variational equation,

-Z

~ d~2

~. =a

.~,~(<«IJ(r)Yi~,l~a
We now introduce new quasi-particles and a vacuum state. They are so defined that they diagonalize the self-consistent deformed potential and monopole pairing but not the cranking term and quadrupole pairing, gz = U=d= - V,d=, g= IBCS)

= 0.

(31) (32)

The coefficiencies u= and va are defined in the same manner as that of the usual BCS theory 2s), U=

=

e Cl + J2

E_

~~ , (33)

The creation and annihilation operators a; , a, of the basis states can be expanded into the quasi-particles g; and g,. This fact makes it possible to express the HFB state I ) which includes cranking and quadrupole pairing as follows : U --- exp (EJ~gz gâ ) . =p

(35)

The coefficients Jap should be determined by the variational eq . (2~. It is very convenient to introduce matrices F and M, (~7~p -- J~p = -Jp=, (M)=p - ( - Fz)~

(36)

M. WAKAI AND A. FAESSLER

94

which give short expressions for

(37)

(1 + M Fl ~ _

M

~d' da) - (1 +M)aa By using eqs . (2~ and (3T,, we get the following equation where X =_ -(FE+EF)+~d~z .~~{A~z~i+~z .~~+F(A~z.~~+B~z .~.i)F-Cz.~~F_FCtz .~~}

The elements of matrices A, B, C, K~ -~ and K~ °~ are defined as follows : (~~.~~) = (,g o. ~)Qa = <ßlf(r)Y,~I~)uzUp, C~II(r)Y~~Ia)Va~a' za (c u.~~) = 2, (K~ =
x

a

sa

a

We define the matrices A, B, C for l = 0 by substituting the constant 1 for f(r) in eq. (40). Since the matrix 8F is antisymmetric, X should satisfy the following equation, X -XT = 0,

(41)

where the symbol T signifies the transpose of the matrix X. We introduce new matrices to get a simpler expression of eq. (41), ~c4. ~ _ ~(Cci . mi + Co. ~~T~ (42) ~~o~ _ ~K~o~+K~oFr)~

The eq. (41) is written as +co(- .3(~co~F_F.9('c°~+F.9fr~-~F).

(43)

95

BACKBENDING

The self-consistency conditions (16)are expressed with the matrices defined in eq. (40) as 2 um Gclid ~

=

Tr rt A um ~ F(1+M) - r +B'~Iw'(1+M) - 1 F+C~ IW " M(1+M) - 1_1

In~n

1=0 4n,

1=2.

The expression of the moment of inertia 9 is 9 = (J~)/co = Tr iK'°'M(1+M)- ' +K'-'F(1+M)- ' ;/cu . (45) When the values of matrix elements of dca .n~lYz~ and cojz are small, we can get the solution F by an iterative procedure expandingf=a as a power series of d~a . .~ and m, taking into consideration relation (17), f=ß =

~

r

( ~ (d~z . ol~~(dlz . r 1)n~(dlz . zl~= .1,~(l, n o , n,, n,).

Ln.n~.nz2_0

(46)

The quadrupole pairing force consists of three parts, corresponding to Iml = 0, 1, 2, and their respective effects on rotational states are quite different. It is interesting to take account of each part separately to investigate their actual effects within the limit of the lowest order of each parameter . In the following parts ofthis section, we will give the expressions for the monopole and quadrupole gaps and the moment of inertia to investigate the effects of each part of the quadrupole force on rotational states . 3 .1 . THE QUADRUPOLE PAIRING Yzo

3.11. Monopole pairing gap d~°" °~. We can get an equation for d~° " °~ by the use of eq. (44) as follows : É_

=

Go

- 2dlo .ôi~ ~_ (aLÎ~(r)Yzo la)+O(co )

(47)

The term d ~a " °~/2dß°' °' on the right-hand side shows that the quadrupole pairing gap changes the effective strength of the monopole pairing interaction "~ aa) and it may have a rather severe effect not only on the excited rotational states but also on the ground state. The term proportional to Wa is identical to that given in ref. a9) . It represents the Coriolis antipairing effect (CAP). 3.1 .2. Qr~upole pairing gap dca, °~. The following equation gives the quadrupole pairing gap d~a" o~ : nG m°, =

dlo, o~ ~ (al ./'(ÉYzol «) _ _

I

_ 2nG~z~ ~ (al ~(')+~ Ißia _ V=y~)z~ (U=Uv \1 =a a +O(coa).

(~)

M. WAKAI AND A. FAESSLER

96

The above equation shows that d~z " °~ is approximately proportional to d~°" °~ . The value of d~z " °~/G~z~ is not zero with no zero monopole pairing gap and it becomes larger by the introduction of the quadrupole pairing force G~z~ . 3 .1 .3. Moment o~ inertia 8. The moment of inertia is given by 8... 9o+d~z .o~e,~ where

(49)

2 ~
(U~U ;:+VpV..)(UÏV,-V-U,) - (UPU  -VaVxU.,U=+V V=)~ . E

P +E Ï

E= +E Ï

The term 9o is the usual expression of the moment of inertia derived by Belyaev' 6) and the second term B' gives the .shift of the moment of inertia due to the quadrupole pairing gap. In conclusion, the m = 0 part influences rotational states through the variation of d~°" °~ and the addition of a new d~ z " °~ proportional term to the moment of inertia. 3.2 . THE QUADRUPOLE PAIRING Y=, 3.2 .1 . Monopole pairing gap d~°" °~ . The gap equation is

~a) 1 2~
(~)

As it will be mentioned later, the value of d~z " ' ~ is small. This fact suggests that the influence of d ~z " ' ~ on the monopole pairing gap is small. 3.2 .1 . Quadrupole pairing gap d cz " '~. The quadrupole pairing gap d ~z " ' ~ is given by 1 d ~z .i~ ~ w~
~ {2nG~z~ - ~. e


E

i+Ez-,)~ß)z e

(U=Ue+VVP)z~-~ .

(51)

We can see from above equation that d cz " ' ~ is proportional to ru, if the dependence of d ~° "°~ on w is neglected.

BACKBENDING

97

With a small value of d ~°" °~, the factor (Ua V~- Va UR) is large for a pair (a, ~ ofwhich one is above the Ferrai surface and the other is below the surface. On the contrary the factor (UQUp+ Va VR ) is small in this case and vice versa. Then, the product of the two factors in the expression (51) becomes smaller with the decrease 3.2.2. Moment o~ inertia. The moment of inertia is given by

(52) gcl~ _ - ~

a. 0

Ea + EO

z -~-_(U=ya-VsUeXUaUe+V=Vp).

The contribution (Ua Vp -V,U~) is known as the Migdal term " .'z) . According to the above discussion, this term is expected to be smaller, when d ~°" °~ decreases due to the CAP. 3.3. THE QUADRUPOLE PAIRING Y==

33 .1. Monopole pairing gap d~°" °~. The effect of the Yzz part is of the order

ofm` and very small. The gap equation is the same with the limit of the order of coz as in the case in which only the Coriolis force is oonsiderod . 3.3 .2. Quadrupole pairing gap d ~z " z~. The following equation gives the value of dy.p dcz.z~ x - ~z ~
(Ea+E,xE~+EpxEa+E~)

x { [(Ea+EgxU,U, +Va V XU;Va -Vg Ua) -(Er +E~xU a Vr -VQ U~XUy Uß+VY Va)] x(U Q U~-V,Va)-(Ea+E~XVaUY -UaVYxV~Ua -U YV~)VVp }.

(53)

Roughly speaking, the quadrupole pairing gap d cz " z' is of the order of cvz and quite small. 3.3.3. Moment of inertia . The moment of inertia is given by 8 x 9o +O(mz).

(54)

In conclusion, the influence of Yzz part is of the order of coz and may be very small. As mentioned above, we considered only the quadrupole force as the field producing interaction. This interaction gives the Nilsson potential which characterized by the deformation parameters ß and y. The effect of the rotation on the nuclear shape is described by the variation of these parameters . It is of the order of cvz and not so large exocpt in the very high-spin region . However, the situation is different, if we take into account also other parts of the

M . WAKAI AND A . FAESSLER

98

residual interactions, such as the Q ' Q part 19 ). The effect of this interaction is of the order of m and seems not to be negligible even at low-spin states . Therefore, we want to give the expression for the moment of inertia for the case when such an interaction is taken into account. We add the following residual interaction to the force given in eq. (4) 2 V - Z~a ~ `~ a, ( T)S~(T)+iGaY ~ ~n, (T)~,e(T), e,, r

n, . r

where (56) If the strength G, is of the same value as that of GQY, the interaction (55) coincides with the lowest order terms of the (Q x Q) x (surface delta interaction) [ref. 3a)] without n-p interaction, which is neglected for the simplicity of the calculation. The residual interaction (55) adds the following terms to eq . (27) ~Go ~ () ~, . r

(57)

+iGsr ~ (
We neglect the m = 0 terms being of order mZ and take into account the m = f 1 part only . For the radial matrix elements we take a constant value
ea=~

a. R

( _

E+E = 9

a

~ aeQQ+seQY, a _

2 1 x ~_1~Go_ ~
+Eß

b9 sY = ( ~

«I([Q x

x -2lGaY - ~ _, o

YZ]i - LQ x

E= +Ee

YZ ]'_, )la)


(

U Va - V= Ue)2 Z

=

YZ ]1 i)I~i2

( U= V0 -Va Ud

z

'

(58)

BACKBENDING

99

The eq. (58) gives the following expression ( -89QQ/Go)

~

~

~

[,, ß

a~sl ß~~ßIQxIa~ (U:Vp- U~V ) ~

E, + EQ

,

(59)

which is similar to that of 9° (49). As is well known, the value of 8° increases with the total angular momentum due to CAP (Coriolis antipairing)' This suggests also that 8®QQ(S9or) has larger values for higher-spin states . " 2"

29)

.

4. N®erical calculations aed results Here we want to show using the example of'6gEr94 the influence of quadrupole pairing and the interplay with the Coriolis force. The procedures of the numerical calculations are the same to that of ref. e). The modifications in the solutions of the HFB eqs. (11) and the minimization of eq . (21) by the quadrupole pairing force (see eq . (4)) are outlined in sect . 2. As the singleparticle basis, we used the oscillator shells N = 4, 5 for protons and N = 5, 6 for neutrons. The single-particle energies are taken from ref. z'). The force con~"ants are 4), in units of MeV, Gp~ = 23/A MeV,

G;,°~ =

X = 73A~'~ 4 MeV,

2

G ;,

~

=

18/A MeV,

tiu~° = 41.2A-} MeV.

The quadrupole pairing strength G~ 2 is fixed to be the same value as G~°~ according to Brogue, Bès and Nilsson In ref. 8), it was shown that the values of the deformation parameters ß, y amd the monopole pairing gap of protons dp°" °~ vary very slowly in the region ofJ 5 24 . Therefore, we kept them to be constant : ß = 0.30, y = 0.0 and dp°" °~ = 0.9 MeV. We should perhaps stress that, as we have tested 8), the ydeformation has practically no influence on the backbending behaviour in ' 62Er. The choice y = 0 which is found by minimizing the energy in the intrinsic system, does not mean that one has a good angular momentum projection Kto the symmetry axis . For higher-angular moments one obtains an appreciable K-mixing by the cranking and the m ~ 0 quadrupole pairing terms . The possibility of aligning nucleons along the rotational axis (
a2)

.

vZ

" ~~

100

M . WAKAI AND A. FAESSLER

N (ROTATIONAL FRl~K,1r~(h11eIV 2 1

Fig . 1 . Twice the moment of inertia of 16 ~Er as a function of the square of rotational frequency for spins from 2 + to 20 + . The experimental vah~es are described by the squares. The open circks indicate the theoretical values obtained with the Yzt , component of the quadrupok pairing foroe, whereas the triangles denote the resuhs obtained without the quadrupole pairing . The inclusion of the Y=t, pairing causes an inaease of about 20 % in the moment of inertia. fl'he quantities (d ; " ~ appearing in brackets here and throughout in the subsoqutnt figures are the unfixed variational parameters in eq. (21).)

The difference between the values of the moments of inertia of the two cases is quite large in the region of low spin. This difference comes from the sculled Migdal term of the moment of inertia' s- ' s). It is interesting to see that the dif%rance becomes smaller in the higher-spin region . It is due to the effect of the monopole pairing gap dé°"°l. Its decrease in that region reduces the influence of the Yzs pairing. The experimental results are also plotted with squares in fig. 1. There is a rather remarkable discrepancy between the theordical and experimental results. The inclusion of the quadrupole pairing gap d oz " ' 1 does not improve the situation. Several years ago, Hamamoto 's) performed ca~ulations on the Migdal terms and showed that the value of such a term is about 30 ~ of the usual cranking term. This value is qualitatively consistent with our result . In her calculation, she took the Nilsson Hamiltonian plus the pairing force as the total Hamiltonian in contrast to our treatment in which a rotational invariant many-body Hamiltonian (2) is usod in calculating the total energy. To see how different results the two treatments are giving in the high-spin region, we made also a calculation by using the Nilsson Hamiltonian plus the pairing forces ignoring the quadrupole interaction. Opposite to the work of Hamamoto ' a) we include the particle number projection . The results are plotted in fig. 2. The two cases give the same results in the low-spin region, I = 2-8. However, the difference appears to be .ret~arkable at high-spin states . It is surprising that the backhanding phenomenon disappears, if we take the Nilsson Hamiltonian instead of the spherical

101

BACKBENDING

single-particle Hamiltonian and the (two-body) quadrupole force as the real Hamiltonian. The discrepancy between our more microscopic approach and the method used by Hamamoto is much smaller, ifone does not project on good particle number. As stressed already earlier' " e) the particle number projection turns out again to be essential for a quantitative description of the backhanding (BB) region . It was clearly shown in ref. e) that the BH phenomena results in ' 62 Er from the alignment of two i,~, ß = } neutrons along the rotational axis . (The single-particle angular momentum projection il is only for J = 0 at co = 0 a good quantum number.) The contribution of these single-particle states to the total angular momentum I as a function of 1 is presented in fig. 3. It is clear that the contribution of this aligned pair to the total angular momentum is very large above the BB region in both cases with and without YZt pairing. The contribution becomes slightly smaller, when we include the quadrupole pairing force Yet , and therefore it has a tendency to reduce backhanding. The effect of YZ t pairing on the BB phenomena is however rather small. Fig. 1 shows that the backhanding begins at almost the same value of m ( x 0.185) in both bases with and without the YZt pairing force. The dependence of the dô° " °I on the total angular momentum I is presented in fig. 4. It is seen that the YZ t pairing increases slightly the value of dô° " °l. .It acts upon the monopole pairing in tie opposite way to the Coriolis force. This fact allows us to determine the sign of the second term in eq . (50) wd ls . tl A to be negative . Since the sign of doZ .'1 is positive, as will be shown later, A has to be negative.

160

rv

00

~

~~

(]t2

(ROTATIONAL FREOIJENCYxtÙz[MeV2

1

Fig . 2 . Twig the moment of inertia of °~Er es a function of the square of the rotational frequency for spins from 2 + to 20* . The open circles give the theoretical values calculated with the Y=t ~ pairing force . The closed circles show the resuhs of similar calculations in which the total energy is evaluated using the Nilsson-Hamiltonian with the monopok and Y= * , pairing forces. A large disaepency between the two results is seen for states with spin higher than 10* .

102

M. WAKAI AND A. FAESSLER

0

4 8 12 16 20 TOTAL ANGULAR MOMENTUM Ilhl Fig. 3. Angular momentum projection along the rotational axis for the quasi-particle state i, 2, n = ~ (neutron) in the canonical representation. The calculated results without the quadrupole pairing force aroshown by the dotted line, whereas the solid line indicates the results obtained with the Y, * , pairing force.

100

~~ 075 4

050

025

0

10 20 TOTAL ANGULAR MOMENTUM I I1ti] Fig. 4. The dependence of the monopole pairing gap of neutron d;°~ °~ on the total angular momentum l. The solid and dotted lines, as in fig. 3, show the results obtained with the Yet , pairing force, and those without the quadrupole pairing force, respectively. For the ground state both results arc the same.

Fig. 5 shows the dependence of the quadrupole gap doz " '' on I. The gap dôZ " t> is zero for the ground state and is proportional to I in the low-spin region . This is consistent with the prediction of eq. (51). A rapid variation of the gap dôs. tl is found in the BB region (see fig. 5). This fact means that the mixing of two bands due to the band crossing a) gives a incoherent contribution to doz" 'l. The effect of quadrupole pairing YZt is twofold : It slows down both rotational alignment (RAL) and the Coriolis antipairing (CAP) effect . Both effects are to the

BACKBENDING

10 3

same mechanism : It is energetically not so costly to transform an angular momentum J = 0 pair into an J = 2 pair. Concluding the discussion of the influence of YZt pairing to backbending, one should point out that it is surprisingly small and could easily be neglected compared with the general reliability of the theory . On the other side there is large influence on the moment of inertia at low angular moments as pointed out already by Migdal t t) and Beliaev t 2). In fig. 6 we demonstrate the dependence of the moment of inertia B with respect to u~2 where only the ~m~ = 2 part of the quadrupole pairing force has been taken into account. This is compared with the moment ofinertia without any quadrupole pairing force. A small discrepancy is seen only at I = 10 and 16. The dependence of d,ß,2 " Zl on the total angular momentum I is shown in fig. 7. The value of d,ß,2 " 2' is proportional to I2, when 1 is small. It shows a drastic variation in the~BB region. In this point, the situation is similar to the case of Yet pairing. Comparing the figs . 6 and 7, one sees that the largè values of d~2 .2i are correlated with the small modification for angular moments I = 10 and 16. The dependence of d~° " °' on I is practically identical to that without the quadrupole pairing force (see fig. 4). According to the discussion in the sect. 3 based on the perturbational approach without particle number projection, we expect that the Y2o pairing plays the most important role in the high-spin region. Fig. 8 confirms this expectation. There, the moment of inertia 8 calculated with the Y2o pairing is shown. The vâlue of B at the low-spin states are considerably reduced and the backbending begins at a larger value of the rotational frequency. In average the agreement with the experimental data is improved . The calculated values ofthe monopole pairing gap dô° " of are plotted in fig. 9 without and with Y2o pairing. The Y2o pairing increases the pairing gap do° " o>.

0

10 20 TOTAL ANGl1L.AR MOhEN'fU+l I Ir1 Fig. 5. The theoretical values of the quadrupote pairing gap of neutron do= " '~ obtained with the Y~ * , pairing force . The results indicate a rapid variation in the bachbending region, I x 10* .

loa

M . WAKAI AND A. FAESSLER

02

-Qt

OO QO4 OO8 O12 (ROTATIONAL FREQFTICY ~Ur1aw21

Fig. 6. Twig the moment of inertia of ' e=Er as a function of the equate of rotational frequency for spins from 2 + to 20 + . As in fig . 1, the experimental results have been described by squares. The open circks and triangles indicate theoretical values calculated with the Y=ti pairing force and without the quadrupole pairing force, respectively . The results for the two cal culations are similar except for the I a 10+ and 16 + states, showing a Gttk difference .

10 20 TOTAL ANGILAR MOhENTUrI I I h 1

Fig. 7 . The quadrupole pairing gap of neutron d ;= " _~ plotted as a function of the total angular momentum L In the bw angular momentum region, the gap d~:. =~ is proportional to the square of the total angular momentum. A drastic variation is seen in the region I = 10* x 18 + .

The increase in dô°" °1 is~partially responsible for the reduction of the moment of inertia at the low-spin states . But from the perturbational result (49) for the moment of inertia with YZ° pairing it is clear that not the whole change of the moment of inertia 8 can be attributed to an increase of dô°" ° 1 in the Beliaev expression 90. Since the YZO pairing gap is markedly different from zero (see fig. 10) one expects a contribution from the second term in eq . (49) which depends on the size of 8'. Since Yz° pairing is not forbidden by selection rules for ro = 0 one finds large values for the gap parameter do2 " ol for small angular momentum states. This is opposite to pairing gaps dôx" '' and dos" sl which increase for small cranking frequencies with w and cuZ, respectively (see eqs. (51) and (53)). The value of d o2 " °1 decreases with increasing I. This reduction corresponds to the antipairing effect in the case of dô°" °I and that is predicted by eq. (48) . Above the BB region, d es " °1 shows an increase with I. This may originate in the increase in components of K = 0 quasi-particle states in the rotational (intrinsic) states due to the higher-order terms of the Coriolis force which may possibly form J = 2, K = 0 pairs. One could argue that the modifications introduced into the backbending behaviour by the YZ° quadrupole pairing may all be simulated by changing the monopole

BACKBENDING

los

40

ao4 o12 ao o.oe (ROTATgNAL FRES~IENLYxhl~leV21

Fig. 8. Twice the moment of inertia of 'b=Er as a function of thesquare of rotational frequency for spins from 2+ to 20* . Experimental data have ban shown by squares. The open circles give the theoretical values calculated with the Ylo pairing force. The triangles indicate the results obtained without the quadrupok pairing force. It is notable that the inclusion of the Y:o pairing gives smaller values of the moment of inertia and makes backbending begin at a larger valueof the rotational frequency.

Fig. 9. The monopole pairing gap of neutron dô°~ °~ as a function of the total angular momentum I. The solid line shows the results for the Y~o pairing force. Whereas the dotted line shows the values obtained without the quadrupok pairing force. It is seen that the Yso pairing increases the value of the pairing gap dé°~ o, .

e 4 o. c~ Q5

10 20 TOTAL AN(Il1LAR MOhET1Tl1rIIIh1 Fig. 10. The dependence ofthequadrupole pairinggap ofneutrond;= " °~ on the total angularmomentum I. The value ofdé'~°~ decreaseswith thetotal angular momentum due to the Coriolis force in the low angular momentum region . The band crossing makes the dependence somewhat complicated in thehigher angular momentum n gion I Z 12* . 0

106

M . WAKAI AND A . FAESSLER

pairing strength. We therefore have reduced the monopole pairing strength keeping the YZ° pairing constant. Reducing thestrength from G = 18/A MeV to G = 17.25/A MeV yields the same monopole gap parameter d;,°" °I as for the original strength G° without YZ° quadrupole pairing. But the moment of inertia is with 29/fi t = 55.9 MeV - ' still smaller than the value (2®/6 2 = 64.1 MeV - ') for the latter case. One needs a reduction ofthe monopole pairing strength to G ° = 14.75/A MeV to fmd in the presence of YZ° quadrupole pairing the same moment of inertia for angular momentum I= 0 than with a pure monopole pairing force with Go = 18/A MeV (see fig. 11). This again confirms eq . (49) which says that the effect of YZ° quadrupole pairing cannot be simulated by an increase of d ~°" °1 alone. On the other side one sees comparing figs. 8 and 11 that the partial improvement due to YZ° pairing especially concerning the critical angular momentum is again removed if the monopole pairing strength is reduced to yield the same moment of inertia for I = 0 than without quadrupole pairing. Nevertheless the YZ° quadrupole pairing shows the largest influence of all gtiadrupole forces on the backhanding behaviour of ' 6 Z Er (see figs . 1, 6, 8 and 11).

EXPERIMENT ~THEORYtllaoi~~nl~lol_14 .751A) aTHE0RY 10i4°I, Gn I°I =18/A ) 0.0 Q04 Q08 0 .12 (ROTATIONAL FREOJENCYxT11Z 1MeW~1 Fig . 11 . Twice the moment of inertia of' 6 ~Er as a function of the square of rotational frequency for spins from 2* to 20* . The squares indicate the experimental values. The triangles give the theoretical values calculated using the monopole pairing force with strength given by G;°' = 18/.! MeV . Results of calculations with the Y=o pairing force are shown by open circles . In this case a little smaller value of G ;°' (G ;°' = 14.75/A MeV, G;= ~ = 18/A MeV) has been used ; such a choice gives the same value of the moment of inertia at spin I = 2* as in the calculations without the quadrupole pairing force.

QO Q04 Q08 0.12 (ROTATIONAL FREOl1FNCYxhIzUdeV~] Fig. 12 . Twice the moment of inertia of °'Er as a function of the squar! of rotational frequency for spins from 2* to 20* . The squares indicate the experimental values. The open circles give the theoretical values calculated with the Y~ ° and Y~ * , pairing forces . The triangle gives the values calculated without any quadrupole pairing forces. The parameters employed in both calculations are G ;°' = Gee' = 18/A MeV . It is seen that the effects of the Y~ ° and Y, * , pairing forces cancel each other for very low spins (2* and 4*).

10 7

BACKBENDING

According to the above results, the YZo and YZ t pairing forces give opposite effects on low-spin states : the YZ t pairing increases the value of moment of inertia and on the other hand the YZo pairing reduces this value. One could expect that the effects of these two types of pairing forces cancel each other if one does a calculation including YZo and YZt pairing simultaneously, Fig. 12 shows the results of such a calculation and confirms our expectation. The value of the moment of inertia at I = 2, 4 are of quite the same values as those of the case with only the monopole pairing. In high-spin region, the effect of the YZ t pairing becomes small, but the YZo pairing still plays an important role. The result is that backbending begins at a larger value of m than in the case with only the monople pairing. A detailed analysis shows that the YZo pairing increases the size of the dôZ ~'' gap in the whole region . This is caused by the increase of do°,°l shown in fig. 9 which is due to the YZo pairing. As was mentioned above, we examined the effects of the type interactions (55) on themoment ofinertia (58) . They may give some correction ST to the deformed Nilsson potential T (13a). However, we neglected such an effec>i and took these residual forces only into account in calculating the energy expectation values . This means that we use the same trial wave functions ~) and do not introduce new parameters . Since the expressions for such expectation values are very similar to that of the quadrupole force, we did not give them here . We can get them by exchanging ~,~ in eq. (24) with S;(A;) and X~X~ with Ga(GQ,.). As the values of the Q~ Q

-~ siEr se sr

EXPERNENT -.- THEORY 10 rolol: (0~01"f~Y2~0Y

--~__ niEaRr

~a..,.-

)_

l0(~~1

40

0.0 0.04 OA9 012 (ROTATIONAL FREOUET~CY" 1ti1 2IMeivZ1

Fig . 13 . Twice the moment of inertia of' 62Er as a function of the square of rotational frequency for spins from 2* to 20* . The squares denote the experimental values . The open circles give the theoretical results obtained with the addition of the c ~ Q+(oY=), ~ (oYz), force to the residual interaction (4), but without the quadrupole pairing force, in the calculation of the total energy. The triangles indicate, for purpose of comparison, the results of the calculation without the quadrupok pairing and the (a ~ a) +(aY2), (vY=), foras . It is remartable that the elFect of the a ~ a+(oY2)1 (aY=), interaction is strong and decreases the value of the moment of inertia in the high-spin region . ~

~

108

M . WAKAI AND A . FAESSLER

constants G, and GQY, we take 25/A [ref. sa )] . Only the m = f 1 part of these interactions are considered, because they give the lowest-order contributions of u~. In fig. 13 the result of such a calculation is plotted. We can see that the effect of the interactions is very small in the low-spin region. The increase of the moment of inertia is 8(29/hz) -0 .64 MeV- ' at I = 2. The additional contribution to the moment of inertia due to this force is about -9.0 MeV - ' for the region I Z 12. The above results indicate a large influence of the Q " Q type interactions only for high-spin states . This result is contrary to the work of Meyer, Speth and Vogeler ' 9). They find in linear response theory, which is valid only for low angular moments, already a value 8(29/ttz) z -7 MeV- ' . As an interaction they used the zero-range Migdal force. If one assumes that the most important level is the neutron i,l state, one can compare the strength of the Q ~ v force of this calculation (25/A = 0.1543 MeV) and of the work of ref. ' 9) (1/4n) Jâ (Po6(r))4rzdr x C = 0.101 MeV). The two numbers are not so different that they could justify the result . Ope could think that the difference might be due to the size of the single-particle basis. We included for each the protons (N = 4, 5) and the neutrons (N = 5, 6) a negative- and a positive-parity oscillator shell. The basis used in refs . "~' 9) is not indicated there. But even ifthe basis would be larger than ours it should not influence the results in such a drastic way : The admixture from the spin-dependent ph force has to be undone by the cranking term, which connects only states within a shell. Contribution from shells farther away from the Ferrai surface should be reduced by the W factors in formula (58). The causes for this discrepancy may be connected with the following fact : the linear response theory gives an average value of the moment of inertia over the range of angular moments in which the term of the density matrix proportional to co is small compared to the term which is proportional to co3. Then, if the second term is still small until the backbending region, an average may yield the result which has been found by Speth and coworkers ' 9). S . Cooclodiaas

In this work we studied the influence of quadrupole pairing and the spin-dependent part of the particle-hole force on backbending. As an example we took the nucleus '6zEr . The intent'of this work was not so much to reproduce the experimental data but to investigate the influence of the different type of forces on backhanding. The main conclusions of this work can be summarized as follows : (i) The quadrupole pairing force Yz, which has till now been considered to be important ' a) for the rotational behaviour of the nucleus plays no role in the backbending region . At low angular moments we found that in agreement to earlierwork (refs. "-' a " '~) this term increases the moment of inertia by about 20 ~ (pp Migdal term). In' 6zEr we find for the 2+ state an increase of 23 ~. But starting with angular momentum 8 + the influence of the Yz, pairing alone can be neglected. Analytical

BACKBENDING

109

formulas which we derived in a pertwbational approach explain these facts. The Migdal term turns out to be proportional to the d « " '~ (and to cv). We furthermore showed that d ~ disappears if d~° " °' the monopole pairing gap goes to zero . In the non-perturbative treatment these relations are not anymore so simple but the general trends are persisting . (ü) The effect of the Y22 quadrupole pairing can be neglected. (iii) The largest influence on backhanding is the YZ° quadrupole pairing. At angular momentum zero it decreases the moment of inertia by 18 ~. This reduction can be undone ifthe monopole pairing is lowered by 18 ~ from Gô°~ = 18/A MeV tô Gô ' = 14.75/A MeV. But an appreciable change in the backhanding region is left over. (iv) A drastic change of the backhanding behaviour is found including both Y2° and Y2 , quadrupole pairing. The moment of inertia for the state with angular momentum 2 is not changed : The increase from the Y2 , pairing is just cancelled by the YZ° pairing. But backhanding starts at a larger value of the square of the rotational frequency ((fiu~) = 0.048 instead of 0.032). This effect one would expect classically from a quadrupole pairing force : If one starts to break a pair coupled to angular momentum zero interacting with a monopole pairing force only, there exists no force which prevents alignment after the pair is broken so that it couples to angular momentum two. Quadrupole pairing even resists alignment if the pairs are not coupled to angular momentum zero . It therefore moves the backhanding point to a higher angular momentum and a higher rotational frequency. The fact that the backhanding angular momentum is still too small may be only in small part connected with the missing higher multipole pairing forces. (v) The spin-dependent ph force (oc v " v and oc (vY2)1 (QY2)1) does not affect the moment of inertia at small angular moments. This is contrary to the results of refs . "" ' which find a reduction by about 15 %. But at the high angular moments above the backhanding region this ph force produces a reduction of the moment of inertia by 13 ~. The reduced value agrees nicely with the experimental data in 'e2Er . One might expect that this ph interaction is the general cause which reduces the moment of inertia above the backhanding to values well below the rigid body value. . (vi) We checked also the influence of using a "self-consistent  Nilsson potential employed by Hamamoto ") instead of the two-body quadrupole-quadrupole force as done in this work . We find an appreciable difference (fig. 2). This difference is not surprising. It has two causes : (a) Ifone uses a Nilsson potential instead of a two-body ph force which induces the deformation, one includes the contributions from the quadrupole-quadrupole force twice in the total energy. (b) We furthermore included particle number projection in contrast to ref. ' It is well known `" s) that particle number projection influences the results in the BB region . ('The curves shown in fig. 2 for both the two methods include particle number projection. The difference should therefore be due only to (a).) ~Z " '

°

+

Z

~

9)

3) .

11 0

M. WAKAI AND A. FAESSLER

>n summary one can say that the quadrupole pairing (Y2o and YZt ) is able to shift BB to higher angular moments and that a spin~ependent ph force improves the results above the BB region. It seems therefore important to take both forces into account when discussing the BB of nuclei. References

1) B. R. Mottelson and J. G. Valatin, Phys . Lett . S (1960) 511 2) A. Faessler, W. Greiner and R K. Sheline, Nucl . Phys. 62 (1965) 241 3) A. Faesskr, L. Lin and F. Wittmann, Phys. Lett . 44B (1973) 127; A. Faessler, F. Grümmer, L. Lin and J. Urbano, Phys. Left. 4SB (1974) 87 4) F. Gnlmmer, K. W. Schmid and A. Faessler, Nucl. Phys . A239 (1975) 289 5) S. Frauendorf, Nucl. Phys. A263 (1976) 150 6) F. S. Stephens and R. S. Simon, Nucl . Phys. A138 (1972) 257 7) B. Banerjee, H. J. Mang and P. Ring, Nucl . Phys. A215 (1973) 366 8) A. Faessler, K. R. Sandhya Devi, F. GrQmmer, K. W. Schmid and R. R. Hilton, Nucl . Phys. AZ56 (1976) 106 9) A. L. Goodman, Nucl . Phys. A266 (1976) 113 10) A. Faesskr, K. R Sandhya Devi and A. Harroso, Nucl. Phys. A286 (1977) 101 11) A. B. Migdal, Nucl . Phys . 13 (1959) 655 12) S. T. Heliaev, Nucl . Phys . 24 (1961) 322 13) I. Hamamoto, Nucl . Phys . A232 (1974) 445 ; Phys . Left . 66B (1977) 222 14) S. G. Nilsson and O. Prior, Mat. Fys. Medd . Dan. Vid. Selsk. 32 no . 16, (1961) 15) O. Prior, F. Boehm and S. G. Nilsson, Nucl. Phys. A110 (1968) 257 16) S. T. Heliaev, Mat. Fys. Meld. Dan. Vid. Selsk. 31 (1959) 11 17) B. L. Birbrair and K. N. Nikolaey, Phys. Lett. 32B (1970) 672 18) T. Kammuri and S. Kusuno, Phys. Lett. 38B (1972) 5 19) J. Meyer, J. Speth and J. H. Vogeler, Nucl. Phys. A193 (1972) 60 20) A. B. Migdal, F'mite Ferrai system theory and properties of atomic nuclei (Nauka, Moscow, 1965) 21) M. Baranger and K. Kuraar, Nucl . Phys . A110 (1968) 490, 529 22) R. A. Broglia, D. R. Hès and B. S. Nilsson, Phys. Lett. SOB (1974) 213 23) A. Bohr and B. R Mottelson, Mat. Fys. Medd . Dan. Vid. Selak. 27 no . 16, (1953) 24) S. G. Nil9son, Mat. Fys. Medd . Dan. Vid. Selsk. 29 no . 16, (1955) 25) J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys . Rev. 108 (1957) 1175, N. N. Bogoliubov, Nuovo Cim. 7 (1958) 794; J. G. Valatin, Nuovo Cim. 7 (1958) 843 26) A. L. Goodman, Nucl. Phys. A230 (1974) 466 27) I. Hamamoto, Nucl . Phys., to be published 28) C. Bloch, A. Messiah, Nucl . Phys . 39 (1962) 95 29) M. Sano, M. Wakai, Nucl. Phys. 67 (1968) 481 ; A97 (1967) 298 30) J. M. Blatt, Theory of superconductivity (Academic Press, NY, 1964) 31) M. Sano, T. Takemasa, M. Wakai and E. Takekoshi, Osaka University Laboratory of Nuclear Studies, OULNS 73-4 (1973) 32) M. Sano, private communication 33) I. M. Green and S. A. Moszkowski, Phys. Rev. 139 (1969) 790 34) B. Castel and I. Hamamoto, Phys . Lett . 6SB (1976) 27