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Even-parity states of 16O in a pairing-plus-quadrupole model
Nuclear Physws A107 (1968) 671--693, (~) North-Holland Pubhshmg Co, Amsterdam
Not to be reproduced by photoprmt or mlcrofilrnwtthout written permlssxonfrom the pubhsher
E V E N - P A R I T Y S T A T E S O F 160 IN A PAIRING-PLUS-QUADRUPOLE
MODEL
L M(JNCHOW and H U JAGER Central Instttute for Nuclear Research, Rossendorf, Germany (GDR)
Received 28 August 1967 Abstract In this paper we give an outhne of the generahzed quas~boson approximation and use this method to investigate the interplay of pairing and Q-Q force m the lowest even-parity states of a°O In particular we discuss the equfllbrmm deformatxon of the self-consistent field m the ground and first excited state and the dependence of the energies and some trans~txon rates on the mteractmn parameters
1 Introduction In the last years lnvestlgatlons have been m a d e on the p r o p e r t i e s o f the even p a n t y states m 160 H e r e the first excited 0 + state at 6 06 M e V Is o f p a r t i c u l a r interest Its energy is a b o u t 15 M e V lower t h a n the u n p e r t u r b e d shell-model energy I n a d & t l o n the s t r o n g E2 t r a n s i t i o n f r o m the first e x o t e d 2 + state at 6 92 M e V to this 0 + state B(E2, 2 + ~ 0~-) = 40__+ 15 e 2 fm 4 a n d the large value o f the m o n o p o l e m a t r i x element
(071 Z r210 )= 3 8rm 2 protons
are r e m a r k a b l e It was shown (see for e x a m p l e refs 1 - 3)) t h a t these p r o p e r t i e s c a n n o t be e x p l a m e d b y & a g o n a h z a t l o n o f a reahstlc H a m d t o m a n in the subspace o f all ( 0 + 2 ) h ~ configurations T h e r e is e x p e r i m e n t a l evidence 4) o f the fact, t h a t the 0 + state at 6 06 M e V is the h e a d o f a r o t a t i o n a l b a n d F r o m H a r t r e e - F o c k calculations it follows, t h a t thls 0 ÷ state s h o u l d have a large 4p-4h c o m p o n e n t B r o w n a n d G r e e n 5) used these argum e n t s a n d m i x e d the lowest d e f o r m e d 2p-2h a n d 4p-4h states w~th fixed d e f o r m a t i o n , respectxvely, a n d the F e r m i g r o u n d state F i t t i n g the energies o f the basic states they c o u l d explain the t r a n s i t i o n p r o b a b l h t l e s between the even p a r i t y states o f 160 Celenza et al 6) also d e t e r m m e d the level p o s i t i o n s m the f r a m e w o r k o f this m o d e l in g o o d a g r e e m e n t with the e x p e r i m e n t I n this p a p e r we shall try to u n d e r s t a n d f r o m a simple m o d e l the influence o f &fferent effects as p a i r i n g interaction, Q-Q i n t e r a c t i o n a n d the m e c h a m s m o f configuration m i x i n g on the p r o p e r t i e s o f 0 ÷ states m d o u b l e - m a g i c light nuclei D u e to 671
672
L
MUNCI-IOW AND H
U
/AGER
the slmphficatlons m the given model only a rough quantltatwe description of experimental results can be expected First we investigate the properties of the p a m n g residual interaction in states with J = 0, T = 1 m the framework of the quaslboson approxtmatxon (QBA) So we get essentially the so-called paxrmg vibration model 7, s) Apart from the simplicity of description this opens the possibility to sketch a very Instructive picture of the effect of pairing m the excited states of double-magic nuclei In a recent work 9) we could show, that a satisfying description of the energy in this model is possible by taking into account also the p-h lnteracUon, which is attractive xn even T states In the vicinity of the crmcal point essentml devlatmns from the simple 2p-2h structure which are not dealt with in the stmple QBA would appear Corrections, which take into account more accurately the Pauh principle and lead to a larger excitation energy, are introduced using the generalization of a method due to Hara ~o) and Ikeda et al ~1) A better descnptmn for these states in the v~cmity of the crmcal point xs possible by an exact dxagonahzatlon of all pairing states for gwen shells. For the case of Identical particles the result of such calculations are represented in the paper of Hogasen 7). In a second step we analyse the simultaneous actmn of the pamng and Q-Q force This reqmres a self-consistent description of the deformations of the strongly correlated ground and excited states Such a program was recently realized by Elchler and M a r u m o n 12), who take into account the long range part of the Q-Q force selfconsistently While these authors gwe a self-consistent descnpUon for deformed correlated 2p and 2h states, we use m our paper an alternatwe way to calculate the self-consistent potential which is felt by the particles and holes in the correlated ground and excited states Solving the self-consistent equatmns and the elgenvalue problem simultaneously we calculate the eqmhbrxum deformatmns for the ground and excited states and the energy surfaces 2. Description of the excited 0 + states in double-magic nuclei
We shall describe the excited 0 + states in double-magtc nuclei as correlated 2p-2h states using a generalized quaslboson approximation Possible p-h components from the 2h~o distant oscillator shell we do not take Into account The correlated 2p-2h states we tmagme as bmlt from 2p states of the A + 2 system and from 2h states of the A - 2 system First we treat onlythe pairing interaction (pairing vibrations 7,8)) in the spherical basis (j, m, z) For our calculations we need the number operators for pamcles and holes N j, Nj
"~-
(1 - 0a,) Z m"¢'
0j ~, m'g
+ C3'm'z', ¢j'm%"
+ Cjm ~ Cjm.t:
(2 1) (2 2)
on a gwenj-shell Here the symbol 0j means the Fermi occupaUon number Further
673
STATES IN 160
we introduce creation operators for 2p states with the quantum numbers J = 0,
T=I,T 3 AfT3
Z ( - - ) / ' - - m ' ( ½ " t " l l z'rzllTa)cym','l t + cy-m'e~ + -- 1--01" , 2x/f2j, m'~'*,'z
(2 3)
and creation operators for 2h states BfT3 -- 0j(--)T3 E ( -- )J +"(½Z, ½%l1 -- T3)Cj,.,, ej_ m,2 2x/f2J ,.r1~2
(2 4)
The symbol f2j stands for the quantity j + ½ The destruction operators for 2p states
Aj.T~ and for 2h states BjT3 are the Hermltlan conjugate of the operators defined by formulae (2 3) and (2 4) The commutation relations [N~,I, [N,,,
* * 3, Aj,2T3] = 23j,,j,~ As,2r
(2 5)
B,~r3] + = 23smBj~r3, +
(2 6)
follow directly from the definitions made above For the non-vamshmg commutators between the pair operators we assumed
[A.r,T3,,Af~Tj =6j, s, fT~,T3,(I_ Nj,,~, 2f2s,1]
EB,,,,3,, B;Tj =
(27)
(1-.N,, ]
(28)
2f2jl]
Here we have retained only the terms diagonal in T3, further we have replaced the operators (1 - 0 y ) Z (1 +2z'T3)cs+m,~,cy,,,~,,
0, Z (1 +2"CTa)Gm~C+,
by Nj, and Nj, respectively This corresponds to partially taking Into account the Pauh principle m contrast to the simple QBA The Hamfltoman for a charge-mvarlant pamng residual interaction in J = 0, T = 1 states may be written as H = Hvac+ Z e.rNy- Z ejNs+H; .... J" j '
=
4 Q y , Qy2Ay,T3A/2T3"} - Z V g2. I2j~ B~, T3 Bj2 T3 JIJ2 _
T3
J 1J'2
+ X(--)T3 E X/hs" f2,(A,+T3B f 7"3+ B,_ T3A,'T3)] JJ
H, ac IS the energy of the Fermi ground state
-
(2 9)
674
L MUNCHOW AND H U JAGER
In this expression we introduce two pairing constants, the value G for pamng wxthm the single-particle levels above or below the Fermi surface and the value xG (0 <=x <=I) for the pmrmg Interaction between partmles and holes This may be more reahstm for the descrlptmn of hght nuclei (for physmal arguments see sect 3) Now we construct correlated 2p states for the A + 2 system 12p, Ta) =
r+T, IOo) =
Z
+
,/l-p,,
- J
( - - ) r ' B , - n ] Ioo),
(2 10)
~/1-p,
whmh obey the Schrodmger equation
HV;r,10o)=
(2 11)
EpFpTa[Oc) " +
The correlated ground state ]0o) of the system with the mass number A should be determined from the condition (2 12)
Fpr~10o) = 0 Using the expressions (2 7), (2 8) and (2 12) and the defimtlons p~, -
<0clN,.10~> 2~,
,
Pj =
(2.13)
<0clN,10o> 2~2j
we get the normahzatlon condmon for the wave function (2 10) m the following simple form Y I~,.I2 - y~ I~,12 = 1 (2 14) J"
j
In an analogous manner we construct correlated 2h states for the A - 2 system 12h, 'ra) =
r+31oc> = ~ / -l ' p-, ,
(--)TaA"-ra + ~ ~/1--p, /~'
B,+~ 1 IOD
(2.15)
with the normahzatlon condition
- Z [/~,.I2+ Z I/~,[~ = t ./'
(2 16)
J
and obeying the Schrodlngm equation
HV+r310o)= EuV+r310o)
(2 17)
For the ground-state vector we have the second reqmrement Fhr3[0c) = 0.
(2 18)
In the framework of our model the state vector
,0%
=
Z (-)~-T3 - r, x/3
F p+T 3 [ h.+_ r 3 [ O ¢ )
(2 19)
675
STATES IN 160
describes an excited 0 +, T = 0 state of the A-partmle system The corresponding energy is (2 20) Eo+ = Ep+Eh-Eo, where E o denotes the energy of the correlated ground state (2 21)
HI0o> = Eo[0=>
In order to evaluate the coefficients ~ and fl of the wave functions (2 10), (2 15) and the appropriate excitation energies
wp = e p - Eo,
wh = Eo - eh
(2 22)
we use the conditions following from the eqs (2 11), (2 17) and (2.21)
Wp(OclA/r312p, Ta>,
(0c[[a~,r3, H]I2P, Ta> =
+ ~,,-T, ~p, T3> < 0 d [ ( - ) I+T~B,-r~, H]I2p, Ta) = Wp(0¢l(-) x+r~'+
(223)
for the A + 2 system and <0o1[(-) ~+r3Af_r3, H]12h, T~> = - W~<0oI(-) ~+r3A}_T~12h, T~>, <0eI[B,T 3 , H]12h, T3> = - l~],<0~lB,r~12h, T3>
(2 24)
for the A - 2 system We calculate the matrix elements of commutators in these equations using of the llneanzatlon procedure of the QBA, for example
Afi, r,~As,~r,~]12p, T3> = <0¢l[A,,r~, Af ,r,~]As,=r,~j2p, Tz> ..~ 6,,j,, 6r3r,~(1-p,,)<0dA,,~r,~12p, Ta>
<0d[Aj,r~,
(2 25)
If we stall take into consideration the commutatmn relations (2 5)-(2 8), the deftnation of the Hamdtonlan (2 9) and the intermediate results <0dAj,r~12p, T3> =
%,~l-pj,,
0 C[~. l: ~ ] ~l+r~A+J ' - - T3 "~h l-Jtl~ T3> =
<0d(-)
l+T3
fl,,~/1-Ps,,
+ B,-r312p, T3> = % ~ 1 - p , ,
<0dB~r~12h, /-3> =
fljx/'l--p~,
(2 26) (227)
we get the following e~genvalue equations
2e,,%,-G~/~,,(1-p,,)[ Z ~/f2s',(1-P,',)%',-x J i
28~%-G.,/a,O-&)[xZ ~/a~,,(1-&,0%,,J'l
Z ~/Q,,(1 - p , , ) % , ] = wp%,, Jl
Z #a,,(1-&~)%,] = %%,
2~,.g.- Gda,.(1-p,.)r Z , / a , . , O - p , . , ) & , - ~ Z a'l
Yl
2ejfls- G~/f2~(1"pj)[x 2 ~/'f2~,,(1-Ps',)flJq- 2 J1
(2 28)
J1
dl
4a,,O- p,,)L,] ~/Z~,(1 -p~,)fl~,]
= w~,.,
= Whflj
(2 29)
676
L
M U N C H O W A N D I-I U
JAGER
The two exgenvalue problems (2 28) and (2 29) are described by the same nonHermman matnx A The corresponding normalization conditions (2 14) and (2 16) are distinguished by the sign If the symbols ~: and 2 stand for the number of the pamcle and hole shells characterized b y j ' and j, respectwely, we are interested in the system of equations A7 = wT, (2 30)
N = Y~ I~j,Iz - ~ I~jI2 = + 1 .1"
(2 32)
,/
Depending on the normahzatxon the solutions are attached to either the 2p system (2 28) or to the 2h system (2 29)
{e
lfN=l
{Wp
=
If N = I
w =
fl
ff N = - 1 '
(2
Wh
33)
ff N = - 1
We shall now investigate a few formal propemes of these solutmns. If we define dl = E ~/f2j,(l - p j , ) y , , ,
d2 = E ~/Oj(1-p,)yj,
J'
d
e(w) = y' Gf2j,(1 - p ~ , ) j" 2e j, -- w
g(w) = Z G f 2 j ( 1 - p j ) j 2ej -- w
(2 34)
the elgenvalue problem (2 30) can be written an the form
-
P") (a, - xd2), 2%, - w
a , / a j ( l - pj) ( x d I - d2),
(2
2ej - w
35)
where we find the conditions d 1 = e(w)(dl-xd2)
,
(2 36)
dz = 9(w)(xdl-d2)
The eIgenvalues w(v) are the zeros of the function f ( w ) = g(1 1 - _ _ e ) [ ( i + g ) ( 1 - e ) + x 2 e g
]
(2 37)
In the VlClmty of these zeros the norm (2 32) of the appropriate state vector is proportional to the derxvatwe of the functlonf(w) N(,) = Gld2l~ df(w)~ w=w(~,"
(2 38)
677
STATES IN 160
A quahtatwe discussion of the functlon f(w) for fixed values p (0 < p < 1) and with the physically estabhshed assumption that es < e~. shows us the following All elgenvalues w(') (v = 1, 2, , K+ 2) are real if the quantmes G and x are sufficiently small In this case we label the elgenvalues so that the condition W (1) >
W (2) ~
:> W (r+2)
is fulfilled Then, corresponding to eqs. (2 33) and (2 38) the elgenvalues w(~) (v = ( v ) + I(~ 1, 2, , to) belong to the 2p system, the appropriate state vectors are U-pr~ J~,/\ The remaining elgenvalues w~) (v = x + 1, . , x + 2) belong to the 2h states F~)+ [0¢) + + In the framework of the normahzatlon condmon (2 32) all operators Fpra and Fhr3 form a orthogonal and complete set If we take into conslderatmn the symmetry propemes of the matrix A we find the equation (
Z/s" ?s' - j"
j
,(v) ys ,(u)'~)
= 0,
and now with the help of (2 33) we get the relations E ~t(v)et(u) --j" -.j, - - ,-, ..(v)^.(u) E ~ J ~J = ~v/z J'
J
J"
.1
Z flo,)fl(u)_s,s"
Z fl(~)fl(u), s = - J , .
J'
(2 39)
j
The inverse representations of the operator defimtlons (2.10) and (2 15) are Aj'+ T3
B J+T 3
=
~-
x/1 - p f [ Z ~"j" "(')r(')+ apT3
x/l-psi-
--
V / ~ (_~r3n(,)I.(,)r~], k ./ /'13" h -
~" ~" ~r~~(v)r(,) ~d ~p-T3 ± 7- ~ / ~ t~(')F(')+l P'J hT3 -]"
L, I--] V~K
(2 40) (2 41)
V>K
In order to solve the problem (2 30) we must still calculate the quantities p j, and pj defined by eqs (2 13) For the correlated ground state we have 10o> = KeSl0r>,
S =
S(j))
4(1- ps,)(1-p j)
t
k~
a~-r3a+ ]
u+
"ad'T3 X"d-T3 "
(242)
The Fermi ground-state vector is denoted by 10v>, the quantity K is a normahzatlon constant and the coefficients SO'j) describe the admixture of p-h components to the Fermi ground state We get from the requirements (2 12) and (2 18) the equations for these coefficients
fl~') = Z [3~v)s(J'J) J
(2 43)
678
L MUNCHOW AND FI U JAGER
Using the relations (2 39) one sees that the second equation follows from the first Each of these equations is sufficient to calculate the xZ coefficients SO'j) Now we employ the Identities (0¢INj,10¢) = (0¢I[N~,, S]10c), (OoIN.,[Oo) ---
which are proved in the paper of Harat°), calculate these commutators and obtain by using the relations (2 40), (2 41), (2 12) or (2 18) and (2 43) the results pj, =
3 ~', (fl(~)" s' ) 2 ~>~ ,
pj =
3 ~ (~))2 ~---~
f2j,
(2 44)
Or
3. Discussion of the results for a spherical nucleus We have solved the system of eqs (2 30)-(2 33) and (2 44) by iteration
A(e, ~, G, x, P~)~+t = w~+11~+i, P,,+t = P(?,,+~),
Po = O,
(3 l )
for the nucleus 160 We take Into consideration the lp and the 2s, ld shell The assumed single-particle energies are as common for p-h calculations in 160 (see ref 13) and table 1) In our first lteratmn step we get the results of the ordinary QBA (0 = O) TABLE 1 Single-particle energies m t60 Level e(MeV)
1P~r --21 8
1pk --15 65
1d{
2s~r
1d{
- - 4 15
--3 27
0 93
A priori we can only fix a certain interval of variation for reasonable values of G They may be of the order of the pairing constants, which are used in calculations in heavier nuclei, where one takes G = 20 .28/A MeV Calculations recently done in the seniority scheme seem to favour the value G = I6 20/A MeV in the oxygenlum region 14) The pairing constant for the Interaction between anyj~ and j2 shell stands for the negative antlsymmetrlc matrix element between the J = 0, T = 1 states of these shells divided by [(2J1 +1)(2J2+ 1)]~ These quantities calculated for the Glllet force t3) are negative for interaction between states with opposite panty and on the average by the factor 0 7 smaller than the remmnlng positive quantities for interaction between states with the same parity Therefore we have used two pamng constants, the quantity
STATES IN 1°O
679
G for pairing within an oscdlator shell and the quantity x G for palnng between the lp and the 2s, ld shell The sign of the quantity x is ms~gmficant in the framework of our model (compare eq (2 37)) It is clear that using x = 1 instead of x < 1 means overestimating the ground state correlation effects In order better to understand the results of our calculatmn we seek the dependence of the lowest excitation energy Wo'l'.mlll = Wp. . . . --Wh . . . .
NOrmm/MeV
× ~
07
20
15
tO
I>
'
GcPJt
'
~/Me V
F i g 1 E x c i t a t i o n e n e r g y o f t h e l o w e s t 0 + s t a t e in d e p e n d e n c e o n t h e p a m n g c o n s t a n t G F o r a c o m p a r i s o n the results of an exact dlagonahzatmn and of the QBA are given (m contrast to the other
results of th~s paper here we take into account only the shells lp~, ld~r, 2s~) on the pairing constant G (see fig 1) The comparison of the QBA with an exact dlagonahzation 1s) of the pairing residual interaction shows two characteristics of the QBA whlch are caused by the violation of the Pauh principle Firstly, the result of the QBA vanishes for a certain value of G (crmcal point), while the exact solution shows only a minimum, secondly in the QBA the states with the lsospln T = 0, 1, 2 are degenerated The QBA is not apphcable m the VlCimty of the crmcal point In this case a possxble low value of the excltanon energy cannot demonstrate the adequacy
680
L MUNCHOW AND H U JAGER
of the approximation The same remark holds for the transition amphtudes, where the QBA yields too large values The dependence of the posmon of the crmcal point on x may be calculated with the help of the functionf(w) defined by eq (2 37), from the conditions
f(wcr,t, Gcr,t, x) ~f
=
O,
= 0
~w
(3 2)
W= Wcrit
Some results are shown m table 2 In fig 2 the dependence of the lowest excitation energy on the pairing constant G is shown for the ordinary QBA and the generalized QBA In the latter case we use the more exact commutation relations (2 7) and (2 8) TABLE 2
Posmon of the critical point in dependence on x x Ger~t (MeV)
0
01
02
03
04
05
07
10
3 36
3 12
2 87
2 67
2 49
2 33
2 08
1 78
WO.mm/MeV
x = 07
I0
I
Get.d
"G/MeV
F i g 2 T h e G-dependence o f the lowest excitation energy c a l c ul a t e d w i t h the o r d i n a r y a n d generahzed QBA
The consideration of pamcles and holes in the correlated ground state (p ~ 0) leads to the result, that according to eqs (2 28) and (2 29) the pairing force becomes less effective in the 2p and 2h states In the vlclmty of the critical point it appears to be an essential difference between the results of the QBA and the generahzed QBA In this paper we describe the excited 0 +, T -- 0 states as superposmon of the 2p and 2h states, each of which is investigated mdependently from the other Therefore
STATES IN 160
681
we have not taken into account the p-h interaction between the 2p and 2h states m the framework of our model In a former paper 9) we showed, that the T = 0 part of the p-h interaction yields an energy shift of about 5 MeV In the following only the results of the generahzed QBA are gxven For the expectation value of the Hamlltonlan (2 9) in the correlated ground state we find Eo =
= H~.o- 3[ E Wp(') E (a~,))2_ Z ~ ' E (fir))2] V~K
v>K
d
(3 3)
J"
Wath increasing p a m n g constants G or xG the correlations m the ground state become larger and the energy of the ground state decreases In fig 3 we demonstrate this effect by the dependence of the p a m c l e and hole number in the ground state (2 13), (2 44) on the pairing constant G
x=07
G=z~,q6
ol_N_.
727J
F72
G=25/16
G=30/I6
lds/2
2Sl/2
td3/2
1PI/2
1.°3/2
Fig 3 D e p e n d e n c e o f the quantities (0c[Nf]0c) a n d (0c[Nsl0e) on the pairing c o n s t a n t G
For the monopole matrix element between the first excited 0 +, T = 0 state and the correlated ground state we get in the subspace of our configurations (0 +, mini ~ protons
r2[0c) = fix/3 ~ o:j~,'n,13sm,~x
(3 4)
m(.o j '
Here we have used the elgenfunctlons of a harmonic oscdlator with hm = 41 A -¢ MeV In table 3 we show the results for the monopole matrix element (3 4) in depend-
682
L MUNCHOW AND H U JAGER
ence on the p m r i n g c o n s t a n t s C o r r e s p o n d i n g to the increase o f the c o r r e l a t i o n s with G a n d x Its value increases too. I n c o m p a r i s o n to the e x p e r i m e n t a l value o f 3 8 fm 2 o u r t h e o r e t i c a l results are t o o small for a spherical nucleus 160 a n d r e a s o n a b l e G and x TABLE 3
The monopole matrLx element (fma) between the first excited 0+, T = 0 state and the ground state m dependence on G and x G(MeV)
1
1 125
1 25
1 57
1 875
x = 07 x ~ 08
0 472 0 544
0 575 0 663
0 693 0 799
1 057 --
1 533 --
4. Consideration of deformation in the excited states T h e d e s c r i p t i o n o f the excited 0 + states d e v e l o p e d In the first t w o sections is Inc o m p l e t e , because it does n o t a c c o u n t for the d e f o r m a t i o n o f the excited states There IS clear e x p e r i m e n t a l evidence 4) for the fact t h a t the first excited 0 + state in 160 is the h e a d o f a r o t a t i o n a l b a n d T h e E2 t r a n s i t i o n p r o b a b i l i t i e s show the typical collective e n h a n c e m e n t I 6 ) . F r o m the studies o f d e f o r m e d nuclei we k n o w t h a t the selfconsistent p a r t o f the Q-Q force c a n be Vlsuahzed as the r e a s o n for d e f o r m a t i o n o f the single-particle field I n the case o f d o u b l e - m a g i c nuclei a d e f o r m a t i o n w o u l d force the m a r e 2p-2h states (which are closest to the F e r m i surface) to get a smaller excitat i o n energy, l e a d i n g also to lower energies o f the c o r r e l a t e d states F o r a q u a n t i t a t i v e d e s c r i p t i o n we a d d to the H a m l l t o n x a n (2 9) a Q-Q force H = Hvac+ ~ e j ' N s ' J'
(Jmz)
~esNa+H'pa,r+HQQ, d
I~ ,
t
2
+
¢
x (J'2m2z2l r Y2-u]Jxm2"c2>cj'lm':',cs'zm'2x'zCjzm:2 aim:,'
(4 1)
a s s u m i n g the s a m e c o n s t a n t Z for the p-p, n-n a n d p - n interaction I n s t e a d o f using the full H a m l l t o n l a n (4 1) we first a p p l y a m o d e l H a m l l t o n l a n / 4 , because we w a n t to t r e a t only the self-consistent p a r t o f the Q - Q force Such a p r o c e d u r e is quite analo g o u s to the one used in d e f o r m a t i o n calculations with a p a i r i n g force 17), Elchler a n d M a r u m o r l 12) also used a similar m e t h o d for describing d e f o r m e d c o r r e l a t e d states F o r the m o d e l H a m l l t o n I a n we have n
+ ' C3m'~C.Im't"~npa,r--Z Z ~,Q~, ,
= Jm~
(4 2)
#
where Qu is the o p e r a t o r of the m a s s q u a d r u p o l e m o m e n t u m
Q~ = 4x/~T-~~ (Jl ml"cl[r2y2.[J2m2"c2)Cf m:l c~2m2~2 (jmz)
(4 3)
683
STATES IN 160
The five parameters ~ are quadrupole parameters specifying the deformation of the potential well Performing as usual a transformation to the body-fixed system and assuming axial symmetry one gets from eq (4 2) /~ + , = E %csm~Csm~+Hpa,r-xflQo, (4 4) ./mr
where fl stands for eo in the body-fixed system and is the deformation parameter Because of the presence of the operator Qo m eq (4 4) the single-particle part of the Hamlltonlan is not diagonal m the chosen sphermal basis By a unitary transformation we go to a new basis (k, y, z) in which the self-consistent part will be diagonal + Cjkr = E
J
+
a k r CkrT •
(4 5)
The quantum number y specifies different states with the same k and z The selfconsistent energies ekr in the deformed basis are elgenvalues of the equations
ekra~r = e./a],r--42fl~/~-~ ~ (J1 kz[, 2y2olJkz)a~,
(4 6)
./l
Here we neglect the m a m x elements of the quantity r2Y20 between different oscillator shells The real transformation coefficmnts a~,r satisfy the orthogonahty relations
E akylSak~, 2, = ~,,r2,
E "kr'""~r'J2= 6j1j 2
./
Y
(4 7)
States distinguished only by the sign of the quantum number k are degenerate, for the coefficients a~r we obtain the relation
a J-k,(--) s-~ = a],~
(4 8)
from the symmetry properties of eq (4 6) In the future we only use positive values k for the classification of the deformed single-particle levels In this sense two protons and two neutrons fill a subshell characterized by (k, ~) Now we introduce pamcles and holes m the deformed basis The symbol 0kr denotes the occupation number in the deformed Fermi ground state We define the number operators for particles and holes Nk,r" = (1--Ok,r,) ~, ( Ck,~,r, + Ck,r,r,+C_k,r,eC_k,~,r,), + r'
Nkr = Ok~'E
(Ck)'r
+ "~-C-Rr~C-kr~) CkTc +
(4 9) (4.10)
r
and the creation operators for 2p or 2h states with the quantum numbers K = 0, T=I,T a +
y,
=
1 ,, , ÷ ÷ (~Z,~Z2IITa)Ck,r,,,,C_k,,,r, ~
(4 11)
1~'11~'2
Bk;r ~ = 0kr(__)*+k+ r~ Z (½Z,½ZZ[1-- 7z)Ckm C-kin flY2
(4 12)
684
L MUNCHOW AND H U JAGER
In this paper we Investigate the double-magic nucleus 160 and take into consideration one oscillator shell above and below the Fermi surface Therefore we can use the orthogonahty relations (4 7) for the transformation of the pairing residual interaction (2 9) We get the model Hamfltonlan/~ in the deformed basis
ffl = 4 ~ Ok.:F,k.~'-~-2 8k'~"Nk'~"-- 2 8k~'NkT+ H'p. . . . k7 ,
=
k' 7"
__
k7
v-
T3
BkI~,t_TaBk272_T
k'l]:' lk'2~'2
(Ak+;,,TaBk;-rad-Bkr-raAk,~,,r3)]
"~ X( --)T3 Z
3
kl)'lk2T2 (413)
k ' )" k~,
The model Hamlltonlan/4 is a part of the full Hamlltonian (4 1) in the body-fixed system H = Hva e -- 2 ~ 0j(23 + 1)~s + HQQ + xflQo + ff-I J
If we use the representation (4 13) and the relation
4 ~., Okre, r = 2 ~ 0~(23 + l)e, k),
.I
for double-magic nuclei, we get
H = Ho + xflQo + H¢~o, Ho = H,,.o+ ~'. ek.r. Nk,r.-- E ekTNl,~+H'p.,r k')"
(4 14)
k~,
The Hamdtoman H o and also the commutators of the number and pair operators (4 9)-(4 12) have the same form in the deformed basis as in the spherical case, if we substitute in the expressions (2 5)-(2 9) ] ~ k, ~,
f2~ -* 1
For this reason we calculate the eigenfunctions and elgenvalues of the Hamlltonian H o with help of the generahzed QBA analogous to what was done in sect 2 The correlated ground state [0c) and the first excited state [0 +, m m ) of the Hamiltonlan H o are again of special interest The corresponding eigenvalues of the full Hamiltonian H are calculated as expectation values like in perturbation theory We obtain expressions for the energy of the correlated ground state (0¢lH[0c> = H . ~ - 3 [ E W..(*)E (a(k;')2 - Z W(*)E (tiff),)2] v<~c
k~'
v>tc
k'7'
+ z/3,
(4 15)
and of the first excited state (0 +, mmlH[0 +, m m ) = w ~ ", p(~'~) - w(m~) Veh / a aU va -
-
e
3[ E w(~) Z (a(k;))2 -- Z 14'h(*) Z U'k~ ' V:t¢~2-1j (~) v
k7
v>~
k'7'
+Zfl(0 +, minlQo[0 +, r a i n ) + ( 0 +, mmIHQQI0+, mln)
(4 16)
S T A T E S I N 160
685
from the eqs (3 3), (2 20) and (2 22) Using a Hartree-Fock type decomposmon for the Q-Q interaction (4 1) m the body-fixed system with axial symmetry and neglecting the exchange terms as usual (0clHaol0¢) = _1•(0¢1Qol0¢)2, (0 +, mmlHaq[0 +, rain) = -½Z(0 +, minlQol0 +, ram) 2
(4 17)
we only need expressions for the expectation values of the quadrupole momentum operator Qo (4 3) to calculate the energies (4 15) and (4 16) In the subspace of our configurations the relation Q0 = 4x/ire[ ~, (k'T'[ r2 Y2olk'7')Nk'r'- Z (kT[ r2 Y2olkT)Nkr] k",,"
(4 18)
k3'
holds The matrix elements of number operators in the correlated ground state (0c[Nk,r.10c) = 6/.., Wk'r', ,
(0¢[Nkrl0¢) = 6 ~ (aCk~))2
V:>K
(4 19)
V~K
are known from the eqs (2 13) and (2 44) Using the exphclte form of the wave functions (2 19), (2 10), (2 15) and the commutators of the number and pair operators we obtain for the matrix elements in the first excited state (0 +, mlnlNk,~,10 +, mln) -- 2[(~k'r') m m 2 --(flk'v') m a x 2 ] + (0olNk'v,10o), (0 +, mmlNk~[0 +, m m ) = 2['(flk~ m a x ) 2 --(Uk~ m , n ) 2 ]+(0¢[Nkrl0o).
(420)
These expectation values of the full Hamfltonlan H depend both directly (see eqs (4 15) and (4 16)) and indirectly (see eq (4 6)) on the parameter fl which is related to Nllsson's is) deformation parameter 6 by
Zfl = ½omco2,
(4 21)
where co IS the oscillator frequency Klssllnger and Sorensen I9) give for the constant Z of the Q-Q interaction (4 l) the estimation 110 1 (~)21VleV Z - 4.4 ( N + ~ ) z We assume again hco = 41 A -~ MeV, the oscdlator quantum number N l s equal to 1 for the hole states and equal to 2 for the particle states Because we in our description have used the same constant Z for particle and hole states we Interpolate Z = 150z,
z - 4A(2+~)2
MeV
(4 22)
In table 4 and fig 4 we show the energy of the correlated ground state (4 15) and of the first excited state (4 16) in dependence on the deformation parameter 6 for given values of the pairing constants G and x The ground-state energy has a minimum for
686
L MUNCHOWAND H
U
JAGER
TABLE 4 T h e energies (4 15), (4 16) m dependence o n the d e f o r m a t i o n p a r a m e t e r t5 (G = 1 MeV, x = 0 7, all energies are given m MeV) G r o u n d state
First excited state
A
ZflO-,o
½ZQ_,2o
energy
14/o+, mla
Zfl0o
½Z0~0
energy 15 976
0
--1 0936
0
0
--1 0936
17 070
0
0
0 05
--1 0971
0 0081
0 0001
--1 0891
16 749
0 655
0 574
15 743
0 10
--1 1081
0 0348
0 0004
--1 0737
15 755
2 655
2 293
15 009
0 15
--1 1281
0 1022
0 0011
--1 0270
14 196
5 388
4 195
14 261
0 20
--1 1592
0 1814
0 0027
- - 0 9805
12 280
8 327
5 637
13 811
0 25
--1 2051
0 3534
00065
- - 0 8582
10 155
11 359
6 713
13 596
0 30 0 35
--1 2725 --1 3690
0 6951 1 4827
0 0175 0 0584
- - 0 5949 0 0553
7 902 5 585
14 557 18 202
7 657 8 796
13 529 13 622
0 40
--1 4733
3 6555
0 2716
1 9106
3 330
23 246
10 983
14 120
T h e energy scale is chosen so that the energy o f t h e spherical g r o u n d state Hvac ~s equal zero By A we denote here t h e quantxty -
3[ Z w. E v~x
Z
k7
v>K
k'~,"
\ / J k ' 7 ' ) J~
t h e s y m b o l Qo m e a n s the expectation value o f q u a d r u p o l e m o m e n t u m m the given state
E/MeV
10
5
01
02
0n
i
Fig 4 T h e energy o f the correlated g r o u n d state a n d o f the first excited state m dependence o n the d e f o r m a t i o n m accordance with table 4
STATES IN 160
687
the spherical shape and depends weakly on the deformation for 6 < 0 2, for larger values of 6 it Is essentially determined by the deformation energy XB(0olaol0o)
-
kx((0ciQol0o))2
The reason for this behavlour is the following The diagonal matrix elements of the quantity r z Y2o are negative for the highest hole levels (lP½, k = ½, lp~, k = {) and positive for the lowest particle levels (ld~, k = ½, ldl, k = {), their absolute values Increase with the deformation Besides, with increasing deformation the energy difference between the lowest particle and highest hole level becomes smaller and therefore also the particle and hole numbers grow Both effects cause in the same direction an Increase of the quadrupole momentum expectation value (compare with the formula (4 18)) On the other slde the ground state energy is lowered by the correlation energy (quantity A in table 4) with Increasing ground-state correlations; this effect is insignificant for great deformation In the first excited state the quadrupole momentum expectation value as larger than an the ground state because of the increased particle and hole numbers For small deformations the energy decrease of the Intrinsic state caused by the more favourable positions of smgle-particle levels is important. Our main interest is just to describe this effect, therefore we used the model Hamlltonlan ~ or H o We should hke to remark that by a different choice of the Q-Q force constant z = 140z instead of (4 22) the ground-state energy in table 4 changes less than 0 02 MeV and the energy of the first excited state less than 0 73 MeV Now it is clear that the energy of the first excited state has a imnlmum for a fixed deformation of the self-conslstent potential From the requirement 0(0 +, mmlH[O +, rain) = 0
a~ follows, that the position of this mlmmum is determined by the condition /~o = (0 +, mmlQol0 +, mln)
(4 23)
analogous to the calculation in the conventional pairing plus Q-Q force case (see for example ref 17)) This is a very instructive self-consistency requirement If the system is equilibrating the deformation fl of the average potential in eq (4 6) should be the same as the deformation of the density distribution of the nucleons With the help of condition (4 23) we are able to understand the position of the minimum in dependence on the pairing constants G and x and on the Q-Q force constant z We show m fig 5 a graphic solution of the self-consistency requirement (4 23) For small values of G only the lowest particle state and the highest hole state are occupied with two particles or holes and give a contribution to the quadrupole momentum expectation value For increasing values of G the pairing force within the particle or hole system raises the contribution of higher particle states and lower hole states, leading to a de-
688
L MUNCHOW AND H U JAGER
ciease of the quadrupole momentum, and on the other hand the pairing between the particle and the hole system hfts mainly nucleons from the highest hole state to the lowest particle state, leading to an increase of the quadrupole momentum The result of these two opposite effects depends on the position of the single-particle levels With increasing deformation the level distance within the particle or hole system grows and the mimmum distance between the particle and hole system decrease In fig 5 we therefore get an intersection of the quadrupole momentum curves which belong to different values of the constant G On the left-hand side of this intersection x
/
///
=07
/
/ / /I
/
--
p:~p
Fig 5 Graphic solution of the self-consistency reqmrement (4 23)
point we get the common competition effect between palnng and Q-Q force, the pairing leading to smaller values of the equfllbrmm deformation On the right the paxrlng force amphfies the equlhbrmm deformation We have computed the reduced E2 transition probability to our first excited state using the formula B(E2, 2 + ~ 0 +) = ~
1
((0 +, mmlQol 0+, ram)) 2
(4 24)
We used the same effectwe charges (ep - 1 5e, e n = 0 5e) as Celenza et al 6) and
STATES IN leO
689
consxdered the symmetry between proton and neutron states in our wave functmns The results are gtven m table 5 m dependence on the pairing constants and on the deformatxon We see, that for the equlhbrlum deformations (underhned values in table 5) our theoretical E2 transmon rate is about the factor 4 smaller than the experimental value We also computed the monopole matrxx element (3 4) between the deformed first excited state and the corresponding correlated ground state of the same deformation, thus we &d not take into account the overlap factor between the deformed and the spherical ground state Table 6 shows that the monopole matrix element increases for greater deformations, but as in the spherical case (table 3) the results are too small compared with the experimental value TABLE 5 The reduced E2 transxtmn p r o b a b d m e s (4 24) In units o f e2fm4
0 G=IMeV,
005
010
015
020
025
030
035
040 1481
x=07
0
077
309
566
760
905
1033
1186
G=
1MeV, x = 0 8
0
076
306
566
767
921
1062
12 41 15 82
G=
1 125MeV, x = 0 7
0
063
262
513
727
897
1055
12 59
--
The underlined values are the nearest to the equilibrium deformation TABLE 6 The m o n o p o l e matrix d e m e n t (fm 2) between the deformed first excited state and the corresponding correlated g r o u n d state o f the same deformation
G= G= G=
1 MeV, x = 0 7 1 MeV, x = 0 8 1 125MeV, x = 0 7
0 05
0 10
0 15
0 20
0 25
0 30
0 35
0 40
047 054 057
045 052 056
045 052 056
047 054 059
052 060 065
062 071 077
080 091 099
1 16 129 --
The posmon of the equlhbrmm deformation depends on the constant g Therefore we also made a Ndsson type determination of the equilibrium deformation That means, mstead of fullfilhng the self-consistency condmon (4 23) we requtre the volume of the nucleus to be the same for any deformatxons (incompressible nucleus) This leads to a deformation dependent oscillator frequency o~(6) = o)o(1 --}62 -- ~-7-v a 6"~a~-"~ /
(4 25)
in the equatmn for the deformed single-particle states ek, ak, j = [ ( N + ~)h~o(3) + e, - ( N + ~)hcoo]3 %,
- 4 ~/~hco(6)6 ~ (jlklre Y2oljk)a~;
(4 26)
690
L MUNCHOW AND H U JAGER
(compare with the formula (4 6)) with the constant term
~l-(N+~)~Oo added, in order to get the empirical single-particle energies for 6 = 0 After computing the deformed single-particle states we start from the Hamiltonian
H = ¼Z 40krek,+ Z ek','Nk'~'-- Z ek~N~,+H'pa,r k7
k' 7'
(4 27)
k~
in the body fixed system and calculate its energies and wave functmns for the ground and excited states in the framework of the generahzed QBA The first term in this Hamdtonlan represents the energy of a deformed Fermi ground state, the factor 3 arises by subtracting the half of the potentml energy from the sum of single-particle energies TABLE 7
The energy of the correlated ground state and of the first excited state for an incompressible nucleus xeO (relative to the spherical Fermi ground state) 6
3 Y Ok,,eke, k~
Correlation energy
G r o u n d state energy
F i rs t excited state Wo+, mln
0 0 0 0 0 0 0 0 0
05 10 15 20 25 30 35 40
0 0 0 1 3 5 9 12 18
207 855 989 669 976 030 987 090
--1 --1 --1 --1 --1 --1 --1 --1 --1
094 096 104 119 143 177 229 302 396
--1 --0 --0 0 2 4 7 11 16
094 889 249 870 526 799 801 685 694
17 16 15 14 12 10 8 6 3
069 768 826 346 529 511 364 142 910
energy 15 15 15 15 15 15 16 17 20
975 879 577 216 055 310 165 827 604
All energies are given in MeV, G = 1 MeV, x = 0 7
In table 7 and fig 6 the energy of the correlated ground state and of the first excited state are gwen in dependence on deformation We get qualitatively sxmilar results as for the previous calculation (table 4, fig 4) Again the correlated ground state has an energy minimum for the spherical shape and the first excited state possesses an energy minimum for a certain deformation Going into details we find differences mainly caused by the Increase of the single-particle energies ek~ (4 26) for large deformations Here we note a stronger rise of the correlated ground-state energy with increasing deformation, and therefore we find the equlhbrmm deformation of the first excited state at smaller values of the parameter 6 An increase of the pairing constants G and x lmphes greater correlations in theground state and a decrease of the excitation energy Wo÷, m,. and leads to smaller equdlbrlum deformations
STATES IN 180
691
5. Conclusions For the double-magic hght nuclei the problem arises to describe strongly correlated states with an essentially different self-consistent potential leading to different effective deformations The co-existence model starts from states with a fixed deformation calculated m a self-consistent procedure and then mixes them in order to get all the correlation effects We use a model with fixed deformation for every state This opens the posslblhty to lnvesngate in detad effects as such the competlnon of pmrmg and Q-Q forces determining the equlllbrmm deformanon in the ground and excited E/MeV
20
15
10
41
~1~ff~
¢-
02
03
or
Fig 6 The energy of the correlated ground state and of the first excited state for an incompressible nucleus m accordance with table 7 states, the influence of these forces on the translt~on properties etc We are able to explain the existence of low-lying strong collective states m double-magic hght nuclei, but quanntatwely the collective and correlation effects are too small From the standpoint of this description the system is m the wcmlty of the critical point, this leads to strong anharmomc effects We took into account only the correcnons needed to overcome the violation of the Pauh prmclple The anharmonlc~ty would lead to stronger 4p-4h components m the first excited state and possible also to a deformed ground state The description of the pairing interaction used is essennally the generahzanon of the B~s method for a magxc nucleus m which case the latter would give only a
692
L
MUNCHOW
AND H
U
JAGER
trivial s o l u t i o n for the gap e q u a U o n if projection is n o t used. Therefore we c a n say t h a t the g r o u n d state is similar to a superflmd state due to the d e f o r m a t i o n of the self-consistent field I n order to take a n h a r m o m c x t y better i n t o a c c o u n t a n a n a l o g o u s calculation with exact pairing wave f u n c t i o n s is n o w i n progress W e t h a n k the m e m b e r s of o u r staff, i n particular H R. Klssener a n d C. Rtedel, for m a n y suggestions F r o m the b e g i n n i n g D r hints a n d helpful attentxon.
K
Muller s u p p o r t e d us by interesting
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
J Sawxckl, Phys Rev 126 (1962) 2231 Y Abgrall, J de Phys 24 (1963) 1113 S S Wong, Phys Lett 20 (1966)188 E B Carter, G E Mitchell and R H Davies, Phys Rev 133B (1964)1421, 1434 G E Brown and A M Green, Nuclear Physics 75 (1966) 401 L S Celenza, R M Drelzler, A Klein and G J Drelss, Phys Lett 23 (1966)241 J Hogasen, Ann de Phys 8 (1963) 697 D R B~s and R A Brogha, Nuclear Physics 80 (1966) 289 H U Jager and L Munchow, Proc Conf on Nuclear reactions with hght nuclei and nuclear structure, ZfK Rossendorf (1966) K Hara, Progr Theor Phys 32 (1964)88 K Ikeda, T Udagawa and H Yamaura, Progr Theor Phys 33 (1965) 22 J Elchler and T Marumon, private commumcatlon V Glllet and N Vmh Mau, Nuclear Physics 54 (1964) 321 F Donau, private commumcatlon H R Klssener, unpubhshed S Gorodetzky et a l , J de Phys 24 (1963) 887, H Fuchs, K Hagemann and C Gaarde, Nuclear Physics 66 (1965) 638 M Baranger and K Kumar, Nuclear Physics 62 (1965) 113 S G Nllsson, Mat Fys Medd Dan Vld Selsk 29, No 16 (1955) L S Klsshnger and R A Sorensen, Mat Fys Medd Dan Vld Selsk 32, No 9 (1960)
Not to be reproduced by photoprmt or mlcrofilrnwtthout written permlssxonfrom the pubhsher
E V E N - P A R I T Y S T A T E S O F 160 IN A PAIRING-PLUS-QUADRUPOLE
MODEL
L M(JNCHOW and H U JAGER Central Instttute for Nuclear Research, Rossendorf, Germany (GDR)
Received 28 August 1967 Abstract In this paper we give an outhne of the generahzed quas~boson approximation and use this method to investigate the interplay of pairing and Q-Q force m the lowest even-parity states of a°O In particular we discuss the equfllbrmm deformatxon of the self-consistent field m the ground and first excited state and the dependence of the energies and some trans~txon rates on the mteractmn parameters
1 Introduction In the last years lnvestlgatlons have been m a d e on the p r o p e r t i e s o f the even p a n t y states m 160 H e r e the first excited 0 + state at 6 06 M e V Is o f p a r t i c u l a r interest Its energy is a b o u t 15 M e V lower t h a n the u n p e r t u r b e d shell-model energy I n a d & t l o n the s t r o n g E2 t r a n s i t i o n f r o m the first e x o t e d 2 + state at 6 92 M e V to this 0 + state B(E2, 2 + ~ 0~-) = 40__+ 15 e 2 fm 4 a n d the large value o f the m o n o p o l e m a t r i x element
(071 Z r210 )= 3 8rm 2 protons
are r e m a r k a b l e It was shown (see for e x a m p l e refs 1 - 3)) t h a t these p r o p e r t i e s c a n n o t be e x p l a m e d b y & a g o n a h z a t l o n o f a reahstlc H a m d t o m a n in the subspace o f all ( 0 + 2 ) h ~ configurations T h e r e is e x p e r i m e n t a l evidence 4) o f the fact, t h a t the 0 + state at 6 06 M e V is the h e a d o f a r o t a t i o n a l b a n d F r o m H a r t r e e - F o c k calculations it follows, t h a t thls 0 ÷ state s h o u l d have a large 4p-4h c o m p o n e n t B r o w n a n d G r e e n 5) used these argum e n t s a n d m i x e d the lowest d e f o r m e d 2p-2h a n d 4p-4h states w~th fixed d e f o r m a t i o n , respectxvely, a n d the F e r m i g r o u n d state F i t t i n g the energies o f the basic states they c o u l d explain the t r a n s i t i o n p r o b a b l h t l e s between the even p a r i t y states o f 160 Celenza et al 6) also d e t e r m m e d the level p o s i t i o n s m the f r a m e w o r k o f this m o d e l in g o o d a g r e e m e n t with the e x p e r i m e n t I n this p a p e r we shall try to u n d e r s t a n d f r o m a simple m o d e l the influence o f &fferent effects as p a i r i n g interaction, Q-Q i n t e r a c t i o n a n d the m e c h a m s m o f configuration m i x i n g on the p r o p e r t i e s o f 0 ÷ states m d o u b l e - m a g i c light nuclei D u e to 671
672
L
MUNCI-IOW AND H
U
/AGER
the slmphficatlons m the given model only a rough quantltatwe description of experimental results can be expected First we investigate the properties of the p a m n g residual interaction in states with J = 0, T = 1 m the framework of the quaslboson approxtmatxon (QBA) So we get essentially the so-called paxrmg vibration model 7, s) Apart from the simplicity of description this opens the possibility to sketch a very Instructive picture of the effect of pairing m the excited states of double-magic nuclei In a recent work 9) we could show, that a satisfying description of the energy in this model is possible by taking into account also the p-h lnteracUon, which is attractive xn even T states In the vicinity of the crmcal point essentml devlatmns from the simple 2p-2h structure which are not dealt with in the stmple QBA would appear Corrections, which take into account more accurately the Pauh principle and lead to a larger excitation energy, are introduced using the generalization of a method due to Hara ~o) and Ikeda et al ~1) A better descnptmn for these states in the v~cmity of the crmcal point xs possible by an exact dxagonahzatlon of all pairing states for gwen shells. For the case of Identical particles the result of such calculations are represented in the paper of Hogasen 7). In a second step we analyse the simultaneous actmn of the pamng and Q-Q force This reqmres a self-consistent description of the deformations of the strongly correlated ground and excited states Such a program was recently realized by Elchler and M a r u m o n 12), who take into account the long range part of the Q-Q force selfconsistently While these authors gwe a self-consistent descnpUon for deformed correlated 2p and 2h states, we use m our paper an alternatwe way to calculate the self-consistent potential which is felt by the particles and holes in the correlated ground and excited states Solving the self-consistent equatmns and the elgenvalue problem simultaneously we calculate the eqmhbrxum deformatmns for the ground and excited states and the energy surfaces 2. Description of the excited 0 + states in double-magic nuclei
We shall describe the excited 0 + states in double-magtc nuclei as correlated 2p-2h states using a generalized quaslboson approximation Possible p-h components from the 2h~o distant oscillator shell we do not take Into account The correlated 2p-2h states we tmagme as bmlt from 2p states of the A + 2 system and from 2h states of the A - 2 system First we treat onlythe pairing interaction (pairing vibrations 7,8)) in the spherical basis (j, m, z) For our calculations we need the number operators for pamcles and holes N j, Nj
"~-
(1 - 0a,) Z m"¢'
0j ~, m'g
+ C3'm'z', ¢j'm%"
+ Cjm ~ Cjm.t:
(2 1) (2 2)
on a gwenj-shell Here the symbol 0j means the Fermi occupaUon number Further
673
STATES IN 160
we introduce creation operators for 2p states with the quantum numbers J = 0,
T=I,T 3 AfT3
Z ( - - ) / ' - - m ' ( ½ " t " l l z'rzllTa)cym','l t + cy-m'e~ + -- 1--01" , 2x/f2j, m'~'*,'z
(2 3)
and creation operators for 2h states BfT3 -- 0j(--)T3 E ( -- )J +"(½Z, ½%l1 -- T3)Cj,.,, ej_ m,2 2x/f2J ,.r1~2
(2 4)
The symbol f2j stands for the quantity j + ½ The destruction operators for 2p states
Aj.T~ and for 2h states BjT3 are the Hermltlan conjugate of the operators defined by formulae (2 3) and (2 4) The commutation relations [N~,I, [N,,,
* * 3, Aj,2T3] = 23j,,j,~ As,2r
(2 5)
B,~r3] + = 23smBj~r3, +
(2 6)
follow directly from the definitions made above For the non-vamshmg commutators between the pair operators we assumed
[A.r,T3,,Af~Tj =6j, s, fT~,T3,(I_ Nj,,~, 2f2s,1]
EB,,,,3,, B;Tj =
(27)
(1-.N,, ]
(28)
2f2jl]
Here we have retained only the terms diagonal in T3, further we have replaced the operators (1 - 0 y ) Z (1 +2z'T3)cs+m,~,cy,,,~,,
0, Z (1 +2"CTa)Gm~C+,
by Nj, and Nj, respectively This corresponds to partially taking Into account the Pauh principle m contrast to the simple QBA The Hamfltoman for a charge-mvarlant pamng residual interaction in J = 0, T = 1 states may be written as H = Hvac+ Z e.rNy- Z ejNs+H; .... J" j '
=
4 Q y , Qy2Ay,T3A/2T3"} - Z V g2. I2j~ B~, T3 Bj2 T3 JIJ2 _
T3
J 1J'2
+ X(--)T3 E X/hs" f2,(A,+T3B f 7"3+ B,_ T3A,'T3)] JJ
H, ac IS the energy of the Fermi ground state
-
(2 9)
674
L MUNCHOW AND H U JAGER
In this expression we introduce two pairing constants, the value G for pamng wxthm the single-particle levels above or below the Fermi surface and the value xG (0 <=x <=I) for the pmrmg Interaction between partmles and holes This may be more reahstm for the descrlptmn of hght nuclei (for physmal arguments see sect 3) Now we construct correlated 2p states for the A + 2 system 12p, Ta) =
r+T, IOo) =
Z
+
,/l-p,,
- J
( - - ) r ' B , - n ] Ioo),
(2 10)
~/1-p,
whmh obey the Schrodmger equation
HV;r,10o)=
(2 11)
EpFpTa[Oc) " +
The correlated ground state ]0o) of the system with the mass number A should be determined from the condition (2 12)
Fpr~10o) = 0 Using the expressions (2 7), (2 8) and (2 12) and the defimtlons p~, -
<0clN,.10~> 2~,
,
Pj =
(2.13)
<0clN,10o> 2~2j
we get the normahzatlon condmon for the wave function (2 10) m the following simple form Y I~,.I2 - y~ I~,12 = 1 (2 14) J"
j
In an analogous manner we construct correlated 2h states for the A - 2 system 12h, 'ra) =
r+31oc> = ~ / -l ' p-, ,
(--)TaA"-ra + ~ ~/1--p, /~'
B,+~ 1 IOD
(2.15)
with the normahzatlon condition
- Z [/~,.I2+ Z I/~,[~ = t ./'
(2 16)
J
and obeying the Schrodlngm equation
HV+r310o)= EuV+r310o)
(2 17)
For the ground-state vector we have the second reqmrement Fhr3[0c) = 0.
(2 18)
In the framework of our model the state vector
,0%
=
Z (-)~-T3 - r, x/3
F p+T 3 [ h.+_ r 3 [ O ¢ )
(2 19)
675
STATES IN 160
describes an excited 0 +, T = 0 state of the A-partmle system The corresponding energy is (2 20) Eo+ = Ep+Eh-Eo, where E o denotes the energy of the correlated ground state (2 21)
HI0o> = Eo[0=>
In order to evaluate the coefficients ~ and fl of the wave functions (2 10), (2 15) and the appropriate excitation energies
wp = e p - Eo,
wh = Eo - eh
(2 22)
we use the conditions following from the eqs (2 11), (2 17) and (2.21)
Wp(OclA/r312p, Ta>,
(0c[[a~,r3, H]I2P, Ta> =
+ ~,,-T, ~p, T3> < 0 d [ ( - ) I+T~B,-r~, H]I2p, Ta) = Wp(0¢l(-) x+r~'+
(223)
for the A + 2 system and <0o1[(-) ~+r3Af_r3, H]12h, T~> = - W~<0oI(-) ~+r3A}_T~12h, T~>, <0eI[B,T 3 , H]12h, T3> = - l~],<0~lB,r~12h, T3>
(2 24)
for the A - 2 system We calculate the matrix elements of commutators in these equations using of the llneanzatlon procedure of the QBA, for example
Afi, r,~As,~r,~]12p, T3> = <0¢l[A,,r~, Af ,r,~]As,=r,~j2p, Tz> ..~ 6,,j,, 6r3r,~(1-p,,)<0dA,,~r,~12p, Ta>
<0d[Aj,r~,
(2 25)
If we stall take into consideration the commutatmn relations (2 5)-(2 8), the deftnation of the Hamdtonlan (2 9) and the intermediate results <0dAj,r~12p, T3> =
%,~l-pj,,
0 C[~. l: ~ ] ~l+r~A+J ' - - T3 "~h l-Jtl~ T3> =
<0d(-)
l+T3
fl,,~/1-Ps,,
+ B,-r312p, T3> = % ~ 1 - p , ,
<0dB~r~12h, /-3> =
fljx/'l--p~,
(2 26) (227)
we get the following e~genvalue equations
2e,,%,-G~/~,,(1-p,,)[ Z ~/f2s',(1-P,',)%',-x J i
28~%-G.,/a,O-&)[xZ ~/a~,,(1-&,0%,,J'l
Z ~/Q,,(1 - p , , ) % , ] = wp%,, Jl
Z #a,,(1-&~)%,] = %%,
2~,.g.- Gda,.(1-p,.)r Z , / a , . , O - p , . , ) & , - ~ Z a'l
Yl
2ejfls- G~/f2~(1"pj)[x 2 ~/'f2~,,(1-Ps',)flJq- 2 J1
(2 28)
J1
dl
4a,,O- p,,)L,] ~/Z~,(1 -p~,)fl~,]
= w~,.,
= Whflj
(2 29)
676
L
M U N C H O W A N D I-I U
JAGER
The two exgenvalue problems (2 28) and (2 29) are described by the same nonHermman matnx A The corresponding normalization conditions (2 14) and (2 16) are distinguished by the sign If the symbols ~: and 2 stand for the number of the pamcle and hole shells characterized b y j ' and j, respectwely, we are interested in the system of equations A7 = wT, (2 30)
N = Y~ I~j,Iz - ~ I~jI2 = + 1 .1"
(2 32)
,/
Depending on the normahzatxon the solutions are attached to either the 2p system (2 28) or to the 2h system (2 29)
{e
lfN=l
{Wp
=
If N = I
w =
fl
ff N = - 1 '
(2
Wh
33)
ff N = - 1
We shall now investigate a few formal propemes of these solutmns. If we define dl = E ~/f2j,(l - p j , ) y , , ,
d2 = E ~/Oj(1-p,)yj,
J'
d
e(w) = y' Gf2j,(1 - p ~ , ) j" 2e j, -- w
g(w) = Z G f 2 j ( 1 - p j ) j 2ej -- w
(2 34)
the elgenvalue problem (2 30) can be written an the form
-
P") (a, - xd2), 2%, - w
a , / a j ( l - pj) ( x d I - d2),
(2
2ej - w
35)
where we find the conditions d 1 = e(w)(dl-xd2)
,
(2 36)
dz = 9(w)(xdl-d2)
The eIgenvalues w(v) are the zeros of the function f ( w ) = g(1 1 - _ _ e ) [ ( i + g ) ( 1 - e ) + x 2 e g
]
(2 37)
In the VlClmty of these zeros the norm (2 32) of the appropriate state vector is proportional to the derxvatwe of the functlonf(w) N(,) = Gld2l~ df(w)~ w=w(~,"
(2 38)
677
STATES IN 160
A quahtatwe discussion of the functlon f(w) for fixed values p (0 < p < 1) and with the physically estabhshed assumption that es < e~. shows us the following All elgenvalues w(') (v = 1, 2, , K+ 2) are real if the quantmes G and x are sufficiently small In this case we label the elgenvalues so that the condition W (1) >
W (2) ~
:> W (r+2)
is fulfilled Then, corresponding to eqs. (2 33) and (2 38) the elgenvalues w(~) (v = ( v ) + I(~ 1, 2, , to) belong to the 2p system, the appropriate state vectors are U-pr~ J~,/\ The remaining elgenvalues w~) (v = x + 1, . , x + 2) belong to the 2h states F~)+ [0¢) + + In the framework of the normahzatlon condmon (2 32) all operators Fpra and Fhr3 form a orthogonal and complete set If we take into conslderatmn the symmetry propemes of the matrix A we find the equation (
Z/s" ?s' - j"
j
,(v) ys ,(u)'~)
= 0,
and now with the help of (2 33) we get the relations E ~t(v)et(u) --j" -.j, - - ,-, ..(v)^.(u) E ~ J ~J = ~v/z J'
J
J"
.1
Z flo,)fl(u)_s,s"
Z fl(~)fl(u), s = - J , .
J'
(2 39)
j
The inverse representations of the operator defimtlons (2.10) and (2 15) are Aj'+ T3
B J+T 3
=
~-
x/1 - p f [ Z ~"j" "(')r(')+ apT3
x/l-psi-
--
V / ~ (_~r3n(,)I.(,)r~], k ./ /'13" h -
~" ~" ~r~~(v)r(,) ~d ~p-T3 ± 7- ~ / ~ t~(')F(')+l P'J hT3 -]"
L, I--] V~K
(2 40) (2 41)
V>K
In order to solve the problem (2 30) we must still calculate the quantities p j, and pj defined by eqs (2 13) For the correlated ground state we have 10o> = KeSl0r>,
S =
S(j))
4(1- ps,)(1-p j)
t
k~
a~-r3a+ ]
u+
"ad'T3 X"d-T3 "
(242)
The Fermi ground-state vector is denoted by 10v>, the quantity K is a normahzatlon constant and the coefficients SO'j) describe the admixture of p-h components to the Fermi ground state We get from the requirements (2 12) and (2 18) the equations for these coefficients
fl~') = Z [3~v)s(J'J) J
(2 43)
678
L MUNCHOW AND FI U JAGER
Using the relations (2 39) one sees that the second equation follows from the first Each of these equations is sufficient to calculate the xZ coefficients SO'j) Now we employ the Identities (0¢INj,10¢) = (0¢I[N~,, S]10c), (OoIN.,[Oo) ---
which are proved in the paper of Harat°), calculate these commutators and obtain by using the relations (2 40), (2 41), (2 12) or (2 18) and (2 43) the results pj, =
3 ~', (fl(~)" s' ) 2 ~>~ ,
pj =
3 ~ (~))2 ~---~
f2j,
(2 44)
Or
3. Discussion of the results for a spherical nucleus We have solved the system of eqs (2 30)-(2 33) and (2 44) by iteration
A(e, ~, G, x, P~)~+t = w~+11~+i, P,,+t = P(?,,+~),
Po = O,
(3 l )
for the nucleus 160 We take Into consideration the lp and the 2s, ld shell The assumed single-particle energies are as common for p-h calculations in 160 (see ref 13) and table 1) In our first lteratmn step we get the results of the ordinary QBA (0 = O) TABLE 1 Single-particle energies m t60 Level e(MeV)
1P~r --21 8
1pk --15 65
1d{
2s~r
1d{
- - 4 15
--3 27
0 93
A priori we can only fix a certain interval of variation for reasonable values of G They may be of the order of the pairing constants, which are used in calculations in heavier nuclei, where one takes G = 20 .28/A MeV Calculations recently done in the seniority scheme seem to favour the value G = I6 20/A MeV in the oxygenlum region 14) The pairing constant for the Interaction between anyj~ and j2 shell stands for the negative antlsymmetrlc matrix element between the J = 0, T = 1 states of these shells divided by [(2J1 +1)(2J2+ 1)]~ These quantities calculated for the Glllet force t3) are negative for interaction between states with opposite panty and on the average by the factor 0 7 smaller than the remmnlng positive quantities for interaction between states with the same parity Therefore we have used two pamng constants, the quantity
STATES IN 1°O
679
G for pairing within an oscdlator shell and the quantity x G for palnng between the lp and the 2s, ld shell The sign of the quantity x is ms~gmficant in the framework of our model (compare eq (2 37)) It is clear that using x = 1 instead of x < 1 means overestimating the ground state correlation effects In order better to understand the results of our calculatmn we seek the dependence of the lowest excitation energy Wo'l'.mlll = Wp. . . . --Wh . . . .
NOrmm/MeV
× ~
07
20
15
tO
I>
'
GcPJt
'
~/Me V
F i g 1 E x c i t a t i o n e n e r g y o f t h e l o w e s t 0 + s t a t e in d e p e n d e n c e o n t h e p a m n g c o n s t a n t G F o r a c o m p a r i s o n the results of an exact dlagonahzatmn and of the QBA are given (m contrast to the other
results of th~s paper here we take into account only the shells lp~, ld~r, 2s~) on the pairing constant G (see fig 1) The comparison of the QBA with an exact dlagonahzation 1s) of the pairing residual interaction shows two characteristics of the QBA whlch are caused by the violation of the Pauh principle Firstly, the result of the QBA vanishes for a certain value of G (crmcal point), while the exact solution shows only a minimum, secondly in the QBA the states with the lsospln T = 0, 1, 2 are degenerated The QBA is not apphcable m the VlCimty of the crmcal point In this case a possxble low value of the excltanon energy cannot demonstrate the adequacy
680
L MUNCHOW AND H U JAGER
of the approximation The same remark holds for the transition amphtudes, where the QBA yields too large values The dependence of the posmon of the crmcal point on x may be calculated with the help of the functionf(w) defined by eq (2 37), from the conditions
f(wcr,t, Gcr,t, x) ~f
=
O,
= 0
~w
(3 2)
W= Wcrit
Some results are shown m table 2 In fig 2 the dependence of the lowest excitation energy on the pairing constant G is shown for the ordinary QBA and the generalized QBA In the latter case we use the more exact commutation relations (2 7) and (2 8) TABLE 2
Posmon of the critical point in dependence on x x Ger~t (MeV)
0
01
02
03
04
05
07
10
3 36
3 12
2 87
2 67
2 49
2 33
2 08
1 78
WO.mm/MeV
x = 07
I0
I
Get.d
"G/MeV
F i g 2 T h e G-dependence o f the lowest excitation energy c a l c ul a t e d w i t h the o r d i n a r y a n d generahzed QBA
The consideration of pamcles and holes in the correlated ground state (p ~ 0) leads to the result, that according to eqs (2 28) and (2 29) the pairing force becomes less effective in the 2p and 2h states In the vlclmty of the critical point it appears to be an essential difference between the results of the QBA and the generahzed QBA In this paper we describe the excited 0 +, T -- 0 states as superposmon of the 2p and 2h states, each of which is investigated mdependently from the other Therefore
STATES IN 160
681
we have not taken into account the p-h interaction between the 2p and 2h states m the framework of our model In a former paper 9) we showed, that the T = 0 part of the p-h interaction yields an energy shift of about 5 MeV In the following only the results of the generahzed QBA are gxven For the expectation value of the Hamlltonlan (2 9) in the correlated ground state we find Eo =
= H~.o- 3[ E Wp(') E (a~,))2_ Z ~ ' E (fir))2] V~K
v>K
d
(3 3)
J"
Wath increasing p a m n g constants G or xG the correlations m the ground state become larger and the energy of the ground state decreases In fig 3 we demonstrate this effect by the dependence of the p a m c l e and hole number in the ground state (2 13), (2 44) on the pairing constant G
x=07
G=z~,q6
ol_N_.
727J
F72
G=25/16
G=30/I6
lds/2
2Sl/2
td3/2
1PI/2
1.°3/2
Fig 3 D e p e n d e n c e o f the quantities (0c[Nf]0c) a n d (0c[Nsl0e) on the pairing c o n s t a n t G
For the monopole matrix element between the first excited 0 +, T = 0 state and the correlated ground state we get in the subspace of our configurations (0 +, mini ~ protons
r2[0c) = fix/3 ~ o:j~,'n,13sm,~x
(3 4)
m(.o j '
Here we have used the elgenfunctlons of a harmonic oscdlator with hm = 41 A -¢ MeV In table 3 we show the results for the monopole matrix element (3 4) in depend-
682
L MUNCHOW AND H U JAGER
ence on the p m r i n g c o n s t a n t s C o r r e s p o n d i n g to the increase o f the c o r r e l a t i o n s with G a n d x Its value increases too. I n c o m p a r i s o n to the e x p e r i m e n t a l value o f 3 8 fm 2 o u r t h e o r e t i c a l results are t o o small for a spherical nucleus 160 a n d r e a s o n a b l e G and x TABLE 3
The monopole matrLx element (fma) between the first excited 0+, T = 0 state and the ground state m dependence on G and x G(MeV)
1
1 125
1 25
1 57
1 875
x = 07 x ~ 08
0 472 0 544
0 575 0 663
0 693 0 799
1 057 --
1 533 --
4. Consideration of deformation in the excited states T h e d e s c r i p t i o n o f the excited 0 + states d e v e l o p e d In the first t w o sections is Inc o m p l e t e , because it does n o t a c c o u n t for the d e f o r m a t i o n o f the excited states There IS clear e x p e r i m e n t a l evidence 4) for the fact t h a t the first excited 0 + state in 160 is the h e a d o f a r o t a t i o n a l b a n d T h e E2 t r a n s i t i o n p r o b a b i l i t i e s show the typical collective e n h a n c e m e n t I 6 ) . F r o m the studies o f d e f o r m e d nuclei we k n o w t h a t the selfconsistent p a r t o f the Q-Q force c a n be Vlsuahzed as the r e a s o n for d e f o r m a t i o n o f the single-particle field I n the case o f d o u b l e - m a g i c nuclei a d e f o r m a t i o n w o u l d force the m a r e 2p-2h states (which are closest to the F e r m i surface) to get a smaller excitat i o n energy, l e a d i n g also to lower energies o f the c o r r e l a t e d states F o r a q u a n t i t a t i v e d e s c r i p t i o n we a d d to the H a m l l t o n x a n (2 9) a Q-Q force H = Hvac+ ~ e j ' N s ' J'
(Jmz)
~esNa+H'pa,r+HQQ, d
I~ ,
t
2
+
¢
x (J'2m2z2l r Y2-u]Jxm2"c2>cj'lm':',cs'zm'2x'zCjzm:2 aim:,'
(4 1)
a s s u m i n g the s a m e c o n s t a n t Z for the p-p, n-n a n d p - n interaction I n s t e a d o f using the full H a m l l t o n l a n (4 1) we first a p p l y a m o d e l H a m l l t o n l a n / 4 , because we w a n t to t r e a t only the self-consistent p a r t o f the Q - Q force Such a p r o c e d u r e is quite analo g o u s to the one used in d e f o r m a t i o n calculations with a p a i r i n g force 17), Elchler a n d M a r u m o r l 12) also used a similar m e t h o d for describing d e f o r m e d c o r r e l a t e d states F o r the m o d e l H a m l l t o n I a n we have n
+ ' C3m'~C.Im't"~npa,r--Z Z ~,Q~, ,
= Jm~
(4 2)
#
where Qu is the o p e r a t o r of the m a s s q u a d r u p o l e m o m e n t u m
Q~ = 4x/~T-~~ (Jl ml"cl[r2y2.[J2m2"c2)Cf m:l c~2m2~2 (jmz)
(4 3)
683
STATES IN 160
The five parameters ~ are quadrupole parameters specifying the deformation of the potential well Performing as usual a transformation to the body-fixed system and assuming axial symmetry one gets from eq (4 2) /~ + , = E %csm~Csm~+Hpa,r-xflQo, (4 4) ./mr
where fl stands for eo in the body-fixed system and is the deformation parameter Because of the presence of the operator Qo m eq (4 4) the single-particle part of the Hamlltonlan is not diagonal m the chosen sphermal basis By a unitary transformation we go to a new basis (k, y, z) in which the self-consistent part will be diagonal + Cjkr = E
J
+
a k r CkrT •
(4 5)
The quantum number y specifies different states with the same k and z The selfconsistent energies ekr in the deformed basis are elgenvalues of the equations
ekra~r = e./a],r--42fl~/~-~ ~ (J1 kz[, 2y2olJkz)a~,
(4 6)
./l
Here we neglect the m a m x elements of the quantity r2Y20 between different oscillator shells The real transformation coefficmnts a~,r satisfy the orthogonahty relations
E akylSak~, 2, = ~,,r2,
E "kr'""~r'J2= 6j1j 2
./
Y
(4 7)
States distinguished only by the sign of the quantum number k are degenerate, for the coefficients a~r we obtain the relation
a J-k,(--) s-~ = a],~
(4 8)
from the symmetry properties of eq (4 6) In the future we only use positive values k for the classification of the deformed single-particle levels In this sense two protons and two neutrons fill a subshell characterized by (k, ~) Now we introduce pamcles and holes m the deformed basis The symbol 0kr denotes the occupation number in the deformed Fermi ground state We define the number operators for particles and holes Nk,r" = (1--Ok,r,) ~, ( Ck,~,r, + Ck,r,r,+C_k,r,eC_k,~,r,), + r'
Nkr = Ok~'E
(Ck)'r
+ "~-C-Rr~C-kr~) CkTc +
(4 9) (4.10)
r
and the creation operators for 2p or 2h states with the quantum numbers K = 0, T=I,T a +
y,
=
1 ,, , ÷ ÷ (~Z,~Z2IITa)Ck,r,,,,C_k,,,r, ~
(4 11)
1~'11~'2
Bk;r ~ = 0kr(__)*+k+ r~ Z (½Z,½ZZ[1-- 7z)Ckm C-kin flY2
(4 12)
684
L MUNCHOW AND H U JAGER
In this paper we Investigate the double-magic nucleus 160 and take into consideration one oscillator shell above and below the Fermi surface Therefore we can use the orthogonahty relations (4 7) for the transformation of the pairing residual interaction (2 9) We get the model Hamfltonlan/~ in the deformed basis
ffl = 4 ~ Ok.:F,k.~'-~-2 8k'~"Nk'~"-- 2 8k~'NkT+ H'p. . . . k7 ,
=
k' 7"
__
k7
v-
T3
BkI~,t_TaBk272_T
k'l]:' lk'2~'2
(Ak+;,,TaBk;-rad-Bkr-raAk,~,,r3)]
"~ X( --)T3 Z
3
kl)'lk2T2 (413)
k ' )" k~,
The model Hamlltonlan/4 is a part of the full Hamlltonian (4 1) in the body-fixed system H = Hva e -- 2 ~ 0j(23 + 1)~s + HQQ + xflQo + ff-I J
If we use the representation (4 13) and the relation
4 ~., Okre, r = 2 ~ 0~(23 + l)e, k),
.I
for double-magic nuclei, we get
H = Ho + xflQo + H¢~o, Ho = H,,.o+ ~'. ek.r. Nk,r.-- E ekTNl,~+H'p.,r k')"
(4 14)
k~,
The Hamdtoman H o and also the commutators of the number and pair operators (4 9)-(4 12) have the same form in the deformed basis as in the spherical case, if we substitute in the expressions (2 5)-(2 9) ] ~ k, ~,
f2~ -* 1
For this reason we calculate the eigenfunctions and elgenvalues of the Hamlltonian H o with help of the generahzed QBA analogous to what was done in sect 2 The correlated ground state [0c) and the first excited state [0 +, m m ) of the Hamiltonlan H o are again of special interest The corresponding eigenvalues of the full Hamiltonian H are calculated as expectation values like in perturbation theory We obtain expressions for the energy of the correlated ground state (0¢lH[0c> = H . ~ - 3 [ E W..(*)E (a(k;')2 - Z W(*)E (tiff),)2] v<~c
k~'
v>tc
k'7'
+ z/3
(4 15)
and of the first excited state (0 +, mmlH[0 +, m m ) = w ~ ", p(~'~) - w(m~) Veh / a aU va -
-
e
3[ E w(~) Z (a(k;))2 -- Z 14'h(*) Z U'k~ ' V:t¢~2-1j (~) v
k7
v>~
k'7'
+Zfl(0 +, minlQo[0 +, r a i n ) + ( 0 +, mmIHQQI0+, mln)
(4 16)
S T A T E S I N 160
685
from the eqs (3 3), (2 20) and (2 22) Using a Hartree-Fock type decomposmon for the Q-Q interaction (4 1) m the body-fixed system with axial symmetry and neglecting the exchange terms as usual (0clHaol0¢) = _1•(0¢1Qol0¢)2, (0 +, mmlHaq[0 +, rain) = -½Z(0 +, minlQol0 +, ram) 2
(4 17)
we only need expressions for the expectation values of the quadrupole momentum operator Qo (4 3) to calculate the energies (4 15) and (4 16) In the subspace of our configurations the relation Q0 = 4x/ire[ ~, (k'T'[ r2 Y2olk'7')Nk'r'- Z (kT[ r2 Y2olkT)Nkr] k",,"
(4 18)
k3'
holds The matrix elements of number operators in the correlated ground state (0c[Nk,r.10c) = 6/.., Wk'r', ,
(0¢[Nkrl0¢) = 6 ~ (aCk~))2
V:>K
(4 19)
V~K
are known from the eqs (2 13) and (2 44) Using the exphclte form of the wave functions (2 19), (2 10), (2 15) and the commutators of the number and pair operators we obtain for the matrix elements in the first excited state (0 +, mlnlNk,~,10 +, mln) -- 2[(~k'r') m m 2 --(flk'v') m a x 2 ] + (0olNk'v,10o), (0 +, mmlNk~[0 +, m m ) = 2['(flk~ m a x ) 2 --(Uk~ m , n ) 2 ]+(0¢[Nkrl0o).
(420)
These expectation values of the full Hamfltonlan H depend both directly (see eqs (4 15) and (4 16)) and indirectly (see eq (4 6)) on the parameter fl which is related to Nllsson's is) deformation parameter 6 by
Zfl = ½omco2,
(4 21)
where co IS the oscillator frequency Klssllnger and Sorensen I9) give for the constant Z of the Q-Q interaction (4 l) the estimation 110 1 (~)21VleV Z - 4.4 ( N + ~ ) z We assume again hco = 41 A -~ MeV, the oscdlator quantum number N l s equal to 1 for the hole states and equal to 2 for the particle states Because we in our description have used the same constant Z for particle and hole states we Interpolate Z = 150z,
z - 4A(2+~)2
MeV
(4 22)
In table 4 and fig 4 we show the energy of the correlated ground state (4 15) and of the first excited state (4 16) in dependence on the deformation parameter 6 for given values of the pairing constants G and x The ground-state energy has a minimum for
686
L MUNCHOWAND H
U
JAGER
TABLE 4 T h e energies (4 15), (4 16) m dependence o n the d e f o r m a t i o n p a r a m e t e r t5 (G = 1 MeV, x = 0 7, all energies are given m MeV) G r o u n d state
First excited state
A
ZflO-,o
½ZQ_,2o
energy
14/o+, mla
Zfl0o
½Z0~0
energy 15 976
0
--1 0936
0
0
--1 0936
17 070
0
0
0 05
--1 0971
0 0081
0 0001
--1 0891
16 749
0 655
0 574
15 743
0 10
--1 1081
0 0348
0 0004
--1 0737
15 755
2 655
2 293
15 009
0 15
--1 1281
0 1022
0 0011
--1 0270
14 196
5 388
4 195
14 261
0 20
--1 1592
0 1814
0 0027
- - 0 9805
12 280
8 327
5 637
13 811
0 25
--1 2051
0 3534
00065
- - 0 8582
10 155
11 359
6 713
13 596
0 30 0 35
--1 2725 --1 3690
0 6951 1 4827
0 0175 0 0584
- - 0 5949 0 0553
7 902 5 585
14 557 18 202
7 657 8 796
13 529 13 622
0 40
--1 4733
3 6555
0 2716
1 9106
3 330
23 246
10 983
14 120
T h e energy scale is chosen so that the energy o f t h e spherical g r o u n d state Hvac ~s equal zero By A we denote here t h e quantxty -
3[ Z w. E v~x
Z
k7
v>K
k'~,"
\ / J k ' 7 ' ) J~
t h e s y m b o l Qo m e a n s the expectation value o f q u a d r u p o l e m o m e n t u m m the given state
E/MeV
10
5
01
02
0n
i
Fig 4 T h e energy o f the correlated g r o u n d state a n d o f the first excited state m dependence o n the d e f o r m a t i o n m accordance with table 4
STATES IN 160
687
the spherical shape and depends weakly on the deformation for 6 < 0 2, for larger values of 6 it Is essentially determined by the deformation energy XB(0olaol0o)
-
kx((0ciQol0o))2
The reason for this behavlour is the following The diagonal matrix elements of the quantity r z Y2o are negative for the highest hole levels (lP½, k = ½, lp~, k = {) and positive for the lowest particle levels (ld~, k = ½, ldl, k = {), their absolute values Increase with the deformation Besides, with increasing deformation the energy difference between the lowest particle and highest hole level becomes smaller and therefore also the particle and hole numbers grow Both effects cause in the same direction an Increase of the quadrupole momentum expectation value (compare with the formula (4 18)) On the other slde the ground state energy is lowered by the correlation energy (quantity A in table 4) with Increasing ground-state correlations; this effect is insignificant for great deformation In the first excited state the quadrupole momentum expectation value as larger than an the ground state because of the increased particle and hole numbers For small deformations the energy decrease of the Intrinsic state caused by the more favourable positions of smgle-particle levels is important. Our main interest is just to describe this effect, therefore we used the model Hamlltonlan ~ or H o We should hke to remark that by a different choice of the Q-Q force constant z = 140z instead of (4 22) the ground-state energy in table 4 changes less than 0 02 MeV and the energy of the first excited state less than 0 73 MeV Now it is clear that the energy of the first excited state has a imnlmum for a fixed deformation of the self-conslstent potential From the requirement 0(0 +, mmlH[O +, rain) = 0
a~ follows, that the position of this mlmmum is determined by the condition /~o = (0 +, mmlQol0 +, mln)
(4 23)
analogous to the calculation in the conventional pairing plus Q-Q force case (see for example ref 17)) This is a very instructive self-consistency requirement If the system is equilibrating the deformation fl of the average potential in eq (4 6) should be the same as the deformation of the density distribution of the nucleons With the help of condition (4 23) we are able to understand the position of the minimum in dependence on the pairing constants G and x and on the Q-Q force constant z We show m fig 5 a graphic solution of the self-consistency requirement (4 23) For small values of G only the lowest particle state and the highest hole state are occupied with two particles or holes and give a contribution to the quadrupole momentum expectation value For increasing values of G the pairing force within the particle or hole system raises the contribution of higher particle states and lower hole states, leading to a de-
688
L MUNCHOW AND H U JAGER
ciease of the quadrupole momentum, and on the other hand the pairing between the particle and the hole system hfts mainly nucleons from the highest hole state to the lowest particle state, leading to an increase of the quadrupole momentum The result of these two opposite effects depends on the position of the single-particle levels With increasing deformation the level distance within the particle or hole system grows and the mimmum distance between the particle and hole system decrease In fig 5 we therefore get an intersection of the quadrupole momentum curves which belong to different values of the constant G On the left-hand side of this intersection x
/
///
=07
/
/ / /I
/
--
p:~p
Fig 5 Graphic solution of the self-consistency reqmrement (4 23)
point we get the common competition effect between palnng and Q-Q force, the pairing leading to smaller values of the equfllbrmm deformation On the right the paxrlng force amphfies the equlhbrmm deformation We have computed the reduced E2 transition probability to our first excited state using the formula B(E2, 2 + ~ 0 +) = ~
1
((0 +, mmlQol 0+, ram)) 2
(4 24)
We used the same effectwe charges (ep - 1 5e, e n = 0 5e) as Celenza et al 6) and
STATES IN leO
689
consxdered the symmetry between proton and neutron states in our wave functmns The results are gtven m table 5 m dependence on the pairing constants and on the deformatxon We see, that for the equlhbrlum deformations (underhned values in table 5) our theoretical E2 transmon rate is about the factor 4 smaller than the experimental value We also computed the monopole matrxx element (3 4) between the deformed first excited state and the corresponding correlated ground state of the same deformation, thus we &d not take into account the overlap factor between the deformed and the spherical ground state Table 6 shows that the monopole matrix element increases for greater deformations, but as in the spherical case (table 3) the results are too small compared with the experimental value TABLE 5 The reduced E2 transxtmn p r o b a b d m e s (4 24) In units o f e2fm4
0 G=IMeV,
005
010
015
020
025
030
035
040 1481
x=07
0
077
309
566
760
905
1033
1186
G=
1MeV, x = 0 8
0
076
306
566
767
921
1062
12 41 15 82
G=
1 125MeV, x = 0 7
0
063
262
513
727
897
1055
12 59
--
The underlined values are the nearest to the equilibrium deformation TABLE 6 The m o n o p o l e matrix d e m e n t (fm 2) between the deformed first excited state and the corresponding correlated g r o u n d state o f the same deformation
G= G= G=
1 MeV, x = 0 7 1 MeV, x = 0 8 1 125MeV, x = 0 7
0 05
0 10
0 15
0 20
0 25
0 30
0 35
0 40
047 054 057
045 052 056
045 052 056
047 054 059
052 060 065
062 071 077
080 091 099
1 16 129 --
The posmon of the equlhbrmm deformation depends on the constant g Therefore we also made a Ndsson type determination of the equilibrium deformation That means, mstead of fullfilhng the self-consistency condmon (4 23) we requtre the volume of the nucleus to be the same for any deformatxons (incompressible nucleus) This leads to a deformation dependent oscillator frequency o~(6) = o)o(1 --}62 -- ~-7-v a 6"~a~-"~ /
(4 25)
in the equatmn for the deformed single-particle states ek, ak, j = [ ( N + ~)h~o(3) + e, - ( N + ~)hcoo]3 %,
- 4 ~/~hco(6)6 ~ (jlklre Y2oljk)a~;
(4 26)
690
L MUNCHOW AND H U JAGER
(compare with the formula (4 6)) with the constant term
~l-(N+~)~Oo added, in order to get the empirical single-particle energies for 6 = 0 After computing the deformed single-particle states we start from the Hamiltonian
H = ¼Z 40krek,+ Z ek','Nk'~'-- Z ek~N~,+H'pa,r k7
k' 7'
(4 27)
k~
in the body fixed system and calculate its energies and wave functmns for the ground and excited states in the framework of the generahzed QBA The first term in this Hamdtonlan represents the energy of a deformed Fermi ground state, the factor 3 arises by subtracting the half of the potentml energy from the sum of single-particle energies TABLE 7
The energy of the correlated ground state and of the first excited state for an incompressible nucleus xeO (relative to the spherical Fermi ground state) 6
3 Y Ok,,eke, k~
Correlation energy
G r o u n d state energy
F i rs t excited state Wo+, mln
0 0 0 0 0 0 0 0 0
05 10 15 20 25 30 35 40
0 0 0 1 3 5 9 12 18
207 855 989 669 976 030 987 090
--1 --1 --1 --1 --1 --1 --1 --1 --1
094 096 104 119 143 177 229 302 396
--1 --0 --0 0 2 4 7 11 16
094 889 249 870 526 799 801 685 694
17 16 15 14 12 10 8 6 3
069 768 826 346 529 511 364 142 910
energy 15 15 15 15 15 15 16 17 20
975 879 577 216 055 310 165 827 604
All energies are given in MeV, G = 1 MeV, x = 0 7
In table 7 and fig 6 the energy of the correlated ground state and of the first excited state are gwen in dependence on deformation We get qualitatively sxmilar results as for the previous calculation (table 4, fig 4) Again the correlated ground state has an energy minimum for the spherical shape and the first excited state possesses an energy minimum for a certain deformation Going into details we find differences mainly caused by the Increase of the single-particle energies ek~ (4 26) for large deformations Here we note a stronger rise of the correlated ground-state energy with increasing deformation, and therefore we find the equlhbrmm deformation of the first excited state at smaller values of the parameter 6 An increase of the pairing constants G and x lmphes greater correlations in theground state and a decrease of the excitation energy Wo÷, m,. and leads to smaller equdlbrlum deformations
STATES IN 180
691
5. Conclusions For the double-magic hght nuclei the problem arises to describe strongly correlated states with an essentially different self-consistent potential leading to different effective deformations The co-existence model starts from states with a fixed deformation calculated m a self-consistent procedure and then mixes them in order to get all the correlation effects We use a model with fixed deformation for every state This opens the posslblhty to lnvesngate in detad effects as such the competlnon of pmrmg and Q-Q forces determining the equlllbrmm deformanon in the ground and excited E/MeV
20
15
10
41
~1~ff~
¢-
02
03
or
Fig 6 The energy of the correlated ground state and of the first excited state for an incompressible nucleus m accordance with table 7 states, the influence of these forces on the translt~on properties etc We are able to explain the existence of low-lying strong collective states m double-magic hght nuclei, but quanntatwely the collective and correlation effects are too small From the standpoint of this description the system is m the wcmlty of the critical point, this leads to strong anharmomc effects We took into account only the correcnons needed to overcome the violation of the Pauh prmclple The anharmonlc~ty would lead to stronger 4p-4h components m the first excited state and possible also to a deformed ground state The description of the pairing interaction used is essennally the generahzanon of the B~s method for a magxc nucleus m which case the latter would give only a
692
L
MUNCHOW
AND H
U
JAGER
trivial s o l u t i o n for the gap e q u a U o n if projection is n o t used. Therefore we c a n say t h a t the g r o u n d state is similar to a superflmd state due to the d e f o r m a t i o n of the self-consistent field I n order to take a n h a r m o m c x t y better i n t o a c c o u n t a n a n a l o g o u s calculation with exact pairing wave f u n c t i o n s is n o w i n progress W e t h a n k the m e m b e r s of o u r staff, i n particular H R. Klssener a n d C. Rtedel, for m a n y suggestions F r o m the b e g i n n i n g D r hints a n d helpful attentxon.
K
Muller s u p p o r t e d us by interesting
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
J Sawxckl, Phys Rev 126 (1962) 2231 Y Abgrall, J de Phys 24 (1963) 1113 S S Wong, Phys Lett 20 (1966)188 E B Carter, G E Mitchell and R H Davies, Phys Rev 133B (1964)1421, 1434 G E Brown and A M Green, Nuclear Physics 75 (1966) 401 L S Celenza, R M Drelzler, A Klein and G J Drelss, Phys Lett 23 (1966)241 J Hogasen, Ann de Phys 8 (1963) 697 D R B~s and R A Brogha, Nuclear Physics 80 (1966) 289 H U Jager and L Munchow, Proc Conf on Nuclear reactions with hght nuclei and nuclear structure, ZfK Rossendorf (1966) K Hara, Progr Theor Phys 32 (1964)88 K Ikeda, T Udagawa and H Yamaura, Progr Theor Phys 33 (1965) 22 J Elchler and T Marumon, private commumcatlon V Glllet and N Vmh Mau, Nuclear Physics 54 (1964) 321 F Donau, private commumcatlon H R Klssener, unpubhshed S Gorodetzky et a l , J de Phys 24 (1963) 887, H Fuchs, K Hagemann and C Gaarde, Nuclear Physics 66 (1965) 638 M Baranger and K Kumar, Nuclear Physics 62 (1965) 113 S G Nllsson, Mat Fys Medd Dan Vld Selsk 29, No 16 (1955) L S Klsshnger and R A Sorensen, Mat Fys Medd Dan Vld Selsk 32, No 9 (1960)