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Ground-state correlations in the yrast levels of deformed nuclei
Nuclear Physics A341 (1980) 229 - 252; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
GROUND-STATE
CORRELATIONS OF DEFORMED
IN THE YRAST NUCLEI
LEVELS
J. L. EGIDO ‘, H. J. MANG and P. RING jhysik-Department,
Technische Unioersitdt Mkhen,
D-8046 Garching, W. Germany
Received 5 November 1977 (Revised 20 February 1980) Abstract: The random phase approximation (RPA) in copjunction with an approximate projection of angular momentum and particle number is used to investigate ground-state correlations in the yrast wave functions of realistic nuclei at high spins. One has to distinguish between two types of correlations, those due to virtual excitation of vibrations and those caused by the removal of the spurious components in the symmetry-violating intrinsic wave function. Both types produce a lowering in the energy of several MeV and both show large anomalies in the backbending region. One observes, however, a cancellation of these anomalies, so that the total correlation energy is a smooth function of angular momentum which can be taken into account in a theory with uncorrelated product states by a simple renormalization of the effective interaction.
1. Introduction
In recent years the yrast line in the high-spin domain of deformed nuclei has been investigated experimentally [for a review see ref. ‘)I and theoretically 2-6) in great detail. So far most of the microscopic descriptions in realistic nuclei use the selfconsistent cranking model (SC?) based on generalized product wave functions of the Hartree-Fock-Bogolyubov (HFB) type. Several authors ’ - ’ 2, have proposed going beyond this first approximation by taking into account the RPA corrections to the self-consistent cranking model. One then finds an attractive correlation energy produced by the virtual excitation of the vibrations. Up to now numerical calculations of this type have been carried out only for rather unrealistic models ‘,s* 12). In such calculations one has, in general, found a considerable improvement of the yrast energies, although there are also some cases in which the RPA breaks down completely. This happened in particular in the region of level crossings where one has a very low-lying two-quasiparticle excitation in the rotating frame. In this region the validity of the cranking approximation itself has become an object of much discussion 13) and it still seems to us an open question as to what extent a level crossing can be described by one product state in the intrinsic frame. The RPA is a small amplitude extension of this static mean field approach. + Work supported by the BMFT (Bundesministerium fur Forschung und Technologie). address: Max-Planck Institut fti Kemphysik, POB 103980, D-6900 Heidelberg 1, W.-Germany. 229
Present
230
J. L. Egido et al. 1 blond-state
cwrelations
In the region of level crossings it has its own problems caused by the violations of the Pauli principle. We therefore have to be very careful in drawing final conclusions from our results in such regions. One often argues that the RPA restores the broken symmetries in the HFB wave functions. This is certainly not true to the extent that the RPA wave function is an eigenstate of the exact angular momentum operator. However, as it shows only small deviations from the underlying HFB function, we consider it as an intrinsic wave function, improved as compared with the simple product state because it contains additional ground-state correlations. To obtain the wave function in the laboratory frame, one has to carry out a projection onto good angular moments. We have therefore determined the intrinsic RPA function from the variational principle after an appropriate projection, This approximation will be discussed in sect. 2. In sect. 3 properties of the RPA wave function are discussed and we show that in order to describe an intrinsic deformed state by the RPA method a restricted class of RPA wave functions has to be used. in sect. 4 we discuss an approximation which is valid as long as the increase of average angular mom~tum caused by the RPA correlations is small compared to the width of the angular momentum distribution in the intrinsic frame. Second-order terms in the angular momentum projection are taken into account in sect. 5. The theory is applied in sect. 6 to the yrast line of 164Erand the influence of the hound-state correlations on the energy is inv~tigated as a function of the angular rnorn~t~, In sect. 7 we summarize our results.
2. Approximate projection of angdar mom~n~rn ?
HFB theory with a ~nstraint on the angular rnorn~t~ .?%can be derived as an approximation to a variation of the intrinsic wave function (within the set of generalized Slater determinants I@)) after projection onto eigenstates of f2 [refs. 15-17)]. To justify this approx~ation, three conditions must be fuElled: (i) Violation of the symmetry in the intrinsic wave function must be strong (@~@/di> w 1.
(1)
(ii) The wave function I@>must have definite signature eis=l@) N I@).
(21
(iii) The deviation from axial symmetry must be weak in the sense (@,l&zJ) * IfI+ 1).
(3)
’ Witbin this paper we treat the problem of p~~ic~e~~~ violation in the HFB and RPA wave functions in the same approximation. In order to keep the formulas reasonably simple, we &all only discuss the case of rotationai symmetry in this section.
J. L. Egiab et al. 1 Ground-statecorrelations
231
In this paper we go beyond the simple product ansatz for the intrinsic wave function and include ground-state correlations of the RPA type. The corresponding wave function will be denoted by 1Y) in contrast to the HFB wave function I@). Angular momentum projection from such functions can be carried out in exactly the same way as from HFB functions. For the convenience of the reader we briefly remember the essential steps of this procedure: The projected energy Er is given by
KK’
kk’
where
and the operator
describes a rotation through Euler angles 61 = (01,/.$y). The hamiltonian overlap (YIHR(61)IY) can be represented expansion is) t (HR(R))
= (a + bLlL, + . . .)( R&2)),
by the Kamlah
(6)
where Ai, = t,- (jx) is the angular momentum operator expressed in terms of Euler angles. For large deformations the expansion (6) can be terminated at first order and we obtain (7) with A.fx = jx,- (.?J. The variation of the projected energy (7) within a suitable set of RPA wave functions IY) leads to &$ = (c?ylHIy)-
In the following we assume that a possible variation within this set of trial functions is ISY) = &Ajx,(Y).
(9)
In the next section it will become clear that this is true for the RPA wave functions ’ In the following all expectation values are taken with the WA wave function IV) if not otherwise specif=d.
J. L. Egiab et al. / Ground-state correlations
232
under consideration.
We then determine the solution of the cranking model IY’,) (6YI&oJ,lrC/)
= 0,
(10)
where o is determined from the subsidiary condition (YIS,lY),
= $0.
(11)
Since ISY) in eq. (9) is an admissible variation, we obtain from eq. (10)
w =
(HAS,) (AZ)
(12)
and find together with eq. (10) that IY,) is a solution of the variational equation (8). We thus find that the cranking wave function IY,) is an approximation to an intrinsic wave function obtained from the variation after projection. This derivation is valid for all wave functions which fulfill conditions (l)(3) and condition (9). An example are HFB wave functions, where, according to the theorem of Thouless, eBd’+D) = I@) + EA.T&D)
(13)
is again a generalized Slater determinant. Another example are the RPA wave functions discussed in the following sections. We then end up with the constrained RPA equations (10) and (11). Before we specify the structure of the RPA functions we go one step further and include quadratic terms in the operators L,, I,,, I., in the expansion (6). The corresponding equation then reads : (HR(G!)) = (a+bA~,+~Ai?+d(d&~~)+eL1,)(R(SZ)).
(14)
The unknown coefficients a, 6, c, d and e are determined from the following set of linear equations : + c(Aj2)
+&A.&f;))
+e(J:),
UO
=a
(HA&)
=
(HAf2)
= a(AJ2)+b(AjxAf2)+c((Af2)2)
+d((Aj~-.@Af2)+e(j~Aj2),
(H&f)
= a(S,2)
+b(Aj,.@
+c(Af2&
+d((A.f;[email protected];:)
+e<.W;>,
(Hj?)
= a(&
+b(AS,j~)
+c(AY2.Q
+d((A.&Qf;)
+e(.Tt).
b( A.?:)
+c(AJ2A~x)+d((A.f~-f~)A~x)+e(.f~A.fx), (15)
We shall return to actual solution of these equations later when we need it. The projected energy is now given in terms of these coefficients as El = CI+ bP(A.fx) + cP(AJ^‘) + dP(A.?z - j$) + eP(jz),
(16)
with
KK’
X”
KK’
(17)
J. L. Egiab et al. / Ground-state correlations
233
or Er =
(H)+bP(d~~)+c(P(dS2)-(df2))
We notice that the cranking model is recovered only when the second-order terms proportional to c, d, e are constant with respect to variations with a constraint on the angular momentum.
3. The RPA approach in the intrinsic wave function In this section we want to specify the structure of the intrinsic wave functions IY) in more detail. In the HFB case we vary among all wave functions I@) which can be written as the vacuum of quasiparticle operators ak, obtained by a general Bogolyubov transformation from some basic fermion operators c., c:. In the RPA case we work in a boson space and allow for all functions 1Y) which can be written as the vacuum of the boson operators B,, B,’ obtained by a general Bogolyubov transformation (19)
from some basic boson operators AK, AK+,In contrast to the fermion case such a transformation contains, besides the usual coefficients XK,, and YKc,constants fp which produce a shift operation. According to a generalization of the Thouless theorem to the boson space, the vacua I@) and IY) with respect to the operators AK and B,
may be connected by the transformation
IY) = exp ($&,ZKKAXCA&+ ; C&)1@),
(21)
with ZJtK’ = (YX_l)KK,,
ct = -(f*x-I),.
(22)
The variation within the set of RPA wave functions with the structure of eq. (21) therefore corresponds to a variation with respect to the hellicients ZKr and CK. This set depends on the underlying basic function 10). In the following we will use a different function I@) for each Z-value, and we leave its precise definition still open at this moment. Let us call the quasiparticle operators based on this vacuum as ak,
234
u: . Within
J. L. Egido
et al. / Ground-state correlations
the quasiboson approximation
we then have
with the index pair K = (k, I), k < 1.
Ai = alat
(23)
In the next step we have to express the operators H and j, in the boson space. It is consistent with the approximation (23) to terminate the boson expansion for H up to second order. We thus end up with H = H’O’+H”‘+H’2’ 3
(24)
with H’O’ = (@IHI@), H(l) = c (H’OAK’+ h.c.), K Hf2’ = ~H&AK+AK,+f KK’
Hi0
=
(@l[A~,H]l@h
f&
=
-<@I[AK,[H,
Hi% =
<@I[AKAK,,
(25) x(H$AK+A&+h.c.), KK'
&]]I@>,
(26)
HIP),
and, using the same notation, jx = j$W+ jii)+ j(2) x ,
(27)
with
(28) .?i" =
c J.&,AK+ KK'
AK,.
A small variation of the wave function l!P) in eq. (21) is then of the form ISY) = (:c 6Z,,J3,+B$ + c 6C,B,+)lY). P PN
(29)
It is evident that d.?XIY) can be written in this form. The set (21) has therefore the property (9), which means that we can use the cranking model for the determination of the intrinsic wave function, if the conditions (l)--(3) fulfilled. Varying the expectation value of H’ = H - cojx with respect to the parameters C,, and Z,,,, in eq. (28) we obtain two equations:
ewp,,
H’IIYY)= 0,
<‘ylC& H’IIW = 0.
OW WW
J. L. Egiab et al. / Grotmd-state correlations
235
This means we can determine the parameters ZKKpby a diagonalization of the twoboson part H’(‘) and the parameters Cx by a shift operation, which annihilates the one-boson part H’(r). The diagonalization leads to the usual RPA equations
for the determination of the RPA amplitudes Xxp and Yx,,. The shift parameters fp are given by
s2,f; =
c Hk”*X,,+
H;f”Yxp.
R
In the case of a broken symmetry in the HFB function I@), as for instance the rotational symmetry around the x-axis, the RPA equation (31) has a zero-frequency mode of the form &). Since H does not depend on the conjugate angle #), it is not necessary to carry out a shift in &(I). The parameter is then given by
c
fJ = - L~TV (H;'q;: + c.c.), K
where $rv will be defined in eq. (37). We now turn to the question of how to choose the HFB function I@), which defines the vacuum of the operators AK in the boson space. Starting from one choice for I@) we can express any other HFB function 15) using the Thouless theorem
IQ =
exp
( C CktdatW). kc1
(33)
In the case that the coefficients Ck1are small, we can use the quasiboson approximation for the operators GL:CL:= Ak: in eq. (33). We then find that 14) is obtained by a shifting of the operators Ax. Since we have already taken into account this degree of freedom in our variational ansatz (21), it is therefore not necessary to vary the HFB function itself in such a case. We only have to be sure that we start from a HFB function I@) which does not differ very much from I$), i.e. the coefficient CK should be small. The optimal choice would certainly be CK = 0 in eq. (21). This can be achieved if we use a cranking solution I@& where o is defined by the angular momentum in the RPA wave function (Yl.Ll!Q,
= J-m.
(34)
In this case I@& minimizes H’ = H -oJ, inthe set of HFB functions, i.e. H'(l) vanishes. Eq. (30b) is automatically fulfilled and the shift parameters fpineq. (32) are zero. If we adopt this choice, it then remains to solve the RPA equations (31). In practice this prescription turns out to be rather complicated, because the
J. L. Egido et al. / Ground-statecorrelations
236
cranking frequency o is determined only after the solution of the RPA equations. For the sake of simplicity we will therefore introduce in the next section an additional approximation, which will lead us to finite shift parameters fp. As discussed in ref. ‘), there exist several spurious solutions connected with the violation of the rotational invariance by the HFB function I@) +. For positive signature there is the zero-frequency mode (35) with the canonically conjugate angle (36) defined by
[H’(2),~~“1 = ~
all’,
[all’, ~~‘)I
=
_ i
(37)
and XTv is the moment of inertia of Thouless and Valatin “). For negative signature the spurious wobbling mode is
at the energy of the cranking frequency CD. The other eigenmodes Ep’ are called normal modes in the following and we use the index p instead of p if we want to specify them in particular. We thus can express the hamiltonian H = H'+dx in terms of the boson operators B,, r, .?i” with the help of the substitution
in eq. (24) and obtain ‘) j(r)2 H=E+CSZ,B,+B,+~ P
+or+r+o(&l'+.p)) 9
(38)
2.fTV
with
t In this discussion we again do not consider spurious modes connected with the particle numbers for protons and neutrons. See, however, sect. 6.
J. L. Egiab et al. / Ground-state correlations ,fsc
231
is the self-consistent moment of inertia in the cranking model
The minimization of the operator H’ with respect to the parameters ZKK, in the ansatz (21) leads to an exact eigenfunction 18) of H' defined by
Bpl!F) = 0,
.pl!F) =
0,
iy!P) = 0.
(41)
This is one definition of the RPA ground state ‘). Since Jil) is hermitian, IY) turns out to have an infinite norm +. As discussed in ref. 14) it is for small angles an approximation to the exact eigenfunction of the angular momentum operators, i.e. to the wave function in the laboratory frame. Since we are interested in a wave function in the intrinsic frame with a large symmetry violation, the definition (41) seems to be not appropriate for our aim because condition (1) is not fulfilled. We need a wave function IY) with large fluctuations in the angular momenta jX and j,, and sharply peaked in the conjugate angles & and &. It should go beyond the HFB wave function 14) only in so far as it contains additional correlations coming from the zero point oscillations in the normal modes B:. The spuriousities connected with violation of rotational symmetry are removed afterwards by the angular momentum projection. Since an unrestricted variation of IY) leads to an eigenstate of ?p) with a small fluctuation (L@), we have to carry out the variation with additional constraints. One possibility would be of course to require lY) to be an eigenfunction of the angle c&’). This, however, leads again to functions with infinite norm since $L1’ is also an hermitian operator. We therefore require )Y) to be an eigenfunction of the boson operator 2 xiy) where x is a linear combination
= 0,
(42)
of 1:‘) and 4:‘)
with [X,X+] = 1. The constant a is determined by the condition (YldJ:lY)
= (@lLls:I@) = (d&,
(44)
i.e. the RPA function )Y) is as strongly deformed as the HFB function I@). Neglecting t In refs. 11*14)a method is presented for calculating all RPA matrix elements independently of this fact.
J. L. Egiab et al. / Ground-state correlations
238
higher terms than .?i” on the 1.h.s. of eq. (44), this condition yields 1 a=m’ which becomes very small for well-deformed nuclei, i.e. IY) is more or less an eigenfunction of the angle $$l’. It is obvious that we also have to modify the third condition in eq.(40). From rlY) = 0 we would obtain
i.e. the conditions (1) and (3) would be violated. We therefore introduce an additional boson operator A
with [n&l+]
= 1,
(48)
and require AIY) = 0.
(49)
The parameter b is determined from the condition (!Pl~;lY)
= (@lc3,21@)= 0;))
which yields in the same approximation b
=
(50)
as in eq. (36)
X5:)
*
(51)
From eq. (39) we obtain (Yl~~l”lY)(Yl~~l”lY)
= $(~lJ&#+*,
(52)
which means that IY) is roughly a minimal wave packet. Using these relations (43), (47) and (52), we can express the hamiltonian H’ in the basis of the boson operators B,, x, A in the following way: H’ =
<4W’I4> - c f&l ; IyKP12 P
+ CQ,B,'B,+
P
+ ~(2(l+~*)n+n+(l-b’Xn’n++nn)), .sC
(53)
J. L. Egirdoet 01. / Ground-state correlations
239
where for simplicity we have neglected the term
This is small because the minimum packet relation (52) is fulfilled already in the HFB function I@) up to factor 2, as we show in table 1. Obviously the wave function IY), defined by the conditions &lY)
= 0,
XIV = 0,
A~Yu>= 0,
(54)
minimizes the expectation value of H’ under the constraints (42) and (49). It is certainly no eigenfunction of H’. Since IYu>is an intrinsic function, this is not necessary. An explicit expression for the wave function IY) determined by eqs. (54) is given in eq. (21) with the coefficients CK = 0,
ZfK’ = (=-%rC,
(55)
where XKc and YKfifor the normal bosons are given by the solution of the RPA equation (31) and
The expectation value of the angular momentum f, is given from eqs. (27) and (34)
(57)
with LIZ =
(Y'&z"Y)=
c
J:&YKrY&,
(58)
pKK’
where the index p runs over all normal modes BP+and the bosons I+ and A+. Finally we obtain for the energy
Besides the I-IFB energy (@/HI@) it contains the correlation energy caused by the normal modes and a correction term which takes into account that I@) has a different expectation value of JI, than IY). Let us conclude this section by summarizing the approximations used so far: (i) We treated the angular momentum projection before the variation in first order of the Kamlah expansion (6), i.e. in the cranking approach,
0.0000 0.0759 0.1328 0.1798 0.2151 0.2388 0.2514 0.2528 0.2475 0.2434 0.2636 0.2921 0.3185 0.3417 0.3616
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
TV
25.64 27.40 32.26 42.31 62.11 103.09 294.12 - 769.23 -285.71 333.33 65.36 65.36 74.07 89.96 100.00
9
0.337 0.655 1.064 1.658 2.691 4.489 3.315 - 2.293 -11.758 - 3.058 -0.681 -0.069 -0.070 -0.76
AI
0.197 0.346 0.467 0.522 0.515 0.704 0.333 -0.997 - 1.542 -0.054 0.032 -0.094 -0.449 -1.411
AL
0.025 0.048 0.075 0.112 0.173 0.275 0.196 -0.137 -0.742 -0.197 -0.044 -0.000 -o.ooo -0.048
d, 1.12 1.29 1.54 1.81 2.02 2.17 2.36 2.79 2.84 2.87 2.77 2.65 2.54 2.44
MPR
-0.113 -0.126 -0.139 -0.142 -0.123 -0.112 -0.125 -0.147 -0.121 -0.123 -0.184 -0.301 - 0.484 -0.699
AZ _._~ -0.141 -0.131 -0.122 -0.102 - 0.072 - 0.055 - 0.061 - 0.078 -0.051 -0.013 -0.008 -0.023 - 0.074 -0.243
AZ,,
-0.031 -0.035 - 0.040 - 0.042 -0.037 - 0.035 - 0.039 - 0.046 - 0.038 -0.041 - 0.066 -0.123 -0.246 -0.627
dz 0.025 0.043 0,062 0.031 -0.204 - 1.563 - 1.630 - 1.728 - 1.263 0.133 0.206 0.190 0.191 0.239
AN
-0.011 -0.002 -0.001 - 0.028 - 0.076 - 0.057 -0.021 - 0.088 -0.154 -0.012 0.055 0.081 0.123 0.217
AN,,
0.000 0.001 0.013 0.001 -0.047 -0.378 -0.407 -0.434 -0.337 0.038 0.065 0.067 0.079 0.137
dN _.-
o. is the cranking frequency determined from condition (60). &v is essentially the moment of inertia of Thouless and Valatin. Since we have to take into account in a realistic calculation the violation of the particle numbers, we can only calculate the tensor ge (see table 2). Here we give the diagonal part ‘/rV = s;:. AI [eq. (SS)] is the difference between the RPA expectation value and the HFB expectation value of J,. Only a small part (d I,,) is produced by the normal modes. The rest A I- A&, is produced by the spurious bosom. The parameter dI of eq. (66) determines the angular momentum correction. MPR = 4(~1~~~)(Q,1J;2j~)/(dij4;I~)* characterizes the validity of the minimal packet relation (5) for the HFB function I@). in complete analogy with the AI, diNM and d, values for the angular momentum in the last six columns we present the values LIZ, AZ,,, $ and AN, AN,,, dN for the correction of proton and neutron numbers. Since we only go to linear order in the correction AI, AZ and AN there are no coupling terms between these operations.
a0
I
The correction of the average angular momentum and particle number as a function of Z(fiist column)
TABLE1
g a s
: a 2 fi 5
G
5 t‘ tJ & 3 e
J. L. Egido et al. 1 Ground-state correlations
241
(ii) we used the RPA ansatz (21) with the restrictions (42) and (49), which guarantee strong deformation, for the variation of the intrinsic wave function, (iii) we te~inated the boson expansion of the hamiltoni~ (24) in second order. It then turned out that the optimal choice of the basic HFB function I@,>is a cranking solution, where the cranking frequency o is determined by the RPA expectation value of jX. In the next section we will modify the last condition somewhat for practical reasons and in sect. 5 we will go beyond the cranking approach for the angular moments projection.
4. The limit of a small angular momentum correction AZ In practice, it is difficult to determine o from eq. (34) because one has to solve the RPA equations (31) in each step of the self-consistent calculation for the wave function IQo). We therefore approximate in the following sections lQw) by l@&. The cranking frequency o. is determined from <@lJXl@>,, = %/m,
(60)
i.e. JQ,,,) is the usual HFB solution of the self-consistent cranking model. In the following we will always work on the basis of quasiparticle operators AK, Ai based on the vacuum [QUO) AKl@io(t>= 0.
(61)
Two solutions ICP,) and I@,,) of the self-consistent cranking model which differ in their angular momentum expectation value by an amount AZ are related in the linear response approximation by “) I@-> = exp (-- ~A~~~))t~~~>,
w
where ~$9) is defined in eq. (36). It generates a shift in the angular frequency by the amount -Aa
= o--w0
= -AZ/,%,,,
(63)
where
(W is given by eq. (58). From eq. (63) it follows that the moment of inertia of Thouless and Valatin &v measures the slope of (@l~Xl@>, as a function of w. In the backbending region it goes to infinity at two points. It is quite different from the selfconsistent moment of inertia .@sc of eq. (40) (see fig. 1).
J. L. Egido et al. / Ground-state correlations
242
200 IMN’) 32
I
160 -
24 -
100 -
i
’
/
’
I
1
16 -
01
02
0.3
0.4
1
I 6
II II I I 16 2L ,
fAMeV) a
b
Fig. 1. (a) The average angular momentum I((& J,I4) = JI(l+ 1)) of the HFB solution in the cranking model as a function of the cranking frequency. (b) The corresponding moments of inertia 4,, = dI/dw and 4, = I/w. The details of the calculation are discussed in sect. 6.
We can also express 4:’ in eq. (62) in terms of the boson x defined in eq. (43). Eq. (62) then reads: lGo) = e- d(r+- “‘l@Jw,),
(65)
with d = A1/2(A@+.
(66)
Obviously the parameter d determines the quality of the linear approximation. Table 1 shows that d is indeed a small parameter with the exception of two singular points in the backbending region. At these points the linear response approximation breaks down. For all other I-values, however, the linear approximation in AZ is well justified. We therefore use this approximation in the following and neglect all terms of higher order in AZ. The RPA wave function IY) is then of the form IY) = exp (4 c ZKK,AK+ A& - iAZ c cp~~AK+)I@,,). KK’
(67)
K
The coefficients ZxK* are determined by the variation of the hamiltonian. H’ = H-c&
= (H-w&)
+ ACL&
(68)
under conditions (42) and (49). Neglecting the term do@, which is of second order in AZ [see eqs. (58) and (63)], we find that the linear term Aw.@) can be eliminated by the transformation (62). For the determination of the coeficients Zxk, we thus obtain once more the RPA equations (31) based on the HFB function I@&. To ensure conditions (42) and (49) we have to choose ZKr as in eqs. (55) and (56).
J. L. Egido et al. / Ground-state correlations
243
We can now calculate the energy
Since with l@& one has the same expectation value of IX as with [!I’), the correction term odl of eq. (59) is now no longer present. The RPA energy obtained so far deviates from the HFB energy only by the correlation energy caused by zero-point fluctuations of the normal modes. Since /!I’> is an intrinsic - symmetry violating - wave function, the energy connected with spurious rotational motions is not yet subtracted. As we shall see in the next section, these spurious parts are obtained in a second-order approximation for the angular momentum projection. Although we are in this paper only interested in the calculation of the yrast levels, we want to remark that very similar arguments also apply to the calculation of excited bands. It is easy to show that the wave functions lp) = Ble-dpW+IY),
(701
where d,, is determined in by the condition + (711 minimize the hamiltonian in RPA order under all the constraints discussed in sect. 3 and under the additional condition of orthogonality to the ground state lY>. The excitation energy of the state }p) is given by the frequency Qp + %,
(721
where oJ;pl is again canceled by the shift (70). That means as long as we are interested only in energy levels, it is sufficient to base the solution of the RPA equations (31) on the solution I@,,,) of the self-consistent cranking model. Its eigenvalues s2, correspond to the excitation energies. This fact was used in ref. ‘il.
5. Second-order terms in the projection
So far we have treated the angular moments projection only in the so-called cranking approximation, i.e. we determined the intrinsic wave function from the variation of the energy expression in eq. (7). For the actual calculation of the projected energy we want to go a step further and take into account the second-order terms in eq. (14) at least for the calculation of the energy.
t For K # 0 bands we have to modify the condition (71) slightly [see ref. ‘I)].
J. L. Egido et al. / Ground-state correlations
244
For this purpose we proceed as discussed in sect. 2. Starting from the ansatz + (HR(S2)) = (a+bA~,+cA&+dL:,+eL2,)(R(SZ))
(73)
and using the hamiltonian of eq. (38) we obtain from eq. (54) a linear set of equations, which corresponds to eq. (15). We have to evaluate matrix elements like (HA?& (H4.f:) . . . and find expressions of the form
With the approximation
(.fj”‘(A .f$)
i, k = x, y, z,
(dj&lj$),
”
the equations simplify enormously
(74)
= 0, 1,2,
n
and we find the solution 1
C-
1
d=e=---
,
(75)
2,fsc ’
2.fTV
a = (H)-
(1; +&, (76)
b = co. For the projected energy we finally obtain
Er = GO- L(Aji)2tfTV
$
(j;+.T:)
c SC
+ oP(A.fx,) + ~ l
P(AJ:)+ $
29Tv
f@ + 2). c SC
(77)
If condition (3) is fulfilled we have p(d.L) 2 Jm-
(Q,
(78)
and for cranking solutions we thus obtain Et = Of)-
1 2CaTv(Ax$-
z)-
O,Z+jZ)+
p(S,Z+ .?Z). (79)
* SC
The last term is usually very small and will be neglected in the following tt. Thus
+ The expectation values (. . .) have to be calculated with the RPA function Ivl). Note that the coeffkients c, d, e have a somewhat different definition than in eq. (14). ++ For HFB wave functions we have P( J:) Z 0.5 and P(z) Z (t) [ref. “‘)I.
J. L. Egiab et al. / Ground-state correlations our
245
result is very similar to the result obtained in ref. ‘l)
-J- (Yyld~lY)-
EI = (W#W With the approximation
2/Tv
introduced in sect. 4 we find t
where I@) is the HFB function at frequency w. which fulfills the constraint (60). We thus obtain with “second-order projection” just the energies connected with the spurious rotational motion. Approximation (74) is certainly only a first step. In principle one can calculate all matrix elements exactly and solve a linear system of equations for the coefficients a, b, c, d and e. One then obtains corrections to the simple expression (81).
6. Numerical applications In ref, ‘l) the RPA equation (31) was solved for the high-spin region of 164Er and the corresponding excited bands were discussed in great detail. Now we use these calculations for the investigation of correlations in the yrast wave function. The underlying interaction is a pairing plus quadrupole force. We use the same configuration space and the same force parameters as we used in ref. 21) and all details and formulas can be found there. We start from the hamiltonian
H=
Ho-hj2Q:Q,-GP’P,
(82)
where Ho is a spherical Nilsson potential, QQ are quadrupole operators and P* creates a Cooper pair. In a first step the HFB wave function I ) of the cranking model is calculated. We use the approximation of sect. 4 and determine it from a variation of the energy without exchange terms
Eo =-~xKQo>~ +
= dm);
GO-‘>*
(83)
as well proton and neutron nnmber
(N,) = 2,
(N,)
= A-Z.
(84)
+ This expression for the energy coicides essentially with the one obtained by Marsbalek l’) in a boson expansion. The only difference is a term fw which has to be added in ref. “), because the cranking wave function I@) has been solved for the constraint (@lf,lcP) = I instead of eq. (60).
J. L. Egido et al. 1 Ground-state correlations
246
In addition to E. the HFB energy (H) EEX
=
-4~
in eq. (81) contains the exchange terms
(AQ~A~~)-G(A~+A~)
i
(85)
ii=-2
with
AQ,, = Qr-(Q&v The fluctuations of the quadrupole
operators
AP = P-(P).
(86)
are of the form
= k;,lQt’,“,l’ = Tr (Q:Qp~-Qp~Q;lfo-QruQ~u*), (87)
where Qz” is the two-quasiparticle part of the operator Q,.,and p, K are the normal and anormal density of the HFB function 1). The first term in eq. (87) comes from the single-particle operator contained in Q: Qa, the second corresponds to the Fock term in the self-consistent field and. the third has its origin in the contribution of the quadrupole force to the pairing field. The fluctuation of the pairing operator (AP+AP)
= kFllP$‘12 = -$Tr(Pp*Pp)
(88)
corresponds to the contribution of the pairing force to the self-consistent field. These terms are of second order in the sense of a boson expansion of the hamiltonian (82) [see ref. ‘)I and are therefore neglected in the variational determination of the HFB function. Since we go in this paper up to RPA order we have to take them into account in the calculation of the energy. Fig. 2 shows the fluctuations of the operators & and P. They are not small, but show a very smooth behaviour as a function of angular momentum. The same is true for the resulting energies of eq. (85). The contribution of the exchange terms is shown in fig. 5 +. One notes that over the angular momentum range between I = 0 and I = 28 EEX changes by roughly 500 keV. Altogether the total yrast energy contains four terms [compare eqs. (69) and (80)] Er = E~+EEx+ENM+&u.
(8%
ENS corresponds to the correlations caused by the zero point vibration of the normal modes with the energy ENM
=
-
~~p~&P12 P
K
and EoM corresponds. to the Goldstone modes, i.e. to the spurious motion in the symmetry violating intrinsic function (see eq. 80). ’ We start with I = 2 becausefor I = 0 the axial symmetry causes degeneracies and some numerical problems arise.
J. L. Egiab et al. / Ground-state correlations I
I
I
1
I
247
I
IAI?p2
,21._.----.,._._._
(At’)2N
8&I
0
II
1
I
I
,
lJ2
24:._.
-,_.
-.-‘-.
16 -
IA< g2
80
)
2&z.
-.--._(
-.-.
I2 t&Q?2 ---_ _ !A0 12 -P.
16----------________ 8-
I
I
28
Fig. 2. The fluctuation of the quadrupole operators Q, and of the pairing operators for protons P, and neutrons P, in the HFB wave function 14) as a functron of the average angular momentum. The units arc b4 for the quadrupole operators (b is the oscillator length) and the pairing operators are dimensionless.
E&‘d
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
IMeVl
- 2.5-
-2.o16 I6 - 1.5 El 12 IO -1.0
-
6” I=“z
- 0.51
I
80
lw
240
320
I n
400
Fig. 3. Convergence properties of the correlation energy (91) resulting from the lowest n normal modes.
248
J. L. Egido et al. / Ground-state correlations
For the calculation of the exact value of ENMone needs all eigenvalues of the RPA matrix (31) and the corresponding eigenvectors. The dimension of this matrix is of the order of several thousands. In ref. 21) the eigenvalue problem was rewritten into a dispersion relation and its low-lying modes were determined by a root search. In the same way we now determine the lowest 400 eigenvalues of the RPA matrix. Thus we were able to investigate the convergence properties of the expression ENMb)
=
-
i p=1
fi,
c 1&,I2
(91)
K
as a function of n. Fig. 3 shows the values E&n) for I 5 20. The convergence is rather slow, such that n = 400 is still probably not large enough to achieve absolute convergence. However, the curves for different I-values are approximately parallel. We therefore hope that a further increase of the absolute value of ENMwill not strongly influence the excitation energies E, - EI= ,,. In the following we therefore use the value ENM = ENM(FI = 80). Fig. 5 shows the correction ENM as a function of I. In the backbanding region at I- 16 we find a very large contribution to the correlation energy. It has its origin in the very low-lying “second” band in this region [see ref. 2’), fig. 1 band a]. We now turn to the investigation of the term EGM. Besides the spurious modes connected with the violation of rotational symmetry, which we have discussed in sect. 5; we also have to take into account the spurious modes connected with the numbers of protons and neutrons. We then have to vary the projected energy (YIHPN,PN,P1lY). The calculation is carried out in complete analogy with sects. 3-5. We shall therefore give only the results. Instead of eq. (38) the hamiltonian takes the form H’ = const + c f&B,+ B, + OPT P
+:g&“’
+ :g,,&l)’
+:gz&)’
+g.&?fi(,l)
We obtain for the energy contribution
+gJz~~l’Ab”+gNZ,~~)~a).
(92)
of the spurious modes
with E,,
= -i-g&I.&,
EJN = -gJ.ddJ%Jr
E NN
=
-+NN’k@)r
EJZ = -g,,@&,h
Ezz = -:gzz(A@)
(94)
The tensor gik is calculated as described in ref. 9). Table 2 gives its actual values for I 5 28. Table 3 shows the corresponding fluctuations. In fig. 4 we present the contributions (94) to the energy. The essential feature is the behaviour of the term
J. L. Egido et al. / Ground-statecorrelations
249
TABLE2
The tensor ge in eq. (92) I
gJJ
0
0.0390 0.0365 0.0310 0.0236 0.0161 0.0097 0.0034 -0.0013 -0.0035 0.0030 0.0153 0.0153 0.0135 0.0115 0.0100
2 4 6 8 10 12 14 16 18 20 22 24 26 28
QNN
szz
0.1920 0.1914 0.1896 0.1859 0.1798 0.1773 0.1907 0.1692 0.1278 0.0512 0.0663 0.1159 0.1533 0.2057 0.3781
0.3245 0.3230 0.3188 0.3118 0.3035 0.2980 0.2976 0.3002 0.3042 0.3091 0.3114 0.3073 0.2969 0.2887 0.3887
gJZ
gJN
-o.oooo
-0.0000
-0.0023 -0.0044 -0.0074 -0.0121 -0.0175 -0.0144 0.0041 0.0227 0.0410 0.0154 0.0047 0.0014 0.0003 0.0008
-0.ooo6
-0.0001 0.0017 0.0037 0.0046 0.0037 0.0037 0.0056 0.0095 0.0067 0.0057 0.0058 0.0066 0.0083
gZN
0.0033 0.0033 0.0033 0.0035 0.~ 0.0050 0.0095 0.0171 0.0214 0.0155 0.0173 0.0233 0.0266 0.0281 0.0297
EJJ. In the backbending region it becomes rather small and even positive because of the strange behaviour of gJJ, which is roughly proportional to l/Yrv, in this region (see table 1). Fig. 5 summarizes different contributions to the correlation energy. ENMand EGO behave in the backbending region in opposite senses. The resulting sum of all correlation energies is therefore again a very smooth function of I. It is not constant, but increases from Z = 2 to I = 28 by more than 2 MeV. If we were to add these terms to the energy E, of the HFB cranking model without exchange terms, we would obtain a considerable stretching of the spectrum. Since the parameters of the pairing plus quadlMeVI
f
I
I
f
I
I
Fig. 4. The contributions E, of the spurious modes in eq. (94) to the energy. ENzvanishes identically and E,, is smaller than 1 keV. c is the sum of all contributions with positive signature c = E,, + ENN+ Ezz +-Em.
J. L. Egiab et al. / Ground-state correlations
250
TABLE 3
The fluctuations of the operators f,, f,, f,, fiP and fin and the coupling terms A &A IV,,,A fXA I’?”in the cranking wave function I) = [4),,, I
0
44.30 45.11 47.12 50.35 54.82 60.27 66.40 71.20 70.12 62.79 60.29 60.96 62.07 63.08 63.70
2 4 6 8 10 12 14 16 18 20 22 24 26 28
6.47 6.37 6.15 5.80 5.34 4.80 4.27 4.00 3.97 3.63 3.00 2.52 2.03 1.47 0.76
3.36 3.31 3.21 3.07 2.90 2.74 2.61 2.55 2.52 2.48 2.27 1.93 1.49 0.96 0.31
(A J,AN,)
0.0
0.0
0.03 0.06 0.12 0.24 0.44 0.40 -0.56 -1.72 -1.44 -0.56 -0.28 -0.17 -0.10 -0.05
0.01 0.01 0.00 -0.02 -0.03 - 0.04 -0.05 -0.05 -0.06 -0.07 -0.08 -0.10 -0.11 -0.06
-__44.30 44.58 45.13 46.00 47.26 48.97 51.01 52.49 52.68 51.52 50.22 49.84 49.91 50.22 50.66
rupole force were adjusted to reproduce the experimental yrast line as well as possible, we would certainly not obtain an improvement in agreement with the data. To have a complete description we should have used the energies EI in eq. (89) for the adjustment of the force parameters. IMenI)
21I-
IS)-
0
-l.C I
I
I
L
8
12
Fig. 5. The different contributions to the correlation ENMthe term coming from the normal modes p = modes (93) and c is the sum of these three terms 1 I = 2, where we have Em = -6.95 MeV, E&n
I
16
I
20
I
24 ,
energy in eq. (89). &x are the exchange terms (85), 1. . .80 in eq. (91), &,,, results from the Goldstone = EEx+ &,,, + EOM.All energies are normalized to = 80) = -0.899 MeV and EOM= -2.742 MeV.
J. L. Egiab et al. / Ground-state correlations
251
The fact that the anomalous behaviour in the functions &M(Z) and E&Z) have canceled out in the final result ,Y of lig. 5 can be understood qualitatively: We only have to use the RPA equation (31) and find c =
EEX+J%+ENM
=
+fcf$--
P
CEK), K
(95)
i.e. the correlation energy is just the difference between the sum of all correlated energies K&,and the sum of all uncorrelated two-quasiparticle energies EK = Ek + El. The peaks in the functions Z&Mand ENMare obviously caused by the level crossing in the backbending region, i.e. by a low-lying two-quasiparticle energy EK, or the corresponding low-lying collective energy Q,,. As we see from fig. 1 in ref. ‘I), EK1 and Qpl are very close to each other, i.e. the difference (95) is a smooth function in the region of the level crossing.
7. Conclusion The self-consistent cranking model based on generalized Slater determinants in the many fermion space used as a description of the yrast line of well-deformed nuclei has been extended: Ground-state correlations in the intrinsic function are taken into account in the RPA scheme and an approximate projection in angular momentum and particle number up to second order has been carried out. According to the different degrees of freedom in the system we find two types of correlations: The first type is connected with the zero- point fluctuations of vibrations in the intrinsic frame; the second type has its origin in the Goldstone modes in the intrinsic function connected with the violation of angular momentum and particle number symmetry. They are removed by the projection. Up to second order the formula for the energy is identical to that obtained by Marshalek ’ ‘) in a boson expansion in a rotating frame. The wave function calculated with the latter method, however, is not normalizable. It presents only for small angles an approximation to the exact eigenstate of the system. It is an advantage of our method that it yields a projected wave function which can be normalized and which yields after projection a “reasonable” approximation to the exact eigenstate in the entire space, not just for very small angles. These theoretical considerations were applied to the nucleus ’ 64Er. The correlations in the yrast wave function were taken into account on the basis of RPA calculations discussed in ref. “). In particular we calculated the exchange terms EEL and correlation energies ENM and &M. Their behaviour as a function of I is discussed. For Z = 0 the three terms together produce a lowering of the yrast energy by roughly 10.5 MeV. With increasing angular momentum the lowering caused by the exchange terms and by the spurious modes is reduced, whereas the correlation energy coming from the normal modes increases. In the backbending region we have found several
252
J. L. Egiab et al. / Ground-state correlations
irregularities: Because of the level crossing one observes a peak in the correlation energy ENM.On the other hand the spurious energy E,, has a dip there because of the singular behaviour of the parameter &v which measures the slope of the angular momentum as a function of the cranking frequency. Both irregularities cancel to a large extent in such a way that the resulting energy correction has a smooth behaviour which can simply be absorbed in a proper adjustment of the effective interaction. We thus understand why a simple calculation in the self-consistent cranking model based on HFB functions with adjustable parameters in the effective interaction is able to reproduce the experimental yrast line to a rather high degree of accuracy. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
14) 15) 16) 17) 18) 19) 20) 2!) 22)
R. M. Lieder and H. Ryde, Adv. Nucl. Phys. 10 (1978) 1 P. Ring, R. Beck and H. J. Mang, Z. Phys. 231 (1970) 10 B. Banerjee, H. J. Mang and P. Ring, Nucl. Phys. A215 (1973) 366 A. L. Goodman, Nucl. Phys. A256 (1976) 113 A. Faessler, K. R. Sandhya Devi, F. Gruemmer, K. W. Schmid and R. Hilton, Nucl. Phys. A256 (1976) 106 J Fleckner, U. Mosel, H. J. Mang and P. Ring, to be published S. Bose, J. Krumlinde and E. R. Marshalek, Phys. Lett. 53B (1974) 136 S. Y. Chu, E. R. Marshalek, P. Ring, J. Krumlinde and J. 0. Rasmussen, Phys. Rev. Cl2 (1975) 1017 E. R. Marshalek, Nucl. Phys. A266 (1976) 317 B. L. Birbrair, Nucl. Phys. A257 (1976) 445; I. N. Mikhailov and D. Janssen, Phys. Lett. 72B (1978) 303 E. R. Marshalek, Nucl. Phys. A275 (1977) 416 J. L. Egido, P. Ring and H. J. Mang, Phys. L.&t. 77B (1978) 123 I. Hammamoto, Nucl. Phys. A263 (1976) 315; Phys. Lett. 66B (1977) 222; E. R. Marshalek and A. L. Goodman, Nucl. Phys. A294 (1978) 92; F. Griimmer, K. W. Schmid and A. Faessler, Nucl. Phys. A308 (1978) 77; S. Cwiok, J. Dudek and Z. Szymanski, Acta Phys. Pol. 99 (1978) 725; R. Hilton, Contributed Paper, Jiilich Conf. on highly excited states in nuclei, Sept. 1975, vol. 1, p. 33 E. R. Marshalek and J. Weneser, Ann. of Phys. 53 (1969) 569; Phys. Rev. C2 (1970) 1682 A. Kamlah, Z. Phys. 216 (1968) 52 R. Beck, H. J. Mang and P. Ring, Z. Phys. 231 (1970) 26 P. Ring, H. J. Mang and B. Banerjee, Nucl. Phys. A225 (1974) 141 H. J. Mang, B. Samadi and P. Ring, Z. Phys. A279 (1976) 325 D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 211 S. Islam, private communication J. L. Egido, H. J. Mang and P. Ring, Nucl. Phys. A334 (1980) 1 D. J. Thouless, Nucl. Phys. 21 (1960) 225
GROUND-STATE
CORRELATIONS OF DEFORMED
IN THE YRAST NUCLEI
LEVELS
J. L. EGIDO ‘, H. J. MANG and P. RING jhysik-Department,
Technische Unioersitdt Mkhen,
D-8046 Garching, W. Germany
Received 5 November 1977 (Revised 20 February 1980) Abstract: The random phase approximation (RPA) in copjunction with an approximate projection of angular momentum and particle number is used to investigate ground-state correlations in the yrast wave functions of realistic nuclei at high spins. One has to distinguish between two types of correlations, those due to virtual excitation of vibrations and those caused by the removal of the spurious components in the symmetry-violating intrinsic wave function. Both types produce a lowering in the energy of several MeV and both show large anomalies in the backbending region. One observes, however, a cancellation of these anomalies, so that the total correlation energy is a smooth function of angular momentum which can be taken into account in a theory with uncorrelated product states by a simple renormalization of the effective interaction.
1. Introduction
In recent years the yrast line in the high-spin domain of deformed nuclei has been investigated experimentally [for a review see ref. ‘)I and theoretically 2-6) in great detail. So far most of the microscopic descriptions in realistic nuclei use the selfconsistent cranking model (SC?) based on generalized product wave functions of the Hartree-Fock-Bogolyubov (HFB) type. Several authors ’ - ’ 2, have proposed going beyond this first approximation by taking into account the RPA corrections to the self-consistent cranking model. One then finds an attractive correlation energy produced by the virtual excitation of the vibrations. Up to now numerical calculations of this type have been carried out only for rather unrealistic models ‘,s* 12). In such calculations one has, in general, found a considerable improvement of the yrast energies, although there are also some cases in which the RPA breaks down completely. This happened in particular in the region of level crossings where one has a very low-lying two-quasiparticle excitation in the rotating frame. In this region the validity of the cranking approximation itself has become an object of much discussion 13) and it still seems to us an open question as to what extent a level crossing can be described by one product state in the intrinsic frame. The RPA is a small amplitude extension of this static mean field approach. + Work supported by the BMFT (Bundesministerium fur Forschung und Technologie). address: Max-Planck Institut fti Kemphysik, POB 103980, D-6900 Heidelberg 1, W.-Germany. 229
Present
230
J. L. Egido et al. 1 blond-state
cwrelations
In the region of level crossings it has its own problems caused by the violations of the Pauli principle. We therefore have to be very careful in drawing final conclusions from our results in such regions. One often argues that the RPA restores the broken symmetries in the HFB wave functions. This is certainly not true to the extent that the RPA wave function is an eigenstate of the exact angular momentum operator. However, as it shows only small deviations from the underlying HFB function, we consider it as an intrinsic wave function, improved as compared with the simple product state because it contains additional ground-state correlations. To obtain the wave function in the laboratory frame, one has to carry out a projection onto good angular moments. We have therefore determined the intrinsic RPA function from the variational principle after an appropriate projection, This approximation will be discussed in sect. 2. In sect. 3 properties of the RPA wave function are discussed and we show that in order to describe an intrinsic deformed state by the RPA method a restricted class of RPA wave functions has to be used. in sect. 4 we discuss an approximation which is valid as long as the increase of average angular mom~tum caused by the RPA correlations is small compared to the width of the angular momentum distribution in the intrinsic frame. Second-order terms in the angular momentum projection are taken into account in sect. 5. The theory is applied in sect. 6 to the yrast line of 164Erand the influence of the hound-state correlations on the energy is inv~tigated as a function of the angular rnorn~t~, In sect. 7 we summarize our results.
2. Approximate projection of angdar mom~n~rn ?
HFB theory with a ~nstraint on the angular rnorn~t~ .?%can be derived as an approximation to a variation of the intrinsic wave function (within the set of generalized Slater determinants I@)) after projection onto eigenstates of f2 [refs. 15-17)]. To justify this approx~ation, three conditions must be fuElled: (i) Violation of the symmetry in the intrinsic wave function must be strong (@~@/di> w 1.
(1)
(ii) The wave function I@>must have definite signature eis=l@) N I@).
(21
(iii) The deviation from axial symmetry must be weak in the sense (@,l&zJ) * IfI+ 1).
(3)
’ Witbin this paper we treat the problem of p~~ic~e~~~ violation in the HFB and RPA wave functions in the same approximation. In order to keep the formulas reasonably simple, we &all only discuss the case of rotationai symmetry in this section.
J. L. Egiab et al. 1 Ground-statecorrelations
231
In this paper we go beyond the simple product ansatz for the intrinsic wave function and include ground-state correlations of the RPA type. The corresponding wave function will be denoted by 1Y) in contrast to the HFB wave function I@). Angular momentum projection from such functions can be carried out in exactly the same way as from HFB functions. For the convenience of the reader we briefly remember the essential steps of this procedure: The projected energy Er is given by
KK’
kk’
where
and the operator
describes a rotation through Euler angles 61 = (01,/.$y). The hamiltonian overlap (YIHR(61)IY) can be represented expansion is) t (HR(R))
= (a + bLlL, + . . .)( R&2)),
by the Kamlah
(6)
where Ai, = t,- (jx) is the angular momentum operator expressed in terms of Euler angles. For large deformations the expansion (6) can be terminated at first order and we obtain (7) with A.fx = jx,- (.?J. The variation of the projected energy (7) within a suitable set of RPA wave functions IY) leads to &$ = (c?ylHIy)-
In the following we assume that a possible variation within this set of trial functions is ISY) = &Ajx,(Y).
(9)
In the next section it will become clear that this is true for the RPA wave functions ’ In the following all expectation values are taken with the WA wave function IV) if not otherwise specif=d.
J. L. Egiab et al. / Ground-state correlations
232
under consideration.
We then determine the solution of the cranking model IY’,) (6YI&oJ,lrC/)
= 0,
(10)
where o is determined from the subsidiary condition (YIS,lY),
= $0.
(11)
Since ISY) in eq. (9) is an admissible variation, we obtain from eq. (10)
w =
(HAS,) (AZ)
(12)
and find together with eq. (10) that IY,) is a solution of the variational equation (8). We thus find that the cranking wave function IY,) is an approximation to an intrinsic wave function obtained from the variation after projection. This derivation is valid for all wave functions which fulfill conditions (l)(3) and condition (9). An example are HFB wave functions, where, according to the theorem of Thouless, eBd’+D) = I@) + EA.T&D)
(13)
is again a generalized Slater determinant. Another example are the RPA wave functions discussed in the following sections. We then end up with the constrained RPA equations (10) and (11). Before we specify the structure of the RPA functions we go one step further and include quadratic terms in the operators L,, I,,, I., in the expansion (6). The corresponding equation then reads : (HR(G!)) = (a+bA~,+~Ai?+d(d&~~)+eL1,)(R(SZ)).
(14)
The unknown coefficients a, 6, c, d and e are determined from the following set of linear equations : + c(Aj2)
+&A.&f;))
+e(J:),
UO
=a
(HA&)
=
(HAf2)
= a(AJ2)+b(AjxAf2)+c((Af2)2)
+d((Aj~-.@Af2)+e(j~Aj2),
(H&f)
= a(S,2)
+b(Aj,.@
+c(Af2&
+d((A.f;[email protected];:)
+e<.W;>,
(Hj?)
= a(&
+b(AS,j~)
+c(AY2.Q
+d((A.&Qf;)
+e(.Tt).
b( A.?:)
+c(AJ2A~x)+d((A.f~-f~)A~x)+e(.f~A.fx), (15)
We shall return to actual solution of these equations later when we need it. The projected energy is now given in terms of these coefficients as El = CI+ bP(A.fx) + cP(AJ^‘) + dP(A.?z - j$) + eP(jz),
(16)
with
KK’
X”
KK’
(17)
J. L. Egiab et al. / Ground-state correlations
233
or Er =
(H)+bP(d~~)+c(P(dS2)-(df2))
We notice that the cranking model is recovered only when the second-order terms proportional to c, d, e are constant with respect to variations with a constraint on the angular momentum.
3. The RPA approach in the intrinsic wave function In this section we want to specify the structure of the intrinsic wave functions IY) in more detail. In the HFB case we vary among all wave functions I@) which can be written as the vacuum of quasiparticle operators ak, obtained by a general Bogolyubov transformation from some basic fermion operators c., c:. In the RPA case we work in a boson space and allow for all functions 1Y) which can be written as the vacuum of the boson operators B,, B,’ obtained by a general Bogolyubov transformation (19)
from some basic boson operators AK, AK+,In contrast to the fermion case such a transformation contains, besides the usual coefficients XK,, and YKc,constants fp which produce a shift operation. According to a generalization of the Thouless theorem to the boson space, the vacua I@) and IY) with respect to the operators AK and B,
may be connected by the transformation
IY) = exp ($&,ZKKAXCA&+ ; C&)1@),
(21)
with ZJtK’ = (YX_l)KK,,
ct = -(f*x-I),.
(22)
The variation within the set of RPA wave functions with the structure of eq. (21) therefore corresponds to a variation with respect to the hellicients ZKr and CK. This set depends on the underlying basic function 10). In the following we will use a different function I@) for each Z-value, and we leave its precise definition still open at this moment. Let us call the quasiparticle operators based on this vacuum as ak,
234
u: . Within
J. L. Egido
et al. / Ground-state correlations
the quasiboson approximation
we then have
with the index pair K = (k, I), k < 1.
Ai = alat
(23)
In the next step we have to express the operators H and j, in the boson space. It is consistent with the approximation (23) to terminate the boson expansion for H up to second order. We thus end up with H = H’O’+H”‘+H’2’ 3
(24)
with H’O’ = (@IHI@), H(l) = c (H’OAK’+ h.c.), K Hf2’ = ~H&AK+AK,+f KK’
Hi0
=
(@l[A~,H]l@h
f&
=
-<@I[AK,[H,
Hi% =
<@I[AKAK,,
(25) x(H$AK+A&+h.c.), KK'
&]]I@>,
(26)
HIP),
and, using the same notation, jx = j$W+ jii)+ j(2) x ,
(27)
with
(28) .?i" =
c J.&,AK+ KK'
AK,.
A small variation of the wave function l!P) in eq. (21) is then of the form ISY) = (:c 6Z,,J3,+B$ + c 6C,B,+)lY). P PN
(29)
It is evident that d.?XIY) can be written in this form. The set (21) has therefore the property (9), which means that we can use the cranking model for the determination of the intrinsic wave function, if the conditions (l)--(3) fulfilled. Varying the expectation value of H’ = H - cojx with respect to the parameters C,, and Z,,,, in eq. (28) we obtain two equations:
ewp,,
H’IIYY)= 0,
<‘ylC& H’IIW = 0.
OW WW
J. L. Egiab et al. / Grotmd-state correlations
235
This means we can determine the parameters ZKKpby a diagonalization of the twoboson part H’(‘) and the parameters Cx by a shift operation, which annihilates the one-boson part H’(r). The diagonalization leads to the usual RPA equations
for the determination of the RPA amplitudes Xxp and Yx,,. The shift parameters fp are given by
s2,f; =
c Hk”*X,,+
H;f”Yxp.
R
In the case of a broken symmetry in the HFB function I@), as for instance the rotational symmetry around the x-axis, the RPA equation (31) has a zero-frequency mode of the form &). Since H does not depend on the conjugate angle #), it is not necessary to carry out a shift in &(I). The parameter is then given by
c
fJ = - L~TV (H;'q;: + c.c.), K
where $rv will be defined in eq. (37). We now turn to the question of how to choose the HFB function I@), which defines the vacuum of the operators AK in the boson space. Starting from one choice for I@) we can express any other HFB function 15) using the Thouless theorem
IQ =
exp
( C CktdatW). kc1
(33)
In the case that the coefficients Ck1are small, we can use the quasiboson approximation for the operators GL:CL:= Ak: in eq. (33). We then find that 14) is obtained by a shifting of the operators Ax. Since we have already taken into account this degree of freedom in our variational ansatz (21), it is therefore not necessary to vary the HFB function itself in such a case. We only have to be sure that we start from a HFB function I@) which does not differ very much from I$), i.e. the coefficient CK should be small. The optimal choice would certainly be CK = 0 in eq. (21). This can be achieved if we use a cranking solution I@& where o is defined by the angular momentum in the RPA wave function (Yl.Ll!Q,
= J-m.
(34)
In this case I@& minimizes H’ = H -oJ, inthe set of HFB functions, i.e. H'(l) vanishes. Eq. (30b) is automatically fulfilled and the shift parameters fpineq. (32) are zero. If we adopt this choice, it then remains to solve the RPA equations (31). In practice this prescription turns out to be rather complicated, because the
J. L. Egido et al. / Ground-statecorrelations
236
cranking frequency o is determined only after the solution of the RPA equations. For the sake of simplicity we will therefore introduce in the next section an additional approximation, which will lead us to finite shift parameters fp. As discussed in ref. ‘), there exist several spurious solutions connected with the violation of the rotational invariance by the HFB function I@) +. For positive signature there is the zero-frequency mode (35) with the canonically conjugate angle (36) defined by
[H’(2),~~“1 = ~
all’,
[all’, ~~‘)I
=
_ i
(37)
and XTv is the moment of inertia of Thouless and Valatin “). For negative signature the spurious wobbling mode is
at the energy of the cranking frequency CD. The other eigenmodes Ep’ are called normal modes in the following and we use the index p instead of p if we want to specify them in particular. We thus can express the hamiltonian H = H'+dx in terms of the boson operators B,, r, .?i” with the help of the substitution
in eq. (24) and obtain ‘) j(r)2 H=E+CSZ,B,+B,+~ P
+or+r+o(&l'+.p)) 9
(38)
2.fTV
with
t In this discussion we again do not consider spurious modes connected with the particle numbers for protons and neutrons. See, however, sect. 6.
J. L. Egiab et al. / Ground-state correlations ,fsc
231
is the self-consistent moment of inertia in the cranking model
The minimization of the operator H’ with respect to the parameters ZKK, in the ansatz (21) leads to an exact eigenfunction 18) of H' defined by
Bpl!F) = 0,
.pl!F) =
0,
iy!P) = 0.
(41)
This is one definition of the RPA ground state ‘). Since Jil) is hermitian, IY) turns out to have an infinite norm +. As discussed in ref. 14) it is for small angles an approximation to the exact eigenfunction of the angular momentum operators, i.e. to the wave function in the laboratory frame. Since we are interested in a wave function in the intrinsic frame with a large symmetry violation, the definition (41) seems to be not appropriate for our aim because condition (1) is not fulfilled. We need a wave function IY) with large fluctuations in the angular momenta jX and j,, and sharply peaked in the conjugate angles & and &. It should go beyond the HFB wave function 14) only in so far as it contains additional correlations coming from the zero point oscillations in the normal modes B:. The spuriousities connected with violation of rotational symmetry are removed afterwards by the angular momentum projection. Since an unrestricted variation of IY) leads to an eigenstate of ?p) with a small fluctuation (L@), we have to carry out the variation with additional constraints. One possibility would be of course to require lY) to be an eigenfunction of the angle c&’). This, however, leads again to functions with infinite norm since $L1’ is also an hermitian operator. We therefore require )Y) to be an eigenfunction of the boson operator 2 xiy) where x is a linear combination
= 0,
(42)
of 1:‘) and 4:‘)
with [X,X+] = 1. The constant a is determined by the condition (YldJ:lY)
= (@lLls:I@) = (d&,
(44)
i.e. the RPA function )Y) is as strongly deformed as the HFB function I@). Neglecting t In refs. 11*14)a method is presented for calculating all RPA matrix elements independently of this fact.
J. L. Egiab et al. / Ground-state correlations
238
higher terms than .?i” on the 1.h.s. of eq. (44), this condition yields 1 a=m’ which becomes very small for well-deformed nuclei, i.e. IY) is more or less an eigenfunction of the angle $$l’. It is obvious that we also have to modify the third condition in eq.(40). From rlY) = 0 we would obtain
i.e. the conditions (1) and (3) would be violated. We therefore introduce an additional boson operator A
with [n&l+]
= 1,
(48)
and require AIY) = 0.
(49)
The parameter b is determined from the condition (!Pl~;lY)
= (@lc3,21@)= 0;))
which yields in the same approximation b
=
(50)
as in eq. (36)
X5:)
*
(51)
From eq. (39) we obtain (Yl~~l”lY)(Yl~~l”lY)
= $(~lJ&#+*,
(52)
which means that IY) is roughly a minimal wave packet. Using these relations (43), (47) and (52), we can express the hamiltonian H’ in the basis of the boson operators B,, x, A in the following way: H’ =
<4W’I4> - c f&l ; IyKP12 P
+ CQ,B,'B,+
P
+ ~(2(l+~*)n+n+(l-b’Xn’n++nn)), .sC
(53)
J. L. Egirdoet 01. / Ground-state correlations
239
where for simplicity we have neglected the term
This is small because the minimum packet relation (52) is fulfilled already in the HFB function I@) up to factor 2, as we show in table 1. Obviously the wave function IY), defined by the conditions &lY)
= 0,
XIV = 0,
A~Yu>= 0,
(54)
minimizes the expectation value of H’ under the constraints (42) and (49). It is certainly no eigenfunction of H’. Since IYu>is an intrinsic function, this is not necessary. An explicit expression for the wave function IY) determined by eqs. (54) is given in eq. (21) with the coefficients CK = 0,
ZfK’ = (=-%rC,
(55)
where XKc and YKfifor the normal bosons are given by the solution of the RPA equation (31) and
The expectation value of the angular momentum f, is given from eqs. (27) and (34)
(57)
with LIZ =
(Y'&z"Y)=
c
J:&YKrY&,
(58)
pKK’
where the index p runs over all normal modes BP+and the bosons I+ and A+. Finally we obtain for the energy
Besides the I-IFB energy (@/HI@) it contains the correlation energy caused by the normal modes and a correction term which takes into account that I@) has a different expectation value of JI, than IY). Let us conclude this section by summarizing the approximations used so far: (i) We treated the angular momentum projection before the variation in first order of the Kamlah expansion (6), i.e. in the cranking approach,
0.0000 0.0759 0.1328 0.1798 0.2151 0.2388 0.2514 0.2528 0.2475 0.2434 0.2636 0.2921 0.3185 0.3417 0.3616
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
TV
25.64 27.40 32.26 42.31 62.11 103.09 294.12 - 769.23 -285.71 333.33 65.36 65.36 74.07 89.96 100.00
9
0.337 0.655 1.064 1.658 2.691 4.489 3.315 - 2.293 -11.758 - 3.058 -0.681 -0.069 -0.070 -0.76
AI
0.197 0.346 0.467 0.522 0.515 0.704 0.333 -0.997 - 1.542 -0.054 0.032 -0.094 -0.449 -1.411
AL
0.025 0.048 0.075 0.112 0.173 0.275 0.196 -0.137 -0.742 -0.197 -0.044 -0.000 -o.ooo -0.048
d, 1.12 1.29 1.54 1.81 2.02 2.17 2.36 2.79 2.84 2.87 2.77 2.65 2.54 2.44
MPR
-0.113 -0.126 -0.139 -0.142 -0.123 -0.112 -0.125 -0.147 -0.121 -0.123 -0.184 -0.301 - 0.484 -0.699
AZ _._~ -0.141 -0.131 -0.122 -0.102 - 0.072 - 0.055 - 0.061 - 0.078 -0.051 -0.013 -0.008 -0.023 - 0.074 -0.243
AZ,,
-0.031 -0.035 - 0.040 - 0.042 -0.037 - 0.035 - 0.039 - 0.046 - 0.038 -0.041 - 0.066 -0.123 -0.246 -0.627
dz 0.025 0.043 0,062 0.031 -0.204 - 1.563 - 1.630 - 1.728 - 1.263 0.133 0.206 0.190 0.191 0.239
AN
-0.011 -0.002 -0.001 - 0.028 - 0.076 - 0.057 -0.021 - 0.088 -0.154 -0.012 0.055 0.081 0.123 0.217
AN,,
0.000 0.001 0.013 0.001 -0.047 -0.378 -0.407 -0.434 -0.337 0.038 0.065 0.067 0.079 0.137
dN _.-
o. is the cranking frequency determined from condition (60). &v is essentially the moment of inertia of Thouless and Valatin. Since we have to take into account in a realistic calculation the violation of the particle numbers, we can only calculate the tensor ge (see table 2). Here we give the diagonal part ‘/rV = s;:. AI [eq. (SS)] is the difference between the RPA expectation value and the HFB expectation value of J,. Only a small part (d I,,) is produced by the normal modes. The rest A I- A&, is produced by the spurious bosom. The parameter dI of eq. (66) determines the angular momentum correction. MPR = 4(~1~~~)(Q,1J;2j~)/(dij4;I~)* characterizes the validity of the minimal packet relation (5) for the HFB function I@). in complete analogy with the AI, diNM and d, values for the angular momentum in the last six columns we present the values LIZ, AZ,,, $ and AN, AN,,, dN for the correction of proton and neutron numbers. Since we only go to linear order in the correction AI, AZ and AN there are no coupling terms between these operations.
a0
I
The correction of the average angular momentum and particle number as a function of Z(fiist column)
TABLE1
g a s
: a 2 fi 5
G
5 t‘ tJ & 3 e
J. L. Egido et al. 1 Ground-state correlations
241
(ii) we used the RPA ansatz (21) with the restrictions (42) and (49), which guarantee strong deformation, for the variation of the intrinsic wave function, (iii) we te~inated the boson expansion of the hamiltoni~ (24) in second order. It then turned out that the optimal choice of the basic HFB function I@,>is a cranking solution, where the cranking frequency o is determined by the RPA expectation value of jX. In the next section we will modify the last condition somewhat for practical reasons and in sect. 5 we will go beyond the cranking approach for the angular moments projection.
4. The limit of a small angular momentum correction AZ In practice, it is difficult to determine o from eq. (34) because one has to solve the RPA equations (31) in each step of the self-consistent calculation for the wave function IQo). We therefore approximate in the following sections lQw) by l@&. The cranking frequency o. is determined from <@lJXl@>,, = %/m,
(60)
i.e. JQ,,,) is the usual HFB solution of the self-consistent cranking model. In the following we will always work on the basis of quasiparticle operators AK, Ai based on the vacuum [QUO) AKl@io(t>= 0.
(61)
Two solutions ICP,) and I@,,) of the self-consistent cranking model which differ in their angular momentum expectation value by an amount AZ are related in the linear response approximation by “) I@-> = exp (-- ~A~~~))t~~~>,
w
where ~$9) is defined in eq. (36). It generates a shift in the angular frequency by the amount -Aa
= o--w0
= -AZ/,%,,,
(63)
where
(W is given by eq. (58). From eq. (63) it follows that the moment of inertia of Thouless and Valatin &v measures the slope of (@l~Xl@>, as a function of w. In the backbending region it goes to infinity at two points. It is quite different from the selfconsistent moment of inertia .@sc of eq. (40) (see fig. 1).
J. L. Egido et al. / Ground-state correlations
242
200 IMN’) 32
I
160 -
24 -
100 -
i
’
/
’
I
1
16 -
01
02
0.3
0.4
1
I 6
II II I I 16 2L ,
fAMeV) a
b
Fig. 1. (a) The average angular momentum I((& J,I4) = JI(l+ 1)) of the HFB solution in the cranking model as a function of the cranking frequency. (b) The corresponding moments of inertia 4,, = dI/dw and 4, = I/w. The details of the calculation are discussed in sect. 6.
We can also express 4:’ in eq. (62) in terms of the boson x defined in eq. (43). Eq. (62) then reads: lGo) = e- d(r+- “‘l@Jw,),
(65)
with d = A1/2(A@+.
(66)
Obviously the parameter d determines the quality of the linear approximation. Table 1 shows that d is indeed a small parameter with the exception of two singular points in the backbending region. At these points the linear response approximation breaks down. For all other I-values, however, the linear approximation in AZ is well justified. We therefore use this approximation in the following and neglect all terms of higher order in AZ. The RPA wave function IY) is then of the form IY) = exp (4 c ZKK,AK+ A& - iAZ c cp~~AK+)I@,,). KK’
(67)
K
The coefficients ZxK* are determined by the variation of the hamiltonian. H’ = H-c&
= (H-w&)
+ ACL&
(68)
under conditions (42) and (49). Neglecting the term do@, which is of second order in AZ [see eqs. (58) and (63)], we find that the linear term Aw.@) can be eliminated by the transformation (62). For the determination of the coeficients Zxk, we thus obtain once more the RPA equations (31) based on the HFB function I@&. To ensure conditions (42) and (49) we have to choose ZKr as in eqs. (55) and (56).
J. L. Egido et al. / Ground-state correlations
243
We can now calculate the energy
Since with l@& one has the same expectation value of IX as with [!I’), the correction term odl of eq. (59) is now no longer present. The RPA energy obtained so far deviates from the HFB energy only by the correlation energy caused by zero-point fluctuations of the normal modes. Since /!I’> is an intrinsic - symmetry violating - wave function, the energy connected with spurious rotational motions is not yet subtracted. As we shall see in the next section, these spurious parts are obtained in a second-order approximation for the angular momentum projection. Although we are in this paper only interested in the calculation of the yrast levels, we want to remark that very similar arguments also apply to the calculation of excited bands. It is easy to show that the wave functions lp) = Ble-dpW+IY),
(701
where d,, is determined in by the condition + (711 minimize the hamiltonian in RPA order under all the constraints discussed in sect. 3 and under the additional condition of orthogonality to the ground state lY>. The excitation energy of the state }p) is given by the frequency Qp
(721
where oJ;pl is again canceled by the shift (70). That means as long as we are interested only in energy levels, it is sufficient to base the solution of the RPA equations (31) on the solution I@,,,) of the self-consistent cranking model. Its eigenvalues s2, correspond to the excitation energies. This fact was used in ref. ‘il.
5. Second-order terms in the projection
So far we have treated the angular moments projection only in the so-called cranking approximation, i.e. we determined the intrinsic wave function from the variation of the energy expression in eq. (7). For the actual calculation of the projected energy we want to go a step further and take into account the second-order terms in eq. (14) at least for the calculation of the energy.
t For K # 0 bands we have to modify the condition (71) slightly [see ref. ‘I)].
J. L. Egido et al. / Ground-state correlations
244
For this purpose we proceed as discussed in sect. 2. Starting from the ansatz + (HR(S2)) = (a+bA~,+cA&+dL:,+eL2,)(R(SZ))
(73)
and using the hamiltonian of eq. (38) we obtain from eq. (54) a linear set of equations, which corresponds to eq. (15). We have to evaluate matrix elements like (HA?& (H4.f:) . . . and find expressions of the form
With the approximation
(.fj”‘(A .f$)
i, k = x, y, z,
(dj&lj$),
”
the equations simplify enormously
(74)
= 0, 1,2,
n
and we find the solution 1
C-
1
d=e=---
,
(75)
2,fsc ’
2.fTV
a = (H)-
(1; +&, (76)
b = co. For the projected energy we finally obtain
Er = GO- L(Aji)2tfTV
$
(j;+.T:)
c SC
+ oP(A.fx,) + ~ l
P(AJ:)+ $
29Tv
f@ + 2). c SC
(77)
If condition (3) is fulfilled we have p(d.L) 2 Jm-
(Q,
(78)
and for cranking solutions we thus obtain Et = Of)-
1 2CaTv(Ax$-
z)-
O,Z+jZ)+
p(S,Z+ .?Z). (79)
* SC
The last term is usually very small and will be neglected in the following tt. Thus
+ The expectation values (. . .) have to be calculated with the RPA function Ivl). Note that the coeffkients c, d, e have a somewhat different definition than in eq. (14). ++ For HFB wave functions we have P( J:) Z 0.5 and P(z) Z (t) [ref. “‘)I.
J. L. Egiab et al. / Ground-state correlations our
245
result is very similar to the result obtained in ref. ‘l)
-J- (Yyld~lY)-
EI = (W#W With the approximation
2/Tv
introduced in sect. 4 we find t
where I@) is the HFB function at frequency w. which fulfills the constraint (60). We thus obtain with “second-order projection” just the energies connected with the spurious rotational motion. Approximation (74) is certainly only a first step. In principle one can calculate all matrix elements exactly and solve a linear system of equations for the coefficients a, b, c, d and e. One then obtains corrections to the simple expression (81).
6. Numerical applications In ref, ‘l) the RPA equation (31) was solved for the high-spin region of 164Er and the corresponding excited bands were discussed in great detail. Now we use these calculations for the investigation of correlations in the yrast wave function. The underlying interaction is a pairing plus quadrupole force. We use the same configuration space and the same force parameters as we used in ref. 21) and all details and formulas can be found there. We start from the hamiltonian
H=
Ho-hj2Q:Q,-GP’P,
(82)
where Ho is a spherical Nilsson potential, QQ are quadrupole operators and P* creates a Cooper pair. In a first step the HFB wave function I ) of the cranking model is calculated. We use the approximation of sect. 4 and determine it from a variation of the energy without exchange terms
Eo =
GO-‘>*
(83)
as well proton and neutron nnmber
(N,) = 2,
(N,)
= A-Z.
(84)
+ This expression for the energy coicides essentially with the one obtained by Marsbalek l’) in a boson expansion. The only difference is a term fw which has to be added in ref. “), because the cranking wave function I@) has been solved for the constraint (@lf,lcP) = I instead of eq. (60).
J. L. Egido et al. 1 Ground-state correlations
246
In addition to E. the HFB energy (H) EEX
=
-4~
in eq. (81) contains the exchange terms
(AQ~A~~)-G(A~+A~)
i
(85)
ii=-2
with
AQ,, = Qr-(Q&v The fluctuations of the quadrupole
operators
AP = P-(P).
(86)
are of the form
= k;,lQt’,“,l’ = Tr (Q:Qp~-Qp~Q;lfo-QruQ~u*), (87)
where Qz” is the two-quasiparticle part of the operator Q,.,and p, K are the normal and anormal density of the HFB function 1). The first term in eq. (87) comes from the single-particle operator contained in Q: Qa, the second corresponds to the Fock term in the self-consistent field and. the third has its origin in the contribution of the quadrupole force to the pairing field. The fluctuation of the pairing operator (AP+AP)
= kFllP$‘12 = -$Tr(Pp*Pp)
(88)
corresponds to the contribution of the pairing force to the self-consistent field. These terms are of second order in the sense of a boson expansion of the hamiltonian (82) [see ref. ‘)I and are therefore neglected in the variational determination of the HFB function. Since we go in this paper up to RPA order we have to take them into account in the calculation of the energy. Fig. 2 shows the fluctuations of the operators & and P. They are not small, but show a very smooth behaviour as a function of angular momentum. The same is true for the resulting energies of eq. (85). The contribution of the exchange terms is shown in fig. 5 +. One notes that over the angular momentum range between I = 0 and I = 28 EEX changes by roughly 500 keV. Altogether the total yrast energy contains four terms [compare eqs. (69) and (80)] Er = E~+EEx+ENM+&u.
(8%
ENS corresponds to the correlations caused by the zero point vibration of the normal modes with the energy ENM
=
-
~~p~&P12 P
K
and EoM corresponds. to the Goldstone modes, i.e. to the spurious motion in the symmetry violating intrinsic function (see eq. 80). ’ We start with I = 2 becausefor I = 0 the axial symmetry causes degeneracies and some numerical problems arise.
J. L. Egiab et al. / Ground-state correlations I
I
I
1
I
247
I
IAI?p2
,21._.----.,._._._
(At’)2N
8&I
0
II
1
I
I
,
lJ2
24:._.
-,_.
-.-‘-.
16 -
IA< g2
80
)
2&z.
-.--._(
-.-.
I2 t&Q?2 ---_ _ !A0 12 -P.
16----------________ 8-
I
I
28
Fig. 2. The fluctuation of the quadrupole operators Q, and of the pairing operators for protons P, and neutrons P, in the HFB wave function 14) as a functron of the average angular momentum. The units arc b4 for the quadrupole operators (b is the oscillator length) and the pairing operators are dimensionless.
E&‘d
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
IMeVl
- 2.5-
-2.o16 I6 - 1.5 El 12 IO -1.0
-
6” I=“z
- 0.51
I
80
lw
240
320
I n
400
Fig. 3. Convergence properties of the correlation energy (91) resulting from the lowest n normal modes.
248
J. L. Egido et al. / Ground-state correlations
For the calculation of the exact value of ENMone needs all eigenvalues of the RPA matrix (31) and the corresponding eigenvectors. The dimension of this matrix is of the order of several thousands. In ref. 21) the eigenvalue problem was rewritten into a dispersion relation and its low-lying modes were determined by a root search. In the same way we now determine the lowest 400 eigenvalues of the RPA matrix. Thus we were able to investigate the convergence properties of the expression ENMb)
=
-
i p=1
fi,
c 1&,I2
(91)
K
as a function of n. Fig. 3 shows the values E&n) for I 5 20. The convergence is rather slow, such that n = 400 is still probably not large enough to achieve absolute convergence. However, the curves for different I-values are approximately parallel. We therefore hope that a further increase of the absolute value of ENMwill not strongly influence the excitation energies E, - EI= ,,. In the following we therefore use the value ENM = ENM(FI = 80). Fig. 5 shows the correction ENM as a function of I. In the backbanding region at I- 16 we find a very large contribution to the correlation energy. It has its origin in the very low-lying “second” band in this region [see ref. 2’), fig. 1 band a]. We now turn to the investigation of the term EGM. Besides the spurious modes connected with the violation of rotational symmetry, which we have discussed in sect. 5; we also have to take into account the spurious modes connected with the numbers of protons and neutrons. We then have to vary the projected energy (YIHPN,PN,P1lY). The calculation is carried out in complete analogy with sects. 3-5. We shall therefore give only the results. Instead of eq. (38) the hamiltonian takes the form H’ = const + c f&B,+ B, + OPT P
+:g&“’
+ :g,,&l)’
+:gz&)’
+g.&?fi(,l)
We obtain for the energy contribution
+gJz~~l’Ab”+gNZ,~~)~a).
(92)
of the spurious modes
with E,,
= -i-g&I.&,
EJN = -gJ.ddJ%Jr
E NN
=
-+NN’k@)r
EJZ = -g,,@&,h
Ezz = -:gzz(A@)
(94)
The tensor gik is calculated as described in ref. 9). Table 2 gives its actual values for I 5 28. Table 3 shows the corresponding fluctuations. In fig. 4 we present the contributions (94) to the energy. The essential feature is the behaviour of the term
J. L. Egido et al. / Ground-statecorrelations
249
TABLE2
The tensor ge in eq. (92) I
gJJ
0
0.0390 0.0365 0.0310 0.0236 0.0161 0.0097 0.0034 -0.0013 -0.0035 0.0030 0.0153 0.0153 0.0135 0.0115 0.0100
2 4 6 8 10 12 14 16 18 20 22 24 26 28
QNN
szz
0.1920 0.1914 0.1896 0.1859 0.1798 0.1773 0.1907 0.1692 0.1278 0.0512 0.0663 0.1159 0.1533 0.2057 0.3781
0.3245 0.3230 0.3188 0.3118 0.3035 0.2980 0.2976 0.3002 0.3042 0.3091 0.3114 0.3073 0.2969 0.2887 0.3887
gJZ
gJN
-o.oooo
-0.0000
-0.0023 -0.0044 -0.0074 -0.0121 -0.0175 -0.0144 0.0041 0.0227 0.0410 0.0154 0.0047 0.0014 0.0003 0.0008
-0.ooo6
-0.0001 0.0017 0.0037 0.0046 0.0037 0.0037 0.0056 0.0095 0.0067 0.0057 0.0058 0.0066 0.0083
gZN
0.0033 0.0033 0.0033 0.0035 0.~ 0.0050 0.0095 0.0171 0.0214 0.0155 0.0173 0.0233 0.0266 0.0281 0.0297
EJJ. In the backbending region it becomes rather small and even positive because of the strange behaviour of gJJ, which is roughly proportional to l/Yrv, in this region (see table 1). Fig. 5 summarizes different contributions to the correlation energy. ENMand EGO behave in the backbending region in opposite senses. The resulting sum of all correlation energies is therefore again a very smooth function of I. It is not constant, but increases from Z = 2 to I = 28 by more than 2 MeV. If we were to add these terms to the energy E, of the HFB cranking model without exchange terms, we would obtain a considerable stretching of the spectrum. Since the parameters of the pairing plus quadlMeVI
f
I
I
f
I
I
Fig. 4. The contributions E, of the spurious modes in eq. (94) to the energy. ENzvanishes identically and E,, is smaller than 1 keV. c is the sum of all contributions with positive signature c = E,, + ENN+ Ezz +-Em.
J. L. Egiab et al. / Ground-state correlations
250
TABLE 3
The fluctuations of the operators f,, f,, f,, fiP and fin and the coupling terms A &A IV,,,A fXA I’?”in the cranking wave function I) = [4),,, I
0
44.30 45.11 47.12 50.35 54.82 60.27 66.40 71.20 70.12 62.79 60.29 60.96 62.07 63.08 63.70
2 4 6 8 10 12 14 16 18 20 22 24 26 28
3.36 3.31 3.21 3.07 2.90 2.74 2.61 2.55 2.52 2.48 2.27 1.93 1.49 0.96 0.31
(A J,AN,)
0.0
0.0
0.03 0.06 0.12 0.24 0.44 0.40 -0.56 -1.72 -1.44 -0.56 -0.28 -0.17 -0.10 -0.05
0.01 0.01 0.00 -0.02 -0.03 - 0.04 -0.05 -0.05 -0.06 -0.07 -0.08 -0.10 -0.11 -0.06
rupole force were adjusted to reproduce the experimental yrast line as well as possible, we would certainly not obtain an improvement in agreement with the data. To have a complete description we should have used the energies EI in eq. (89) for the adjustment of the force parameters. IMenI)
21I-
IS)-
0
-l.C I
I
I
L
8
12
Fig. 5. The different contributions to the correlation ENMthe term coming from the normal modes p = modes (93) and c is the sum of these three terms 1 I = 2, where we have Em = -6.95 MeV, E&n
I
16
I
20
I
24 ,
energy in eq. (89). &x are the exchange terms (85), 1. . .80 in eq. (91), &,,, results from the Goldstone = EEx+ &,,, + EOM.All energies are normalized to = 80) = -0.899 MeV and EOM= -2.742 MeV.
J. L. Egiab et al. / Ground-state correlations
251
The fact that the anomalous behaviour in the functions &M(Z) and E&Z) have canceled out in the final result ,Y of lig. 5 can be understood qualitatively: We only have to use the RPA equation (31) and find c =
EEX+J%+ENM
=
+fcf$--
P
CEK), K
(95)
i.e. the correlation energy is just the difference between the sum of all correlated energies K&,and the sum of all uncorrelated two-quasiparticle energies EK = Ek + El. The peaks in the functions Z&Mand ENMare obviously caused by the level crossing in the backbending region, i.e. by a low-lying two-quasiparticle energy EK, or the corresponding low-lying collective energy Q,,. As we see from fig. 1 in ref. ‘I), EK1 and Qpl are very close to each other, i.e. the difference (95) is a smooth function in the region of the level crossing.
7. Conclusion The self-consistent cranking model based on generalized Slater determinants in the many fermion space used as a description of the yrast line of well-deformed nuclei has been extended: Ground-state correlations in the intrinsic function are taken into account in the RPA scheme and an approximate projection in angular momentum and particle number up to second order has been carried out. According to the different degrees of freedom in the system we find two types of correlations: The first type is connected with the zero- point fluctuations of vibrations in the intrinsic frame; the second type has its origin in the Goldstone modes in the intrinsic function connected with the violation of angular momentum and particle number symmetry. They are removed by the projection. Up to second order the formula for the energy is identical to that obtained by Marshalek ’ ‘) in a boson expansion in a rotating frame. The wave function calculated with the latter method, however, is not normalizable. It presents only for small angles an approximation to the exact eigenstate of the system. It is an advantage of our method that it yields a projected wave function which can be normalized and which yields after projection a “reasonable” approximation to the exact eigenstate in the entire space, not just for very small angles. These theoretical considerations were applied to the nucleus ’ 64Er. The correlations in the yrast wave function were taken into account on the basis of RPA calculations discussed in ref. “). In particular we calculated the exchange terms EEL and correlation energies ENM and &M. Their behaviour as a function of I is discussed. For Z = 0 the three terms together produce a lowering of the yrast energy by roughly 10.5 MeV. With increasing angular momentum the lowering caused by the exchange terms and by the spurious modes is reduced, whereas the correlation energy coming from the normal modes increases. In the backbending region we have found several
252
J. L. Egiab et al. / Ground-state correlations
irregularities: Because of the level crossing one observes a peak in the correlation energy ENM.On the other hand the spurious energy E,, has a dip there because of the singular behaviour of the parameter &v which measures the slope of the angular momentum as a function of the cranking frequency. Both irregularities cancel to a large extent in such a way that the resulting energy correction has a smooth behaviour which can simply be absorbed in a proper adjustment of the effective interaction. We thus understand why a simple calculation in the self-consistent cranking model based on HFB functions with adjustable parameters in the effective interaction is able to reproduce the experimental yrast line to a rather high degree of accuracy. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
14) 15) 16) 17) 18) 19) 20) 2!) 22)
R. M. Lieder and H. Ryde, Adv. Nucl. Phys. 10 (1978) 1 P. Ring, R. Beck and H. J. Mang, Z. Phys. 231 (1970) 10 B. Banerjee, H. J. Mang and P. Ring, Nucl. Phys. A215 (1973) 366 A. L. Goodman, Nucl. Phys. A256 (1976) 113 A. Faessler, K. R. Sandhya Devi, F. Gruemmer, K. W. Schmid and R. Hilton, Nucl. Phys. A256 (1976) 106 J Fleckner, U. Mosel, H. J. Mang and P. Ring, to be published S. Bose, J. Krumlinde and E. R. Marshalek, Phys. Lett. 53B (1974) 136 S. Y. Chu, E. R. Marshalek, P. Ring, J. Krumlinde and J. 0. Rasmussen, Phys. Rev. Cl2 (1975) 1017 E. R. Marshalek, Nucl. Phys. A266 (1976) 317 B. L. Birbrair, Nucl. Phys. A257 (1976) 445; I. N. Mikhailov and D. Janssen, Phys. Lett. 72B (1978) 303 E. R. Marshalek, Nucl. Phys. A275 (1977) 416 J. L. Egido, P. Ring and H. J. Mang, Phys. L.&t. 77B (1978) 123 I. Hammamoto, Nucl. Phys. A263 (1976) 315; Phys. Lett. 66B (1977) 222; E. R. Marshalek and A. L. Goodman, Nucl. Phys. A294 (1978) 92; F. Griimmer, K. W. Schmid and A. Faessler, Nucl. Phys. A308 (1978) 77; S. Cwiok, J. Dudek and Z. Szymanski, Acta Phys. Pol. 99 (1978) 725; R. Hilton, Contributed Paper, Jiilich Conf. on highly excited states in nuclei, Sept. 1975, vol. 1, p. 33 E. R. Marshalek and J. Weneser, Ann. of Phys. 53 (1969) 569; Phys. Rev. C2 (1970) 1682 A. Kamlah, Z. Phys. 216 (1968) 52 R. Beck, H. J. Mang and P. Ring, Z. Phys. 231 (1970) 26 P. Ring, H. J. Mang and B. Banerjee, Nucl. Phys. A225 (1974) 141 H. J. Mang, B. Samadi and P. Ring, Z. Phys. A279 (1976) 325 D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 211 S. Islam, private communication J. L. Egido, H. J. Mang and P. Ring, Nucl. Phys. A334 (1980) 1 D. J. Thouless, Nucl. Phys. 21 (1960) 225